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Application of non-equilibrium statisticalmechanics to the analysis of problems in
financial markets and economy
Andrey Sokolov
Submitted in total fulfilment of the requirements
of the degree of Doctor of Philosophy
School of Physics
The University of Melbourne
September, 2014
Produced on archival quality paper
Abstract
This thesis contributes to a growing body of work in the emerging inter-
disciplinary field of econophysics, where tools and techniques of statistical
mechanics and other branches of physics are applied to problems in economics
and finance. The thesis examines the following four topics: 1) money flows in
the interbank networks, 2) wealth distributions in multi-agent exchange sys-
tems, 3) variability and dynamics of the foreign exchange markets via lattice
gauge theories, 4) memory loss and patterns of change in Abelian sandpiles
via hidden Markov models.
In economic and financial systems, interactions between the agents oc-
cur on a network, whose properties depend on the system in question. Such
networks are not static but rather dynamic in character, since the links com-
prising the network frequently depend on the actions of the agents as they
interact with one another. Moreover, the links of the network typically repre-
sent flows, e.g. money flows in financial networks, which further complicates
their study. These issues are addressed in the thesis in the case of the financial
networks of money flows between Australian banks.
In most advanced economies, including Australia, high-value transactions
in the banking system are settled in real time via the so called real-time gross
settlement systems, which are controlled by the central banks. Such systems
have been introduced in many countries in the last ten to twenty years in
order to diminish the liquidity risk in the banking system. As such systems
are computerized, the information pertaining to all transactions, including
their source, destination, and value, is recorded by the central banks and can
be used to investigate the properties and dynamics of the interbank flows.
The flows of payments in the interbank networks are not homogeneous but
possess an intricate structure that reflects the nature of various payments. In-
deed, some of the payments recorded by the central bank represent overnight
loans extended from one bank to another and the return payments made on
the following day. The work undertaken in the thesis examines the flows of
overnight loans and the flows of other payments separately. It is shown that
these two kinds of flows are not entirely independent. The flows of overnight
loans appear to counteract the imbalances in the bank’s reserve accounts cre-
ated by the flows of other payments. This is in accord with the dynamics
existing in the interbank money market, where the bank’s with the surplus of
reserves lend the surplus to the banks with depleted reserves.
Agent-based models are gaining popularity in the efforts to understand
economic and financial systems and their dynamics. The strength of these
models lies in the fact that they require the formulation of local rules of interac-
tion only without specifying the global constraints on the system’s behaviour,
which are often poorly understood. The simulations that use agent-based
models reveal the emergent behaviour of the economic and financial systems
that arises as a result of the collective actions of the agents following individ-
ual rules. In particular, agent-based models have been used to address the
problem of the wealth and income distribution in the economies. The work
conducted in the thesis examines one such model, referred to as the giver
scheme, where the rule of exchange stipulates that the transfer amount from
the giver to the receiver is equal to a fixed fraction of the giver’s wealth.
The giver model of asset exchanges is examined in the thesis by means
of multi-agent simulations. The system rapidly evolves to a steady state, in
which the distribution of wealth does not vary with time, even though agents
continue to exchange wealth. This model is amenable to the analysis based on
the master equation, which balances the influx and outflow of agents at every
wealth value. The thesis presents an investigation of the master equation by
means of the Laplace transform. A novel technique for calculating the wealth
distribution in the steady state is introduced and used to investigate wealth
inequality.
The giver model is shown to exhibit two distinct regimes in the steady
state. One of them is reminiscent of exchanges that occur in the economy
where the agents typically exchange a small fraction of their wealth. The other
regime is more akin to gambling and is characterised by exchanges where most
of the giver’s wealth is lost, so that fortunes are made and lost frequently. The
first regime is characterised by relatively low inequality, whereas the second
one is prone to exhibit very large inequality.
In addition to the study of inequality, the thesis investigates applicability
of the Boltzmann entropy as a measure of disorder in the giver model. The
giver model represent a closed system with no sources or sinks of agents or
wealth, i.e. it is conservative and in many respects is similar to an ideal gas.
However, numerical simulations reveal that the Boltzmann entropy does not
evolve monotonically in the giver model and, therefore, is not a faithful mea-
sure of disorder. This paradox is resolved by observing that the exchange rules
of the giver model are not time reversible, i.e. in order to reverse the dynamics
of exchanges a quantitatively different rule of exchange is required.
The foreign exchange market is a complex dynamical system that provides
prodigious amount of financial data, which makes it an attractive subject of
research. One of the approaches to modeling it relies on the lattice gauge
theory, where the lattice is constructed by considering two or more currencies
and discretising time and the gauge refers to the arbitrage on the lattice. A
brief investigation of one such model is undertaken in the thesis. It is shown
that the model is unstable in most realistic situations and thus cannot be used
to model the market behaviour.
Despite wealth of data, the foreign exchange market is difficult to study,
since the internal structure of the market is not well known. The Abelian
sandpile model, a cellular automaton that exhibits self-organised criticality,
is used in the thesis as a toy model that captures some of the features of the
foreign exchange market. The model is used to study temporal correlations in
the observed behaviour and their relation to the underlying internal structure.
A technique based on the site occupancy numbers is proposed in the thesis.
It reveals that the loss of memory in the sandpile model occurs in two distinct
stages, the fast stage characterised by rapid loss of memory and the subsequent
slow stage, during which memory of the initial state is lost at a much reduced
pace. Both stages are shown to be roughly exponential and the scaling of the
time decay is investigated.
The temporal correlations in Abelian sandpiles are also investigated in-
dependently by means of hidden Markov models, which show exceptional ca-
pabilities in detecting patterns in sequences of data. Hidden Markov models
have not been applied to Abelian sandpiles despite their popularity in a broad
range of applications ranging from bioinformatics to speech recognition. It is
demonstrated in the thesis that hidden Markov models do detect patterns in
the temporal variability of avalanche size, consistent with the results based on
the occupancy numbers. However, the connection between these patterns and
the internal structure of the sandpile has not been established. A number of
promising directions to address this problem are proposed.
Declaration
This is to certify that:
1. This thesis comprises only my original work, except where indicated in
the preface.
2. Due acknowledgement has been made in the text to all other material
used.
3. The thesis is less than 100,000 words in length, exclusive of tables, bib-
liographies, and appendices.
Andrey Sokolov
Preface
The majority of the work presented in this thesis is my own. The details of
specific contributions are listed below.
• Chapter 1 is an original review of the key research topics in econophysics,
based on a number of publications quoted in the text.
• Chapter 2 is based very closely on the following paper, co-authored with
Rachel Webster, AndrewMelatos, and Tien Kieu. It uses the data kindly
provided by the Reserve Bank of Australia. The work is original and my
own, including the code I wrote for data analysis and visualisation. The
idea to use the Anderson-Darling test was suggested by an anonymous
referee. Pip Pattison and Andre Gygax provided advice on the network
analysis.
Sokolov, Webster, Melatos, and Kieu (2012)
Loan and nonloan flows in the Australian interbank network.
• Chapter 3 is based very closely on the following publication, co-authored
with Andrew Melatos and Tien Kieu. The work is a result of a close
collaboration with Andrew Melatos who suggested the idea to use the
master equation.
Sokolov, Melatos, and Kieu (2010b)
Laplace transform analysis of a multiplicative asset transfer model.
• Chapter 4 is based very closely on the following publication, co-authored
with Tien Kieu and Andrew Melatos. Tien Kieu provided the gauge
theory expertise for this work.
Sokolov, Kieu, and Melatos (2010a)
A note on the theory of fast money flow dynamics.
• Chapter 5 is based on a paper submitted to Physica A and co-authored
with Andrew Melatos, Tien Kieu, and Rachel Webster. It uses an
Abelian sandpile simulator that I wrote. The idea to use hidden Markov
models can be traced to Hyam Rubinstein.
• Chapter 6 is original work.
Acknowledgements
In 2009, The School of Physics at the University of Melbourne offered a Ph.D.
scholarship in econophysics generously funded by the Portland House Foun-
dation. Tien Kieu, who was Director of Research at the The Portland House
Research Group at the time, played an instrumental role in making the fund-
ing available. The scholarship in econophysics was enthusiastically supported
by Rachel Webster and Andrew Melatos, who along with Tien Kieu became
my supervisors and collaborators. I am grateful to Tien, Rachel, and Andrew
for their support during my studies and research. I am eager to acknowledge
Andrew’s boundless enthusiasm and readiness to get involved, Rachel’s wise
advice, and Tien’s generosity with his time despite his pressing commitments
at The Portland House.
I thank the Portland House Foundation for funding this work.
The work on the interbank network would not be possible without Beth
Webster, who helped us get in touch with Chris Kent and Peter Gallagher
of the Reserve Bank of Australia. I thank the RBA and Peter Gallagher
in particular for making the interbank data available. During this project I
received valuable assistance from Andre Gygax and Pip Pattison.
I thank SIRCA for their timely assistance in obtaining financial data. I
gratefully acknowledge Hyam Rubenstein, Mark Joshi, Omar Foda, and Edda
Claus who advised me in the early stages of my studies. Thanks to David
Jamieson for his continued interest in my work. Many thanks to Sean and the
other members of the IT support group for dealing with my numerous requests.
And thanks to the students and faculty in the Astro group for having a lively
interest in my research.
Contents
Contents viii
List of Figures x
List of Tables xvi
1 Introduction 1
1.1 What is econophysics? . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Progress in econophysics . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Choosing the topics of research . . . . . . . . . . . . . . . . . . 10
1.4 Details of research projects . . . . . . . . . . . . . . . . . . . . 21
2 Loan and nonloan flows in the Australian interbank network 27
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Overnight loans . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Nonloans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5 Loan and nonloan imbalances . . . . . . . . . . . . . . . . . . . 38
2.6 Flow variability . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.7 Net flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3 Laplace transform analysis of a multiplicative asset transfer
model 59
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 Giver scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3 Laplace transform of the master equation . . . . . . . . . . . . 63
3.4 Steady-state wealth distribution by Laplace inversion . . . . . . 66
3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
viii
3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.7 Iterative procedure . . . . . . . . . . . . . . . . . . . . . . . . . 79
4 A note on the theory of fast money flow dynamics 81
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2 Lattice gauge theory and fast money flow dynamics . . . . . . . 82
4.3 Analysis of the Euler-Lagrange equations . . . . . . . . . . . . 86
4.4 Revisiting the action . . . . . . . . . . . . . . . . . . . . . . . . 91
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5 Memory on multiple time-scales in an Abelian sandpile 95
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2 Abelian sandpiles . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3 Site occupancy fraction distributions . . . . . . . . . . . . . . . 100
5.4 Short- and long-term memory in site occupancy . . . . . . . . . 101
5.5 Hidden Markov model . . . . . . . . . . . . . . . . . . . . . . . 104
5.6 HMM analysis of long-term memory . . . . . . . . . . . . . . . 106
5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6 Conclusion 111
6.1 The interbank network . . . . . . . . . . . . . . . . . . . . . . . 111
6.2 Wealth distributions . . . . . . . . . . . . . . . . . . . . . . . . 114
6.3 Critique of fast money flow theory . . . . . . . . . . . . . . . . 117
6.4 On memory in sandpiles . . . . . . . . . . . . . . . . . . . . . . 117
Bibliography 121
List of Figures
2.1 The distribution of transaction values v (in Australian dollars) on
a logarithmic scale, with bin size ∆ log10 v = 0.1; the vertical axis
is the number of transactions per bin. Components of the Gaussian
mixture model are indicated by the dashed curves; the solid curve
is the sum of the two components. The dotted histogram shows the
relative contribution of transactions at different values to the total
value (to compute the dotted histogram we multiply the number
of transactions in a bin by their value). . . . . . . . . . . . . . . . 35
2.2 Hypothetical interest rate rh versus value of the first leg of the
transaction pairs detected by our algorithm, with no restrictions
on value or interest rate. The dotted rectangle contains the trans-
actions that we identify as overnight loans. The least-squares fit is
shown with a solid red line. . . . . . . . . . . . . . . . . . . . . . 36
2.3 (a) The distribution of loan values v on a logarithmic scale. The
vertical axis is the number of loans per bin for bin size ∆ log10 v =
0.25. The dotted line is the same distribution multiplied by the
value corresponding to each bin (in arbitrary units). The date of
the first leg of the loans is indicated. (b) The distribution of loan
interest rates rh. The vertical axis is the number of loans per bin
for bin size ∆rh = 0.01. The date of the first leg of the loans is
indicated. The mean and standard deviation are 6.25% and 0.08%
on Monday (19-02-2007), and 6.26% and 0.07% on the other days. 37
x
2.4 The distribution of nonloan transaction values of the six largest
banks for Monday through Thursday (from left to right); the banks
are selected by the combined value of incoming and outgoing trans-
actions over the entire week. Black and red histograms correspond
to incoming (bank is the destination) and outgoing (bank is the
source) transactions; red histograms are filled in to improve visi-
bility. The banks’ anonymous labels, the combined daily value of
the incoming and outgoing transactions, and the daily imbalance
(incoming minus outgoing) are quoted at the top left of each panel
(in units of A$109). The horizontal axis is the logarithm of value
in A$. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5 Left: loan imbalance ∆l vs nonloan imbalance ∆v for individual
banks and days of the week (in units of A$109). Right: the absolute
value of loan imbalance |∆l| vs nonloan total value (incoming plus
outgoing transactions) for individual banks and days of the week.
Thursday data are marked with crosses. . . . . . . . . . . . . . . 41
2.6 Nonloan flow value pairs on one day (horizontal axis) and the next
(vertical axis). Only flows present on both days are considered.
Flows that do not change lie on the diagonal (red dotted line).
The solid line is the weighted orthogonal least squares fit to the
scatter diagram; the weights have been defined to emphasize points
corresponding to large flows. . . . . . . . . . . . . . . . . . . . . . 43
2.7 As for Figure 2.6 but for loan flows. . . . . . . . . . . . . . . . . . 43
2.8 Loan flow values versus nonloan flow values combined over four
days. Triangles correspond to loan flows with three or more trans-
actions per flow. The solid line is the orthogonal least squares fit
to the scatter diagram; the weighting is the same as in Figure 2.6. 46
2.9 The distribution of values of net nonloan flows (black histogram)
on a logarithmic scale with bin size ∆ log10 v = 0.1. The compo-
nents of the Gaussian mixture model are indicated with the dashed
curves; the solid curve is the sum of the two components. Net loan
flows are overplotted in red. The vertical axis counts the number
of net flows per bin. . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.10 (a) Degree distribution of the net nonloan flow networks (for conve-
nience, in-degrees are positive and out-degrees are negative). The
total value of the net flows corresponding to the specific degrees
is shown with red dots (the log of value in A$109 is indicated on
the right vertical axis). (b) Degree distribution of the net non-
loan flows when the degree data for all four days are aggregated
(in-degrees are circles; out-degrees are triangles). . . . . . . . . . . 49
2.11 (a) Same as Figure 2.10a, but for the net loan flow networks. (b)
Same as Figure 2.10b, but for the net loan flows. . . . . . . . . . . 50
2.12 Network of net nonloan flows on Tuesday, 20-02-2007. White (grey)
nodes represent negative (positive) imbalances. The bank labels
are indicated for each node. The size of the nodes and the thick-
ness of the edges are proportional to the logarithm of value of the
imbalances and the net flows respectively. . . . . . . . . . . . . . 52
2.13 Networks of daily net nonloan flows for D, AV, BP, T, W, BA, AH,
AF, U, AP, P, A, BG. All the other nodes and the flows to and
from them are combined in a single new node called “others”. The
size of the nodes and the thickness of the edges are proportional to
the logarithm of value of the imbalances and the net flows respec-
tively. The value of the flows and the imbalances can be gauged
by referencing a network shown in the middle, where the values of
the flows are indicated in units of A$1 billion. . . . . . . . . . . . 54
2.14 Networks of daily net loan flows. The same nodes as in Fig-
ures 2.13a–2.13d are used. The scale of the loan flows, the im-
balances, and the positions of the nodes are the same as those
used for the nonloan flows in Figures 2.13a–2.13d to simplify vi-
sual comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1 The Laplace transform g(z) for f = 0.1 obtained by solving (3.4) it-
eratively. (a) Re[g(z)] (top left panel), Im[g(z)] (bottom left panel),
and |g(z)| (right panel) versus r along the real axis (dashed curve),
the imaginary axis (solid curve), and the line inclined at θ = 60◦
to the real axis (dotted curve). The variables r and θ are defined
by z = reiθ. (b) A view of the real (top) and imaginary (bottom)
parts of g(z) (values above 1 and below −1 have been cut off). . . 65
3.2 The steady-state wealth probability distribution function ps(w) ob-
tained by inverting the Laplace transform g(z) for the following
values of the transfer fraction: f = 0.5 (bold solid curve), 0.25
(dash-dot curve), 0.1 (dotted curve), 0.05 (dashed curve), and 0.025
(thin solid curve). . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3 The steady-state wealth probability distribution function ps(w) ob-
tained by inverting the Laplace transform g(z) for the following
values of the transfer fraction: f = 0.5 (bold solid curve), 0.6
(dash-dot curve), 0.7 (dotted curve), 0.8 (dashed curve), and 0.9
(thin solid curve). . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.4 The population distribution n(w), shown with crosses, as a function
of wealth w, measured in ficticious monetary units (m.u.) used in
the agent-based simulations. The distribution is computed as the
number of agents in every unit wealth interval after 100 steps in
the simulation of the giver scheme with total number of agents
N = 4 × 105 and transfer parameter (a) f = 0.95 and (b) f =
0.05. The initial distribution is uniform in the wealth interval (a)
[0, 100m.u.] and (b) [0, 500m.u.]. The corresponding solution of
the steady-state master equation for the same f is shown with a
solid curve, with ps(w) scaled to conform with the definition of
n(w) according to Nps(w/〈w〉)/〈w〉 where 〈w〉 is the mean wealth.
Both the agent-based simulations and the master equation predict
oscillations in the wealth distribution in (a) but not in (b). . . . . 69
3.5 (a) Boltzmann entropy Ss of the steady-state distribution as a
function of the variance σ2s = f/(1 − f). The critical values
σ2s = 0.062, 1, and 5.098, corresponding to f = 0.058, 0.5, and
0.836 respectively, are indicated with the dotted lines. (b) Entropy
as a function of time for the initial distribution given by (3.12) with
f = 0.058, computed from the multi-agent simulation of the giver
scheme. For the simulation, the distribution (3.12) was scaled up
to give N = 337123 agents in 0 ≤ w ≤ 1421. To compute the en-
tropy, the population distribution produced by the simulation was
normalized to a probability distribution with unit mean. . . . . . 72
3.6 (a) The limiting probability distribution ξ(w), shown with crosses,
obtained from one realization of the asymmetric random process (3.13)
for f = 0.05 after 106 iterations. The mean is 0.9998 and the vari-
ance is 0.0519 (cf. 0.0526 theoretically from the master equation).
The steady-state distribution ps(w) for the same f obtained by
Laplace inversion is shown as a solid curve for comparison. (b)
The first 1000 values of {wi}. . . . . . . . . . . . . . . . . . . . . . 74
3.7 Gini coefficient of the steady-state distribution ps(w) as a function
of the variance σ2s = f/(1− f). . . . . . . . . . . . . . . . . . . . 76
4.1 Re-creation of Ilinski’s solution of Eqs. (4.9–4.11) given on page 169
of Ilinski (2001) for α1 = 1.5, α2 = 10, C0 = 0, and the initial
conditions: η(0) = 0.2, υ(0) = 0, ρ(0) = 0.5. The factor α1 in
Eq. (4.10) is replaced with unity to match Ilinski’s Euler-Lagrange
equations. The displayed quantities are as follows: ρ− 1/2 (solid),
υ + η (dashed), η (dot-dashed), υ (dotted). . . . . . . . . . . . . . 84
4.2 The solution of Eqs. (4.9–4.11) for the same parameters and initial
conditions as in Fig. 4.1. The factor α1 in Eq. (4.10) is restored. . 85
4.3 The solution of Eqs. (4.9–4.11) for the same parameters and initial
conditions as in Fig. 4.2, except α1 = 0. . . . . . . . . . . . . . . . 87
4.4 The solution of Eqs. (4.9–4.11) for the same parameters and initial
conditions as in Fig. 4.2, except with C0 = 0.1 instead of C0 = 0.
The curves are coded as in Figs. 4.1 and 4.2. . . . . . . . . . . . . 89
4.5 As for Fig. 4.4, with C0 = −0.1. . . . . . . . . . . . . . . . . . . . . 90
4.6 The trading volume V (bold solid curve) and the return R (bold
dot-dashed curve) for the same parameters as in Fig. 4.2. For
comparison, we also display ρ− 1/2 (thin solid curve) and η (thin
dot-dashed curve). . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.7 Lattice diagram for the intra-day foreign exchange trading in two
currencies. Interest rates are ignored. . . . . . . . . . . . . . . . . 92
5.1 Probability density function p(fk) of the fraction fk of sites with
charge k = 0, 1, 2, 3 (left to right) for two sandpile simulations on
a square grid with N = 32 (blue) and N = 64 (red), based on
samples consisting of 103N2 steps in the recurrent regime. The
curves show the normal distributions for the corresponding values
of the mean and standard deviation; they agree well with the data.
The first 5N2 steps of the simulations are discarded to eliminate
transient configurations. . . . . . . . . . . . . . . . . . . . . . . . 100
5.2 Site occupancy analysis of an Abelian sandpile. (a) Ensemble-
averaged site occupancy fractions 〈δk〉 versus the time delay t− t0,where δk is the fraction of sites with charge k = 0, 1, 2, 3 in the
absolute difference matrix Dij = |zij(t) − zij(t0)|. The ensemble-
averaged curves are based on 104 trials (one representative trial
is shown in black). The standard error of the ensemble-averaged
values, which is ∼ 0.001, is not shown. The dashed lines represent
the expected values dk for t → ∞ and N = 32. (b) Approach of
〈δk〉 to the expected values dk shown on a logarithmic scale. . . . 103
5.3 Hidden Markov analysis of an Abelian sandpile. (a) Emission prob-
abilities Bij from hidden state i = 0 (red) and hidden state i = 1
(blue) to observed states j = 0, 1, 2 versus the averaging time-scale
Ta for the 32×32 sandpile. A discrete HMM with two hidden states
and three observed states is used. Observed states are defined by
binning the avalanche size into high, medium, and low bins, such
that there are equal number of samples in each bin for each aver-
aging interval. (b) Results of a control experiment where the same
input data is shuffled randomly before being fed into the HMM. . 107
List of Tables
2.1 The number of transactions (volume) and their total value (in units
of A$109) for each day. . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2 Mean 〈u〉, variance σ2u, and mixing proportion P of the Gaussian
mixture components shown in Figure 2.1 (u = log10 v, where v is
value). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 Statistics of the overnight loans identified by our algorithm: the
number of loans (volume), the total value of the first leg of the loans
(in units of A$109), and the fraction of the total value of the loans
(first legs only) with respect to the total value of all transactions
on a given date. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4 Loan and nonloan imbalances for the six largest banks (in units of
A$109). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.5 Mean 〈u〉, variance σ2u, and mixing proportion P of the Gaussian
mixture components appearing in Figure 2.9 (u = log10 v). . . . . . 46
5.1 Mean values and standard deviations for the samples shown in
Figure 5.1. The bottom row gives the analytical estimates of prob-
abilities pk, obtained in the limit of an infinitely large sandpile. . . 101
5.2 Transition probabilities Aij (left) between the hidden states i = 0, 1
(H0,H1) and emission probabilities Bij (right) from the hidden
states to observed states j = 0, 1, 2 (O0, O1, O2) for the averaging
time-scale Ta = 32. . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
xvi
Chapter 1
Introduction
1.1 What is econophysics?
Econophysics was established in 1995 as an interdisciplinary science with close
association to physics on one hand and economics/finance on the other. It is
conducted by physicists who apply data analysis and multi-agent simulations
to various problems in economics and finance.
The advent of econophysics coincides with availability of large arrays of
financial/economic data due to the implementation of various computer sys-
tems for tracking transactional data, which itself is part of “big data” ex-
plosion (Sagiroglu and Sinanc, 2013). For instance, such data is recorded on
various stock exchanges such as NYSE and NASDAQ, while Thomson Reuters
and Bloomberg provide a wealth of financial information on prices of stocks,
derivatives, foreign exchange, and other securities.
In 1995, a group of physicists gathered in Kolkata to discuss results of their
research on several topics traditionally associated with economics and finance
rather than physics. At this conference, Stanley coined the term econophysics
to refer to the research in economics and finance conducted by physicists us-
ing various techniques borrowed from statistical mechanics and other relevant
disciplines in physics. It is similar in etymology to geophysics and biophysics,
which are concerned with understanding geological and biological phenom-
ena from the physics-oriented point of view using the methods developed in
relevant fields of physics such as solid state physics, hydrodynamics, thermo-
dynamics, statistical mechanics, and quantum mechanics.
The difference between physics and economics is wider than between physics
and, say, biology, for one cannot apply the laws of physics to economic systems
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1. Introduction
directly. Therefore, the emphasis of the studies at present is empirical, where
many techniques developed in physics for analysing the data can be applied
to the economic systems directly. Econophysics is different from economet-
rics, which is a branch of economics also concerned with analysing data, since
physicists analyse data in order to attempt to uncover the underlying laws
governing the behaviour of the economic and financial systems, whereas the
econometrist views data in terms of already established economic theories, i.e.
economics is not empirically based and mostly interprets economic data in
terms of axiomatically postulated theories.
Economic and financial systems typically involve a large number of inter-
acting agents and are complex by nature. Even when the agents act according
to a set of simple rules, the systems demonstrate emergent properties char-
acteristic of various complex chaotic systems encountered in physics, such as
weather. It is natural to address such systems with multi-agent simulations,
which are frequently invoked in the studies of various complex systems. So,
econophysics emerges as a multi-disciplinary science that studies economic
and financial systems empirically and via multi-agent simulations by means
of various techniques developed by physicists for analysing natural systems.
Several factors contributed to the emergence of econophysics in the 1990s,
such as data availability, widespread use of computers in research, and poor
record of the standard economic theory in making accurate quantitative pre-
dictions. Arguably, the most important factor is the availability of economics
and financial data, which became possible thanks to the introduction of various
computer systems in stock (and other) exchanges, foreign exchange market,
private and central banking, and governments. At present, most financial
transactions are conducted using computers, which record and store trans-
actional data and make it easily available for analysis. Most of this data is
proprietary and is not publicly available, but some of it is available or can be
acquired upon request. Stock exchanges, where shares of public companies
are traded, provide the most comprehensive source of data, since each trans-
action must be reported (both its volume, number of shares, and the price
are thus made available). The price of various securities, such as bonds and
derivatives, and foreign exchange is recorded in real time and is available for
future analysis. Interbank transactions in many countries are conducted via
computer platforms provided by the central banks, where detailed records of
each high-value interbank transaction are stored (see for instance Gallagher
et al. (2010)). These data are not publicly available but have been made
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1. Introduction
available in some form to various research groups.
Another important reason that contributed to the advent of econophysics
is the inability of the traditional economic theory to make accurate quantita-
tive predictions. Most economic analysis at present is conducted on the basis
of a theory usually referred to as neoclassical economics, which is also domi-
nant in economics education as evidenced by most standard textbooks, while
other economic theories are marginalised (see Keen (2011) for a critical anal-
ysis). Neoclassical economics puts emphasis on exchange of goods between
various agents in the economy. It makes the assumption that the market is
in equilibrium, where the supply of goods matches the demand, and that the
economic agents act rationally and pursue the goal of maximising their util-
ity. The same assumptions are made in modeling the financial system, while
the banking system is usually assumed to play an auxiliary role in facilitating
exchanges. These assumptions are challenged by heterodox economic theories
such as Post-Keynesian economics and by econophysics (Keen, 2011), where
economic and financial systems are typically found in non-equilibrium states
and agents may act irrationally. The economy and finance are treated as dy-
namical systems and the actions of the economic agents are not necessarily
directed towards utility maximisation since their judgement is based at best
on incomplete data even if they act rationally.
Even though most research in econophysics occurred in the last twenty
years, some results, which are treated as part of modern econophysics, date
back to the turn of the twentieth century. In a pioneering study, economist
Vilfredo Pareto (1848–1923) examined data on wealth and income distribu-
tion from different countries and time periods and found that the number of
persons as a function of wealth/income follows the power-law distribution,
also referred to as Pareto distribution when applied to economic systems (see
(Richmond et al., 2006) for a historical account). The empirical fact that it
was not a Gaussian was unexpected; it suggested that the social laws may be
fundamentally different from physical laws known at the time, e.g. thermody-
namics, where bell-shaped curves predominated.
Another pioneering contribution was delivered by mathematician Louis
Bachelier (1870–1946) who in his PhD thesis made use of a Gaussian random
walk, which is a common stochastic process also known as Brownian walk,
to describe the evolution of stock option prices (Bachelier (2011)). Benoit
Mandelbrot (1924–2010) investigated price changes in some markets and found
that they do not follow a Gaussian distribution, with many more price changes
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1. Introduction
in excess of what might be expected from a Gaussian distribution (Mandelbrot,
1967). The distribution of price changes exhibits what is known as fat tails
characteristic of power-law distributions.
There is a methodological overlap between econophysics and some hetero-
dox economic theories and approaches that use dynamical models or multi-
agent simulations to model the economy. Heterodox economics encompasses a
large number of diverse economic theories such as Austrian economics, in-
stitutional economics, Marxian economics, Post-Keynesian economics, and
many others (see Keen (2011) for a brief review). Various branches of post-
Keynesian economics are of particular interest in the context of macroeco-
nomics, for they devote particular attention to monetary economics by mod-
eling money and banking and their effect on the economy explicitly.
The assumption frequently made by these theories is that money is en-
dogenous, i.e. it is produced by the banking system in response to prevailing
economic conditions in the market. In this view, money and credit become
important independent variables in describing the state of the economy via
a set of differential equations. An equilibrium is possible but may be unsta-
ble and there could be multiple equilibria, or the system could be in a state
of stable perpetual oscillations. Using dynamical equations for modeling the
economy may be described as the top-down approach. In contrast, multi-agent
simulations start with the micro level by establishing the rules that govern the
individual behaviour of the agents. Here, the agents can be persons, groups of
people, companies, etc. The emergent properties of the system on the macro
level arise as a result of action of the agents.
There are also certain connections between econophysics and financial en-
gineering. Financial engineering provides mathematical (analytical) and com-
putational support to the finance industry in the form of price assessment of
various securities, risk assessment, and technical analysis for predicting price
changes. In particular, the Black-Scholes option pricing model introduced in
Black and Scholes (1973) has gained significant prominence in finance indus-
try; similar statistical models have been applied to other securities (Joshi,
2003). Just as Bachelier’s model, Black-Scholes’ model relies on Brownian
motion to describe stochastic properties of the price changes. However, this
assumption is inconsistent with the observed statistical properties of price
changes where fat tails are typical, and therefore the model ignores rare event
that can nonetheless have a significant impact on the market due to large price
changes Nassim (2007). Despite strong criticism expressed in Haug and Taleb
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1. Introduction
(2011), Black-Scholes model is widely used in academic research.
Risk analysis is an important part of an investment strategy and also
plays a significant role in banking, where limited reserves make the banks
vulnerable to external shocks. A commonly used measure of financial risk is
the so-called value-at-risk measure, which is widely used in banking and has
been analysed by econophysicists (Bouchaud and Potters, 2003). Technical
analysis is used in financial trading of stocks and other securities and tries
to predict the changes in the price of a security based on the past behaviour
by utilising various techniques such as trend-following (momentum) and mean
reversion (Taylor and Allen, 1992), which are of interest to econophysicists,
e.g. Tanaka-Yamawaki and Tokuoka (2007).
1.2 Progress in econophysics
Econophysics celebrated its 15th anniversary in 2010 and a large number of
reviews of its recent progress has been written in the past few years. Some of
these reviews are outlined below.
In a recent interview (Gangopadhyay, 2013), Stanley tells that the name
econophysics first appeared at the STATPHYS–Kolkata II conference held in
Kolkata in 1995, which was probably the first conference with significant focus
on econophysics. Stanley coined the term econophysics to refer to research
based on the methodology adopted in physics and largely done by physicists
who address problems in economics and finance, in the same sense biophysics
or geophysics refer to the application of physics to problems in biology and
geology. Stanley highlights the difference in culture between physicists, who
adopt an empirical approach guided by the data, and economists, who follow
a mathematical (axiomatic) approach guided by theories.
In a thesis piece published in Nature (Buchanan, 2013), Buchanan lists
a number of lasting contributions made by econophysicists to economics and
finance. Econophysics is partially responsible for: 1) Precise empirical facts
about financial markets such as power-law distribution of large price move-
ments and its self-similar scaling properties (Borland et al., 2005). 2) Instruc-
tive links between markets and other natural phenomena such as Omori law for
earthquake aftershocks and market behaviour following a large crash (Weber
et al., 2007), which suggests that market dynamics may be governed by some
general dynamic principles not specific to financial markets. 3) More realistic
models of markets represented as an ecology of interactive adaptive agents,
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1. Introduction
giving rise to fat-tailed distributions (see Hommes (2002) for a review), 4) New
qualitative features of markets, one of which is that market dynamics depends
on the diversity of participants’ strategic behaviour (Johnson et al., 2012).
Markets operate smoothly when the agents use many diverse strategies but
break down if the strategies become similar. In view of recent international
financial crises, the focus of many econophysics studies on market instability
acquires new significance. Buchanan lists the following findings: 1) A market,
where participants compete with one another by increasing leverage, is inher-
ently unstable (Thurner et al., 2012). 2) Widespread risk sharing in a network
of financial institutions raises systemic risk since a contagion can spread too
easily through an over-connected network (Battiston et al., 2012a). 3) Market
efficiency as it becomes more complete in terms of the range and availability
of financial instruments brings with it inherent market instability (Caccioli
and Marsili, 2010). Buchanan concludes by mentioning DebtRank (Battiston
et al., 2012b; Thurner and Poledna, 2013), a network measure developed by
econophysicists that assesses the fragility of a particular institution embedded
in a complex network of mutual financial dependencies encompassing a large
number of financial institutions.
The importance of collaboration between hard sciences like physics and
social (soft) sciences, which includes sociology and economics, is emphasised
in the review by Chakrabarti and Chakraborti (2010), which introduces a
special issue of Science and Culture devoted to econophysics and published in
2010 (commemorating 15 years of econophysics). In particular, with regard
to fostering these collaborative links the review points out such centres of
interdisciplinary research as the Indian Statistical Institute (India), which fo-
cuses on application of statistics to natural and social sciences, and the Santa
Fe Institute (US), which focuses on the study of complex systems, including
physical, evolutionary, and social systems. The authors view econophysics as a
new research field, whose objective is to turn economics into a natural science.
According to the authors, important developments in econophysics include the
following: 1) Deviation from Gaussian statistics in financial markets (Stan-
ley, Mantegna, Bouchaud, Farmer) building on the earlier work of Mandelbrot
and Fama. 2) Correlations among different stocks/sectors (Mantegna, Marsili,
Kertesz, Kaski, Iori, Sinha). 3) Income and wealth distributions and the rel-
evant kinetic models (Redner, Souma, Yakovenko, Chakrabarti, Chakraborti,
Richmond, Patriarca, Toscani). 4) Behavioural models of market bubbles and
crashes (Bouchaud, Lux, Stauffer, Gallegati, Sornette, Kaizoji). 5) Learning
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1. Introduction
in multi-agent game models and minority games in particular (Zhang, Marsili,
Savit, Kaski).
Bouchaud says in an opinion essay (Bouchaud, 2008) that compared with
physics the quantitative success of economics has been disappointing as it
demonstrates recurrent inability to predict and avert crises. Economists have
no framework for understanding ‘wild’ markets, which are not efficient and
where judgement errors get amplified leading to crashes. These characteris-
tics of financial markets are better addressed in the framework of complex
chaotic systems, the study of which is largely neglected by economists who
instead adopt a number of axiomatic assumptions such as the rationality of
economic agents who maximise their profits, the ‘invisible hand’ of the market
that gives the best outcome for society as a whole even as agents pursue their
own interests, and market efficiency, which states that market prices perfectly
reflect all known information about the assets. According to Bouchaud, sta-
tistical regularities should emerge in the behaviour of large populations in the
same way statistical laws of ideal gases emerge as a result of chaotic motion
of molecules.
An interesting perspective on the progress of econophysics is given by
Roehner (2010) who emphasises the importance of studying social phenom-
ena and intereactions in general rather than pure economic phenomena. In
view of the recent trend in physics to establish and promote departments
focusing on complex systems (colloids, polymers, sandpiles, traffic jams, neu-
ral networks, colonies of social insects, stock markets, financial derivatives),
Roehner reminds that the success of physics and chemistry has been his-
torically predicated on the search for simplicity rather than complexity, by
breaking down and simplifying complex systems and phenomena to isolate
the fundamental laws. In physics, spurious exogenous effects are eliminated
or isolated as much as possible in order to get to the fundamental properties.
According to Roehner the same approach may prove fruitful when examining
economic data, where the big players (corporations, media, etc) that can affect
the system on a macro level need to be taken into account.
Mathematical economist Rossner who works in nonlinear economic dy-
namics (using chaos and complexity theory) provides another perspective on
econophysics (Rosser Jr, 2006). Even though classical physics has long been
influencing ideas and models used in economics, Rossner notes that the inter-
action between economics and biology must not be underestimated, especially
in the context of evolutionary theory, as is witnessed by the appearance of
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1. Introduction
ecological economics (Daly and Farley, 2010). Similar sentiment is echoed by
Carbone et al. (2007), who introduces a special issue of Physica A devoted to
papers from the conference on “Application of Physics in Financial Analysis”
(2006). It is noted that early developments in economics (utility maximi-
sation, equilibrium) were strongly influenced by Newtonian physics, but the
interaction between physics and economics ceased around the 1930s with the
development of non-equilibrium thermodynamics. Since the 1930s, scientists
were engaged in the study of many-body systems in far from equilibrium con-
ditions, which frequently demonstrate emergent behaviour characteristic of
complex adaptive systems, where nonlinear interactions between a large num-
ber of individual particles (or agents) engender a dynamic behaviour of the
system on the macroscopic level, i.e. macroscopic behaviour emerges from mi-
croscopic interactions. Traditional models used in economics are not suitable
for representing these dynamic features, which led to proliferation of new dis-
ciplines such as econophysics, behavioural economics, evolutionary economics,
and neuroeconomics.
A number of heterodox economists sympathetic to econophysics discuss
its strengths and weaknesses in (Gallegati et al., 2006). Among the most
important contributions of econophysics, the authors see 1) the work on fat
tails of asset price changes, which has been established as a universal feature of
financial markets, and 2) the discovery that the empirical correlation matrix of
price changes of different assets and classes of assets is poorly determined, i.e.
dominated by noise (Laloux et al., 1999). The latter finding undermines the
capital asset pricing model (Sharpe, 1964), which is still highly regarded by
many economists, while the former is in conflict with the assumptions of the
Black-Scholes model of option pricing (Black and Scholes, 1973), a popular tool
used frequently in financial trading. The critique of econophysics by Gallegati
et al. (2006) is mostly aimed at models of income and wealth distributions,
most of which assume that money is conserved (i.e. money is akin to energy
in physical systems). The authors points out that this assumption is not valid
in the modern economy where the amount of money grows over time and
production plays as important a role as exchange.
In response to the issues raised by Gallegati et al. (2006), McCauley (2006)
points out that conservation laws in physics follow from space-time symme-
tries, but only one such symmetry (related to arbitrage) has been identified
in empirical analysis of price distributions, while money is not a conserved
quantity in the modern economy. One of the main points McCauley makes is
8
1. Introduction
that financial markets are non-stationary in the statistical sense (McCauley,
2008). Therefore any model that relies on stationarity, be it general equi-
librium theory or an agent-based model, is necessarily going to be invalid
over the relevant time scales. The difference between economics and econo-
physics from a philosophical and methodological perspective is considered by
Schinckus (2010), who argues in favour of the independence between eco-
nomics and econophysics. Economics adopts certain a priori postulates about
the real world, e.g. about the rationality of the economic agents or Gaussian
distribution of financial series returns, without adequate empirical verification.
A balanced view on the relationship between economics and econophysics
is taken in (Chen and Li, 2012), which introduces a number of papers from
Econophysics Colloquium 2010 held in Taipei and published in a special issue
of International Review of Financial Analysis. The authors document a long
history of connection between economics and classical physics, e.g. energy and
oscillations in physics versus utility and business cycles in economics. These
developments come under the rubric classical econophysics, which mostly con-
cerns 19th and early 20th centuries, or earlier. The authors trace the mul-
tidisciplinary character of modern econophysics, which they divide into four
main streams: (1) nonlinear dynamics (macroeconomic dynamics, non-linear
time series), (2) distributions of income, firm size, asset returns, etc, (3) so-
cial interactions (agent-based computational economics, Ising model, master
equation, cellular automata, kinetic and percolation model, minority games),
(4) complex networks and their statistical properties. The review sees econo-
physics as extending beyond statistical physics and finance, the focus of most
work to date.
A comprehensive review of the evolution of econophysics is presented in
(Ghosh and Chakrabarti, 2014). It provides succinct biographies of main
contributors to classical and modern econophysics, detailing their major con-
tributions. Both physicists and economists whose work can be classified as
econophysics are included; in all, over one hundred biographies. Besides in-
dividual scientists and research groups they lead, there are several institutes
that devote significant part of their activities to econophysics. These include:
Indian Statistical Institute, The Santa Fe Institute, and Institute for New Eco-
nomic Thinking. Other important centres of econophysics research include:
Boston University (Eugene Stanley), Saha Institute of Nuclear Physics (Bikas
Chakrabarti), Ecole Central Paris (Anirban Chakraborti, Damien Challet),
University of Maryland (Victor Yakovenko), University of Palermo (Rosario
9
1. Introduction
Mantegna), Kyoto University (Hideaki Aoyama), Leiden University (Diego
Garlaschelli), University of Houston (Joseph McCauley), and several others.
The review lists eighteen books on econophysics and eleven special journal
issues devoted to econophysics.
Progressive institutionalisation of econophysics is documented in (Gingras
and Schinckus, 2012) which uses bibliometric methods. The authors identi-
fied 242 econophysics papers published in the period from 1980 to 2008; 147
papers are published in refereed journals, 90 of which appeared in Physica
A, 27 in European Physical Journal B, and the others in Physical Review E,
Quantitative Finance, and Journal of Economic Behaviour & Organization.
Unsurprisingly, most of these journals have at least one prominent econo-
physicist in their board of editors. Besides these journals, the authors find the
following two economics journals citing econophysics papers: Journal of Eco-
nomic Dynamics and Control and Macroeconomic Dynamics, which are both
macroeconomics journals. A related study (Ghosh and Chakrabarti, 2014)
uses Google Scholar to gather statistics of papers that mention econophysics
to illustrate graphically the growth of econophysics since the term’s inception
in 1995. The study reveals that the growth in the number of papers has been
roughly linear with only a few papers in 1998 and nearly 1000 papers in 2012.
A similar analysis of papers that mention sociophysics also reveals a roughly
linear growth from about 2000, with over 200 papers published in 2012.
The first dedicated conference in econophysics was held in 1998 in Bu-
dapest. Since then, a number of econophysics conferences have been held in
various locations. There are two prominent conferences on econophysics held
annually: Econophysics Colloquium (first held in 2005 at The Australian Na-
tional University) and Econophysics-Kolkata (first held also in 2005). Closely
related to econophysics is The Workshop on Economics with Heterogeneous
Interacting Agents (first held in 1996 at University of Ancona). The Inter-
national Conference on Statistical Physics, which also runs annually, includes
econophysics as one of its topics.
1.3 Choosing the topics of research
The economy is a complex phenomenon which involves agents from countries
and companies down to individuals who are engaged in social and economic
interactions via production and consumption of goods and services over an
intricate network. The great majority of exchanges in the economy are facili-
10
1. Introduction
tated by the use of money. The origin and nature of money are controversial
and poorly understood despite its overwhelming presence in everyday life.
Money is usually described in economic texts through its properties as a unit
of account, a store of value, and a medium of exchange. Through credit, it
is also a driving force behind innovation and other entrepreneurial activity as
has been long recognised (Schumpeter, 1934). The origin of money is tradi-
tionally traced to primitive societies where money appears as some commonly
accepted commodity that simplifies direct exchange of goods (barter). This
widespread view is challenged by anthropological and historical studies that
find little evidence to corroborate it (Graeber, 2011). The view that equates
ancient money with precious metal coins emphasises the role of money as
the medium of exchange and store of value, whereas an alternative view that
money is based on credit and debt depends on the function of money as the
unit of account, which provides a record of mutual debt obligations between
members of an economic network. In modern economy, money is inextricably
linked with credit, whose role as a driver for innovation (and production) is
overshadowed by its more recent role as a driver of consumption and financial
speculation.
Money in modern economies is created by the central banks and the com-
mercial banking system via their credit facilities. Understanding its origin
and dynamics is essential for understanding modern economic and financial
systems. Unfortunately, a commonly accepted framework for understanding
and modeling money in econophysics has not been developed yet. Physical
concepts such as energy, which play crucial role in physics, fail to serve as a
viable theoretical representation of money that captures its endogenous char-
acter. Instead, research efforts have been concentrated on certain aspects and
manifestations of money via simplified modeling or data analysis of particular
segments of the economy where data is available. The data for such research
can come potentially from the banking system via tracking the majority (by
value) of monetary transfers in the economy and the records of credit is-
suance. Some indirect data are delivered by the foreign exchange market via
the exchange rate of various currencies, which is one of the richest sources of
data at present and can provide a valuable insight into the nature of money,
since exchange rates depend on the monetary policies of individual countries.
Dynamical models and multi-agent simulations of these systems provides ad-
ditional ways of addressing these issues.
Data availability determines in part the range of topics selected for re-
11
1. Introduction
search in this thesis. The foreign exchange data is provided by SIRCA via
the Thomson Reuters Tick History database (TRTH). SIRCA was established
in 1997 by a number of Australian and New Zealand universities (including
the University of Melbourne) as a not-for-profit company to provide finan-
cial data for research at the member institutions. SIRCA provides access to
TRTH database, which is one of the premier sources of data used by most lead-
ing financial institutions. A sample of high-value transactional data between
Australian banks has been kindly provided by the Reserve Bank of Australia
(RBA), which is the central bank in Australia. RBA administers a real-time
gross settlement systems (called The Reserve Bank Information and Transfer
System or RITS) for settling high-value transactions between banks in Aus-
tralia. The data settled in RITS mostly originate from the SWIFT payment
delivery system and Austraclear securities settlement system and therefore it
mostly concerns financial transactions (purchase of various securities and for-
eign exchange). Small-value transactions of mostly non-financial nature are
settled by the banks on a daily basis and do not involve RITS. The dataset
provided by the Reserve Bank includes value, source, and destination of each
transfer and thus provides an exciting opportunity to investigate the dynam-
ics of monetary flows in the interbank network in Australia, as well as the
statistical and topological properties of the network itself.
Networks
The study of networks in mathematics is known as graph theory. A graph
is a representation of a set of objects (nodes or vertices) some of which are
connected by links (edges). This subject has a long history and originates
with a paper written by Leonhard Euler (1707–1783) on the Seven Bridges
of Konigsberg (1736). Networks are frequently employed in computer science
to represent communication routes, data structures, flow of computations,
and so on. They are also used extensively in linguistics, chemistry, physics,
ecology, and many other disciplines. The significance of networks in social and
economic contexts is well recognised through early examples of case studies in
sociology prior to the nineties and subsequent modeling of network formation
and games on networks in the last twenty years (Jackson and Zenou, 2013). In
an economic context, important questions concerning networks that have been
addressed in research are 1) network formation and growth by using random
or strategic assembly rules and 2) games on a network, which study the effects
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1. Introduction
of network structure on the interaction between the nodes (Jackson, 2008a;
Goyal, 2012).
The properties of networks can be characterised by the average distance
between the nodes (the distance is the minimal number of edges that connect
two nodes), and maximum distance, which is also called the diameter of the
network, the degree distribution, where the degree of a node gives the number
of its neighbours, the assortativity, which measures the similarity (in terms
of the degree) between the neighbouring nodes, and a range of other charac-
teristics such as node clustering and centrality. Random networks, where the
degree of most nodes is close to the average degree of the network and the
overall degree distribution is binomial, played a significant role in the devel-
opment of graph theory (Bollobas, 1998). However, many real-world networks
frequently possess a small number of nodes with elevated degree. Many such
networks have been found to have a scale-free degree distribution (Caldarelli,
2007), i.e. the distribution is described by a power law without a characteristic
scale besides the upper cut-off. One of the earliest examples of such a network
was provided by the World Wide Web (Albert et al., 1999). Financial and
economic networks and in particular networks of transactional flows between
banks demonstrate similar characteristics. In these cases, the distribution of
distances is frequently peaked around small values, a phenomenon known as
the small-world effect.
If the links between the nodes of the network possess some directional
information (e.g. a web page has a hyperlink to another page, or one paper
cites another), the network is known as a directed network. In the context
of the banking networks, the edges correspond to transactional flows between
the banks, so that two edges pointing in the opposite directions are possible
between a single pair of banks. The banking networks are flow networks
with edges serving as conduits for payments (another important example of
a flow network is a transportation network). Flow networks are dynamic in
character; the flow characteristics are variable. Moreover, the structure of the
flow network can change with time when old nodes and edges are removed or
new ones are added.
In countries where the central bank operates a real-time gross settlement
(RTGS) system for settling large-value transactions between banks, the pay-
ments entered into RTGS are settled by the central bank by adjusting the
accounts of the participating banks held by the central bank. The member
banks are required to maintain a certain amount of funds in these accounts to
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1. Introduction
ensure real-time gross settlement. This contrasts with the deferred net settle-
ment applied to small payments made by individuals, which are tallied at the
end of a business day. Payment flows between banks cause the reserves of some
banks to increase, while those of the other banks decrease. Since RTGS oper-
ate in real time, these changes can be monitored by the central bank in real
time, which enables it if necessary to extend credit to banks whose reserves
are low, thereby improving the overall stability of the banking system.
Smooth functioning of RTGS is vital for the bank’s liquidity and systemic
stability of the banking system. It also has an effect on the conduct of mone-
tary policy by the central bank, since problems with liquidity in RTGS can lead
to rising interest rates in the money market and greater volatility and, on the
other hand, increased volatility on the money market may cause some banks
to stop submitting their payments through RTGS due to concerns over credit
risk. There has been a significant effort to analyse the properties of RTGS sys-
tems by means of standard neoclassical theoretical tools and also by means of
agent-based simulations. The latter have certain advantages over the former
as pointed out in (Alexandrova-Kabadjova et al., 2012), since 1) payment flows
do not follow any nice theoretical distribution, 2) each bank’s decision affects
the other banks so that a bank cannot solve its liquidity demand problem in
isolation, 3) the system design, which is becoming increasingly sophisticated,
strongly influences the participant’s incentives, 4) there is a high degree of
heterogeneity between the participants. Alexandrova-Kabadjova et al. (2012)
notes the lack of comprehensive empirical studies of these systems and the
participant’s behaviour in particular.
Banks whose reserves increase as a results of the flow of payments can
loan the excess in the overnight market to banks with depleted reserves at
the interest rate which is more favourable than the rate attached to their
accounts with the central bank. On the other hand, banks with depleted
reserves are anxious to restore them to a more comfortable level and therefore
seek funds in the overnight market. Thanks to overnight loans, the balance
of reserves in the banking system is restored. The flow of overnight loans
forms a network complementary to the flow network of other payments. This
creates an interesting causal relationship between the flows of loans and other
payments.
The topology of the interbank network is a significant factor in the overall
stability of the banking system, with important policy implications (Haldane
and May, 2011b). For instance, one of the recommendations is that capital
14
1. Introduction
and liquid asset requirements take into account the systemic importance of the
bank, characterised by its position in the network, with higher limits for banks
that are more interconnected and therefore can contribute to fast spread of a
financial contagion due to bad loans.
Agent-based simulations
A simulation is an approximate representation of a natural process or phe-
nomenon by means of a specific physical or computational model. Thanks
to computers, simulations have become an important part of the scientific
method, complementing deductive and inductive reasoning. Inductive reason-
ing uses experiments and observations to infer simple reproducible behaviours,
i.e. natural laws. Deductive reasoning starts with axioms (to be tested) or laws
(empirically verified) and uses logic to work out the physical consequences.
Simulations are employed when deductive reasoning alone is not sufficient to
derive the consequences that follow from the axioms and laws. Agent-based
simulations rely on the laws that govern the actions of individual autonomous
agents as they interact with one another and their environment. A cellular
automaton is a simplified example of an agent-based model, where the agents
have fixed positions on the grid and can only interact with their adjacent
neighbours, but even such a simple system can exhibit extremely complex
behaviour as exemplified by, for example, Conway’s game of life (Gardner,
1970).
Since the laws that govern the dynamics of social systems may be difficult
to infer from data, which are often sparse, agent-based simulations can play
an important role in sociology, psychology, politics, and economics. The pi-
oneering agent-based models used simple assumptions about the agents such
as zero-intelligence agents (Gode and Sunder, 1993) and swarm intelligence,
e.g. boids introduced by Reynolds (1987) to model bird flocks. Sugarscape
(Epstein, 1996) (see Gilbert and Troitzsch (2005) for an elementary introduc-
tion), which is used for simulating growing artificial societies, initiated another
set of agent-based models, where the agents demonstrate some goal-directed
behaviour. In the modern models, the agents can possess learning and plan-
ning and other advanced capabilities. Cooperation (e.g., in the prisoner’s
dilemma), reciprocity, prejudice and social influence have been investigated
via agent-based simulations (Axelrod, 1997). In sociophysics, agent-based
simulations have been used to address the universal features of group decision
15
1. Introduction
making, including democratic voting (Galam, 2012).
One of the well known applications of agent-based techniques in econo-
physics is the so-called minority game (Challet and Zhang, 1997), which is
based on the El Farol Bar problem (Arthur, 1994). Agents choose at each
step between two options, which can serve as a proxy for buying and selling
in a financial market, and those agents who adopted a less common option
(the minority view) gain, while the other agents lose. The agents have limited
memory that extends several steps into the past and they use adaptive strate-
gies from a limited pool of available strategies. The strategies are assessed
and selected based on their past performance. Under certain conditions, i.e.
when the number of agents is large enough compared to the number of avail-
able strategies, the asymptotic behaviour of the system depends strongly on
the initial conditions (Moro, 2004). The agents keep adapting their strate-
gies without ever settling on a particular strategy and no detailed balance is
observed. The minority game model is simple enough to address analytically
using techniques of statistical physics, but it does not explain some of the ob-
served facts about financial markets such as fat tails and volatility clustering
(the so-called stylised facts). The minority game model has become popular
in econophysics due to the fact that it shares many basic characteristics, such
as phase transitions, with other physical models.
The minority game in its simple form does not account for the stylised
facts mentioned previously, which requires more realistic agent-based models
of financial markets. For instance, Lux and Marchesi (1999) use two kinds
of agents: fundamentalists, who have a stabilising effect on the market, and
chartists, or noise traders, which can be either optimists or pessimists and
who are responsible for creating bubbles and crashes. The noise traders are
susceptible to herding behaviour. The agents can switch between these two
kinds depending on the market conditions; a random fluctuation of the mar-
ket price can cause many agents to become chartists, which increases volatility
thanks to an increased number of random buy and sell orders. Another no-
table agent-based model that simulates herding (Cont and Bouchaud, 2000)
is inspired by percolation theory, with clusters representing groups of agents
making the same decisions. This model and its subsequent variations explain
many features of financial markets; see Samanidou et al. (2007) for a review.
Other notable examples include the Santa Fe Artificial Stock Market model
(LeBaron, 2002), which uses heterogeneous agents who adapt their strategy
according to certain classifiers based on fundamentalist and chartist rules; the
16
1. Introduction
agent-based model for the stock market crash of October 19, 1987 (Kim and
Markowitz, 1989), which finds the cause of the crash in homogenisation of
algorithmic hedging strategies.
Asset exchange simulations
An asset exchange simulation is an agent-based simulation, where agents are
involved in the exchange of a certain asset, which can be identified with money,
for instance. The asset exchange simulations are of interest as they address
the question of income and wealth distribution and in particular wealth in-
equality characteristic of many human societies. Even though specific money
transfers among people occur for a variety of complex reasons, the aggregate
effect of these exchanges on the wealth distribution is simple enough, so that
simple exchange rules may be sufficient. Different rules of exchange can be
distinguished by their effect on the shape of the wealth distribution and the
degree of inequality.
The first multi-agent model of asset exchanges was developed in the 1980s
(Angle, 1986, 2002) in order to provide an explanation for the observed wealth
distributions, which can be closely approximated by the gamma distribution
(with the exception of the Pareto tail). In this model, the exchange amount
depends multiplicatively on the wealths of the agents. Various additive and
multiplicative models of asset exchanges were developed independently in (Is-
polatov et al., 1998); in the additive models, the amount exchanged does not
depend on the wealth of the agents. Besides multi-agent simulations, (Ispola-
tov et al., 1998) study the system analytically using a kinetic approach based
on the master equation, which balances the influx and outflow of agents at
specific wealths.
The model of Dragulescu and Yakovenko (2000) is conceptually the closest
to statistical mechanics of ideal gases. In this model, the exchange amount is
either fixed or equal to a random fraction of the total amount of money which
is conserved, the number of agents is fixed, and the pairing of agents in the
exchanges is completely random, provided that the agents have enough money
for an exchange. The behaviour of the agents in this model is reminiscent of
the behaviour of atoms in the ideal gas studied in the microcanonical ensem-
ble. Dragulescu and Yakovenko (2000) argue that the distribution of wealth
resulting after many additive exchanges must be given by the exponential
Boltzmann-Gibbs function, where the temperature parameter is related to
17
1. Introduction
the total wealth and the number of agents. Multi-agent simulations confirm
this supposition and also show that the Boltzmann entropy increases steadily
as expected in the ideal gas.
Chakraborti and Chakrabarti (2000) describe a multi-agent model of asset
exchanges in which agents are characterised by a propensity to save a certain
fraction of their wealth, while the rest of their wealth is made available for
random exchanges. This model is similar to the other multiplicative models
of asset exchange and the simulations based on this model yield in the steady-
state a distribution of wealth closely approximated by a gamma distribution.
Chatterjee et al. (2004) extended this model by using agents whose propensity
to save is assigned randomly from a uniform distribution. Once assigned, the
saving propensity of a given agent is fixed; agents with low propensity make
most of their wealth available for random exchanges, whereas agents with high
propensity save most of their wealth. Multi-agents simulations based on this
model yield a distribution of wealth which exhibits a power-law tail at high
wealths populated by agents with very high propensity to save who release
only a small fraction of their wealth for exchanges.
Foreign exchange
Foreign exchange markets are rich sources of data. However, the dynamics
of exchange rates is not well understood. The available data consist of the
indicative buy and sell prices offered by the so-called market makers (banks
and other financial institutions), while the records of actual transactions (their
value, source, and destination) are not available. Moreover, the structure of
the market is difficult to discover, since there is no central exchange where
transactions are processed and recorded. The total value of transactions on
the foreign exchange market is significant and plays a significant role in the
global economy.
Some insight into the structure and significance of the foreign exchange
market is provided by the Triennial Central Bank Survey prepared by the
Bank for International Settlement (BIS) every three years since 1989, with the
latest survey conducted in April 2013. According to the survey, the turnover
in foreign exchange markets averaged $5.3 trillion per day, of which swaps (for-
eign exchange transactions that reverse after a period of time) accounted for
$2.2 trillion per day and spots, which unlike swaps do not reverse, $2.0 trillion
per day. Reporting dealers accounted for 39% of turnover, smaller banks that
18
1. Introduction
are not participating in the survey as reporting dealers accounted for 24% of
turnover, pension funds, insurance companies, and the like contributed 11%,
while hedge funds and proprietary trading firms accounted for 11%. Trading
with non-financial customers (e.g. industrial firms) accounted for only 9% of
the turnover. The US dollar remains the dominant traded currency, followed
by the euro and Japanese yen. Most trading is done in the primary financial
centres such as London, New York, Singapore, and Tokyo. The survey found
that voice execution accounted for 43% of foreign exchange transactions, the
rest taken up by electronic execution, which includes single banks proprietary
trading systems, Thomson Reuters Matching and EBS platforms, as well as
other electronic communication networks such as Currenex, FX Connect, FX-
all, and Hotspot FX.
Patterns in foreign exchange series can be analysed in a number of ways by
means of the standard techniques of technical analysis routinely employed by
traders and by more exotic means such as artificial neural networks (ANN),
hidden Markov models (HMM), or dynamical models. One of the dynamical
model approaches was developed by Ilinski in (Ilinski, 2001) and relies on the
lattice gauge theory where the gauge is associated with arbitrage. On the
other hand, ANN and HMM do not rely on any underlying theory.
The model developed by Ilinski (2001) is based on the lattice-gauge theory,
where the gauge transformation is associated with the rescaling of currencies.
Indeed, whether the currency is expressed as a certain amount of dollars or
a certain amount of cents should not have any effect on the dynamics of the
exchange, i.e. invariance with respect to price dilation. The lattice represents
investors in 1) the two currencies in a given currency pair and 2) discrete time
steps. The curvature on the lattice depends on the arbitrage along elemen-
tary circular plaquettes (closed loops involving two consecutive time steps).
A similar model based on the arbitrage as a curvature of the gauge connection
was developed in (Young, 1999). The model can be extended to multiple cur-
rencies, or financial securities in general. The lattice geometry introduced by
Ilinski was used in (Dupoyet et al., 2010) to develop and perform numerical
simulations based on the Markov chain Monte Carlo technique, which yield the
distribution of price increments in agreement with the NASDAQ one-minute
price data over four orders of magnitude. Kostanjcar et al. (2011) derived the
discrete dynamics of asset price relations from the minimum arbitrage princi-
ple using Ilinki’s lattice geometry. Zhou and Xiao (2010) rewrite the model in
the form of a partial differential equation on a fibre bundle in covariant form.
19
1. Introduction
Sandpiles
The idea that the foreign exchange markets are hierarchically organised with
different traders contributing to the market dynamics on different time scales
(and possibly on different scale of trade value) leads to the notion of informa-
tion cascade in the market akin to the energy cascade observed in turbulence
Voit (2003). Similarly, the fact that local activity can precipitate a global
movement in the market in the right circumstances suggests that the behaviour
of avalanches in stochastic systems such as sandpiles may approximate some
of the processes occurring in the foreign exchange markets.
A sandpile is a cellular automaton that models some of the properties of
a pile of sand onto which grains of sand are dropped at random locations.
As grains accumulate, the local slope of the pile may become too steep to
support the grains, in which case an avalanche occurs, the grains in unstable
locations are redistributed in the pile until a new equilibrium position is found.
The sandpile is usually assumed to be located on a square grid, such that
the unstable grains at the edges of the pile can be assumed to drop off the
edges. Over the long term, the influx of grains due to random grain drops
is counterbalanced by the efflux of grains falling off the edges, so that the
sandpile enters stochastic equilibrium as soon as a certain number of grains is
deposited in the pile. Some drops that cause small avalanches do not result in
any efflux of grains from the pile, because the avalanches are contained inside
the pile. In this case, generally the internal stress of the sandpile configuration
increases, which means that a large avalanche is more likely to happen.
In terms of the relationship between the sandpiles and the foreign exchange
market, one can think of the grain drops as individual trades while avalanches
can be associated with the movements of the exchange rate. Then, a small
avalanche may be associated with a small random movement, while a large
avalanche is more indicative of a large systematic change in the exchange rate.
A sandpile simulation on a Sierpinski gasket has been used to represent market
behaviour (Ausloos et al., 2002; Ausloos, 2006).
A sandpile is a simplistic representation of the foreign exchange market,
but it is valuable in the following sense. The foreign exchange data available
for analysis consists of the time series of the exchange rate, which can be
easily converted into a series of the exchange rate differences, increments and
decrements. The time series of the exchange rate differences can be analysed in
order to discover any patterns in the series, for instance by applying a hidden
20
1. Introduction
Markov model to the series, which probabilistically associates an unobserved
(hidden) Markov chain to the observed series. However, whatever pattern is
discovered it is difficult to relate it to any structural properties of the market
since its structure is not available for analysis. On the other hand, the same
technique applied to a time series of avalanche power in a sandpile offers a
better chance of relating any pattern discovered by the HMM to the internal
structure of the sandpile, which is known precisely at all times.
1.4 Details of research projects
The chapters of the thesis are based on the four papers accepted (or submit-
ted) for publication in Physica A and The European Physical Journal B. The
following sections provide a brief introduction to the papers.
Australian interbank network
The research on the Australian interbank network is described in Chapter 2.
The data for the analysis of the interbank network originates from the real-
time gross settlement system operated by the Reserve Bank of Australia. It
covers five consecutive days in February 2007, when the financial markets were
relatively stable, and consists of records of all interbank transactions settled
through the RTGS (note that low-value transactions are settled on a net basis
daily and therefore are not present in the sample) including the amount, the
bank of the origin (the payer), and the bank of destination (the payee). The
dataset contains 123078 transactions with the combined value in excess of
A$848 billion.
Some of these transactions constitute overnight loans, which are payments
that are returned on the next day with the same amount plus some interest.
The interest can be expected to be close to the target rate set by the RBA,
which suggests a procedure to detect overnight loans and separate them from
the other payments. This procedure yields 897 overnight loans over the four
day period with the combined value of over A$42 billion (Friday is excluded
since the following business day is not present in the sample and, therefore,
the overnight loan detection procedure cannot be applied). Once loans and
other payments (nonloans) are separated, a detailed statistical analysis of the
two sets can be conducted. Furthermore, as the source and destination of
each loan and nonloan transaction are known, statistical properties of sent
21
1. Introduction
and received loans and nonloans can be analysed for each bank in the sample.
In particular, loan and nonloan daily imbalance, the difference between sent
and received payments, for each bank can be determined and the interplay
between loan and nonloan imbalances can be investigated.
The payments between the banks can be thought of as flows in the net-
work, whose vertices are banks. Typically, there are two flows going in the
opposite directions between large banks. The difference between them yields
the net flow between the banks. The flows can be measured by the number of
transactions or the total amount per day (in principle, intraday flows can be
similarly estimated, but precise time of transactions is not given in our sam-
ple). Naturally, the largest flows occur between big banks, whereas there are
no transactions and consequently no flows between many small banks. The
total number of nonloan flows is about 800 on a given day, out of 2970 possible
flows for the 55 banks present in the sample (there are about 470 net nonloan
flows per day). On the other hand, the number of loan flows is about 75 per
day (about 60 net loan flows per day). Daily variability of loan and nonloan
flows is investigated.
The properties of the network of net flows, both loan and nonloan, are
investigated by computing the degree distribution for in- and out-degree sepa-
rately, and the assortativity of the networks is estimated. Furthermore, visual
analysis of the topology of the net flows between the largest 12 banks is pre-
sented and the daily variability of the networks is discussed, with the view to
provide visual clues to the structure and stability of the banking network on
the basis of a limited data sample.
Asset transfers
In Section 3, a “giver scheme” of asset transfers is analysed. A large number
of agents exchange assets according to a rule which stipulates that the transfer
amount is proportional to the total amount of assets in possession of the giver
and is independent of the receiver. A numerical simulation of the giver scheme
reveals that the distribution of assets, i.e. the number of agents in a given asset
(or wealth) interval, quickly approaches a stable distribution regardless of the
initial distribution. Unfortunately, agent-based simulations are not sufficient
to accurately constrain the properties of the distribution of wealth, especially
in the tail of the distribution where the number of agents is small.
A novel procedure for computing the distribution of wealth in the steady
22
1. Introduction
state is presented in Chapter 3. In the steady state, the number of agents
arriving in each wealth interval is balanced by the number of agents departing
the interval. The distribution in the steady state can thus be analysed by
means of the master equation. The master equation for the giver scheme has
not been solved analytically, except for several special cases. In Chapter 3 a
new method to solve it numerically is presented. It relies on the Laplace trans-
form of the master equation, which gives a functional equation in the complex
plane that can be solved numerically by iterations. To obtain the wealth dis-
tribution, the numerical inverse Laplace transform has to be applied to the
solution of the functional equation. This technique is investigated for usabil-
ity for a wide range of the giver schemes and is applied to two representative
cases, which correspond to small and large constants of proportionality in the
exchange formula. The small value of the constant yields behaviour reminis-
cent of monetary exchanges, which correspond to buying and selling goods and
services in the economy. On the other hand, the large value is more appropri-
ate for monetary exchanges in gambling, where large sums of money change
hands frequently. Accurate numerical estimate of the steady-state distribu-
tion of wealth makes it possible to address the question of wealth inequality
in the giver scheme more precisely than it is possible to do with numerical
simulations.
The total amount of assets is conserved in the giver scheme and there are
no sources of sinks of assets, i.e. the system is conservative and the transfers
are similar to the exchange of energy in collisions between atoms in an ideal
gas. By analogy with the physical system, one expects the entropy, which
measures the level of disorder in the system, to increase until it reaches the
maximum when the agent-based simulation of asset transfer reaches the steady
state. The actual behaviour of the Boltzmann entropy in the giver scheme is
investigated by conducting numerical simulations of the asset transfer system.
In addition, a simple Markov chain random process that mimics some of
the features of the giver scheme is introduced. The Markov chain is discrete in
time and continuous in the state variable. It is asymmetric in the sense that
the increment is additive when it is positive and multiplicative otherwise. This
asymmetry corresponds to the asymmetry in the giver scheme where the loss
of the asset is proportional to the giver’s wealth while the gain can originate
from any agent in the whole population.
23
1. Introduction
Money flow dynamics
Chapter 4 is devoted to the critique of the gauge theory of arbitrage which
was introduced by Ilinski (2001) and applied to fast money flow dynamics.
The theory of fast money flow dynamics attempts to model the evolution of
currency exchange rates and stock prices on short, e.g. intraday, time scales.
It has been used to explain some of the heuristic trading rules, known as
technical analysis, that are used by professional traders in the equity and
foreign exchange markets. The study presented in chapter 4 demonstrates that
the choice of the input parameters used in Ilinski (2001) results in sinusoidal
oscillations of the exchange rate, in conflict with the results presented in Ilinski
(2001), because of an algebraic error in the derivation of the Euler-Lagrange
equations. It is also found that the dynamics predicted by the theory are
generally unstable in most realistic situations, with the exchange rate tending
to zero or infinity exponentially. Critical analysis of the action that governs
the dynamics of fast money flows is also given in chapter 4.
Temporal patterns in Abelian sandpiles
Abelian sandpile simulations on square two-dimensional grids of various sizes
are used to investigate memory effects in a sandpile. The objective is to see
if the observable pattern of “avalanches” in a slowly driven system with local
interactions like a sandpile can be related to its internal states. The sandpile
as a dynamical system can then provide a template, in the context of a toy
model, for analysing internal states and their time evolution in other complex
dynamical systems like the FX market where the internal states cannot be
observed.
A sandpile configuration (or state) at a given moment is represented as
a two-dimensional matrix of heights (or charges), which can also be thought
of as the number of grains in a given location in the sandpile. A grain drop
at a random location in the pile and a possible subsequent avalanche change
the configuration of the pile. However, most of the heights in the pile remain
unchanged after a single drop. As the number of drops and the avalanches
increases, the sandpile’s configuration gradually evolves away from the initial
configuration, which is assumed to be a recurrent one. In other words, the
sandpile is expected to gradually lose memory of the initial configuration.
Memory loss can be described by computing the difference between two
configurations, the initial configuration and a configuration at some later time,
24
1. Introduction
i.e. after a certain number of grain drops. The difference matrix represents
a detailed imprint of memory loss, which depends not only on the initial
configuration but also on the history of random grain drops. Since the latter
has nothing to do with the internal structure of the sandpile, the effect of
random grain drops can be diminished by computing the mean characteristics
of the difference matrix, such as the total number of sites with a given value
of the difference. The time dependence of such mean characteristics can then
be used as a convenient measure of memory loss. The effect of random drops
can be completely eliminated by combining a large number of simulations and
computing ensemble averaged mean quantities as a function of time delay from
the initial configuration in each simulation.
A complementary approach to the memory loss problem is to make use of
the hidden Markov modeling, which is effective at capturing temporal patterns
in the succession of observed states. In the case of Abelian sandpiles, many
properties can serve as observed states, e.g. the avalanche size (or power).
As the avalanche size at a particular moment in time depends on where a
randomly dropped grain lands in the pile, the randomness can be diminished
by using time averaged characteristics as the input for the HMM analysis. For
instance, the sequence of avalanche sizes can be averaged by computing the
mean of all avalanches that occurred during a contiguous sequence of grain
drops of certain fixed length, the averaging period. By comparing the results
of the HMM analysis based on the sequences obtained for different averaging
periods one can assess the effects of memory loss at different time-scales. These
results are described in chapter 5.
25
Chapter 2
Loan and nonloan flows in the
Australian interbank network
High-value transactions between banks in Australia are settled in the Reserve
Bank Information and Transfer System (RITS) administered by the Reserve
Bank of Australia. RITS operates on a real-time gross settlement (RTGS)
basis and settles payments and transfers sourced from the SWIFT payment
delivery system, the Austraclear securities settlement system, and the inter-
bank transactions entered directly into RITS. In this paper, we analyse a
dataset received from the Reserve Bank of Australia that includes all inter-
bank transactions settled in RITS on an RTGS basis during five consecutive
weekdays from 19 February 2007 inclusive, a week of relatively quiescent mar-
ket conditions. The source, destination, and value of each transaction are
known, which allows us to separate overnight loans from other transactions
(nonloans) and reconstruct monetary flows between banks for every day in
our sample. We conduct a novel analysis of the flow stability and examine
the connection between loan and nonloan flows. Our aim is to understand
the underlying causal mechanism connecting loan and nonloan flows. We find
that the imbalances in the banks’ exchange settlement funds resulting from
the daily flows of nonloan transactions are almost exactly counterbalanced by
the flows of overnight loans. The correlation coefficient between loan and non-
loan imbalances is about −0.9 on most days. Some flows that persist over two
consecutive days can be highly variable, but overall the flows are moderately
stable in value. The nonloan network is characterised by a large fraction of
persistent flows, whereas only half of the flows persist over any two consec-
utive days in the loan network. Moreover, we observe an unusual degree of
27
2. Loan and nonloan flows in the Australian interbank network
coherence between persistent loan flow values on Tuesday and Wednesday. We
probe static topological properties of the Australian interbank network and
find them consistent with those observed in other countries.
2.1 Introduction
Financial systems are characterised by a complex and dynamic network of
relationships between multiple agents. Network analysis offers a powerful way
to describe and understand the structure and evolution of these relationships;
background information can be found in Kolaczyk (2009), Jackson (2008b),
and Caldarelli (2007). The network structure plays an important role in de-
termining system stability in response to the spread of contagion, such as
epidemics in populations or liquidity stress in financial systems. The im-
portance of network studies in assessing stability and systemic risk has been
emphasised in Schweitzer et al. (2009) in the context of integrating economic
theory and complex systems research. Liquidity stress is of special interest in
banking networks. The topology of a banking network is recognised as one of
the key factors in system stability against external shocks and systemic risks
Haldane and May (2011a). In this respect, financial networks resemble eco-
logical networks. Ecological networks demonstrate robustness against shocks
by virtue of their continued survival and their network properties are thought
to make them more resilient against disturbances May et al. (2008). Often
they are disassortative in the sense that highly connected nodes tend to have
most of their connections with weakly connected nodes (see Newman (2003)
for details). Disassortativity and other network properties are often used to
judge stability of financial networks.
There has been an explosion in empirical interbank network studies in
the last years thanks largely to the introduction of electronic settlement sys-
tems. One of the first, reported in Boss et al. (2004), examines the Austrian
interbank market, which involves about 900 participating banks. The data
are drawn from the Austrian bank balance sheet database (MAUS) and the
major loan register (GKE) containing all high-value interbank loans above
e0.36× 106; smaller loans are estimated by means of local entropy maximisa-
tion. The authors construct a network representation of interbank payments
for ten quarterly periods from 1999 to 2003. They find that the network ex-
hibits small-world properties and is characterised by a power-law distribution
of degrees. Specifically, the degree distribution is approximated by a power
28
2. Loan and nonloan flows in the Australian interbank network
law with the exponent −2.01 for degrees & 40. This result, albeit with differ-
ent exponents, holds for the in- and out-degree distributions too (the exponent
is −3.1 for out-degrees and −1.7 for in-degrees). A recent study of transac-
tional data from the Austrian real-time interbank settlement system (ARTIS)
reported in Kyriakopoulos et al. (2009) demonstrates a strong dependence
of network topology on the time-scales of observation, with power-law tails
exhibiting steeper slopes when long time-scales are considered.
The network structure of transactions between Japanese banks, logged
by the Bank of Japan Financial Network system (BOJ-NET), is analysed in
Inaoka et al. (2004). The authors consider several monthly intervals of data
from June 2001 to December 2002 and construct monthly networks of inter-
bank links corresponding to 21 transactions or more, i.e. one or more trans-
action per business day on average. Truncating in this way eliminates about
200 out of 546 banks from the network. The resulting monthly networks have
a low connectivity of 3% and a scale-free cumulative distribution of degrees
with the exponent −1.1.
More than half a million overnight loans from the Italian electronic broker
market for interbank deposits (e-MID), covering the period from 1999 to 2002,
are analysed in De Masi et al. (2006). There are about 140 banks in the
network, connected by about 200 links. The degree distribution is found to
exhibit fat tails with power-law exponent 2.3 (2.7 for in-degrees and 2.15
for out-degrees), the network is disassortative, with smaller banks staying
on its periphery. In a related paper Iori et al. (2007), the authors make
use of the same dataset to uncover liquidity management strategies of the
participating banks, given the reserve requirement of 2% on the 23rd of each
month imposed by the central bank. Signed trading volumes are used as a
proxy for the liquidity strategies and their correlations are analysed. Two
distinct communities supporting the dichotomy in strategy are identified by
plotting the correlation matrix as a graph. The two communities are mainly
composed of large and small banks respectively. On average, small banks serve
as lenders and large banks as borrowers, but the strategies reversed in July
2001, when target interest rates in the Euro area stopped rising and started
to decrease. The authors also note that some mostly small banks tend to
maintain their reserves through the maintenance period. The evolution of the
network structure over the monthly maintenance period is examined in Iori
et al. (2008).
A study of the topology of the Fedwire network, a real-time gross settle-
29
2. Loan and nonloan flows in the Australian interbank network
ment (RTGS) system operated by the Federal Reserve System in the USA, is
reported in Soramaki et al. (2007). The study covers 62 days in the 1st quarter
of 2004, during which time Fedwire comprised more than 7500 participants
and settled 3.45× 105 payments daily with total value $1.3 trillion. It reveals
that Fedwire is a small-world network with low connectivity (0.3%), moderate
reciprocity (22%), and a densely connected sub-network of 25 banks responsi-
ble for the majority of payments. Both in- and out-degree distributions follow
a power law for degrees & 10 (exponent 2.15 for in-degrees and 2.11 for out-
degrees). The network is disassortative, with the correlation of out-degrees
equal to −0.31. The topology of overnight loans in the federal funds market
in the USA is examined in Bech and Atalay (2010), using a large dataset
spanning 2415 days from 1999 to 2006. It is revealed that the overnight loans
form a small-world network, which is sparse (connectivity 0.7%), disassorta-
tive (assortativity ranging from −0.06 to −0.28), and has low reciprocity of
6%. The reciprocity changes slowly with time and appears to follow the target
interest rate over the period of several years. A power law is the best fit for the
in-degree distribution, but the fit is only good for a limited range of degrees.
A negative binomial distribution, which requires two parameters rather than
one for a power law, fits the out-degree distribution best.
A comprehensive survey of studies of interbank networks is given in Imakubo
and Soejima (2010). The number of interbank markets being analysed con-
tinues to increase. For example, a study of the interbank exposures in Brazil
for the period from 2004 to 2006 was reported in Cajueiro and Tabak (2008).
A topological analysis of money market flows logged in the Danish large-value
payment system (Kronos) in 2006 was reported in Rørdam and Bech (2008),
where customer-driven transactions are compared with the bank-driven ones.
Empirical network studies have been used to guide the development of a net-
work model of the interbank market based on the interbank credit lending
relationships Li et al. (2010).
Establishing basic topological features of interbank networks is essential
for understanding these complex systems. Fundamentally, however, interbank
money markets are flow networks, in which links between the nodes correspond
to monetary flows. The dynamics of such flows has not been examined in depth
in previous studies, which mostly viewed interbank networks as static or slowly
varying. But the underlying flows are highly dynamic and complex. Moreover,
monetary flows are inhomogeneous; loan flows are fundamentally different
from the flows of other payments. Payments by the banks’ customers and the
30
2. Loan and nonloan flows in the Australian interbank network
banks themselves cause imbalances in the exchange settlement accounts of the
banks. For some banks, the incoming flows exceed the outgoing flows on any
given day; for other banks, the reverse is true. Banks with excess reserves lend
them in the overnight money market to banks with depleted reserves. This
creates interesting dynamics: payment flows cause imbalances, which in turn
drive compensating flows of loans. Understanding this dynamic relationship
is needed for advancing our ability to model interbank markets effectively.
In this paper, our objective is to define empirically the dynamics of inter-
bank monetary flows. Unlike most studies cited above, we aim to uncover the
fundamental causal relationship between the flows of overnight loans and other
payments. We choose to specialise in the Australian interbank market, where
we have privileged access to a high-quality dataset provided by the Reserve
Bank of Australia (RBA). Our dataset consists of transactions settled in the
period from 19 to 23 February 2007 in the Australian interbank market. We
separate overnight loans and other payments (which we call nonloans) using a
standard matching procedure. The loan and nonloan transactions settled on a
given day form the flow networks, which are the main target of our statistical
analysis. We compare the topology and variation of the loan and nonloan
networks and reveal the causal mechanism that ties them together. We inves-
tigate the dynamical stability of the system by testing how individual flows
vary from day to day. Basic network properties such as the degree distribution
and assortativity are examined as well.
2.2 Data
High-value transactions between Australian banks are settled via the Reserve
Bank Information and Transfer System (RITS) operated by the RBA since
1998 on an RTGS basis (Gallagher et al., 2010). The transactions are settled
continuously throughout the day by crediting and debiting the exchange set-
tlement accounts held by the RBA on behalf of the participating banks. The
banks’ exchange settlement accounts at the RBA are continuously monitored
to ensure liquidity, with provisions for intra-day borrowing via the intra-day
liquidity facility provided to the qualifying banks by the RBA. This obviates
the need for a monthly reserve cycle of the sort maintained by Italian banks
as discussed in Iori et al. (2008). The RITS is used as a feeder system for
31
2. Loan and nonloan flows in the Australian interbank network
Date Volume Value (A$109)
19-02-2007 19425 82.250620-02-2007 27164 206.102321-02-2007 24436 161.973322-02-2007 25721 212.135023-02-2007 26332 184.9202
Table 2.1: The number of transactions (volume) and their total value (in unitsof A$109) for each day.
transactions originating from SWIFT1 and Austraclear for executing foreign
exchange and securities transactions respectively. The member banks can also
enter transactions directly into RITS. The switch to real-time settlement in
1998 was an important reform which protects the payment system against
systemic risk, since transactions can only be settled if the paying banks pos-
sess sufficient funds in their exchange settlement accounts. At present, about
3.2 × 104 transactions are settled per day, with total value around A$168
billion.
The data comprise all interbank transfers processed on an RTGS basis by
the RBA during the week of 19 February 2007. During this period, 55 banks
participated in the RITS including the RBA. The dataset includes transfers
between the banks and the RBA, such as RBA’s intra-day repurchase agree-
ments and money market operations. The real bank names are obfuscated
(replaced with labels from A to BP) for privacy reasons, but the obfuscated
labels are consistent over the week. The transactions are grouped into separate
days, but the time stamp of each transaction is removed.
During the week in question, around 2.5 × 104 transactions were settled
per day, with the total value of all transactions rising above A$2 × 1011 on
Tuesday and Thursday. The number of transactions (volume2) and the total
value (the combined dollar amount of all transactions) for each day are given
in Table 2.1. Figure 2.1 shows the distribution of transaction values on a
logarithmic scale. Local peaks in the distribution correspond to round values.
The most pronounced peak occurs at A$106.
In terms of the number of transactions, the distribution consists of two
1Society for Worldwide Interbank Financial Telecommunication2The term “volume” is sometimes used to refer to the combined dollar amount of trans-
actions. In this paper, we only use the term “volume” to refer to the number of transactionsand “total value” to refer to the combined dollar amount. This usage follows the one adoptedby the RBA Gallagher et al. (2010).
32
2. Loan and nonloan flows in the Australian interbank network
Date Component 1 Component 2〈u〉 σ2u P 〈u〉 σ2u P
19-02-2007 4.00 1.12 0.81 6.68 0.68 0.1920-02-2007 3.55 0.72 0.43 5.73 1.49 0.5721-02-2007 3.66 0.86 0.55 5.86 1.43 0.4522-02-2007 3.87 1.01 0.68 6.42 1.07 0.3223-02-2007 3.82 0.87 0.61 6.12 1.19 0.39
Table 2.2: Mean 〈u〉, variance σ2u, and mixing proportion P of the Gaussianmixture components shown in Figure 2.1 (u = log10 v, where v is value).
approximately log-normal components, with lower-value transactions being
slightly more numerous. The standard entropy maximisation algorithm for a
Gaussian mixture model with two components (McLachlan and Peel, 2000)
produces a satisfactory fit with the parameters indicated in Table 2.2. The
lower- and higher-value components are typically centred around A$104 and
A$106 respectively. The high-value component is small on Monday (19-02-
2007) but increases noticeably on subsequent days, while the low-value com-
ponent diminishes. By value, however, the distribution is clearly dominated
by transactions above A$106, with the highest contribution from around A$2×108.
2.3 Overnight loans
The target interest rate of the RBA during the week of our sample was rt =
6.25% per annum. If the target rate is known, it is easy to extract the overnight
loans from the data by identifying reversing transactions on consecutive days.
A hypothetical interest rate can be computed for each reversing transaction
and compared with the target rate. For instance, suppose a transaction of
value v1 from bank A to bank B on day 1 reverses with value v2, from bank
B to bank A, on day 2. These transactions are candidates for the first and
second legs of an overnight loan from A to B. The hypothetical interest rate
for this pair of transactions is given by rh = 100% × 365 × (v2 − v1)/v1; note
that the quoted target rate is per annum. Since large banks participate in
many reversing transactions that can qualify as loans, we consider all possible
hypothetical pairs and prefer the one that gives rh closest to the target rate.
The algorithm for loan extraction is applied from Monday to Thursday; loans
issued on Friday cannot be processed since the next day is not available. A
33
2. Loan and nonloan flows in the Australian interbank network
Date Volume Value (A$109) Loan fraction
19-02-2007 185 7.50 9.12%20-02-2007 221 9.18 4.45%21-02-2007 226 11.08 6.84%22-02-2007 265 14.93 7.04%
Table 2.3: Statistics of the overnight loans identified by our algorithm: thenumber of loans (volume), the total value of the first leg of the loans (in unitsof A$109), and the fraction of the total value of the loans (first legs only) withrespect to the total value of all transactions on a given date.
similar procedure was pioneered by Furfine Furfine (2003); see also Ashcraft
and Duffie (2007).
The application of the above algorithm results in the scatter diagram
shown in Figure 2.2. There is a clearly visible concentration of the revers-
ing transaction pairs in the region v > 2 × 105 and |rt − rh| < 0.5% (red
box). We identify these pairs as overnight loans. Contamination from non-
loan transaction pairs that accidentally give a hypothetical rate close to the
target rate is insignificant. By examining the adjacent regions of the diagram,
i.e. v > 2× 105 and rh outside of the red box, we estimate the contamination
to be less than 2% (corresponding to ≤ 5 erroneous identifications per day).
It is also possible that some genuine loans fall outside our selection criteria.
However, it is unlikely that overnight interest rates are very different from the
target rate; and the lower-value transactions (below A$104), even if they are
real loans, contribute negligibly to the total value.
We identify 897 overnight loans over the four days. A daily breakdown is
given in Table 2.3. Here and below, we refer to the first leg of the overnight
loans as simply loans and to all other transactions as nonloans. The loans
constitute less than 1% of all transactions by number and up to 9% by value
(cf. Tables 2.1 and 2.3). The distribution of loan values and interest rates is
shown in Figures 2.3a and 2.3b. The interest rate distribution peaks at the
target rate 6.25%. The mean rate is within one basis point (0.01%) of the
target rate, while the standard deviation is about 0.07%. The average interest
rate increases slightly with increasing value of the loan; a least-squares fit
yields rh = 6.248 + 0.010 log10(v/A$106).
The same technique can be used to extract two-day and longer-term loans
(up to four-day loans for our sample of five consecutive days). Using the
same selection criteria as for the overnight loans, our algorithm detects 27,
34
2. Loan and nonloan flows in the Australian interbank network
log10v
23-02-2007
log10v
22-02-2007
21-02-2007
20-02-2007
19-02-2007
01
23
45
67
89
10
01
23
45
67
89
10
01
23
45
67
89
10
01
23
45
67
89
10
01
23
45
67
89
10
0
500
1000
15000
500
1000
15000
500
1000
1500
Figure
2.1:
Thedistribution
oftran
sactionvaluesv(inAustralian
dollars)on
alogarithmic
scale,
withbin
size
∆log10v=
0.1;
thevertical
axisisthenumber
oftran
sactionsper
bin.Com
pon
ents
oftheGau
ssianmixture
model
areindicated
bythedashed
curves;thesolidcu
rveisthesum
ofthetw
ocompon
ents.Thedottedhistogram
show
stherelative
contribution
oftran
sactions
atdifferentvalues
tothetotalvalue(tocompute
thedottedhistogram
wemultiply
thenumber
oftran
sactionsin
abin
bytheir
value).
35
2. Loan and nonloan flows in the Australian interbank network
r h
log10 v
2 3 4 5 6 7 8 9
4.5
5
5.5
6
6.5
7
7.5
8
Figure 2.2: Hypothetical interest rate rh versus value of the first leg of thetransaction pairs detected by our algorithm, with no restrictions on value orinterest rate. The dotted rectangle contains the transactions that we identifyas overnight loans. The least-squares fit is shown with a solid red line.
67, and 24 two-day loans, with total values A$1.3, A$2.2, and A$1.4 billion,
on Monday, Tuesday, and Wednesday, respectively. The total value of the
two-day loans is 1.5%, 1.0%, and 0.9% of the total transaction values on these
days respectively.
2.4 Nonloans
We display the distributions of the incoming and outgoing nonloan transac-
tions, for which the bank is the destination and the source respectively, for
the six largest banks in Figure 2.4. The distributions are similar to the to-
tal distribution shown in Figure 2.1, with the notable exception of BA (see
below). There is also an unusually large number of A$106 and A$400 transac-
tions from W to T on Monday. Note that the daily imbalance for each bank
is mostly determined by the highest value transactions; large discrepancies
between incoming and outgoing transactions at lower values are less relevant.
The distribution for BA is clearly bimodal; it contains an unusually high
proportion of transactions greater than A$106. Moreover, below A$106, in-
coming transactions typically outnumber outgoing ones by a large amount.
BA is also involved in many high value transactions that reverse on the same
day. These transactions probably correspond to the central bank’s repurchase
agreements, which facilitate intra-day liquidity of the banks (Campbell, 1998).
The banks shown in Figure 2.4 are also the largest in term of the number
36
2. Loan and nonloan flows in the Australian interbank network
log10v
22-02-2007
21-02-2007
20-02-2007
19-02-2007
45
67
89
10
45
67
89
10
45
67
89
10
45
67
89
10
0
10
20
30
400
10
20
30
400
10
20
30
400
10
20
30
40
(a)
r h
66.25
6.5
66.25
6.5
66.25
6.5
66.25
6.5
0
20
40
600
20
40
600
20
40
600
20
40
60
(b)
Figure
2.3:
(a)Thedistribution
ofloan
valuesvon
alogarithmic
scale.
Thevertical
axisisthenumber
ofloan
sper
bin
forbin
size
∆log10v=
0.25.Thedottedlineis
thesamedistribution
multiplied
bythevaluecorrespon
dingto
each
bin
(inarbitrary
units).Thedateof
thefirstlegof
theloan
sis
indicated
.(b)Thedistribution
ofloan
interest
ratesr h.Thevertical
axis
isthe
number
ofloan
sper
bin
forbin
size
∆r h
=0.01.Thedateof
thefirstlegof
theloan
sis
indicated
.Themeanan
dstan
dard
deviation
are6.25%
and0.08%
onMon
day
(19-02-2007),an
d6.26%
and0.07%
ontheother
days.
37
2. Loan and nonloan flows in the Australian interbank network
of transactions, with the exception of BA. The rank order by the number of
transactions matches that by value. For D, which is the largest, the number
of nonloan transactions reaches 48043 over the week. By the number of trans-
actions, the order of the top twelve banks is D, BP, AV, T, W, AH, AF, U,
AP, BI, BA, P. By value, the order is D, BP, AV, BA, T, W, BG, U, A, AH,
AB, BM. The situation is similar when considering the overnight loans. By
value, AV, D, BP, and T dominate. For these four banks, weekly total loans
range from A$11.5 to A$18 billion and number from 254 to 399. For the other
banks the total loan value is less than A$3 billion.
In view of the discussion above, it is noteworthy that Australia’s retail
banking system is dominated by four big banks (ANZ, CBA, NAB, andWBC)3
that in February 2007 accounted for 65% of total resident assets, according to
statistics published by Australian Prudential Regulation Authority (APRA);
see http://www.apra.gov.au for details. The resident assets of the big four
exceeded A$225 billion each, well above the next largest retail bank, St George
Bank Limited4 (A$93 billion). The distinction between the big four and the
rest of the banks in terms of cash and liquid assets at the time was less clear,
with Macquarie Bank Limited in third position with A$8 billion. According
to APRA, cash and liquid assets of the big four and Macquarie Bank Limited
accounted for 56% of the total.
2.5 Loan and nonloan imbalances
In order to maintain liquidity in their exchange settlement accounts, banks
ensure that incoming and outgoing transactions roughly balance. However,
they do not control most routine transfers, which are initiated by account
holders. Therefore, the imbalances arise. On any given day, the nonloan
imbalance of bank i is given by
∆vi = −∑
j
∑
k
vk(i, j) +∑
j
∑
k
vk(j, i), (2.1)
where {vk(i, j)}k is a list of values of individual nonloan transaction from bank
i to bank j, settled on the day. The nonloan imbalances are subsequently
compensated by overnight loans traded on the interbank money market. The
3Australia and New Zealand Banking Group, Commonwealth Bank of Australia, Na-tional Australia Bank, and Westpac Banking Corporation.
4In December 2008, St George Bank became a subsidiary of Westpac Banking Corpora-tion.
38
2. Loan and nonloan flows in the Australian interbank network
19-02-2007 20-02-2007 21-02-2007 22-02-2007nonloans loans nonloans loans nonloans loans nonloans loans
D −0.51 +0.12 −0.28 +0.12 −0.76 +0.45 −1.25 +1.44BP +2.08 −1.64 +0.80 −0.59 +1.38 −0.85 +0.16 +0.82AV −0.32 −0.17 +1.39 −0.79 +0.55 −0.65 +1.08 +0.25BA +0.03 −0.19 −0.10 −0.31 −0.32 −0.05 −1.53 −0.64T −0.76 +1.10 −0.75 +0.68 −0.62 +0.64 −0.21 +0.27W −0.09 +0.07 +0.08 +0.26 −0.36 +0.41 +0.36 −0.37
Table 2.4: Loan and nonloan imbalances for the six largest banks (in units ofA$109).
loan imbalances are defined in the same way using transactions corresponding
to the first leg of the overnight loans. Note that we do not distinguish between
the loans initiated by the banks themselves and those initiated by various
institutional and corporate customers. For instance, if the funds of a corporate
customer are depleted, this customer may borrow overnight to replenish the
funds. In this case, the overnight loan is initiated by an account holder,
who generally has no knowledge of the bank’s net position. Nevertheless, the
actions of this account holder in acquiring a loan reduce the bank’s imbalance,
provided that the customer deposits the loan in an account with the same
bank.
The loan and nonloan imbalances for the six largest banks are given in
Table 2.4. The data generally comply with our assumption that the overnight
loans compensate the daily imbalances of the nonloan transactions. The most
obvious exception is for BA on Thursday (22-02-2007), where a large negative
nonloan imbalance is accompanied by a sizable loan imbalance that is also
negative. Taking all the banks together, there is a strong anti-correlation
between loan and nonloan imbalances on most days. We see this clearly in
Figure 2.5. The Pearson correlation coefficients for Monday through Thursday
are −0.93, −0.88, −0.95, −0.36. It is striking to observe that many points
fall close to the perfect anti-correlation line. The anti-correlation is weaker on
Thursday (crosses in Figure 2.5), mostly due to BA and AV.
A correlation also exists between the absolute values of loan imbalances
and the nonloan total values (incoming plus outgoing nonloan transactions);
the Pearson coefficients are 0.74, 0.75, 0.66, 0.77 for Monday through Thurs-
day. This confirms the intuitive expectation that larger banks tolerate larger
loan imbalances.
39
2. Loan and nonloan flows in the Australian interbank network
W/34/+
0.3
6W
/20/-0
.36
W/27/+
0.0
8W
/7/-0
.09
T/30/-0
.21
T/21/-0
.62
T/29/-0
.75
T/15/-0
.76
BA/30/-1
.53
BA/30/-0
.32
BA/33/-0
.10
BA/23/+
0.0
3
AV/51/+
1.0
8AV/37/+
0.5
5AV/51/+
1.3
9AV/20/-0
.32
BP/57/+
0.1
6BP/41/+
1.3
8BP/57/+
0.8
0BP/17/+
2.0
8
D/66/-1
.25
D/51/-0
.76
D/60/-0
.28
D/15/-0
.51
01
23
45
67
89
01
23
45
67
89
01
23
45
67
89
01
23
45
67
89
0
50
100
150 0
100
200 0
20
40 0
100
200 0
100
200 0
100
200
300
Figu
re2.4:
Thedistrib
ution
ofnon
loantran
sactionvalu
esof
thesix
largestban
ksfor
Mon
day
throu
ghThursd
ay(from
leftto
right);
theban
ksare
selectedbythecom
bined
valueof
incom
ingan
dou
tgoingtran
sactionsover
theentire
week
.Black
andred
histogram
scorresp
ondto
incom
ing(ban
kisthedestin
ation)an
dou
tgoing(ban
kisthesou
rce)tran
sactions;red
histogram
sare
filled
into
improve
visib
ility.Theban
ks’an
onymou
slab
els,thecom
bined
daily
valueof
theincom
ingan
dou
tgoingtran
sactions,
andthedaily
imbalan
ce(in
comingminusou
tgoing)
arequoted
atthetop
leftof
eachpan
el(in
units
ofA
$109).
Thehorizon
talax
isis
thelogarith
mof
valuein
A$.
40
2. Loan and nonloan flows in the Australian interbank network
log10|∆
l|
log10 v
∆l
∆v
6 7 8 9 10 11−2 −1 0 1 25.5
6
6.5
7
7.5
8
8.5
9
9.5
−2
−1
0
1
2
Figure 2.5: Left: loan imbalance ∆l vs nonloan imbalance ∆v for individualbanks and days of the week (in units of A$109). Right: the absolute value ofloan imbalance |∆l| vs nonloan total value (incoming plus outgoing transac-tions) for individual banks and days of the week. Thursday data are markedwith crosses.
2.6 Flow variability
For each individual source and destination, we define the nonloan flow as
the totality of all nonloan transactions from the given source to the given
destination on any given day. The value of the flow is the sum of the nonloan
transaction values and the direction is from the source to the destination. On
any given day, the value of the flow from bank i to bank j is defined by
vflow(i, j) =∑
k
vk(i, j), (2.2)
where {vk(i, j)}k is a list of values of individual nonloan transaction from i to
j on the day. For example, all nonloan transactions from D to AV on Monday
form a nonloan flow from D to AV on that day. The nonloan transactions
in the opposite direction, from AV to D, form another flow. A flow has zero
value if the number of transactions is zero. Typically, for any two large banks
there are two nonloan flows between them. The loan flows are computed in a
similar fashion.
Nonloan flows
There are 55 banks in the network, resulting in Nflow = 2970 possible flows.
The actual number of flows is much smaller. The typical number of nonloan
flows is ∼ 800 on each day (the actual numbers are 804, 791, 784, 797). Even
though the number of nonloan flows does not change significantly from day to
day, we find that only about 80% of these flows persist for two days or more.
The other 20% are replaced by different flows, i.e. with a different source
41
2. Loan and nonloan flows in the Australian interbank network
and/or destination, on the following day. Structurally speaking, the network
of nonloan flows changes by 20% from day to day. However, persistent flows
carry more than 96% of the total value.
Even when the flow is present on both days, its value is rarely the same.
Given that 80% of the network is structurally stable from day to day, we as-
sess variability of the network by considering persistent flows and their values
on consecutive days. Figure 2.6 shows the pairs of persistent flow values for
Monday and Tuesday, Tuesday and Wednesday, and Wednesday and Thurs-
day. If the flow values were the same, the points in Figure 2.6 would lie on
the diagonals. We observe that the values of some flows vary significantly, es-
pecially when comparing Monday and Tuesday. Moreover, there is a notable
systematic increase in value of the flows from Monday to Tuesday by a factor
of several, which is not observed on the other days. For each pair of days
shown in Figure 2.6, we compute the Pearson correlation coefficient, which
gives 0.53 for Monday and Tuesday, 0.70 for Tuesday and Wednesday, and
0.68 for Wednesday and Thursday.
To characterize the difference between the flows on different days more
precisely, we compute the Euclidean distance between normalised flows on
consecutive days. We reorder the adjacency matrix {vflow(i, j)}ij of the flow
network on day d as an Nflow-dimensional vector vd representing a list of all
flows on day d (d = 1, 2, . . . , 5). For each pair of consecutive days we compute
the Euclidean distance between normalized vectors vd/|vd| and vd+1/|vd+1|,which gives 0.62, 0.50, 0.50 for all flows and 0.61, 0.49, 0.49 for persistent
flows (the latter are computed by setting non-persistent flows to zero on both
days). Since the flow vectors are normalized, these quantities measure random
flow discrepancies while systematic deviation such as between the flows on
Monday and Tuesday are ignored. For two vectors of random values uniformly
distributed in interval (0, 1), the expected Euclidean distance is 0.71 and the
standard deviation is 0.02 for the estimated number of persistent nonloan flows
of 640. So the observed variability of the nonloan flows is smaller than what
one might expect if the flow values were random.
Loan flows
Variability of the loan flows is equally strong. The number of loan flows varies
from 69 to 83 (actual numbers are 69, 75, 77, 83). Only about 50% of these
flows are common for any two consecutive days. Moreover, persistent flows
42
2. Loan and nonloan flows in the Australian interbank network
log10von22-02-2007
log10von21-02-2007
log10von21-02-2007log10von20-02-2007
log10von20-02-2007
log10von19-02-2007
01
23
45
67
89
10
01
23
45
67
89
10
01
23
45
67
89
10
0123456789
10
0123456789
10
0123456789
10
Figure
2.6:
Non
loan
flow
valuepairs
onon
eday
(horizon
talax
is)an
dthenext(verticalax
is).
Only
flow
spresenton
bothdays
areconsidered
.Flowsthat
donot
chan
gelieon
thediagonal
(red
dottedline).Thesolidlineis
theweigh
tedorthogon
alleast
squares
fitto
thescatterdiagram
;theweigh
tshavebeendefi
ned
toem
phasizepoints
correspon
dingto
largeflow
s.
log10von22-02-2007
log10von21-02-2007
log10von21-02-2007
log10von20-02-2007
log10von20-02-2007
log10von19-02-2007
56
78
910
56
78
910
56
78
910
56789
10
56789
10
56789
10
Figure
2.7:
AsforFigure
2.6butforloan
flow
s.
43
2. Loan and nonloan flows in the Australian interbank network
carry only about 65% of the total value of the loan flows on any given day,
cf. 80% of nonloan flows. For persistent loan flows, the Pearson correlation
coefficients are 0.63, 0.90, and 0.76 for the consecutive pairs of days starting
with Monday and Tuesday. The correlation is generally similar to that of the
nonloan flows, with the notable exception of the loan flows on Tuesday and
Wednesday, when the sub-network of persistent loan flows appears to be more
stable.
The Euclidean distances between the normalized loan flows for each pair
of consecutive days are 0.85, 0.68, 0.73 for all flows and 0.63, 0.44, and 0.44
for persistent flows. For two vectors of random values uniformly distributed
in interval (0, 1), the expected Euclidean distance is 0.7 and the standard
deviation is 0.1 for the estimated number of persistent loan flows of 40. So
the observed variability of the persistent loan flows is much smaller than what
one might expect if the flow values were random.
Relation between nonloan and loan flows
Some loan flows do not have corresponding nonloan flows between the same
nodes on the same day. These flows carry about 14% of loan value on Mon-
day, and about 7% on Tuesday through Thursday. Nonloan flows that have
corresponding loan flows account for 35% to 48% of all nonloan flows by value,
even though the number of these flows is less than 10% of the total.
To improve the statistics, we aggregate the flows on all four days. Fig-
ure 2.8 shows nonloan and corresponding loan flow values. We fail to find any
significant correlation between loan and nonloan flows (Pearson coefficient is
0.3). The correlation improves if we restrict the loan flows to those consisting
of three transactions or more; such flows mostly correspond to large persis-
tent flows. In this case the Pearson coefficient increases to 0.6; banks that
sustain large nonloan flows can also sustain large loan flows, even though the
loan flows on average are an order of magnitude lower than the corresponding
nonloan flows. The lack of correlation when all loans are aggregated is due to
the presence of many large loans that are not accompanied by large nonloan
transactions, and vice versa.
44
2. Loan and nonloan flows in the Australian interbank network
2.7 Net flows
The net flow between any two banks is defined as the difference of the opposing
flows between these banks. The value of the net flow equals the absolute value
of the difference between the values of the opposing flows. The direction of the
net flow is determined by the sign of the difference. If vflow(i, j) > vflow(j, i),
the net flow value from i to j is given by
vnet(i, j) = vflow(i, j) − vflow(j, i). (2.3)
For instance, if the flow from D to AV is larger than the flow in the opposite
direction, then the net flow is from D to AV.
General properties
The distributions of net loan and nonloan flow values are presented in Fig-
ure 2.9. The parameters of the associated Gaussian mixture models are quoted
in Table 2.5. The distribution of net nonloan flow values has the same general
features as the distribution of the individual transactions. However, unlike in-
dividual transactions, net flow values below A$104 are rare; net flows around
A$108 are more prominent.
There are on average around 470 net nonloan flows each day. Among these,
roughly 110 consist of a single transaction and 50 consist of two transactions,
mostly between small banks. At the other extreme, net flows between the
largest four banks (D, BP, AV, T) typically have more than 103 transactions
per day each. Overall, the distribution of the number of transactions per net
flow is approximated well by a power law with exponent α = −1.0± 0.2:
Nnet(n) ∝ nα, (2.4)
whereNnet(n) is the number of net nonloan flows that consist of n transactions
(n ranges from 1 to more than 1000). This is consistent with the findings for
Fedwire reported in Bech and Atalay (2010) (see right panel of Fig. 14 in Bech
and Atalay (2010)).
There are roughly 60 net loan flows each day. As many as 40 consist of
only one transaction. On the other hand, a single net loan flow between two
large banks may comprise more than 30 individual loans. The distribution
of the number of transactions per net loan flow is difficult to infer due to
poor statistics, but it is consistent with a power law with a steeper exponent,
45
2. Loan and nonloan flows in the Australian interbank network
log10l
log10 v
5 6 7 8 9 105
6
7
8
9
10
Figure 2.8: Loan flow values versus nonloan flow values combined over fourdays. Triangles correspond to loan flows with three or more transactions perflow. The solid line is the orthogonal least squares fit to the scatter diagram;the weighting is the same as in Figure 2.6.
Date Component 1 Component 2〈u〉 σ2u P 〈u〉 σ2u P
19-02-2007 5.14 1.88 0.60 7.51 0.36 0.4020-02-2007 5.70 2.17 0.51 7.82 0.50 0.4921-02-2007 5.73 1.97 0.52 7.72 0.44 0.4822-02-2007 5.78 2.06 0.57 7.86 0.45 0.43
Table 2.5: Mean 〈u〉, variance σ2u, and mixing proportion P of the Gaussianmixture components appearing in Figure 2.9 (u = log10 v).
−1.4± 0.2, than that of the nonloan distribution. There are no net loan flows
below A$105 or above A$109. Comparing net loan and nonloan flows, it is
obvious that net loan flows cannot compensate each and every net nonloan
flow. Not only are there fewer net loan flows than nonloan flows, but the total
value of the former is much less than the total value of the latter.
Net loan and net nonloan flows are not correlated; the correlation coeffi-
cient is 0.3. Restricting net loan flows to those that have three transactions or
more does not improve the correlation. If a net loan flow between two banks
46
2. Loan and nonloan flows in the Australian interbank network
log10v
22-02-2007
log10v
21-02-2007
20-02-2007
19-02-2007
01
23
45
67
89
10
01
23
45
67
89
10
05
10
15
20
25
3005
10
15
20
25
30
Figure
2.9:
Thedistribution
ofvalues
ofnet
non
loan
flow
s(black
histogram
)on
alogarithmic
scalewithbin
size
∆log10v=
0.1.
Thecompon
ents
oftheGau
ssianmixture
model
areindicated
withthedashed
curves;thesolidcu
rveis
thesum
ofthetw
ocompon
ents.Net
loan
flow
sareoverplotted
inred.Thevertical
axis
counts
thenumber
ofnet
flow
sper
bin.
47
2. Loan and nonloan flows in the Australian interbank network
was triggered to a significant degree by the magnitude and the direction of
net nonloan flow between these bank, one expects a correlation between net
loan and nonloan flows. Our examination shows that in this respect loan flows
are decoupled from nonloan flows. The connection between them is indirect.
Namely, nonloan flows cause an imbalance in the account of each bank, which
is subsequently compensated by loan flows, which are largely unrelated to the
nonloan flows that caused the imbalance.
Degree distribution and assortativity
We define the in-degree of node i as the number of net flows that terminate
at i, i.e. the number of net flows with destination i, and the out-degree as
the number of net flows that originate from i, i.e. the number of net flows
with source i. The degree distribution of the nonloan networks is shown in
Figure 2.10a. Node BA has the highest in-degree of 37 on Monday, but on the
other days it drops to 15 on average, while the out-degree is 11.75 on average
for this node. The highest in-degrees are usually found among the four largest
banks (D, BP, AV, T); the only exception is Monday, when AF’s in-degree of
22 is greater than AV’s 21, and BA has the highest in-degree. The highest
out-degrees are usually achieved by D, BP, AV, T, W, and AH; the exceptions
are Monday, when D’s out-degree of 17 is less than AR’s and AP’s 18, and
Thursday, when AV’s out-degree of 16 is less than P’s 18.
It is difficult to infer the shape of the degree distribution for individual
days due to poor statistics. The two-sample Kolmogorov-Smirnov (KS) test
does not distinguish between the distributions on different days at the 5%
significance level. With this in mind, we combine the in- and out-degree data
for all four days and graph the resulting distributions in Figure 2.10b. We
find that a power law distribution does not provides a good fit for either in- or
out-degrees. Visually, the distribution is closer to an exponential. However,
the exponential distribution is rejected by the Anderson-Darling test.
The degree distribution conceals the fact that flows originating or termi-
nating in nodes of various degrees have different values and therefore provide
different contributions to the total value of the net flows. Nodes with lower
degrees are numerous, but the flows they sustain are typically smaller than
those carried by a few high-degree nodes. In particular, for the nonloan flows,
nodes with in-degree d ≤ 10 are numerous, ranging from 35 to 37, but their
outgoing net flows carry about 20% of the value on average. On the other
48
2. Loan and nonloan flows in the Australian interbank network
d
22-02-2007
21-02-2007
20-02-2007
19-02-2007
05
10
15
20
25
30
35
40
95059950599505995059
−9
−5059
−9
−5059
−9
−5059
−9
−5059
(a)
d
100
101
100
101
(b)
Figure
2.10:(a)Degreedistribution
ofthenet
non
loan
flow
networks(for
convenience,in-degrees
arepositivean
dou
t-degrees
arenegative).Thetotalvalueof
thenet
flow
scorrespon
dingto
thespecificdegrees
isshow
nwithreddots(thelogof
valuein
A$10
9isindicated
ontherigh
tvertical
axis).
(b)Degreedistribution
ofthenet
non
loan
flow
swhen
thedegreedataforallfour
daysareaggregated
(in-degrees
arecircles;
out-degrees
aretriangles).
49
2. Loan and nonloan flows in the Australian interbank network
d
05
10
15
9 5 0 5 9 9 5 0 5 9 9 5 0 5 9 9 5 0 5 9
−9
−5 0 5 9
−9
−5 0 5 9
−9
−5 0 5 9
−9
−5 0 5 9
(a)
d
100
101
100
101
(b)
Figu
re2.11:
(a)Sam
eas
Figu
re2.10a,
butfor
thenet
loanflow
netw
orks.
(b)Sam
eas
Figu
re2.10b
,butfor
thenet
loanflow
s.
50
2. Loan and nonloan flows in the Australian interbank network
hand, nodes with d ≥ 17 are rare, but their flows carry 50% of the value. The
same effect is observed for the out-degrees.
The degree distribution of the network of net loan flows is shown in Fig-
ure 2.11a (we ignore the nodes that have zero in- and out- degrees over four
days). Similarly to nonloan flows, the KS test does not distinguish between
the distributions on different days at the 5% significance level. The combined
distribution is shown in Figure 2.11b.
To probe assortativity of the net flow networks, we compute the in-assortativity
defined in Piraveenan et al. (2010) as the Pearson correlation coefficient be-
tween the in-degrees of sources and destinations of the net flows (out-assortativity
is computed similarly using the out-degrees). The net nonloan flow network
is disassortative, with in-assortativity of −0.39, −0.37, −0.38, −0.37 and out-
assortativity of −0.35, −0.38, −0.39, −0.37 on Monday, Tuesday, Wednesday,
and Thursday, respectively. The net loan flow network is less disassortative;
the in-assortativity is −0.16, −0.26, −0.18, −0.19 and the out-assortativity
is −0.03, −0.10, 0.02, −0.20 for the same sequence of days. In biological
networks, the tendency of out-assortativity to be more assortative than in-
assortativity has been noted in Piraveenan et al. (2010).
Topology of the net flows
Given the source and destination of each net flow, we can construct a network
representation of the net flows. An example of the network of net nonloan flows
is shown in Figure 2.12. The size of the nodes and the thickness of the edges
are proportional to the net imbalances and net flow values respectively (on a
logarithmic scale). We use the Fruchterman-Reingold algorithm to position
the nodes Fruchterman and Reingold (1991); the most connected nodes are
placed in the centre, and the least connected nodes are moved to the periphery.
The core of the network is dominated by the four banks with the largest total
value and the largest number of transactions: D, BP, AV, and T. The other
big banks, such as AF, AH, and W, also sit near the core. It is interesting to
note the presence of several poorly connected nodes (Q, V, BF, and especially
X) that participate in large incoming and outgoing flows, which produce only
negligible imbalances in the banks themselves.
The sub-network consisting of D, BP, AV, BA, T, W, U, A, AH, AF, AP,
and P is fully connected on all five days, i.e. every node is connected to every
other node. The sub-network of D, AV, and BP is fully connected, even if we
51
2. Loan and nonloan flows in the Australian interbank network
Figure 2.12: Network of net nonloan flows on Tuesday, 20-02-2007. White(grey) nodes represent negative (positive) imbalances. The bank labels areindicated for each node. The size of the nodes and the thickness of the edgesare proportional to the logarithm of value of the imbalances and the net flowsrespectively.
restrict the net flows to values above A$108.
In Figure 2.12, the flows between the largest nodes are difficult to discern
visually, because the nodes are placed too close to each other in the image.
We therefore employ the following procedure to simplify the network. We
consider the fully connected sub-network of twelve nodes, plus node BG, and
combine all other nodes into a new node called “others” in such a way that
the net flows are preserved (BG is included because it usually participates
in large flows and is connected to almost every node in the complete sub-
52
2. Loan and nonloan flows in the Australian interbank network
network). The result of this procedure applied to the daily nonloan networks
is presented in Figures 2.13a–2.13d. For these plots, we employ the weighted
Fruchterman-Reingold algorithm, which positions the nodes with large flows
between them close to each other. The imbalances shown in Figure 2.13b are
the same as those of the full network in Figure 2.12. The daily networks of
net loan flows for the same nodes are shown in Figures 2.14a–2.14d.
We observe that the largest flows on Monday (19-02-2007) were signifi-
cantly lower than the flows on the subsequent days. The largest nodes (D,
BP, AV, T, W) are always placed close to the center of the network, because
they participate in the largest flows. The topology of the flows is complex and
difficult to disentangle, even if one concentrates on the largest flows (above
A$5 × 108). For instance, on Monday, probably the simplest day, the flow of
nonloans is generally from BG to “others” to D to BP. There are also siz-
able flows from T to AV and from AV to “others” and BP. However, lower
value flows (below A$5 × 108) cannot be neglected completely because they
are numerous and may contribute significantly to the imbalance of a given
node.
Nodes D, T, BP, AV, and W form a complete sub-network of net loan flows
on Monday, Tuesday, and Wednesday. This sub-network is almost complete on
Thursday too, except for the missing link between BP and W. The appearance
of the net loan network is different from that of the nonloan network, since the
same nodes participate in only a few loan flows. Therefore, the position of a
node in the network image is strongly influenced by the number of connections
of that node. Some of the poorly connected nodes are placed at the periphery
despite the fact that they possess large flows. The four largest nodes (D, T,
BP, AV) are always positioned at the center of the network.
Network variability
The net nonloan flow network is extremely volatile in terms of flow value and
direction. For example, a A$109 flow from D to BP on Monday transforms
into a A$3.2×109 flow in the same direction on Tuesday, only to be replaced by
a A$6.3 × 108 flow in the opposite direction on Wednesday, which diminishes
further to A$2.5×109 on Thursday. Nodes T and BP display a similar pattern
of reversing flows between Tuesday and Wednesday. On the other hand, the
net flow between T and AV maintains the same direction, but the flow value
is strongly fluctuating. In particular, a moderate A$4.8×108 flow on Monday
53
2. Loan and nonloan flows in the Australian interbank network
(a) 19-02-2007 (b) 20-02-2007
(c) 21-02-2007 (d) 22-02-2007
Figure 2.13: Networks of daily net nonloan flows for D, AV, BP, T, W, BA,AH, AF, U, AP, P, A, BG. All the other nodes and the flows to and from themare combined in a single new node called “others”. The size of the nodes andthe thickness of the edges are proportional to the logarithm of value of theimbalances and the net flows respectively. The value of the flows and theimbalances can be gauged by referencing a network shown in the middle,where the values of the flows are indicated in units of A$1 billion.
54
2. Loan and nonloan flows in the Australian interbank network
(a) 19-02-2007 (b) 20-02-2007
(c) 21-02-2007 (d) 22-02-2007
Figure 2.14: Networks of daily net loan flows. The same nodes as in Fig-ures 2.13a–2.13d are used. The scale of the loan flows, the imbalances, andthe positions of the nodes are the same as those used for the nonloan flows inFigures 2.13a–2.13d to simplify visual comparison.
55
2. Loan and nonloan flows in the Australian interbank network
rises to A$1.9×109 on Tuesday, then falls sharply to A$2×108 on Wednesday
and again rises to A$2.2× 109 on Thursday.
Considering any three nodes, we observe that circular and transitive flows
are present on most days, the latter being more common. The most obvious
example is a circular flow between D, T, and BP on Thursday and a transitive
flow involving BG, T, and AV on the same day. The circular flows are unstable
in the sense that they do not persist over two days or more.
The net loan flow network exhibits similar characteristics. Few net loan
flows persist over the four days. For example, the flow from AV to T has the
same direction and is similar in value on all four days. Circular loan flows are
also present, as the flow between AV, T, and BP on Thursday demonstrates.
2.8 Conclusions
In this paper, we study the properties of the transactional flows between Aus-
tralian banks participating in RITS. The value distribution of transactions
is approximated well by a mixture of two log-normal components, possibly
reflecting the different nature of transactions originating from SWIFT and
Austraclear. For the largest banks, the value distributions of incoming and
outgoing transactions are similar. On the other hand, the central bank dis-
plays a high asymmetry between the incoming and outgoing transactions, with
the former clearly dominating the latter for transactions below A$106.
Using a matching algorithm for reversing transactions, we successfully sep-
arate transactions into loans and nonloans. For overnight loans, we estimate
the identification rate at 98%. The mean derived interest rate is within 0.01%
of the central banks’ target rate of 6.25%, while the standard deviation is
about 0.07%. We find a strong anti-correlation between loan and nonloan im-
balances (Pearson coefficient is about 0.9 on most days). A likely explanation
is that nonloan flows create surpluses in some banks. The banks lend the
surplus to banks in deficit, creating loan flows that counteract the imbalances
due to the nonloan flows. Hence, loan and nonloan imbalances of individual
banks are roughly equal in value and opposite in sign on any given day.
The flow networks are structurally variable, with 20% of nonloan flows
and 50% of loan flows replaced every day. Values of persistent flows, which
maintain the same source and destination over at least two consecutive days,
vary significantly from day to day. Some flow values change by several orders
of magnitude. Persistent flows increase in value several-fold between Monday
56
2. Loan and nonloan flows in the Australian interbank network
and Tuesday. Individual flow values can change by several orders of magnitude
on the following day. Overall, there is a reasonable correlation between the
flow values on consecutive days (Pearson coefficient is 0.65 for nonloans and
0.76 for loans on average). We also find that larger banks tend to sustain
larger loan flows, in accord with the intuitive expectations. However, there is
no correlation between loan and nonloan flows.
We examine visually the topology of the net loan and nonloan flow net-
works. The centre of both networks is dominated by the big four banks.
Twelve banks form a complete nonloan sub-network, in which each bank is
connected to every other bank in the sub-network. The three largest banks
form a complete sub-network even if the net flows are restricted to values
above A$108. Our examination reveals that the network topology of net flows
is complicated, with even the largest flows varying greatly in value and direc-
tion on different days.
Our findings suggest a number of avenues for future research on interbank
networks. Firstly, the relationships we uncovered can be used to constrain
analytical models and numerical simulations of interbank flows in financial
networks. In particular, our explanation of the link between the loan and
nonloan imbalances needs to be tested in numerical simulations. Secondly, it
is necessary to analyse interbank markets in other countries to establish what
elements of our results are signatures of general dynamics and what aspects
are specific to the epoch and location of this study. Even when high qual-
ity data are available, most previous studies concentrate on analysing static
topological properties of the networks or their slow change over time. The
internal dynamics of monetary flows in interbank networks has been largely
ignored. Importantly, one must ask whether the strong anti-correlation be-
tween loan and nonloan imbalances is characteristic of RTGS systems whose
institutional setup resembles the Australian one or whether it is a general
feature. For instance, in Italy a reserve requirement of 2% must be observed
on the 23rd of each month, which may encourage strong deviations between
loan and nonloan imbalances on the other days.
57
Chapter 3
Laplace transform analysis of
a multiplicative asset transfer
model
We analyze a simple asset transfer model in which the transfer amount is
a fixed fraction f of the giver’s wealth. The model is analyzed in a new
way by Laplace transforming the master equation, solving it analytically and
numerically for the steady-state distribution, and exploring the solutions for
various values of f ∈ (0, 1). The Laplace transform analysis is superior to
agent-based simulations as it does not depend on the number of agents, en-
abling us to study entropy and inequality in regimes that are costly to address
with simulations. We demonstrate that Boltzmann entropy is not a suitable
(e.g. non-monotonic) measure of disorder in a multiplicative asset transfer sys-
tem and suggest an asymmetric stochastic process that is equivalent to the
asset transfer model.
3.1 Introduction
A vibrant research theme in econophysics is the analysis of asset exchange
models. In these models, a large number of agents iteratively exchange assets,
typically representing monetary amounts. In the simplest model that has been
considered, the transfer amount is constant and independent of the agent’s
wealth, producing an exponential wealth distribution in the steady state (see
Yakovenko and Rosser Jr. (2009) for a review). More complicated fractional
59
3. Laplace transform analysis of a multiplicative asset transfer model
exchange models have also been considered by several authors Chatterjee and
Chakrabarti (2007); Hayes (2002), in which the size of each transfer is a linear
function of the wealths of the agents involved in the exchange.
It has been found both analytically and numerically that the steady-state
wealth probability distribution function ps(w) in fractional exchange models
depends strongly on the parameters that characterize the exchange Matthes
and Toscani (2008). Certain parameter values or exchange rules yield a
strongly peaked distribution with an exponential tail, while other values yield
a broad distribution with Pareto-like qualities. The dichotomy is exemplified
by two simple models. If the transfer amount is a fixed fraction f of the
giver’s wealth (the giver is the agent who surrenders the asset in the transfer),
then the resulting steady-state distribution is strongly peaked and decays ex-
ponentially in the tail. If, on the other hand, the transfer amount is a fixed
fraction f of the poorer agent’s wealth, then one finds a broad steady-state
distribution, which can be fitted well by a power law with exponent −1 across
a broad interval of wealths. In this paper, we refer to these two models as the
giver scheme and the poorer scheme respectively.1
In some of the asset exchange models considered in Chakraborti and
Chakrabarti (2000) and several other studies Chatterjee and Chakrabarti
(2007); Chatterjee et al. (2005), the fractional exchange amount is a random
linear combination of the wealths of the participating agents. The controlling
parameter is the saving propensity, λ, which determines the fraction of the
agents’ wealths that they do not offer to exchange. Comparing the output of
simulations for the giver scheme and the exchange schemes based on the sav-
ing propensity, one observes that the schemes are closely related, with f ≈ 0
corresponding to λ ≈ 1. If the saving propensity is the same for all the agents,
then the resulting steady-state distributions are similar to those obtained for
the giver scheme. On the other hand, if the saving propensity is uniformly
distributed, the steady-state distribution is a power law, ps(w) ∝ w−2. In
a recent study based on numerical simulations Saif and Gade (2007), it was
found that a combination of the poorer and giver schemes in one simulation
results in a power-law wealth distribution whose exponent depends on the
relative contributions of the two schemes. The more agents follow the giver
scheme, the greater the exponent.
Asset exchange models can be treated analytically via a kinetic or mas-
1They are also known as the theft-and-fraud and yard-sale models respectively (see Hayes(2002)).
60
3. Laplace transform analysis of a multiplicative asset transfer model
ter equation, which tracks the rate of change of the number of agents at any
given wealth. In particular, the master equation for the giver scheme has been
derived in Ispolatov et al. (1998). These authors found an expression for the
second moment in the steady state, which agrees with the expression found
in Angle (2006) for a similar model by assuming the gamma distribution of
wealth. The standard deviation converges to its steady-state value exponen-
tially on a time scale ∼ [f(1− f)]−1. The authors also found the asymptotic
behaviour of the wealth distribution at small values of wealth. The master
equation for the poorer scheme was derived recently Moukarzel et al. (2007),
but its solutions have not yet been studied. In Chatterjee et al. (2005), the au-
thors derived the kinetic equation for the case of uniformly distributed saving
propensity. They demonstrated that the solution follows a power law with the
same exponent as in the simulations. The kinetic equation approach was also
used in Slanina (2004) to analyze self-similar solutions of a non-conservative
asset exchange system. The author found a closed-form solution in the limit
of continuous trading by means of the Laplace transform and observed that
the distribution exhibits power-law behaviour at large wealths.
The dependence of the relaxation time on the exchange parameters has
been investigated numerically in Patriarca et al. (2007) for the models con-
sidered in Chatterjee and Chakrabarti (2007); Chatterjee et al. (2005). The
relaxation time-scale was found numerically to scale as ∼ (1 − λ)−1. This is
consistent with the values found analytically in the giver scheme for the stan-
dard deviation. The authors also considered how the relaxation time depends
on the number of agents but failed to find any significant trend.
Recently, much effort has been directed profitably at developing more so-
phisticated and realistic multi-agent models to be analyzed by means of numer-
ical simulations. In the present paper we take the opposite tack and return
instead to one of the simplest multiplicative models, the giver scheme. We
show that its master equation can be solved efficiently by a Laplace transform
technique. Armed with this new tool backed by multi-agent simulations, we
identify the following new properties of the system. (1) We get precise val-
ues of various quantities such as the steady-state entropy as a function of the
model parameter f , independently of the number of agents. (2) We explore
the thinly studied regime 1/2 < f < 1 and identify its unusual properties,
e.g. oscillations in ps(w). (3) Using multi-agent simulations, we investigate
how the Boltzmann entropy evolves with time as the system approaches equi-
librium and argue that the Boltzmann entropy is not a suitable entropy for
61
3. Laplace transform analysis of a multiplicative asset transfer model
the giver scheme, even though the system is closed and conservative. (4) We
propose a simple asymmetric stochastic process that is equivalent to the giver
scheme. (5) We investigate how the degree of inequality, characterized by
the Gini coefficient, depends on f . (6) Finally, we apply phase-space tech-
niques from statistical mechanics to the giver scheme in order to illuminate
the difficulties and opportunities that this asset transfer model presents.
3.2 Giver scheme
We consider a simple asset transfer model, in which the transfer amount is
equal to a fixed fraction of the giver’s wealth.2 If wg is the giver’s wealth and
wr is the receiver’s wealth prior to the transfer, then their wealths after the
transfer are given by wg−∆w and wr+∆w respectively, with ∆w = fwg and
f ∈ (0, 1). The model comprises a large number of agents, who are assigned
wealths initially according to some distribution. The transfers are assumed to
take place over a fixed time interval ∆t. At each discrete time ti, the agents
are divided randomly into pairs and the transfer formula is applied to each
pair. The transfers are complete by the time ti+1 = ti + ∆t and the process
repeats at the time ti+1. The probability of drawing any pair is the same.
In each pair, the giver is assigned randomly regardless of the wealths of the
agents.
The master equation for this system was derived in Ispolatov et al. (1998)
and is given by
∂p(w, t)
∂t= −p(w, t)+ 1
2(1 − f)p
(
w
1− f, t
)
+1
2f
∫ w
0dw′ p
(
w − w′
f, t
)
p(w′, t).
(3.1)
It is easy to verify that the mean of the distribution, µ1 =∫∞0 dw wp(w, t),
does not depend on time. Upon integrating by parts, one arrives at the evo-
lution equationdµ2(t)
dt= −f(1− f)µ2 + fµ21 (3.2)
for the second moment, µ2(t) =∫∞0 dw w2p(w, t), first reported in Ispolatov
et al. (1998). This equation can be solved for the variance
σ2(t) = µ2(t)− µ21 =
(
µ2(0) −µ21
1− f
)
e−f(1−f)t +fµ211− f
. (3.3)
2The giver is also called the payer or the loser in the literature.
62
3. Laplace transform analysis of a multiplicative asset transfer model
In the steady state, one has σs = σ(t→ ∞) = µ1[f/(1−f)]1/2. For simplicity,
we assume henceforth that the mean of p(w, t) equals unity.3 In the following
sections, we are mostly concerned with the steady-state distribution ps(w) =
p(w, t → ∞).
3.3 Laplace transform of the master equation
The Laplace transform of the master equation in the steady state is given by
g(z) =1
2g(z − fz) +
1
2g(z)g(fz), (3.4)
with g(z) =∫∞0 dw e−zwps(w). Note that the functional equation (3.4) applies
to any integral transform whose kernel depends only on the product of the
arguments of the function and its transform. For f = 1/2, the functional
equation has a closed form solution
g(z) =1
1 + Cz, (3.5)
where C is a complex-valued constant. Using the definition of the transform,
we have g(0) = 1 from the normalization of ps(w) and g′(0) = −1 from the
assumption that ps(w) has unit mean, which gives C = 1. Applying the
inverse Laplace transform to this solution gives the exponential distribution,
ps(w) = e−w, which was obtained in Ispolatov et al. (1998) by substituting
simple “test” functions into the master equation. No closed-form solutions
have been found for other values of f ∈ (0, 1).
The Taylor expansion of g(z) at z = 0 can be derived by substituting the
expansion in (3.4) and using g(0) = 1 and g′(0) = −1. For the first four terms
of the expansion, this procedure gives
g(z) = 1− z +1
2(1− f)z2 − 1 + f
6(1− f)2z3 +O(z4). (3.6)
In general, for g(z) =∑∞
n=0 an(−z)n/n!, a0 = 1, and a1 = 1, we obtain
an =
n−1∑
k=1
(
n
k
)
fkakan−k
1− fn − (1− f)nfor n > 1. (3.7)
Since g(−z) is the moment-generating function for the distribution ps(w),
the n-th moment of the distribution µn equal an, i.e. all moments of the
3If the mean µ1 6= 1, one can consider the function q(x) = µ1p(µ1x), which is normalizedand has unit mean.
63
3. Laplace transform analysis of a multiplicative asset transfer model
steady-state wealth distribution can be computed for any f using the recursive
formula (3.7).
Using the Taylor expansion, one has an → 1 as f → 0 and hence g(z) →e−z. Note that the functional equation (3.4) becomes an identity for f = 0
and g(0) = 1. Taking the inverse Laplace transform of g(z) = e−z, formally
one gets ps(w) = δ(w− 1). However, this wealth distribution is never reached
because the relaxation time scale tr = [f(1 − f)]−1 determined from (3.3)
tends to infinity in this limit. Indeed, ps(w) is equal to the initial distribution
if f = 0. On the other hand, the Taylor expansion does not have a limit as
f → 1. The functional equation (3.4) has the solution g(z) = 1 when f = 1,
but it does not satisfy the condition g′(0) = −1. It appears that ps(w) does
not have a proper limit as f → 1. Note also that the relaxation time tends to
infinity as f approaches unity as well.
The asymptotic behaviour of g(z) at infinity can also be deduced readily
from the functional equation. In the directions in the complex plane for which
one has g(z) → 0 as |z| → ∞, the equation
g(z − fz) = 2g(z) (3.8)
must be approximately true for large enough |z|. Assuming a power-law shape
|g(z)| ∝ |z|−α as |z| → ∞, equation (3.8) gives
α =−1
log2(1− f). (3.9)
By Watson’s lemma Davies (2002), this is consistent with the asymptotic
behaviour p(w) ∝ wα−1 as w → 0 that was found in Ispolatov et al. (1998) by
the method of dominant balance.
The functional equation (3.4) can be solved iteratively when it is cast in
the form
gi+1(z) =gi(z − fz)
2− gi(fz), (3.10)
where gi(z) is the i-th iteration. Experimentation shows that the choice
g0(z) = 1/(1 + z) works well for all f . A detailed description of the com-
putational procedure is given in 3.7; there are some subtleties involved in the
choice of grid and interpolation method. An example of the numerical solution
for f = 0.1 is presented in Figures 3.1a and 3.1b. The power-law behaviour
at large |z|, with the exponent given by (3.9), is confirmed numerically for
f = 0.1 (right panel of Figure 3.1a) and a range of other values. The itera-
tions converge in the negative half-plane, Re(z) < 0, despite the complicated
64
3. Laplace transform analysis of a multiplicative asset transfer model
θ=
90◦
θ=
60◦
θ=
0◦
log10|g(z)|
log10
r
Im[g(z)]
log10
r
Re[g(z)]
log10
r
−2
−1
01
23
−2−1
01
23
−2−1
01
23
−16
−14
−12
−10
−8
−6
−4
−20
−1
−0.50
0.51
−1
−0.50
0.51
(a)
Im
(z)
Re(z)
Im
[g(z)]
Im
(z)
Re(z)
Re[g
(z)]
−5
05
10
−5
05
10
−10
−5
05
10
−10
−5
05
10
−1
−0.50
0.51
−1
−0.50
0.51
(b)
Figure
3.1:
TheLap
lace
tran
sformg(z)forf=
0.1ob
tained
bysolving(3.4)iteratively.
(a)Re[g(z)]
(top
left
pan
el),
Im[g(z)]
(bottom
left
pan
el),
and|g(z)|
(rightpan
el)versusralon
gthereal
axis
(dashed
curve),theim
aginaryax
is(solid
curve),an
dthelineinclined
atθ=
60◦to
thereal
axis
(dottedcu
rve).Thevariab
lesran
dθaredefi
ned
byz=re
iθ.(b)A
view
ofthe
real
(top
)an
dim
aginary(bottom)parts
ofg(z)(values
above1an
dbelow
−1havebeencu
toff
).
65
3. Laplace transform analysis of a multiplicative asset transfer model
structure of g(z), illustrated in Figure 3.1b, as it gradually approaches e−z
for decreasing values of f . The convergence does not depend on the initial
function g0(z); e.g. g0(z) = e−z works just as well for small values of f .
3.4 Steady-state wealth distribution by Laplace
inversion
The steady-state probability distribution function ps(w) can be obtained by
inverting its Laplace transform g(z) numerically. A number of inversion al-
gorithms were reviewed recently in Hassanzadeh and Pooladi-Darvish (2007)
and Abate and Whitt (2006). The reviewers advised that at least two different
algorithms should be used as a cross-check, because different algorithms work
well for specific classes of functions and none of the algorithms is universally
accurate. Fortunately, the algorithms are easy to implement. We test four (re-
ferred to as the Euler, Talbot, Stehfest, and Zakian algorithms in Hassanzadeh
and Pooladi-Darvish (2007); Abate and Whitt (2006)) and find that the first
two give accurate results over a wider range of w. Euler has an additional ad-
vantage over Talbot: it samples g(z) in the positive half-plane only, where the
function g(z) has a simpler structure, as one sees in Figure 3.1b. The results
of the inversion are presented in Figures 3.2a and 3.2b for 0.025 ≤ f ≤ 0.5
and Figures 3.3a and 3.3b for 0.5 ≤ f ≤ 0.9. The exponential analytic solu-
tion is recovered numerically for f = 0.5. From the output of the inversion
algorithms, we compute the moments of the distribution (µ0, µ1, and µ2)
and find agreement with the analytical results to 8 significant digits. In Fig-
ures 3.4a and 3.4b for f = 0.95 and f = 0.05 respectively, we compare the
wealth distributions obtained from the Laplace transform (curves) and from
the agent-based simulations (crosses). We find excellent agreement between
these two methods for all values of f that we consider.
The algorithms that perform the inverse Laplace transform suffer from
truncation errors. Maximum precision is achieved near the peak of ps(w). In
the tail, the precision decreases until the results are completely dominated
by the truncation errors below a threshold value of ps(w). For example, we
perform all computations with 16 significant digits and achieve ∼ 8 significant
digits of precision at the peak of ps(w), but the algorithms break down at
ps(w) . 10−8.
The wealth distributions that we find for f < 1/2 are characterized by
66
3. Laplace transform analysis of a multiplicative asset transfer model
f=
0.025
f=
0.05
f=
0.1
f=
0.25
f=
0.5
ps(w)
w
00.5
11.5
22.5
30
0.51
1.52
2.53
(a)linearscale
f=
0.025
f=
0.05
f=
0.1
f=
0.25
f=
0.5
log10ps(w)
w
01
23
45
67
8−
7
−6
−5
−4
−3
−2
−101
(b)log-linearscale
Figure
3.2:
Thesteady-state
wealthprobab
ilitydistribution
functionps(w
)ob
tained
byinvertingtheLap
lace
tran
sformg(z)
forthefollow
ingvalues
ofthetran
sfer
fraction
:f
=0.5(boldsolidcu
rve),0.25
(dash-dot
curve),0.1(dottedcu
rve),0.05
(dashed
curve),an
d0.025(thin
solidcu
rve).
67
3. Laplace transform analysis of a multiplicative asset transfer model
f=
0.9
f=
0.8
f=
0.7
f=
0.6
f=
0.5
log10 ps(w)
log10w
−3
−2.5
−2
−1.5
−1
−0.5
00.5
11.5
2−
5
−4
−3
−2
−1 0 1 2
(a)log-lo
gsca
le
f=
0.9
f=
0.8
f=
0.7
f=
0.6
f=
0.5
log10 ps(w)
w
05
10
15
20
25
30
35
40
45
50
−5
−4
−3
−2
−1 0 1
(b)log-lin
earsca
le
Figu
re3.3:
Thestead
y-state
wealth
prob
ability
distrib
ution
function
ps (w
)ob
tained
byinvertin
gtheLap
lacetran
sformg(z)
forthefollow
ingvalu
esof
thetran
sferfraction
:f=
0.5(bold
solidcu
rve),0.6
(dash
-dot
curve),
0.7(dotted
curve),
0.8(dash
edcu
rve),an
d0.9
(thin
solidcu
rve).
68
3. Laplace transform analysis of a multiplicative asset transfer model
log10n(w)
log10w
(m.u.)
−1
−0.5
00.5
11.5
22.5
33.5
4−
10123456
(a)f=
0.95
log10n(w)w
(m.u.)
0100
200
300
400
500
600
700
−1
−0.50
0.51
1.52
2.53
3.54
(b)f=
0.05
Figure
3.4:
Thepop
ulation
distribution
n(w
),show
nwithcrosses,
asafunctionof
wealthw,measuredin
ficticiousmon
etary
units(m
.u.)
usedin
theagent-based
simulation
s.Thedistribution
iscomputedas
thenumber
ofagents
ineveryunit
wealth
interval
after100step
sin
thesimulation
ofthegiverschem
ewithtotalnumber
ofagentsN
=4×
105an
dtran
sfer
param
eter
(a)f=
0.95
and(b)f=
0.05.Theinitialdistribution
isuniform
inthewealthinterval
(a)[0,100
m.u.]an
d(b)[0,500
m.u.].
Thecorrespon
dingsolution
ofthesteady-state
masterequationforthesamefis
show
nwithasolidcu
rve,
withps(w
)scaled
toconform
withthedefi
nitionofn(w
)accordingto
Nps(w/〈w〉)/〈w〉where〈w
〉is
themeanwealth.Boththeagent-based
simulation
san
dthemasterequationpredictoscillationsin
thewealthdistribution
in(a)butnot
in(b).
69
3. Laplace transform analysis of a multiplicative asset transfer model
power-law behaviour at w ≪ 1, in accord with the analytical results, and
approximately exponential tails at large w. A careful examination of the tails
confirms that the asymptotic behaviour at large wealths is not exactly ex-
ponential. However, we have not been able to find a closed-form expression
for it. The distribution becomes tightly concentrated around its peak as f
decreases; the peak of the distribution gradually shifts towards w = 1. On
the other hand, the peak shifts towards w = 0 as f increases; the distribu-
tions eventually turns into an exponential function for f = 1/2. This overall
behaviour is similar to that observed in the asset exchange models based on
the saving propensity with 0 < λ < 1 Chatterjee and Chakrabarti (2007).
The structure of the steady-state solutions for f > 1/2 is very different.
The asymptotic approximation ps(w) ∝ wα−1, with α = −1/ log2(1 − f), is
valid for f > 1/2 as well and indicates that the distribution diverges at w = 0
(as α < 1). For values of f sufficiently close to 1, the wealth distribution
acquires a shape that is akin to a power law, ps(w) ∝ w−1, with overlaid
oscillations that become more prominent as f increases. This power-law be-
haviour cuts off exponentially at some critical wealth that increases slowly as
f approaches 1. At the same time the exponential drop-off at large wealths
becomes shallower as evident from Figure 3.3b. The oscillations of ps(w) ap-
pear to be periodic on a logarithmic scale, with the period depending on f .
For example, the periods for f = 0.9 and f = 0.99 are roughly one and two
decades respectively. This is directly related to the fact that all givers retain
1% of their wealth for f = 0.99 and 10% for f = 0.9.
3.5 Discussion
We now use the Laplace transform tools developed in Section 3.4 to address
two questions that are costly to explore with agent-based simulations: the
nature of disorder (entropy) and its evolution in the giver scheme, and the
degree of inequality in the steady state.
Entropy and the approach to equilibrium
According to Boltzmann, states with higher entropy are more probable be-
cause they correspond to a larger number of microscopic configurations of the
system. A closed system evolves to a state of maximum entropy, i.e. maxi-
mum disorder, which is characterized by the Boltzmann-Gibbs distribution.
70
3. Laplace transform analysis of a multiplicative asset transfer model
The exponential distribution observed in models where the transfer amount
is fixed and constant (see section II.C in Yakovenko and Rosser Jr. (2009))
is thus consistent with entropy maximization ideas. On the other hand, it is
argued in Yakovenko and Rosser Jr. (2009) that multiplicative asset exchanges
may lead to non-exponential distributions because of the broken time-reversal
symmetry, whereas, in models with fixed additive exchanges, the time-reversal
symmetry is preserved. Despite this, the entropy maximization technique has
been applied in Chakraborti and Patriarca (2008) to an asset exchange model
described by a Hamiltonian quadratic in wealth variables. It predicts a gamma
distribution of wealth, but ps(w) in multiplicative asset exchange models is
not a gamma distribution in general.
In this section, we explore the applicability of the Boltzmann entropy to
the giver scheme. Using the Laplace transformed solutions of the master
equation (3.1), we compute the steady-state Boltzmann entropy4
Ss = −∫ ∞
0dw ps(w) log[ps(w)] (3.11)
for several values of f in the range 0.01 ≤ f ≤ 0.9 and plot the results
in Figure 3.5a. Entropy maximization arguments Kapur (1989) imply that
the entropy defined by (3.11) leads uniquely to the exponential distribution,
p(w) = e−w with Ss = 1, if the only condition is that the mean of the dis-
tribution is fixed to µ1 = 1. In our model, however, the transfer fraction f
places additional constraints on how the distribution of wealth evolves with
time. Therefore, it is not surprising that we observe a range of steady-state
entropies Ss 6= 1 corresponding to different values of f . The exponential distri-
bution for f = 1/2 appears to be the most disordered state of the system with
the highest entropy Ss = 1. For all other values of f , the entropy is smaller
and it becomes negative as f approaches 0 or 1. Reading off the graph, we
find that Ss is negative for 0 < f < 0.058 and 0.836 < f < 1. The entropy
tends to negative infinity as f → 0 or f → 1, which is in accord with the
behaviour of ps(w) in these limits.
4We stress that this definition applies to a normalized distribution with unit mean. Un-like the case of discrete probability distributions, continuous entropy can be negative andit is not invariant with respect to the change of variable. It may be more appropriate toconsider the Kullback-Leibner divergence Ds =
∫∞
0dw ps(w) log[ps(w)/m(w)], which is a
measure of the divergence between ps(w) and the reference distribution m(w). It is conve-nient to take m(w) = e−w, in which case the divergence essentially reverts to Boltzmannentropy because Ds = 1− Ss.
71
3. Laplace transform analysis of a multiplicative asset transfer model
Ss
σ2s
10−
210−
1100
101
−1
−0.8
−0.6
−0.4
−0.2 0
0.2
0.4
0.6
0.8 1
(a)stea
dy-sta
teen
tropy
S(t)
t(step
s)
010
20
30
40
50
60
70
80
90
100
0
0.0
5
0.1
0.1
5
0.2
0.2
5
(b)en
tropyevolutio
n
Figu
re3.5:
(a)Boltzm
ann
entrop
ySsof
thestead
y-state
distrib
ution
asafunction
ofthevarian
ceσ2s=f/(1
−f).
The
criticalvalu
esσ2s=
0.062,1,
and5.098,
correspon
dingtof=
0.058,0.5,
and0.836
respectively,
areindicated
with
thedotted
lines.
(b)Entrop
yas
afunction
oftim
efor
theinitial
distrib
ution
givenby(3.12)
with
f=
0.058,com
puted
fromthemulti-
agentsim
ulation
ofthegiver
schem
e.For
thesim
ulation
,thedistrib
ution
(3.12)was
scaledupto
giveN
=337123
agents
in0≤w
≤1421.
Tocom
pute
theentrop
y,thepop
ulation
distrib
ution
produced
bythesim
ulation
was
norm
alizedto
aprob
ability
distrib
ution
with
unitmean
.
72
3. Laplace transform analysis of a multiplicative asset transfer model
Negative values of Ss are already a warning that the Boltzmann entropy
may not be a faithful measure of disorder in a multiplicative asset transfer
system like the giver scheme. However, the situation worsens when we look
at how S(t) = −∫∞0 dw p(w, t) log[p(w, t)] evolves with time by conducting
multi-agent simulations. In many cases, it decreases instead of increasing. For
example, if we choose p(w, 0) = e−w initially, S(t) decreases with time for all
f 6= 1/2 and remains constant for f = 1/2. Moreover, we can easily find
realistic situations where S(t) does not change monotonically with time, as
the experiment described below shows. Consider an initial distribution of the
form
p(w, 0) =
p1, 0 ≤ w ≤ 1,
p2, 1 < w ≤ w2,
0, otherwise,
(3.12)
with the parameters p1, p2, and w2 chosen to give S(0) = 0 (p1 ≈ 0.296, p2 ≈1.669, and w2 ≈ 1.421). The evolution of entropy for this initial distribution
with f = 0.058 is plotted in Figure 3.5b. The entropy grows initially but
after about ten steps in the simulation it turns over and begins to decrease,
eventually reaching Ss = 0 as expected for f = 0.058. This is in marked
contrast to the behaviour of S(t) in an ideal gas, where one has dS(t)/dt ≥ 0
according to Boltzmann’s H-theorem.
The population distribution is determined by dividing the wealth axis into
small bins and computing the number of agents that fall in each bin. One can
define the multiplicityW as the number of permutations of the agents between
different wealth bins such that the occupation numbers of the bins do not
change. The definition of entropy, Ss = logW , leads to the expression (3.11) in
the continuous limit. Entropy maximization under the assumption that total
wealth is conserved gives an exponential distribution. However, this ignores
the global constraints on the probability distribution ps(w) imposed by the
transfer fraction f . The maximization procedure must take these constraints
into account to derive the steady-state wealth distribution appropriate to the
giver scheme. Unfortunately, at the time of writing, we have been unable to
derive these additional constraints, and they do not appear in the literature.
73
3. Laplace transform analysis of a multiplicative asset transfer model
log10 ξ(w)
w
00.5
11.5
22.5
3−
5
−4
−3
−2
−1 0 1
(a)
wi
i
0100
200
300
400
500
600
700
800
900
1000
0
0.2
0.4
0.6
0.8 1
1.2
1.4
1.6
1.8 2
(b)
Figu
re3.6:
(a)Thelim
itingprob
ability
distrib
ution
ξ(w),show
nwith
crosses,ob
tained
fromon
erealization
oftheasy
mmetric
random
process
(3.13)for
f=
0.05after
106iteration
s.Themean
is0.9998
andthevarian
ceis
0.0519(cf.
0.0526theoretically
fromthemaster
equation
).Thestead
y-state
distrib
ution
ps (w
)for
thesam
efob
tained
byLap
laceinversion
isshow
nas
asolid
curve
forcom
parison
.(b)Thefirst
1000valu
esof{w
i }.
74
3. Laplace transform analysis of a multiplicative asset transfer model
Random process
The evolution of wealth in the giver scheme can be analyzed in terms of a
random process defined by
wi+1 = wi +∆wi, (3.13)
where w1 = 1 and ∆wi = +f or ∆wi = −fwi with equal probability. This
process is asymmetric, i.e. multiplicative in the negative direction and addi-
tive in the positive. To illuminate the relationship between the giver scheme
and the random process we note that (1) an agent’s loss of wealth is always
proportional to his wealth, i.e. it is multiplicative, and (2) an agent’s gain
of wealth can originate from any other agent in the population and therefore
equals f〈w〉 on average, or simply f if we set 〈w〉 = 1. We compute the lim-
iting distribution ξ(w) of this process by applying (3.13) a sufficiently large
number of times and then constructing a histogram of all {wi}. By the ergodic
assumption, this is equivalent to computing a large number of realizations of
this random process and using the final values in each realization to find the
limiting distribution.
Figures 3.6a and 3.6b display one particular realization of the random
process (3.13) and the corresponding limiting distribution. Given the close
link between the transfer model and the random process, it is not surprising
that the limiting distribution ξ(w) of the random process (3.13) appears to be
identical to the steady-state distribution ps(w) of the giver scheme. We obtain
similar results for other values of f ≪ 1. Note that the slight discrepancy
between ξ(w) and ps(w) at w > 2 is due to insufficient sampling of large
wealths by the random process. The agreement improves as the number of
iterations increases.
Inequality of wealth
A traditional measure of inequality in economic systems is the Gini coefficient,
defined asG = 1−2∫ 10 L(X)dX, where L(X) is the Lorenz curve. For a contin-
uous distribution, we have L(w) =∫ w0 dw′ w′ps(w
′), X(w) =∫ w0 dw′ ps(w
′),5
and hence
Gs = 1− 2
∫ ∞
0dw ps(w)
∫ w
0dw′ w′ps(w
′), (3.14)
5 Note that we have L(X) ≤ X for all X because L(0) = 0, L(1) = 1, and dL/dX is amonotonically increasing function.
75
3. Laplace transform analysis of a multiplicative asset transfer model
Gs
σ2s
10−2 10−1 100 1010
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 3.7: Gini coefficient of the steady-state distribution ps(w) as a functionof the variance σ2s = f/(1− f).
such that Gs = 0 corresponds to perfect equality, and Gs = 1 to perfect
inequality.
In Figure 3.7 we plot Gs versus the steady-state variance σ2s = f/(1 − f)
for 0.01 ≤ f ≤ 0.9. As expected, Gs increases monotonically with σ2s , since
both quantities are measures of dispersion. Interestingly, however, there is
an inflection point in the Gs(σ2s) curve at σ2s = 1, Gs = 1/2, corresponding
to the exponential distribution (i.e. f = 1/2). For f → 0, we have ps(w) →δ(w− 1), which corresponds to perfect equality since all agents have the same
wealth. On the other hand, for f → 1, the distribution ps(w) becomes sharply
peaked near w = 0, while the standard deviation approaches infinity. This
corresponds to the situation where most agents have zero wealth, except for
one who has everything, i.e. perfect inequality.
We can understand the evolution towards inequality in terms of the state
vector of the system. Consider N agents whose wealths are characterized
by random variables wi, i = 1, 2, . . . , N . The state of the system can be
described by the phase-space vector w = (w1, w2, . . . , wN ). The constraints
that define the phase-space are (1) 0 ≤ wi ≤ 1 for all i, and (2)∑N
i=1 wi = 1
(we assume for convenience that the total wealth is unity). These constraints
76
3. Laplace transform analysis of a multiplicative asset transfer model
define a segment of the (N − 1)-dimensional hyperplane embedded in N -
dimensional space. Without any additional constraints, entropy maximization
Kapur (1989) gives gi(wi) = (N−1)(1−wi)N−2 for the probability distribution
of the wealth wi of the i-th agent, with mean 〈wi〉 = 1N and variance σ2wi
=N−1
N2(N+1). However, in our system, the asset transfer process and the value of
the parameter f place additional restrictions on the evolution of w. For the
increment of the phase-space vector w from time tk to time tk+1, i.e. after one
generation of asset transfers, we have
|∆w|2 =N∑
i=1
[wi(tk+1)− wi(tk)]2. (3.15)
The terms in the sum on the right hand side can be split into two groups,
associated with the givers and the receivers. Since the transfer amount is
proportional to the giver’s wealth, we get
|∆w|2 = 2f2∑
i∈givers
[wi(tk)]2. (3.16)
Therefore the following inequality must always be satisfied:
|∆w| ≤ 21/2f |w|. (3.17)
In addition, we have
N−1/2 ≤ |w| ≤ 1 (3.18)
due to the restrictions of the phase-space itself. Note that the state wi = 1/N
for all i is the nearest point to the origin.
When f is small, the norm of the increment |∆w| is also small compared
with the maximum linear extent of the phase space (which equals 21/2); the
evolution of w is gradual. Furthermore, |∆w| is also constrained by |w|, whichcan be very small if N is large and all agents are clustered as close to the origin
as possible. So, if the dispersion in wealth is modest, w moves slower through
the phase space than if there is great inequality. On the other hand, w changes
more rapidly on average if |w| is close to unity, which corresponds to large
inequality. The states of equality are therefore more probable, which explains
why the steady-state distribution tends towards a delta function for f → 0.
Even if the initial wealth distribution is very unequal, w drifts quickly towards
the states of near equality.
When f is close to 1, |∆w| can be comparable to the size of the phase
space. Since f is large, the gains in wealth of the individual agents can be
77
3. Laplace transform analysis of a multiplicative asset transfer model
large as well. This leads to the situation where a few agents own most of the
wealth. These agents retain their large wealth for a short time only (typically a
few time steps) before they become givers and pass their large wealth to other
agents. In the extreme case f = 1, one agent possesses all the wealth at any
instant, while all the other agents have zero wealth. This maximum wealth is
passed from agent to agent frequently. This corresponds to w jumping from
one corner of the phase space to another. For f ≈ 1, w evolves similarly, with
ps(w) peaking strongly at zero wealth.
3.6 Conclusions
We develop a new technique for computing the steady-state probability dis-
tribution of a multiplicative asset transfer model, which we call the giver
scheme, by Laplace transforming the associated master equation to give a
functional equation for the characteristic function of the distribution. In the
giver scheme, the transfer amount fwg is proportional to the giving agent’s
wealth wg, so the model depends on a single parameter f ∈ (0, 1). We develop
an efficient iterative method to solve the functional equation for any f , and
we employ several Laplace inversion algorithms to recover the steady-state
distribution ps(w).
We comprehensively explore the dependence of the wealth distribution on
the value of f , especially the thinly studied regime 1/2 ≤ f < 1. We find
a stark qualitative difference between the distributions for f ≈ 0 (sharply
peaked distribution centred around the mean wealth) and f ≈ 1 (broad dis-
tribution of approximately power-law shape with overlaid oscillations). These
two extremes correspond to near-perfect equality and inequality respectively,
as characterized by the Gini coefficient. Both extremes are also characterized
by negative Boltzmann entropy. While the regime f ≈ 0 is generally thought
to represent to some extent the exchange processes occuring in the real econ-
omy, the regime f ≈ 1 is probably less applicable to realistic economic systems,
except perhaps in situations involving extreme leverage. The regime f ≈ 1
may also be relevant to the analysis of gambling, where transitory fortunes
are made and lost frequently.
We show that the Boltzmann entropy is unlikely to be a faithful measure of
disorder in a multiplicative asset transfer system, since it does not vary mono-
tonically as a function of time, assuming the second law of thermodynamics.
This is an important and counterintuitive result, because the system in the
78
3. Laplace transform analysis of a multiplicative asset transfer model
giver scheme is closed and the microscopic transfer rules conserve wealth, in a
manner reminiscent of the microcanonical ensemble in statistical mechanics.
In a multiplicative transfer system, the correlations between various subsys-
tems (e.g. subclasses corresponding to a particular historical sequence of giving
and receiving) and the time-reversal asymmetry of the microscopic rules are
crucial to the system’s dynamics and, therefore, cannot be ignored.
3.7 Iterative procedure
We assume that the computations are carried out with 16 significant digits.
For a given complex argument z, define a uniform grid u = {uk}K1 that covers
the interval [−4, log10(|z|)]. The approximation (3.6) gives sufficient precision
for |z| < 10−4 for computations with 16 significant digits. Choose the num-
ber of points K such that there are a large number of points in every unit
interval, say, 103 logarithmic grid points per decade in [10−4, |z|]. Define two
auxiliary grids, u(f) = log10(f) + u and u(1−f) = log10(1 − f) + u, and ini-
tialize the iterations with g0(10uz/|z|) = 1/(1 + 10uz/|z|). For a given set of
values gi(10uz/|z|), defined on the grid u, find the corresponding values on
the auxiliary grids by performing a spline interpolation or using the approxi-
mation (3.6) where appropriate. Then use these values in equation (3.10) to
find gi+1(10uz/|z|). Continue iterating until the convergence criterion is met
(we find that the convergence spreads gradually from zero to |z|). Typically
the convergence requires a few dozens of iterations for |z| ∼ 100 in the posi-
tive half-plane. Once the convergence is reached, apply a spline interpolation
to find g(z′) for any z′ along the same direction in the complex plane as z,
provided that |z′| < |z|.The obvious disadvantage of the procedure outlined above is that it relies
on interpolation. Its precision is therefore limited by the number of points in
the grid u, i.e. the discretization of the interval [0, |z|]. It is possible, however,to avoid interpolation altogether by defining a special non-uniform grid that
is invariant with respect to division by f and (1 − f). This gives rise to an
alternative procedure for computing the iterations.
Define a grid rk,m = fk(1 − f)m with 0 ≤ k ≤ K and 0 ≤ m ≤ M ,
where K = ⌈log(10−4/|z|)/ log(f)⌉ and M = ⌈log(10−4/|z|)/ log(1 − f)⌉ are
defined such that |z|rK,0 < 10−4 and |z|r0,M < 10−4. The function g(z) on
the grid zk,m = rk,mz, defined according to gk,m = g(rk,mz), has the following
properties: g(fzk,m) = g(zk+1,m) = gk+1,m and g[(1 − f)zk,m] = g(zk,m+1) =
79
3. Laplace transform analysis of a multiplicative asset transfer model
gk,m+1. Therefore the iteration rule becomes
gk,m =gk,m+1
2− gk+1,m, (3.19)
for 0 ≤ k ≤ K − 1 and 0 ≤ m ≤ M − 1. For gK,M one can use the ap-
proximation (3.6). In fact, the Taylor expansion can be used for any point
|zk,m| < 10−4. Thus, no interpolation is required and the iterations can be
computed more efficiently. However, unlike the approach based on interpola-
tion, this procedure must be repeated for different arguments even if they lie
in the same direction in the complex plane.
80
Chapter 4
A note on the theory of fast
money flow dynamics
The gauge theory of arbitrage was introduced by Ilinski in Ilinski (1997) and
applied to fast money flows in Ilinskaia and Ilinski (1999); Ilinski (2001). The
theory of fast money flow dynamics attempts to model the evolution of cur-
rency exchange rates and stock prices on short, e.g. intra-day, time scales. It
has been used to explain some of the heuristic trading rules, known as tech-
nical analysis, that are used by professional traders in the equity and foreign
exchange markets. A critique of some of the underlying assumptions of the
gauge theory of arbitrage was presented by Sornette in Sornette (1998). In
this paper, we present a critique of the theory of fast money flow dynamics,
which was not examined by Sornette. We demonstrate that the choice of the
input parameters used in Ilinski (2001) results in sinusoidal oscillations of the
exchange rate, in conflict with the results presented in Ilinski (2001). We
also find that the dynamics predicted by the theory are generally unstable in
most realistic situations, with the exchange rate tending to zero or infinity
exponentially.
4.1 Introduction
Fast money flows are analyzed in Ilinskaia and Ilinski (1999); Ilinski (2001) in
terms of the lattice gauge theory of arbitrage developed in Ilinski (1997). The
main idea of the theory is that the dynamics should only depend on gauge
invariant quantities rather than the exchange rates themselves. Changing the
units in which stocks of currency are denominated obviously changes the nom-
81
4. A note on the theory of fast money flow dynamics
inal exchange rate. However, it is obvious that such changes of scale, i.e. gauge
transformations, should have no effect on its dynamics. Some assumptions of
the theory have been criticized in Sornette (1998); for example, the lack of
justification for the exponential form of the weight of a given market config-
uration. However, the results of the theory reported in Ilinskaia and Ilinski
(1999); Ilinski (2001) seem impressive, reproducing in particular some of the
phenomenological rules of technical trading employed by professional traders.
Hence the theory appears to be a promising tool for analyzing the markets.
In this note, we present our analysis of the theory of fast money flow dy-
namics and re-examine the results presented in Ilinskaia and Ilinski (1999);
Ilinski (2001). In Sect. 4.2, we present the derivation of the dynamical equa-
tions of the theory. In Sect. 4.3, we examine the dynamics predicted by the
theory for various initial conditions. We highlight certain inconsistencies in
the theory, the unstable dynamics for most realistic values of the parameters
and initial conditions, and the resulting problems in applying the theory to
technical trading. In Sect. 4.4, we revisit the action and demonstrate that the
expression used in Ilinskaia and Ilinski (1999); Ilinski (2001) is inconsistent
with the evolution operator resulting from the lattice formulation.
4.2 Lattice gauge theory and fast money flow
dynamics
In analogy with quantum electrodynamics, Ilinski identified the exchange rate
S between two currencies with the field and the trading agents with matter.
In general, the exchange rate dynamics depends on the interest rates of the
underlying currencies. However, since we are interested in intra-day dynamics
only, we consider the special case of zero interest rates. Ilinski tacitly assumed
that the interest rates of the two currencies are identical, i.e r1 = r2. In this
paper we set r1 = r2 = 0 and assume that transaction costs are zero.
The part of the action s1 that describes the dynamics of the field on its
own is formulated by identifying arbitrage on the lattice with the curvature,
which gives
s1 = − 1
2σ2
∫ T
0dt
(
dy
dt
)2
. (4.1)
In Eq. (4.1), T is the investment horizon and σ2 is the volatility (presumed
to be constant in the interval 0 ≤ t ≤ T ). This expression is equivalent to a
Gaussian random walk in y = lnS.
82
4. A note on the theory of fast money flow dynamics
The effect of the field y on “matter”, i.e. the trading agents, is described
by the Hamiltonian
H(ψ1, ψ+1 , ψ2, ψ
+2 ) = H21ψ
+1 ψ2 +H12ψ
+2 ψ1, (4.2)
where ψ+k and ψk are creation and annihilation operators for agents in currency
k (k = 1, 2), and the coefficients H21 and H12 depend on y. According to
Ilinski, H21 = heβy and H12 = he−βy, where h and β are constants (we
discuss the motivation behind these formulas in Sect. 4.4). Following the
standard treatment of a quantum harmonic oscillator (see, e.g. Slavnov and
Faddeev (1980)), Ilinski Ilinski (2001) derived a path-integral expression for
the evolution operator in terms of the coherent states ψ1 and ψ2, which are
the eigenstates of the annihilation operators ψ1 and ψ2 respectively. From the
evolution operator one can obtain the expression for the part of the action s2
that represents the field’s effect on matter:
s2 =
∫ T
0dt
[
ψ1dψ1
dt+ ψ2
dψ2
dt+H(ψ1, ψ1, ψ2, ψ2)
]
, (4.3)
where the overbar denotes complex conjugation.
Finally, departing from the electrodynamics analogy, Ilinski introduced
Farmer’s term F to describe the effect of matter on the field. As a result, the
action s1 is replaced by
s1F = − 1
2σ2
∫ T
0dt
[
d(y + F )
dt
]2
, (4.4)
where
F =f
M(ψ1ψ1 − ψ2ψ2), (4.5)
M is the total number of agents, and f is a constant (Ilinski (2001) uses α in
place of f).
The total action is given by
s = s1F + s2. (4.6)
Following Ilinski, we introduce new variables η = βy and τ = ht, and replace
complex-valued ψk with φk and ρk, defined by ψk = (Mρk)1/2e−iφk (k = 1, 2)
and ρ1 + ρ2 = 1. Ilinski identifies Mρk with the number of agents in currency
k; the total number of agents is conserved. The action can be written as
s =M
∫ hT
0dτ L, (4.7)
83
4. A note on the theory of fast money flow dynamics
0 1 2 3 4 5 6 7 8 9 10−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Figure 4.1: Re-creation of Ilinski’s solution of Eqs. (4.9–4.11) given on page 169of Ilinski (2001) for α1 = 1.5, α2 = 10, C0 = 0, and the initial conditions:η(0) = 0.2, υ(0) = 0, ρ(0) = 0.5. The factor α1 in Eq. (4.10) is replaced withunity to match Ilinski’s Euler-Lagrange equations. The displayed quantitiesare as follows: ρ− 1/2 (solid), υ + η (dashed), η (dot-dashed), υ (dotted).
where the Lagrangian L is given by
L = −(2α2)−1(
η′ + α1ρ′)2
+ ρυ′ + φ′2+
+ 2[ρ(1 − ρ)]1/2 cosh(υ + η), (4.8)
with α1 = 2βf , α2 =Mβ2σ2/h, ρ = ρ1, υ = φ1−φ2. A prime denotes a deriva-
tive with respect to τ . Due to the unique structure of the Lagrangian (4.8),
the resulting Euler-Lagrange equations can be simplified to the following first
84
4. A note on the theory of fast money flow dynamics
0 1 2 3 4 5 6 7 8 9 10−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Figure 4.2: The solution of Eqs. (4.9–4.11) for the same parameters and initialconditions as in Fig. 4.1. The factor α1 in Eq. (4.10) is restored.
order differential equations:
η′ = α2(1/2 − ρ)−− 2α1[ρ(1− ρ)]1/2 sinh(υ + η) + C0,
(4.9)
υ′ = 2ρ− 1)[ρ(1 − ρ)]−1/2 cosh(υ + η)+
+ 2α1[ρ(1− ρ)]1/2 sinh(υ + η),(4.10)
ρ′ = 2[ρ(1 − ρ)]1/2 sinh(υ + η). (4.11)
However, some of the second-order nature of the Euler-Lagrange equations is
retained in the constant C0 = η′(0) + α1ρ′(0) + α2[ρ(0) − 1/2], whose value
depends explicitly on the derivatives ρ′(0) and η′(0). The equation for φ2 is
85
4. A note on the theory of fast money flow dynamics
trivial and we omit it. To solve Eqs. (4.9–4.11), one needs to specify the initial
conditions η(0), υ(0), ρ(0), and η′(0), which uniquely determine C0 (note that
ρ′(0) is given by Eq. (4.11)). Alternatively, one can set η(0), υ(0), ρ(0), and
C0, which uniquely determine η′(0).
4.3 Analysis of the Euler-Lagrange equations
Missing coefficient
By introducing new variables, ρ = ρ − 1/2 and η = υ + η, and linearizing
(|ρ| ≪ 1, |η| ≪ 1), we obtain η = ρ′ and
ρ′′ + (α2 − 4)ρ = C0. (4.12)
For α2 > 4, the general solution is
ρ = A sin(2πνt+ θ) + C0(α2 − 4)−1, (4.13)
η = 2πνA cos(2πνt+ θ), (4.14)
with ν = (α2−4)1/2/2π (A and θ are found from the initial conditions). This is
inconsistent with the solutions presented in Ilinskaia and Ilinski (1999); Ilinski
(2001), which exhibit oscillations decaying slowly with time. The origin of this
inconsistency can be traced to a simple algebraic mistake in the derivation of
the equations of motion given in Ilinskaia and Ilinski (1999); Ilinski (2001).
On page 168 of Ilinski (2001), the second term on the right-hand side of the
equation for υ′ is missing a factor α1. The same coefficient is also missing in the
equations given in Ilinskaia and Ilinski (1999). This is essentially equivalent
to replacing α1 in our Eq. (4.10) with unity, while keeping α1 in our Eq. (4.9)
intact.
We verify the above by numerically solving Eqs. (4.9–4.11) in their incor-
rect form (with α1 missing from one of the equations as in Ilinskaia and Ilinski
(1999); Ilinski (2001)) and in their correct form derived in this paper. We are
able to perfectly reproduce1 the plots presented on page 169 of Ilinski (2001)
by solving the incorrect equations (see Fig. 4.1). Note that we have α1 = 1.5
and α2 = 10 for the parameters used in Ilinski (2001). Ilinski claimed to set
η′(0) = 0 (dy(0)/dt = 0 in his notation), but this is obviously incorrect; the
solutions he presented are obtained for C0 = 0, which gives η′(0) ≈ −0.3020.
1In the caption of figure 7.2 in Ilinski (2001), it is claimed that one of the quantitiesdisplayed is η (y in Ilinski’s notation), but actually η + υ is plotted.
86
4. A note on the theory of fast money flow dynamics
0 1 2 3 4 5 6 7 8 9 10−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Figure 4.3: The solution of Eqs. (4.9–4.11) for the same parameters and initialconditions as in Fig. 4.2, except α1 = 0.
As anticipated by the linearized analysis, the correct nonlinear equations of
motion do not show any decay in the oscillation amplitude (see Fig. 4.2).
Furthermore, we do not observe any enhancement of oscillations for smaller
values of α1, as Farmer’s term becomes less important. In fact, the solutions
for α1 = 0 plotted in Fig. 4.3 are only slightly different from those for α1 = 1.5
(cf. the plots given on page 171 of Ilinski (2001)). After some exploration, we
conclude that Farmer’s term does not have any critical effect on the dynamics
of the system; it only affects the amplitude of oscillations of η and υ, and their
phase shift from ρ.
87
4. A note on the theory of fast money flow dynamics
Unstable solutions
In Sect. 4.3, we explored the dynamics of η = η + υ in the case C0 = 0.
However, there is no a priori reason why the initial conditions should conspire
to give C0 = 0. In this section, we briefly examine the dynamics of η = β lnS
in the more general case C0 6= 0.
Linearizing Eqs. (4.9) and (4.10) gives
η′ = −α2ρ− α1η + C0, (4.15)
υ′ = 4ρ+ α1η. (4.16)
We find that the solutions for η and υ are also harmonic oscillations plus an
extra term linear in time. The average value of η changes linearly with time
at a rate −4C0(α2−4)−1, while the average of υ changes at the same rate but
with the opposite sign. This behaviour is illustrated in Figs. 4.4 and 4.5 (note
that ρ and η remain small, so the linearization assumption is not broken).
Thus, for C0 > 0, the exchange rate S decays exponentially to zero, whereas
for C0 < 0, it grows exponentially. In both cases the exponential time-scale is
given by τc = 0.25(α2 − 4)|C0|−1.
Technical trading
Ilinski justified certain rules employed in technical trading (see Ilinskaia and
Ilinski (1999) and pages 170–173 of Ilinski (2001)), e.g., the use of positive
and negative volume indices (PVI and NVI respectively), by appealing to the
solutions of the equations of motion. The relevant figures are presented in
Ilinski (2001) on pages 170 (figure 7.3) and 172 (figure 7.7). We identify the
trading volume V with |ρ′| and the return R with η′/β = S′/S. In Ilinski
(2001), the derivative of η = υ+ η is used incorrectly instead of η to compute
the return (see also footnote 1). For comparison, we plot the volume and the
return curves in Fig. 4.6, computed using the correct equations of motion and
C0 = 0. The quantities plotted in figure 7.7 of Ilinski (2001) are not specified,
nor are the parameters and initial conditions, so we do not comment on that
figure’s validity.
Ilinski used the trading volume and the return curves to construct con-
tinuous2 versions of PVI and NVI. The details of the construction are left
unspecified. However, the PVI and NVI are usually computed from daily re-
turns, not from continuous intra-day variables. In any event, the resulting
2In technical trading, these quantities are discrete and defined by recursive formulas.
88
4. A note on the theory of fast money flow dynamics
0 1 2 3 4 5 6 7 8 9 10−1
−0.5
0
0.5
1
Figure 4.4: The solution of Eqs. (4.9–4.11) for the same parameters and initialconditions as in Fig. 4.2, except with C0 = 0.1 instead of C0 = 0. The curvesare coded as in Figs. 4.1 and 4.2.
construction must depend strongly on the time-scale that is chosen, since the
indices are defined recursively. Examining figure 7.7 in Ilinski (2001), one
observes that, for instance, the continuous PVI is constant if the trading vol-
ume V is decreasing with time and changes linearly if V is increasing, with a
slope of +1 where the return curve R is positive and −1 where R is negative.
However, this simple trend is inconsistent with the recursive definitions of the
PVI and NVI employed in technical trading.
Moreover, the constant-amplitude solutions we employed in this section
only exist for C0 = 0. In all other cases, the exchange rate converges to
89
4. A note on the theory of fast money flow dynamics
0 1 2 3 4 5 6 7 8 9 10−1
−0.5
0
0.5
1
Figure 4.5: As for Fig. 4.4, with C0 = −0.1.
zero or diverges to infinity exponentially on a short time scale. The condition
C0 = 0 requires precise alignment between the initial values ρ(0), v(0), η(0),
and η′(0). There is no reason to expect that such a precise alignment will
be observed at any time in the real market. Therefore, the lattice gauge
model predicts unrealistic behaviour (e.g., exponential divergence if C0 < 0)
of the exchange rate under most circumstances. Given the issues raised in
this section, it is premature to conclude that the technical trading schemes
employed by market participants can be justified by the lattice gauge model.
90
4. A note on the theory of fast money flow dynamics
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Figure 4.6: The trading volume V (bold solid curve) and the return R (bolddot-dashed curve) for the same parameters as in Fig. 4.2. For comparison, wealso display ρ− 1/2 (thin solid curve) and η (thin dot-dashed curve).
4.4 Revisiting the action
We conclude by re-examining the derivation of the action s given by Eq. 4.6.
Consider two currencies, referred to as currency 1 and currency 2, linked by
an exchange rate S(t) that depends on time t, such that the amount C2 of
currency 2 at time t corresponds to the amount C1 = S(t)C2 of currency 1. We
assume that the currencies can only be exchanged at the discrete times tn =
n∆t (n = 0, . . . , N) and define Sn = S(tn). At any given time tn, an agent can
decide to either exchange his stock of currency for the counterpart currency
91
4. A note on the theory of fast money flow dynamics
or keep his position, in which case his stock of currency remains unchanged
(recall that we neglect interest rates completely since we are interested in the
intra-day dynamics). We display these possibilities in Fig. 4.7, showing part
of the lattice from time tn to time tn+1.
✲Sn
✛S−1n
✲Sn+1
✛S−1
n+1
✻
❄
✻
❄
Currency 2 Currency 1
tn
tn+1
Figure 4.7: Lattice diagram for the intra-day foreign exchange trading in twocurrencies. Interest rates are ignored.
The returns on arbitrage along the closed loops of the elementary plaquette
shown in Fig. 4.7 are given by S−1n Sn+1−1 for the clockwise loop and SnS
−1n+1−
1 for the counter-clockwise loop. The total return SnS−1n+1 + S−1
n Sn+1 − 2 is
identified in Ilinski (1997) with the curvature on the lattice and, therefore, the
corresponding discrete action is given by
A1 =N∑
n=0
an(SnS−1n+1 + S−1
n Sn+1 − 2). (4.17)
Assuming that for any n we have an∆t → 1/2σ2 in the limit ∆t → 0, we
obtain the continuous action s1 given by (4.1). No justification is given in
Ilinski (1997, 2001) for why the limit of an∆t must be finite. The expression
for Farmer’s term was derived in Ilinski (2001), but we omit it because its
inclusion has no critical effect on the dynamics (see Sect. 4.3).
In order to derive the Hamiltonian given by Eq. (4.2) and the expressions
for the coefficients H12 and H21, Ilinski considered the case of a single trader
first and then generalized to multiple traders by using creation and annihi-
lation operators. In the case of a single trader, Ilinski postulated that the
probability of a given path Q through the lattice from t0 to tN is exponen-
tially weighted with respect to s(Q) = ln(U1U2 . . . UJ), where {Uj} are the
parallel transport coefficients on the lattice (note that J > N for most paths).
Thus, for a given path Q, the probability is given by
P (Q) ∼ eβs(Q). (4.18)
92
4. A note on the theory of fast money flow dynamics
Depending on the path, a given Uj can be Sn, S−1n , or unity (note that Ilinski
introduces a new gauge, under which the exchange rates remain unchanged,
except at t0 and tN where they equal unity; see pages 131–132 of Ilinski (2001)
for more details). The state of the trader is characterized by the probabilities
p1 and p2 of being in currency 1 and currency 2 respectively. The evolution
of the state vector ( p1p2 ) can be described by the transition matrix
P (tn; tn−1) =
(
1 Sβn
S−βn 1
)
, (4.19)
which Ilinski essentially identifies3 with the discrete version of the continuous
evolution operator U(t, t′) that satisfies
∂U
∂t= HU, (4.20)
where H is the Hamiltonian and U(0, 0) is the identity matrix. Ilinski claim
that the expression for the transition matrix (4.19) and the formula (4.20)
result in
H =1
∆t
(
0 Sβ
S−β 0
)
. (4.21)
Finally, identifying the parameter h with 1/∆t, we obtain the expressions for
H12 and H21, the Hamiltonian H given by (4.2), and the action s2.
In deriving the action Ilinski considered a more general case of non-zero
interest rates, but this does not nullify the two issues pointed out below.
Firstly, we note that the Hamiltonian given by (4.21) becomes infinite in
the limit ∆t → 0. It is stated in Ilinski (2001) that ∆t in the continuous-
time calculations “stands for the smallest time-scale of the theory, the time
cut-off” (see page 133). However, if ∆t is retained in the finite form in the
Hamiltonian and, therefore, the action s2, it must also appear in the finite
form in the expression for the action s1 for consistency. Secondly, we observe
that the transition matrix P (tn; tn−1) is degenerate; its determinant is zero.
Therefore, it cannot possibly be identified with the evolution operator. We
conclude that the justification provided for the Hamiltonian (4.2) in Ilinski
(2001) is insufficient.
3 In the case of non-zero interest rates, P (tn; tn−1) is related to U(tn; tn−1) by a simplematrix transform (see page 132 of Ilinski (2001)); however, P (tn; tn−1) = U(tn; tn−1) ifr1 = r2 = 0 and the transaction costs are zero.
93
4. A note on the theory of fast money flow dynamics
4.5 Conclusions
We have examined the theory of fast money flow dynamics developed in Ilin-
skaia and Ilinski (1999); Ilinski (2001) and uncovered errors in 1) the derivation
and the analysis of the equations of motion based on the theory, and 2) the
justification of the action based on the lattice gauge formalism.
The equations of motion presented in Ilinskaia and Ilinski (1999); Ilin-
ski (2001) are missing the coefficient α1 in one term, crucially modifying the
dynamics of the system. We also find that most of the solutions of the equa-
tions of motion, in their correct form derived in this paper, are unstable with
respect to the initial conditions, resulting in unrealistic behaviour of the ex-
change rate. We show that the justification of the technical trading given in
Ilinski (2001) is based on an erroneous interpretation of the variables related
to the exchange rate and on the stability predicted by the incorrect equations
of motion.
The theory of fast money flows relies on a particular form of the Hamilto-
nian that describes the effect of the exchange rate on the actions of the agents.
We demonstrate that this form is not consistent with the lattice gauge formu-
lation and diverges in the continuum limit.
94
Chapter 5
Memory on multiple
time-scales in an Abelian
sandpile
We report results of a numerical analysis of the memory effects in two-dimen-
sional Abelian sandpiles. It is found that a sandpile forgets its instantaneous
configuration in two distinct stages: a fast stage and a slow stage, whose du-
rations roughly scale as N and N2 respectively, where N is the linear size
of the sandpile. We confirm the presence of the longer time-scale by an in-
dependent diagnostic based on analysing emission probabilities of a hidden
Markov model applied to a time-averaged sequence of avalanche sizes. The
application of hidden Markov modeling to the output of sandpiles is novel. It
discriminates effectively between a sandpile time series and a shuffled control
time series with the same time-averaged event statistics and hence deserves
further development as a pattern-recognition tool for Abelian sandpiles.
5.1 Introduction
The Abelian sandpile is a simple open dynamical system governed by deter-
ministic rules, which demonstrates interesting emergent behaviour. Its charac-
teristic feature is the presence of avalanches, whose scale-free size distribution
is bounded above by the size of the sandpile. Sandpiles have been studied ex-
tensively in the past two decades both computationally and analytically, but
many of their properties remain unexplained Dhar (2006); Pruessner (2012).
95
5. Memory on multiple time-scales in an Abelian sandpile
Most research has concentrated on the microstructure and properties of the
avalanches or on long-term average properties of the recurrent configurations.
The intermediate time-scale, that covers many avalanches but not so many as
to reach the long-term averages, has not been investigated as thoroughly due
to its complexity. In this paper we present the results of numerical experiments
on Abelian sandpiles on the intermediate time-scale.
Starting from an arbitrary configuration in the recurrent regime, subse-
quent grain drops and the avalanches they cause gradually modify the dis-
tribution of charges that defines the starting configuration. This results in a
gradual loss of the memory of the starting configuration. Some features are
lost quickly while others persist for a long time until eventually all memory of
the starting configuration is lost. Even though the basic features of memory
loss are known Dhar (2006), the details of this process are not well understood.
For instance, we do not know how the rate of memory loss changes with time,
nor do we know which features of the starting configuration are responsible
for maintaining memory on different time scales. These issues are important
for developing a deeper understanding of the dynamical properties of Abelian
sandpiles. We explore these issues by conducting numerical simulations of a
two-dimensional sandpile and analysing the output in two independent ways:
(i) the distribution of occupation numbers in absolute difference maps, and
(ii) emission probabilities in a hidden Markov model (HMM). Hidden Markov
models Rabiner (1989) owe their success to ease of implementation and ef-
fectiveness in capturing temporal patterns. They have become one of the
standard tools in such diverse areas as speech recognition Pieraccini (2012)
and bioinformatics Durbin (1998). Since hidden Markov models incorporate
Markov chains in their setup, they allow one to examine statistically the suc-
cession of events, or states, an important capability missing in standard tech-
niques of statistical analysis. To our knowledge, this technique has not been
applied to sandpiles. We show here by way of a control experiment that it
captures hidden patterns in the succession of time-averaged avalanche sizes.
We discuss the relevant properties of Abelian sandpiles and introduce site
occupancy fractions in Section 5.2; their statistical properties are briefly ex-
plored in Section 5.3. Section 5.4 defines absolute difference maps and presents
results on the dual-time-scale evolution of sandpile memory. Hidden Markov
models are introduced in Section 5.5, where we emphasize their pattern recog-
nition and classification capacities. Their application to time-averaged se-
quences of avalanche sizes as a memory diagnostic is described in Section 5.6.
96
5. Memory on multiple time-scales in an Abelian sandpile
5.2 Abelian sandpiles
A sandpile is a cellular automaton introduced in Bak et al. (1987) as an exam-
ple of a slowly driven system with dissipative boundaries that spontaneously
evolves to a critical state, which is characterised by scale-free distributions. It
does not require any parameter adjustment to achieve criticality, unlike (say)
the Ising model. Hence this system and its numerous derivatives demonstrate
what has become known as self-organised criticality Jensen (1998). The orig-
inal model described in Bak et al. (1987) is closely based on its prototype,
an actual pile of sand onto which grains of sand are dropped. The slope of
the pile increases, as grains are added, until an avalanche occurs, whereupon
the slope decreases. The time of the next avalanche is unpredictable and the
distribution of avalanche sizes is approximated by a power law. A sandpile
is an open dynamical system with random driving and deterministic rules for
toppling.
In Dhar (1990), a generalisation of the original toppling rules was proposed,
which has the property that the outcome of an avalanche does not depend on
the order in which the critical sites topple. Most research in the last two
decades has concentrated on this model known as the Abelian sandpile rather
than the original model where an avalanche does depend on the order of
the toppling. The properties of Abelian sandpiles have been investigated by
means of computer simulations and analytical approaches including branching
processes and spanning trees, Abelian operators, loop-removed random walks,
mean-field theory, renormalisation techniques and the logarithmic conformal
field theory (see Pruessner (2012) for a comprehensive review).
Toppling rules
In a typical simulation of a two-dimensional sandpile one considers a square
grid of sites described by a matrix zij , i, j = 1, . . . , N , where N is frequently
chosen to be a multiple of two. The values of zij are variously known as
height, slope, or charge. Initially the sandpile is assumed to be empty, with
zij = 0. At each step of the evolution, the charge of a randomly selected site
is increased by one, representing a grain drop at this site. That is, the charge
of a site can be thought as the number of grains located at the site. A site
is considered stable if zij < 4 and unstable or critical if zij ≥ 4. If a grain
drop creates an unstable site, an avalanche begins and further drops are halted
97
5. Memory on multiple time-scales in an Abelian sandpile
until the avalanche finishes. A critical site contains at least four grains and
it topples by distributing four of its grains into the four neighbouring sites,
i.e. if (i, j) is a critical site, then it topples according to the following rule:
zij → zij − 4, zi±1,j → zi±1,j + 1, and zi,j±1 → zi,j±1 + 1. If a critical site
is at the edge of the sandpile, it lacks one or two neighbours and the grains,
which would have gone to those neighbouring sites, are removed (i.e. they fall
off the edges of the pile). The avalanche continues until there are no critical
sites left, after which regular evolution of the sandpile consisting of random
grain drops resumes. The number of topples in an avalanche is referred to as
the size (or power) of the avalanche.
Site occupancy
If the sandpile starts from an empty configuration, initially there are no or few
critical sites and the charge increases steadily at most sites. The sandpile is in
the transient regime. After about 2.125N2 steps the mean charge reaches the
expected value of 2.125, after which the influx of grains is balanced on average
by the efflux over the edges due to avalanches. At this stage the sandpile is
in the recurrent regime. Once the sandpile is in a recurrent configuration, it
can only move to another recurrent configuration; given enough time any re-
current configuration is recreated eventually. The so-called burning algorithm
Dhar (1990) can be used to establish whether a configuration is recurrent or
transient. It recursively removes every site, whose charge is equal to or greater
than the number of its neighbours, starting from the edges of the sandpile.
The number of recurrent configurations is estimated to be ∼ 3.21N2
Dhar
(2006), which for a large sandpile is a vanishingly small fraction of the total
number of configurations 4N2
.
The problem of determining the probability pk that a randomly chosen
site has charge k in a recurrent configuration has been solved analytically in
the limit N → ∞ using various techniques such as uniform spanning trees
Priezzhev (1994); Caracciolo and Sportiello (2012), loop-erased random walks
Kenyon and Wilson (2011), and the dimer model Poghosyan et al. (2011). The
analytical expressions give p0 ≈ 0.0736, p1 ≈ 0.1739, p2 ≈ 0.3063, p3 ≈ 0.4462.
The number of sites nk with charge k = 0, 1, 2, 3 can be easily computed for
a given configuration zij. Then, fk = nk/N2 gives the fraction of sites with
charge k = 0, 1, 2, 3, and one has∑3
k=0 fk = 1. Considering recurrent configu-
rations only, the ensemble-averaged quantities 〈fk〉 approach the probabilities
98
5. Memory on multiple time-scales in an Abelian sandpile
pk as N → ∞. For finite sandpiles, the ensemble-averaged fractions 〈fk〉 devi-ate from pk, and the discrepancy increases as N decreases. The mean sandpile
charge can be estimated numerically by∑3
k=1 k〈fk〉 for finite N and is given
by∑3
k=1 kpk = 2.125 in the limit N → ∞.
Toppling waves
Avalanches in an Abelian sandpile can be viewed as waves of toppling Ivashke-
vich et al. (1994), where each wave is defined by allowing all critical sites, bar
the original critical site, to topple until there are no critical sites left, at which
point the original critical site is toppled again thereby launching the next
wave. The region encompassed by a wave consists of a set of contiguous sites
(without holes) each of which has toppled once. After a wave stops, the sites
whose charge differs from the original charge before the wave passes through
are located at the boundary of this region, while all internal sites regain their
original charge. The boundary itself consists of two thin layers; the wave
causes the charge of the sites at the outer layer to increase and the charge of
the sites at the inner layer to decrease. Avalanches that consist of multiple
waves can be thought of as a series of consecutive waves; subsequent waves
can be larger or smaller than preceding waves, and there is no restriction on
the number of waves in a given avalanche (note that the number of waves
is equal to the number of times the trigger site topples Fey et al. (2010)).
The structure of a multi-wave avalanche is therefore more complex than that
of a single-wave avalanche. Furthermore, the interaction of the waves in a
multi-wave avalanche is likely to be a factor in the observed deviation in two
dimensions of the avalanche size distribution from the power-law predicted by
the mean-field approximation Abdolvand and Montakhab (2010).
Temporal correlations
In this paper, we concentrate on the problem of sandpile memory, as mani-
fested in the temporal sequence of avalanche sizes. Detailed, time-averaged
statistics of output sequences have not received much attention in the litera-
ture. Most of the emphasis has been on the power spectral density of sandpile
“noise”. Indeed, the original motivation for studying sandpiles was to model
the long-range temporal correlations with a 1/f -type power spectrum exhib-
ited by some dynamical systems. Further studies revealed that the power
spectrum of the sandpile’s temporal activity has 1/f2 behaviour (see Jensen
99
5. Memory on multiple time-scales in an Abelian sandpile
0.0 0.1 0.2 0.3 0.4 0.5fk, k = 0, 1, 2, 3
100
101
102
p(f
k)
Figure 5.1: Probability density function p(fk) of the fraction fk of sites withcharge k = 0, 1, 2, 3 (left to right) for two sandpile simulations on a squaregrid with N = 32 (blue) and N = 64 (red), based on samples consisting of103N2 steps in the recurrent regime. The curves show the normal distributionsfor the corresponding values of the mean and standard deviation; they agreewell with the data. The first 5N2 steps of the simulations are discarded toeliminate transient configurations.
(1998) for details). Studies of temporal and spatial correlations in sandpiles
typically use some modification of the standard rules of the Abelian sandpile,
e.g. Davidsen and Paczuski (2002); Barrat et al. (1999), or confine the in-
vestigation to sandpiles consisting of narrow strips (a quasi-one-dimensional
geometry), e.g. Yadav et al. (2012); Maslov et al. (1999). As loss of memory in
sandpiles is related to relaxation due to grain movement caused by avalanches,
we also note the study of residence time of grains Dhar and Pradhan (2004)
where the scaling function of probability distribution of residence time is de-
rived explicitely for a simplified one dimensional model and a constraint is
obtained in the general case. In this paper, in contrast, we use the standard
Abelian sandpile model to investigate memory effects concerned with specific,
time-ordered changes in the global structure of a given sandpile configuration,
as the sandpile evolves away from it.
5.3 Site occupancy fraction distributions
The simulations we perform follow the standard recipe for a two-dimensional
Abelian sandpile on a square grid withN2 sites. We start each simulation from
the zero configuration zij = 0 (the empty sandpile). A single grain is added
at each step at a random position; no grains are added during an avalanche.
100
5. Memory on multiple time-scales in an Abelian sandpile
N µ1 µ2 µ3 µ4 σ1 σ2 σ3 σ432 0.0790 0.1844 0.3110 0.4256 0.0062 0.0102 0.0142 0.010564 0.0763 0.1793 0.3090 0.4354 0.0030 0.0050 0.0071 0.0051∞ 0.0736 0.1739 0.3063 0.4462
Table 5.1: Mean values and standard deviations for the samples shown inFigure 5.1. The bottom row gives the analytical estimates of probabilities pk,obtained in the limit of an infinitely large sandpile.
Dissipation happens at the edges of the sandpile, where grains fall out of the
system. The first 5N2 steps are excluded from the statistical analysis to avoid
transient configurations. Following each grain drop the sandpile is allowed to
relax, if the drop causes an avalanche. The sandpile configuration at time-step
t is given by the matrix zij after t grain drops and relaxations.
The first set of simulations explores the baseline behaviour of the fractions
fk introduced earlier. The probability density function p(fk) for each k is
presented in Figure 5.1 for a simulation consisting of 103N2 time-steps in
the recurrent regime. The numerical output can be fitted reasonably well
(see Figure 5.1) by Gaussian distributions with the parameters recorded in
Table 5.1. The standard deviations decrease by approximately a factor of
two as N doubles. As N increases the mean values µk = 〈fk〉 approach the
analytical estimates pk obtained in the limit N → ∞.
5.4 Short- and long-term memory in site
occupancy
One way to study memory in an Abelian sandpile is to measure the difference
between two configurations separated by a certain time interval. The details
of this approach are as follows. For a given t0 > 5N2 we compute the absolute
difference matrix Dij(t, t0) = |zij(t)−zij(t0)| between the initial configuration
zij(t0) at time-step t0 and a subsequent configuration zij(t) at t > t0; this
approach is somewhat reminiscent of how damage is measured in (Stapleton
et al., 2004) even though in our case no damage is introduced but two con-
figuration at different times are compared. Since for each position (i, j) we
have zij(t) ∈ {0, 1, 2, 3} at any time-step t, entries in the matrix Dij(t, t0) are
restricted to the same set of values, and we have Dij(t0, t0) = 0. Dij(t, t0)
can be used to describe the rate and dynamics of how the sandpile forgets
the configuration at t0 as t increases, but it contains too much information to
101
5. Memory on multiple time-scales in an Abelian sandpile
be practical. It is more manageable to consider the fractions δk of sites with
value k in the absolute difference matrix Dij . This is the same calculation
applied previously to the matrix zij to find fk but now applied to Dij to find
δk. The fractions δk are functions of the time delay t − t0 and can change
significantly after an avalanche. As we do not wish to be biased towards a
particular avalanche history, we consider the ensemble-averaged fractions 〈δk〉based on a large sample of trials.
As a control experiment, one can compute the fractions δk for the absolute
difference matrix of two sandpile configurations taken randomly from the set
of recurrent configurations. Given a large enough sample of such configura-
tions, one obtains distributions for δk and mean values 〈δk〉, which can also
be estimated analytically in the limit N → ∞ as
d0 = p20 + p21 + p22 + p23, d1 = 2(p0p1 + p1p2 + p2p3),
d2 = 2(p0p2 + p1p3), d3 = 2p0p3, (5.1)
with approximate values d0 ≈ 0.3285, d1 ≈ 0.4055, d2 ≈ 0.2003, d3 ≈ 0.0657.
For instance, a site (i, j) with Dij = 3 arises only if the corresponding location
in the two configurations used to compute the absolute difference has 0 in
one configuration and 3 in the other, or vice versa. Since the probabilities
to observe charge 0 and 3 at a given location are given by p0 and p3, the
probability to find 0 at a given location in one configuration and 3 at the same
location in the other configurations is given by p0p3, which implies d3 = 2p0p3.
The ensemble-averaged fractions 〈δk〉 converge to dk once enough time elapses,
such that the configuration at t loses all memory of the initial configuration
at t0 as t→ ∞. Note that for a relatively small sandpile such as 32× 32, it is
more relevant to use the mean values µk in place of pk to compute dk, since
the probabilities pk are determined in the limit N → ∞.
To test for this loss of memory, we conduct a second set of simulations,
where we compute δk as a function of t− t0 for a large number of trials. We
use several sizes ranging from N = 16 to N = 128 and run each trial for at
least N2 time-steps, after which the value of t0 is reset to the current time-
step t for the next trial. The number of trials varies from 105 for N = 16
to 100 for N = 128. The results for N = 32 with 104 trials are shown in
Figures 5.2a and 5.2b. We observe that the fractions 〈δk〉 converge to dk on
the time-scale ∼ N2, which is close to the time-scale over which a sandpile
starting with zij = 0 approaches the recurrent regime. Note that the ensemble-
averaged curves shown in Figure 5.2 eliminate most of the jitter characteristic
102
5. Memory on multiple time-scales in an Abelian sandpile
100
101
102
103
t−
t 0
0.0
0.2
0.4
0.6
0.8
1.0
〈δk〉
k=0
k=1
k=2
k=3
(a)
0200
400
600
800
1000
t−t 0
10−4
10−3
10−2
10−1
100
log10|〈δk〉−dk|
k=0
k=1
k=2
k=3
(b)
Figure
5.2:
Siteoccupan
cyan
alysisof
anAbeliansandpile.
(a)Ensemble-averagedsite
occupan
cyfraction
s〈δ
k〉v
ersusthetime
delay
t−t 0,whereδ k
isthefraction
ofsiteswithchargek=
0,1,2,3in
theab
solute
differen
cematrixD
ij=
|z ij(t)−z ij(t
0)|.
Theen
semble-averagedcu
rves
arebased
on10
4trials
(onerepresentative
trialis
show
nin
black).
Thestan
darderrorof
the
ensemble-averagedvalues,whichis
∼0.001,
isnot
show
n.Thedashed
lines
representtheexpectedvaluesdkfort→
∞an
dN
=32.(b)Approachof
〈δk〉t
otheexpectedvaluesdkshow
non
alogarithmic
scale.
103
5. Memory on multiple time-scales in an Abelian sandpile
of individual trials, the level of which (estimated visually to be ∼ 0.1) gives
∼ 0.1/√104 = 0.001 for the standard error of the ensemble-averaged curves.
Interestingly, Figure 5.2a reveals that the memory of a given configuration is
lost in two stages: a fast stage on the time-scale ∼ N , characterised by rapid
changes in 〈δk〉, and a slow stage on the long time-scale, where the fractions
〈δk〉 vary slowly. The transition from the fast stage to the slow stage is followed
by a small bump or depression in 〈δk〉 near t − t0 ≈ 100 superimposed over
approximately exponential curves. Moreover, 〈δ2〉 approaches d2 noticeably
quicker than the other fractions.
The loss of memory during the fast and slow stages is roughly exponential,
and fitting the sum of two exponentials (Koehler, 2012) to each memory curve
shown in Figure 5.2b yields the following decay times: τ0 = 21, τ1 = 20,
τ2 = 22, τ3 = 24 for the fast stage and τ0 = 190, τ1 = 200, τ2 = 100,
τ3 = 230 for the slow stage, where the subscript refers to the charge k. The
large uncertainty in the exponential fits does not allow us to infer accurate
scaling dependence of the decay times, but we find it broadly consistent with
the approximate scaling dependence of the fast and slow stages, i.e. ∼ N and
∼ N2.
At the beginning of the fast stage, each avalanche contributes fully (or
nearly fully) to the departure from the original configuration and therefore to
rapid memory loss. Even though some of the sites affected by an avalanche
may have already been changed by previous avalanches, they constitute a
small fraction of the affected sites. However, the fraction of such sites tends to
increase with each avalanche, until eventually the majority of sites affected by
a new avalanche have already been changed by previous avalanches. At this
stage, each avalanche contributes marginally to memory loss and therefore
the memory loss slows down. We hypothesise that this corresponds to the
transition from the fast stage to the slow stage.
5.5 Hidden Markov model
The short- and long-term memory loss displayed in Figure 5.2 pertains to
global quantities, such as the site occupancy fractions 〈δk〉. Another global
characteristic, which can be used to address the problem of memory loss, is the
sequence of avalanches and their associated sizes. By averaging avalanche size
over a fixed time interval we obtain a quantity describing the average activity
of the sandpile during that period. The patterns in the sequence of average
104
5. Memory on multiple time-scales in an Abelian sandpile
sizes contain information about sandpile memory. To access this information,
we now conduct an experiment using hidden Markov methods. The method
is described in Section 5.5 and the results of the experiments in Section 5.6.
Given a sequence of observed states characterising some system, a hidden
Markov model seeks to represent the observed sequence in terms of 1) an un-
derlying finite-state Markov chain that cannot be observed and 2) an emission
model, which gives the probability of each observed state when the Markov
chain is in a given hidden state (see Rabiner (1989)). In this paper we only
consider HMMs with discrete observed and hidden states. At each time-step
the system is described by one of the observed states and is assumed to be
in one of the hidden states. The meanings (i.e. the physical significance) of
the hidden states are not known a priori. The Markov chain is described by a
set of transition probabilities, which form an n × n matrix A, where n is the
number of hidden states. The emission model is described by a set of emission
probabilities, which form an n ×m matrix B, where m > n is the number of
observed states. To complete the HMM, one also needs to define a vector π of
initial probabilities of the hidden states. The set {A,B, π} defines the HMM.
The Baum-Welch forward-backward algorithm is a likelihood-maximisation
technique that recursively adjust the probabilities of the HMM in such a way
that the probability of the observed sequence increases Rabiner (1989). The
recursive adjustment, also referred to as training (or learning), continues until
the probability of the observed sequence ceases to increase, and the HMM pa-
rameters stabilise. To start training, one specifies some initial HMM, i.e. some
values for transition probabilities, emission probabilities, and initial probabili-
ties of the hidden states. The rate of convergence and the limit of convergence
depend strongly on the initial specification. There is no guarantee of conver-
gence to an optimal model, so one performs training for a range of initial
HMMs. The training algorithm is easy to implement and its description can
be found in e.g. Rabiner (1989); Li et al. (2000).
To illustrate the classification potential of an HMM consider the following
example Cave and Neuwirth (1980); Stamp (2004). A text in English can be
treated as a sequence of 27 states, consisting of 26 characters of the alphabet
and the symbol for space, after all punctuation marks and other symbols are
removed. An HMM with two hidden states, whose meanings are not known
a priori, can be trained on a given text. Remarkably, the resulting emission
probabilities reveal that one hidden state, say, H0, corresponds to vowels and
the other, say, H1, to consonants. Specifically, if the system is in the hidden
105
5. Memory on multiple time-scales in an Abelian sandpile
H0 H1
H0 0.32 0.68H1 0.65 0.35
O0 O1 O2
H0 0.00 0.32 0.68H1 0.65 0.35 0.00
Table 5.2: Transition probabilities Aij (left) between the hidden states i =0, 1 (H0,H1) and emission probabilities Bij (right) from the hidden states toobserved states j = 0, 1, 2 (O0, O1, O2) for the averaging time-scale Ta = 32.
state H0 at a certain position in the text, the probability of the observed
state in that position is high for vowels and low for consonants. The reverse
is true for the hidden state H1, i.e. the probability is high for consonants
and low for vowels. Since an HMM strives to capture the foremost statistical
properties of the succession of observed states, one concludes that the most
prominent statistical feature of an arbitrary English text in terms of the two-
state classification is the dichotomy between vowels and consonants.
5.6 HMM analysis of long-term memory
We harness the classifying power of HMMs to investigate memory loss in a
sandpile. At each step a sandpile can be characterised by the output size of
the released avalanche; if no avalanche is initiated, the size is zero. The power
released by an avalanche after each drop depends in an unpredictable way on
where exactly the dropped grain lands. Since the grains are dropped randomly,
this translates into shot noise in the observed avalanche size, which has little
to do with the intrinsic internal structure of the sandpile. To suppress the
shot noise, we average the avalanche size over an averaging interval Ta and
bin the average size into three bins (low, medium, high), such that the number
of samples in each bin is the same.
We apply an HMM to the average size time-series generated by the 32 ×32 sandpile (N = 32), specifically to a sequence of 103N3 time-steps in the
recurrent regime (the first 5N2 steps are discarded). We consider several
averaging time-scales spaced logarithmically in the range N ≤ Ta ≤ N3.
For each averaging time-scale we feed a sequence of 103 samples of average
avalanche size into the HMM; that is, the observed sequence contains 103
terms for each averaging time-scale Ta. The observed sequence consists of
three states. We assume there are two hidden states. For each Ta we use 20
random initial HMMs and train each model for several thousand steps, which
is sufficient for the best models to stabilise.
106
5. Memory on multiple time-scales in an Abelian sandpile
3226
2728
291024
211
212
213
214
215
Ta
0
0.250.5
0.751
Bij
(a)sandpilesequen
ce
3226
2728
291024
211
212
213
214
215
Ta
0
0.250.5
0.751
Bij
(b)shuffled
sandpilesequen
ce
Figure
5.3:
Hidden
Markovan
alysisof
anAbeliansandpile.
(a)Emission
probab
ilitiesB
ijfrom
hidden
statei=
0(red
)an
dhidden
statei=
1(blue)
toob
served
statesj=
0,1,2versustheaveragingtime-scaleTaforthe32×32
sandpile.
Adiscrete
HMM
withtw
ohidden
states
andthreeob
served
states
isused.Observed
states
aredefi
ned
bybinningtheavalan
chesize
into
high,med
ium,an
dlow
bins,
such
that
thereareequal
number
ofsamplesin
each
bin
foreach
averaginginterval.(b)Results
ofacontrol
experim
entwherethesameinputdatais
shuffled
random
lybeforebeingfedinto
theHMM.
107
5. Memory on multiple time-scales in an Abelian sandpile
The transition and emission probabilities of the final HMM with the high-
est likelihood are shown in Table 5.2 for Ta = 32. We observe that the hidden
state i = 0 is biased towards the lower values of the averaged size and i = 1
is biased towards the higher values. In other words, when the sandpile is in
the hidden state i = 0 for the duration of the averaging period Ta = 32, the
avalanches have higher power on average (loud state) as compared to when
the sandpile is in the hidden state i = 1, when avalanches have lower power
on average (quiet state). We also observe that a transition from one to the
other hidden state is roughly twice as likely as the lack of a transition.
The emission probabilities Bij of the best HMM at the end of training
for other values of Ta are displayed in Figure 5.3a. The values Bij are fairly
stable in the range N ≤ Ta ≤ 2N2, where N = 32, and the same applies to the
transition probabilities. This indicates that there is no fundamental difference
in the statistics of the average size sequence on these time-scales. However,
for Ta > 2N2 the final HMMs are qualitatively different. Loud and quiet
states are no longer distinguishable, as can be seen in Figure 5.3a, and the
likelihood of a transition between the hidden states increases markedly. This
result is consistent with the memory horizon observed in the memory plots
discussed previously (Figure 5.2). It is unlikely that any memory will persist
over a time interval longer than ∼ N2 time steps. Indeed, a square sandpile
with N2 sites takes about 2.125N2 steps to reach the recurrent regime from
an empty configuration, where 2.125N2 is the expected mean number of sand
grains in the sandpile. In the recurrent regime, one might expect the sandpile
to migrate to a completely different configuration on the same time-scale.
The results of a control experiment are shown in Figure 5.3b, where we
apply the same procedure as above to randomly shuffled sequences of sizes, i.e.
the temporal pattern is shuffled without changing the probability distribution
of sizes. We find that the final HMM resembles that obtained in Figure 5.3a
for Ta > 2N2. That is, shuffling the sequence of sizes completely destroys the
classification pattern found by the HMM on the original unshuffled sequences
of average sizes.
5.7 Conclusion
Our simulations indicate that an Abelian sandpile forgets a given configuration
in two stages: a fast stage on the time-scale ∼ N and a slow stage on the
time-scale ∼ N2. The details of memory loss are embedded in the particular
108
5. Memory on multiple time-scales in an Abelian sandpile
sequence of grain drops and avalanches, which is reflected in the behaviour of
site occupancy fractions as functions of time. By taking the ensemble averages
of the site occupancy curves over many configurations, we eliminate the details
of a particular sequence of random drops and derive the smooth memory curves
characteristic of a given Abelian sandpile. We find that memory loss is roughly
exponential during the fast and slow stages, with the site occupancy fraction
for charge 2 decaying much faster than the other fractions. We also observe a
hint of an oscillation in the memory curves following the transition from the
fast to the slow stage.
An independent analysis based on hidden Markov modeling confirms that
memory extends up to the time scale ∼ N2. The analysis identifies the hidden
states with quiet and loud periods in the sandpile’s evolution, during which
the avalanche power is respectively low or high on average. We note a re-
markable consistency in the output of the hidden Markov models obtained
from avalanche size sequences averaged on different time-scales ranging from
N to N2. This indicates that the statistical properties of the succession of the
observed states, and by implication the hidden states too, are similar across a
broad range of time-scales. As the waiting time between individual avalanches
is exponential, the temporal patterns detected by the hidden Markov model
must be driven by longer term structural changes in the sandpile. The tem-
poral patterns disappear if the observed sequence is shuffled, even though the
distribution of avalanche sizes remains the same.
Our HMM analysis focuses on the avalanche size, since this is the simplest
and most prominent characteristic of the sandpile evolution. The patterns
detected by the HMM are interesting but difficult to interpret without further
insight into the underlying structure of the sandpile. Of course, the sandpile
possesses many other characteristics that can be fed, after averaging, into an
HMM. Future studies could address and exploit sandpile properties like the
waiting time between avalanches, number of waves per avalanche, the mean
charge of the sandpile, or the occupancy numbers, just to name a few. More-
over, one can combine several characteristics into a single observed quantity to
take advantage of correlations between variables. We do not know at present
which combination of parameters will be the most successful in detecting the
underlying patterns of the sandpile, but future numerical experiments should
be able to address these questions.
Other promising avenues for future studies include 1) improving statisti-
cal significance of the reported results by increasing the number of samples
109
5. Memory on multiple time-scales in an Abelian sandpile
used for computing the memory curves and in the hidden Markov analysis, 2)
extending the simulations to larger two-dimensional and higher-dimensional
sandpiles, as well as other sandpile models such as Manna and Oslo models
(Manna, 1991; Christensen et al., 1996), 3) developing a dynamical model of
memory loss, and possibly based on the mean-field approximation or other
analytical techniques, which explains the evolution of the ensemble-averaged
site occupancy fractions.
110
Chapter 6
Conclusion
The main objective of this work is to contribute to our understanding of
the economy and financial system by conducting a number of projects in the
emerging field of econophysics, which uses techniques developed in statistical
physics to model and analyse economic and financial systems. The thesis ad-
dresses the following issues: 1) empirical investigation of transactional flows
between commercial banks and their network properties, 2) analytical and
agent-based investigation of wealth distributions and income inequality, 3) an
analysis of lattice-gauge theories of fast money flows on the foreign exchange
market, 4) numerical study of memory effects in stochastic dynamical systems
and the application of hidden Markov models to discover temporal patterns
in such systems. An important component of this work was to create a foun-
dation for future research in the field of econophysics at the University of
Melbourne. This was achieved by exploring a broad range of diverse issues of
interest and methods employed in econophysics research such as multi-agent
simulations and network science. The over-arching theme that ties the issues
explored in the thesis is the nature and properties of monetary instruments
and their role in the modern economy.
6.1 The interbank network
Empirical investigation of the Australian interbank transactional flows and
their network properties described in chapter 2 contributes to the growing
body of literature on the interbank network properties in other countries,
their network topologies, and properties of the flows. It is based on the data
provided by the Reserve Bank of Australia through their real-time gross set-
111
6. Conclusion
tlement system that captures all high-value transactions between banks in
Australia. The study offers a unique view of the structure and variability of
daily monetary flows in the Australian banking system and is the first study
to report on the Australian interbank network and its variability. Another
unique feature of this study is simultaneous investigation of the transactional
flows due to overnight loans and the flows due to other (nonloan) payments.
The sample of interbank transactions provided by the RBA consists of all
payments including overnight loans; the loans have been found by following
Furfine (2003). The procedure involves comparing transactions on two con-
secutive days and detecting those transactions that reverse on the next day
with the same amount plus interest, which closely matches the central bank’s
target rate. Just under 900 overnight loans have been identified over the
course of four days (a week in February 2007) out of over 95000 transactions
over the same period. The overnight loans account for less than 1% of all
transactions and about 6.5% of value of all payments. The interest rate of the
overnight loans is found to lie within 0.1% per annum of the target rate set
by the RBA, which was 6.5% per annum during this period. The distribution
of the nonloan transactions is well represented by a mixture of two lognormal
components, which are likely to correspond to the transactions arising from
the SWIFT and Austraclear feeds of the gross settlement system.
A major finding of the study is strong anti-correlation (whose value is
about −0.9 on most days) between the daily imbalances of overnight loans
and nonloans. The daily imbalance is computed for each bank and represents
the cumulative change in the reserve account of the bank, i.e. it is equal to
the difference in value of all incoming and outgoing transactions on a given
day. The payments recorded by the RBA correspond to financial and other
transactions between the customers of commercial banks and to a smaller
extent between the banks themselves. The daily stream of such payments is
largely random and results in an unpredictable change in the reserve accounts
of the commercial banks, with some banks’ accounts increasing in value while
the others decreasing. The money held in the reserve accounts attracts smaller
interest than the target rate set by the RBA, which encourages the banks with
the positive imbalances to lend the excess in the short term money market. At
the same time, banks whose reserves are depleted seek to eliminate the deficit,
and the implied liquidity risk, by acquiring overnight loans. This creates
an interesting dynamics in the interbank network, whereby flows of nonloan
transactions create imbalances of the banks’ reserves, which in turn engender
112
6. Conclusion
flows of overnight loans to remove the imbalances. The study reported in
Chapter 2 confirms this dynamics and provides empirical constraints on the
extent of the connection between loan and nonloan flows.
Comparing the interbank networks on different days during the week re-
veals that about 80% of nonloan flows persist for two days or more, although
the amount of persistent flows can change significantly. For the overnight loan
flows, only 50% of the flows persist. Furthermore, persistent loan flows carry
about 65% of the total value of the loan flows, whereas persistent nonloan
flows account for as much as 96% of the total value of nonloan flows. There
is no significant correlation between individual loan and nonloan flows, i.e.
overnight loans and other payments are linked via imbalances only.
As for the net flows, the number of transactions per net flow is approx-
imated well by a power law with the exponent α = −1 for nonloans and
α = −1.4 for overnight loans. Out of about 470 net nonloan flows per day
there are 110 flows that consist of a single transaction (usually these occur be-
tween small banks); more than 1000 transactions can be present in a net flow
between two large banks. Similarly, out of about 60 loan flows per day there
are as many as 40 flows that consist of a single transaction (between small
banks); loan flows between large banks can consist of 30 individual loans or
more. Given the small number of commercial banks in Australia compared
to other countries, where the interbank networks have been studied, it is not
surprising that the shape of the degree distribution of the Australian inter-
bank network is difficult to infer. It is inconsistent with a power law, which
has been observed in many countries, and is close to an exponential distribu-
tion, although this could not be rigorously confirmed. The loan and nonloan
networks are disassortative with an assortativity coefficient of about −0.4 for
nonloans and −0.1 for loans on average. The topology of the net flows is
found to be highly variable, with many circular and transitive flow structures
present.
The study of the interbank network based on five days provides a valuable
insight into the dynamics of the interbank network. However, to understand
statistical properties of the network and its variability requires follow-up stud-
ies based on longer sequences of data. A longer sequence of data will also allow
to constrain the contribution from interbank loans with two-day maturity or
longer. In addition, the intraday timing of transactions will allow to inves-
tigate the dynamics at more finely grained time scales, which is significant
for understanding the interbank network’s reaction to external events such
113
6. Conclusion
as changes in target interest rates or other relevant economic news. These
future studies are essential for uncovering the dynamical laws that govern the
dynamics of the interbank flows composed of high-value transactions, particu-
larly given that the internal dynamics of monetary flows in interbank networks
has been neglected so far. The relationship between loan and nonloan flows via
the daily imbalances of the banks that was reported in Chapter 2 is a natural
one but it has not been addressed quantitatively in the studies of the banking
networks in other countries. Therefore, there is a need for similar studies in
order to confirm this relationship in other countries where the properties and
the institutional design of the banking system may be different.
Another area of future studies that could stem from research reported in
Chapter 2 concerns multi-agent numerical simulations of transactional flows
in banking systems. These simulations are of particular interest as they al-
low one to probe various mechanism that determine bank’s behaviour in the
overnight loan market in response to changes in their reserve accounts. An
appropriate model for nonloan transactions also needs to be investigated. A
naive approach where nonloan payments are assumed to be randomly taken
from a suitable distribution may be incorrect. Indeed, a significant reduction
in the reserve account of a bank indicates the preponderance of outgoing pay-
ments (by value) over the incoming payments. If this occurs, it may reduce
the likelihood of further outgoing payments, since the funds of the bank’s
customers have been reduced. Similarly, a significant increase in the reserve
account of a bank may increase the likelihood of outgoing payments. Since
the overnight loans have to be paid back (with interest) on the next day (or
the day after next in case of two-day loans), the largely random dynamics
of nonloan payments has an effect on the following days and therefore the
dynamics on a given day cannot be considered independent of previous days.
This may have serious implications for stability of the interbank network and
the continuous operation of the real time gross settlement system.
6.2 Wealth distributions
The studies reported in Chapter 3 seek to refine our understanding of asset
exchange systems, and the giver scheme in particular, by proposing an efficient
technique for numerically computing the shape of the wealth distribution in
the steady state. The technique also provides an interesting example of em-
ploying a numerical inverse Laplace transform to solve the master equation
114
6. Conclusion
that describes the detailed balance of the giver scheme by matching influx and
outflow of agents at every value of wealth. In the steady state, the Laplace
transform of the master equation yields a functional equation that is amenable
to analysis, e.g. by Taylor series, which gives a recursive expression for the
moments of the wealth distribution. The functional equation is solved numer-
ically by iterations in the complex plane; numerical experiments reveal that
the convergence does not depend on the shape of the initial approximation or
details of the grid. The solution of the functional equation thus obtained is
then fed into the numerical inverse Laplace transform, which yields the wealth
distribution for the specified value of the transfer parameter.
The procedure for computing the wealth distribution in the giver scheme
compares favourably with the direct approach of estimating the distribution by
running a multi-agent simulation of the asset exchanges. It is computationally
faster and provides much better precision than estimating the distribution
by computing the histogram of the agents’ wealth. In particular, it gives a
handle on the asymptotic behaviour of the distribution function at extreme
values of wealth where the number of agents is small. It is confirmed that the
asymptotic behaviour is not exponential, even though a closed-form expression
for the tail has not been found. The dependence of the wealth distribution on
the value of the transfer parameter and the corresponding changes in wealth
inequality are investigated in Chapter 3. The Gini coefficient, a common
measure of income inequality, is computed for a range of different values of
the transfer parameter.
The wealth distributions obtained for small values of the transfer parame-
ter are found to be qualitatively different from those for values close to unity,
which corresponds to the case when the givers concede most of their wealth in
a single exchange (the gambling scenario). Namely, the former distributions
(small values of the transfer parameter) are peaked around the mean of the
distribution and are characterised by low level of inequality (Gini coefficient
close to zero), whereas the latter (values close to unity) are approximately
power-law in shape with overlaid oscillations and show high level of inequality
(Gini coefficient close to unity). This unexpected oscillatory pattern in the
wealth distribution is found to be approximately periodic on logarithmic scale
with the period inversely proportional to the fraction of wealth retained by
the givers.
The giver scheme is a closed system with constant amount of total wealth
and no friction. It can be expected to exhibit the usual traits of Boltzmann
115
6. Conclusion
entropy, which measures the level of disorder in the system, i.e. the entropy
is expected to rise steadily as the system evolves towards the steady state.
However, multi-agent simulations reported in Chapter 3 demonstrate that the
entropy of the system evolves in a non-monotonic fashion, in stark conflict
with the expectations based on its behaviour in physical systems such as an
ideal gas. The Boltzmann H-theorem, according to which entropy cannot
decrease, is not applicable to the giver scheme since its microscopic rules of
exchange are not symmetric with respect to time reversal. Indeed, the exact
rules that reconstruct past behaviour of system in the reverse time order can
be worked out easily but they are different from the rules describing exchange
when time order is normal. Therefore, Boltzmann entropy is not a faithful
measure of disorder in a multiplicative asset transfer system.
The study reported in Chapter 3 raises a number of questions, which could
inform future research efforts in this area. Firstly, the technique of using the
master equation and the Laplace transform is applicable to asset transfer sys-
tems with rules different from those employed in the giver scheme. Secondly,
since Boltzmann entropy fails to behave properly in the giver scheme, other
measures on entropy, e.g. Tsallis or Renyi entropy, need to be investigated
in the context of asset exchange systems that lack time symmetry. Thirdly,
an intriguing area of future research concerns exchange systems where the
total amount of money in the systems is not conserved and, moreover, can be
produced endogenously by the system’s participants.
The giver scheme and most other exchange models investigated by econo-
physicists view money as a commodity that cannot be created by the agents,
which is consistent with the perspective commonly accorded to consumers
and companies in mainstream economics. According to the mainstream view,
base money (currency) is created by the central bank and is multiplied by
commercial banks through fractional reserves, such that the banks can loan
the deposited money as long as they retain a certain fraction of all deposits
(the reserve). However, this picture conflicts with how money is actually cre-
ated by commercial banks, which create new money when they make loans
(McLeay and Radia, 2014). The actual process of money creation is endoge-
nous and is the opposite of what is implied by the money multiplier theory.
In practise, loans created by the bank engender new deposits, which in turn
raise the amount of base money as the central banks attempts to accommo-
date the demand for cash in the economy. Modeling the banking system and
its impact on the economy in this light presents a timely opportunity to make
116
6. Conclusion
a contribution which is both theoretically interesting and rich with significant
social implications in terms of the institutional design of the banking system.
6.3 Critique of fast money flow theory
The objective of Chapter 4 is to analyse the lattice gauge model of fast money
flow dynamics. The assumptions of the theory are reviewed; a careful deriva-
tion of the Euler-Lagrange equations that determine the model dynamics are
given. It is found that the dynamics of the model reported by Ilinski (2001)
is inconsistent with the Lagrangian derived there. For instance, Ilinski (2001)
observes that the oscillations in exchange rate are slowly decaying with time
for a certain combination of initial values of the model variables. However, it
is shown in Chapter 4 that the oscillations persist indefinitely with no decay
in this case. The inconsistency is traced to an algebraic error in the deriva-
tion of the Euler-Lagrange equations given in Ilinski (2001). Furthermore, it
is shown that the constraint on the initial values of model variables used by
Ilinski (2001) is unrealistic. If the constraint is relaxed, the dynamics become
unstable with the exchange rate either growing exponentially or decaying ex-
ponentially to zero. In light of these results, the implications for technical
analysis are re-evaluated and it is found that the model provides no support
for technical trading.
Furthermore, it is observed in Chapter 4 that the continuous form of the
action has not been sufficiently motivated. The part of the action that de-
scribes the dynamics of the exchange rate, which is identified with a field, is
obtained by taking the limit ∆t → 0, whereas the part that represents the
effect of the exchange rate on the number of agents in each currency uses a
finite value of ∆t as one of the input parameters of the model. The transition
from discrete evolution of the state vector, which represents the number of
agents in each currency at each time step, to continuous evolution described
by a Hamiltonian is unjustified, since the transition matrix can be degenerate
and therefore its action cannot be identified with the evolution operator.
6.4 On memory in sandpiles
In chapter 5, the Abelian sandpile is introduced as a toy model that captures
some of the features of financial markets in order to explore the connection
between observed changes in the market, e.g. price changes in the foreign ex-
117
6. Conclusion
change (FX) market, and underlying structural features of the market, which
are not observed. The emphasis of the study is on analysing Abelian sandpiles
rather than establishing a plausible model of the FX market. To that end,
chapter 5 investigates memory effects in the sandpile. It examines the hidden
structural changes resulting from grain drops and avalanches and relates those
changes to the observed avalanches by analysing the sandpile evolution with
a hidden Markov model to capture patterns in the average intensity of the
avalanches.
The investigation reported in chapter 5 concerns two-dimensional Abelian
sandpiles on a square grid. The quantitative measure of memory loss employed
in the study relies on computing site occupancy fractions, which are equal to
the number of sites occupied by a specific charge (ranging from 0 to 3 in
a two-dimensional sandpile) normalised by the total number of sites. Site
occupancy fractions are closely related to the probabilities that a given site
has a specific charge; the two concepts coincide in the limit of the infinite
sandpile in a recurrent configuration. Each of the four occupancy fractions
varies as the sandpile evolves. The distribution of each occupancy fraction
computed from a large number of configurations is found to be approximated
well by a Gaussian.
The memory loss is determined by comparing sandpile configurations sep-
arated by a certain time delay measured in grain drops. Occupancy fractions
are computed for the absolute difference of these configurations and the frac-
tion’s variation with time delay is investigated. As time delay increases, the
fractions approach constant values characteristic of the absolute difference
maps of unrelated configurations. The analysis described in chapter 5 shows
that the most common value in the absolute difference map is 1, followed by
0 and then 2, with 3 being the least common. It is interesting to note that
as far as the sandpile configuration is concerned the most common charge is
3, followed by 2, 1, and 0, in order of frequency. The dynamics of memory
loss is revealed most conveniently by comparing the fractions for the absolute
difference maps of time-delayed configurations with their expected values for
the absolute difference maps of unrelated configurations, for which time delay
is effectively infinite.
The simulations show that the memory of a given configuration is lost
in two stages, each characterised by approximately exponential decay as mea-
sured by the rate of approach of the fractions to the expected values. The first
stage is characterised by a faster decay; its time-scale is proportional to the
118
6. Conclusion
linear size of the sandpile. The second stage scales with the number of sites
in the sandpile (linear size squared) and consequently lasts much longer than
the first stage. It is not clear at present what is responsible for the two-stage
pattern of memory loss. It is left to future research to uncover the detailed
mechanisms responsible for this behaviour and develop an analytical model of
the fractions as a function of the time delay from the initial configuration.
Chapter 5 also attempts to answer the following question: can memory
loss be detected in the sequence of observed quantities that characterise the
sandpile’s activity, e.g. avalanche size (which in the financial context may cor-
respond to large exchange rate movements in the foreign exchange market, for
example). To eliminate “shot noise” due to random grain drops, the average
avalanche size over a specific time window is fed as a sequence into a hidden
Markov model. The hidden Markov model looks for patterns in the time series
of average avalanche activity; the analysis is repeated for a range of averaging
periods. The hidden Markov output demonstrates that the sandpile retains
memory on time scales that are less than the long time scale. On longer time
scales, the output depends randomly on the time scale, which is qualitatively
similar to the output obtained from the same time series shuffled randomly.
In other words, no pattern is detected by the hidden Markov model on time
scales exceeding the long time scale. The hidden Markov analysis demon-
strates that all memory is lost on long time scales, in accord with the previous
study based on the occupation fractions.
The work presented in chapter 5 contributes to the line of research devoted
to analysing temporal correlations in the Abelian sandpiles by introducing the
innovation of hidden Markov analysis. It can be extended easily to other sand-
pile models like the Manna model or the Oslo model, and conceivably to other
dynamical systems that exhibit self-organised criticality. Even in the context
of the standard Abelian sandpile model, it is desirable to extend the study’s
results by considering larger sandpiles, longer sequences of data, and higher-
dimensional sandpiles. An important direction for future research is to identify
the internal elements of the sandpile that are responsible for the dynamics of
memory loss exhibited by ensemble-averaged fractions. Due to averaging over
many realisations this dynamics is independent of random driving by grain
drops and therefore genuinely reflects the internal structure of the sandpile.
The new idea of analysing sandpile dynamics with hidden Markov models mer-
its future development. Such models can be used in a number of ways, one of
which is to feed to the model other sandpile characteristics besides avalanche
119
6. Conclusion
size, or even combinations of two or more quantities.
Unlike the study based on the occupation fractions, the hidden Markov
analysis is agnostic with respect to the internal structure of the dynamical
system it is applied to, yet it is sensitive to the memory effects, which depend
on the internal structure of the system, as the above results demonstrate.
Therefore, hidden Markov models can in principle be used as a diagnostic tool
of the internal structure of financial systems, where the internal structure is
hidden as in the case of the FX markets.
120
Bibliography
J. Abate and W. Whitt. A unified framework for numerically inverting Laplace
transforms. INFORMS Journal on Computing, 18(4):408, 2006.
Amir Abdolvand and Afshin Montakhab. Scaling and complex avalanche dy-
namics in the abelian sandpile model. The European Physical Journal B,
76(1):21–30, 2010.
Reka Albert, Hawoong Jeong, and Albert-Laszlo Barabasi. Internet: Diameter
of the world-wide web. Nature, 401(6749):130–131, 1999.
Biliana Alexandrova-Kabadjova, Serafin Martinez-Jaramill, Alma Lilia
Garcia-Almanza, and Edward Tsang. Simulation in computational finance
and economics. IGI Global, 2012.
J. Angle. The inequality process as a wealth maximizing process. Physica A:
Statistical Mechanics and its Applications, 367:388–414, 2006.
John Angle. The surplus theory of social stratification and the size distribution
of personal wealth. Social Forces, 65(2):293–326, 1986.
John Angle. The statistical signature of pervasive competition on wage and
salary incomes. Journal of Mathematical Sociology, 26(4):217–270, 2002.
W Brian Arthur. Inductive reasoning and bounded rationality. The American
economic review, 84(2):406–411, 1994.
A.B. Ashcraft and D. Duffie. Systemic illiquidity in the federal funds market.
The American economic review, 97(2):221–225, 2007.
Marcel Ausloos. Econophysics of stock and foreign currency exchange markets.
Econophysics and Sociophysics: Trends and Perspectives, pages 249–278,
2006.
121
Bibliography
Marcel Ausloos, Kristinka Ivanova, and Nicolas Vandewalle. Crashes: symp-
toms, diagnoses and remedies. In Empirical Science of Financial Fluctua-
tions, pages 62–76. Springer, 2002.
Robert M Axelrod. The complexity of cooperation: Agent-based models of
competition and collaboration. Princeton University Press, 1997.
Louis Bachelier. Louis Bachelier’s theory of speculation: the origins of modern
finance. Princeton University Press, 2011.
Per Bak, Chao Tang, Kurt Wiesenfeld, et al. Self-organized criticality: An
explanation of 1/f noise. Physical Review Letters, 59(4):381–384, 1987.
Alain Barrat, Alessandro Vespignani, and Stefano Zapperi. Fluctuations and
correlations in sandpile models. Physical Review Letters, 83(10):1962, 1999.
Stefano Battiston, Domenico Delli Gatti, Mauro Gallegati, Bruce Greenwald,
and Joseph E Stiglitz. Liaisons dangereuses: Increasing connectivity, risk
sharing, and systemic risk. Journal of Economic Dynamics and Control, 36
(8):1121–1141, 2012a.
Stefano Battiston, Michelangelo Puliga, Rahul Kaushik, Paolo Tasca, and
Guido Caldarelli. Debtrank: Too central to fail? financial networks, the fed
and systemic risk. Scientific reports, 2, 2012b.
Morten L. Bech and Enghin Atalay. The topology of the federal funds market.
Physica A: Statistical Mechanics and its Applications, 389(22):5223–5246,
2010.
Fischer Black and Myron Scholes. The pricing of options and corporate lia-
bilities. The journal of political economy, pages 637–654, 1973.
Bela Bollobas. Modern graph theory, volume 184. Springer, 1998.
L Borland, J-P Bouchaud, J-F Muzy, G Zumbach, et al. The dynamics of
financial markets-mandelbrot’s cascades and beyond. Wilmott Magazine,
2005.
M. Boss, H. Elsinger, M. Summer, and S. Thurner. Network topology of the
interbank market. Quantitative Finance, 4(6):677–684, 2004. ISSN 1469-
7688.
122
Bibliography
Jean-Philippe Bouchaud. Economics needs a scientific revolution. Nature, 455
(7217):1181–1181, 2008.
Jean-Philippe Bouchaud and Marc Potters. Theory of financial risk and
derivative pricing: from statistical physics to risk management. Cambridge
University Press, 2003.
Mark Buchanan. What has econophysics ever done for us? Nature Physics, 9
(6):317–317, 2013.
Fabio Caccioli and Matteo Marsili. On information efficiency and financial
stability. arXiv preprint arXiv:1004.5014, 2010.
D.O. Cajueiro and B.M. Tabak. The role of banks in the Brazilian interbank
market: Does bank type matter? Physica A: Statistical Mechanics and its
Applications, 387(27):6825–6836, 2008.
Guido Caldarelli. Scale-free networks: complex webs in nature and technology.
OUP Catalogue, 2007.
Frank Campbell. Reserve Bank Domestic Operations under RTGS. Reserve
Bank of Australia Bulletin, November:54, 1998.
Sergio Caracciolo and Andrea Sportiello. Exact integration of height prob-
abilities in the abelian sandpile model. Journal of Statistical Mechanics:
Theory and Experiment, 2012(09):P09013, 2012.
Anna Carbone, Giorgio Kaniadakis, and Antonio M Scarfone. Where do we
stand on econophysics? Physica A: Statistical Mechanics and its Applica-
tions, 382(1):xi–xiv, 2007.
Robert L Cave and Lee P Neuwirth. Hidden markov models for English. In
Hidden Markov Models for Speech, pages 8–15. IDA–CRD, 1980.
Bikas K Chakrabarti and Anirban Chakraborti. Fifteen years of econophysics
research. Science and Culture, 76:293–295, 2010.
A. Chakraborti and M. Patriarca. Gamma-distribution and wealth inequality.
Pramana, 71(2):233–243, 2008.
123
Bibliography
Anirban Chakraborti and Bikas K Chakrabarti. Statistical mechanics of
money: how saving propensity affects its distribution. The European Phys-
ical Journal B-Condensed Matter and Complex Systems, 17(1):167–170,
2000.
Damien Challet and Y-C Zhang. Emergence of cooperation and organization
in an evolutionary game. Physica A: Statistical Mechanics and its Applica-
tions, 246(3):407–418, 1997.
A. Chatterjee and BK Chakrabarti. Kinetic exchange models for income and
wealth distributions. The European Physical Journal B-Condensed Matter
and Complex Systems, 60(2):135–149, 2007.
A. Chatterjee, B.K. Chakrabarti, and R.B. Stinchcombe. Master equation
for a kinetic model of a trading market and its analytic solution. Physical
Review E, 72(2):26126, 2005.
Arnab Chatterjee, Bikas K Chakrabarti, and SS Manna. Pareto law in a ki-
netic model of market with random saving propensity. Physica A: Statistical
Mechanics and its Applications, 335(1):155–163, 2004.
Shu-Heng Chen and Sai-Ping Li. Econophysics: Bridges over a turbulent
current. International Review of Financial Analysis, 23:1–10, 2012.
Kim Christensen, Alvaro Corral, Vidar Frette, Jens Feder, and Torstein
Jøssang. Tracer dispersion in a self-organized critical system. Physical
review letters, 77(1):107, 1996.
Rama Cont and Jean-Philipe Bouchaud. Herd behavior and aggregate fluctu-
ations in financial markets. Macroeconomic dynamics, 4(02):170–196, 2000.
Herman E Daly and Joshua Farley. Ecological economics: principles and
applications. Island Press, 2010.
Jorn Davidsen and Maya Paczuski. 1/fα noise from correlations between
avalanches in self-organized criticality. Physical Review E, 66(5):050101,
2002.
B. Davies. Integral transforms and their applications. Springer Verlag, 2002.
G. De Masi, G. Iori, and G. Caldarelli. Fitness model for the Italian interbank
money market. Physical Review E, 74(6):066112, 2006.
124
Bibliography
Deepak Dhar. Self-organized critical state of sandpile automaton models.
Physical Review Letters, 64(14):1613, 1990.
Deepak Dhar. Theoretical studies of self-organized criticality. Physica A:
Statistical Mechanics and its Applications, 369(1):29–70, 2006.
Deepak Dhar and Punyabrata Pradhan. Probability distribution of residence
times of grains in sand-pile models. Journal of Statistical Mechanics: Theory
and Experiment, 2004(05):P05002, 2004.
Adrian Dragulescu and Victor M Yakovenko. Statistical mechanics of money.
The European Physical Journal B-Condensed Matter and Complex Systems,
17(4):723–729, 2000.
B Dupoyet, HR Fiebig, and DP Musgrove. Gauge invariant lattice quantum
field theory: Implications for statistical properties of high frequency finan-
cial markets. Physica A: Statistical Mechanics and its Applications, 389(1):
107–116, 2010.
Richard Durbin. Biological sequence analysis: probabilistic models of proteins
and nucleic acids. Cambridge university press, 1998.
Joshua M Epstein. Growing artificial societies: social science from the bottom
up. Brookings Institution Press, 1996.
Anne Fey, Lionel Levine, and Yuval Peres. Growth rates and explosions in
sandpiles. Journal of Statistical Physics, 138(1-3):143–159, 2010.
T.M.J. Fruchterman and E.M. Reingold. Graph drawing by force-directed
placement. Software - Practice and Experience, 21(11):1129–1164, 1991.
C.H. Furfine. Interbank Exposures: Quantifying the Risk of Contagion. Jour-
nal of Money, Credit & Banking, 35(1):111–129, 2003.
Serge Galam. Sociophysics: a physicist’s modeling of psycho-political phenom-
ena. Springer, 2012.
Peter Gallagher, Jon Gauntlett, and David Sunner. Real-time Gross Settle-
ment in Australia. Reserve Bank of Australia Bulletin, September:61, 2010.
Mauro Gallegati, Steve Keen, Thomas Lux, and Paul Ormerod. Worrying
trends in econophysics. Physica A: Statistical Mechanics and its Applica-
tions, 370(1):1–6, 2006.
125
Bibliography
Kausik Gangopadhyay. Interview with Eugene H. Stanley. SAGE, 2(2):73–78,
2013.
Martin Gardner. Mathematical games: The fantastic combinations of john
conway’s new solitaire game ”life”. Scientific American, 223(4):120–123,
1970.
Asim Ghosh and Anindya S Chakrabarti. Econophysics and sociophysics:
Problems and prospects. In Econophysics of Agent-Based Models, pages
287–297. Springer, 2014.
Nigel Gilbert and Klaus G Troitzsch. Simulation for the social scientist.
McGraw-Hill International, 2005.
Yves Gingras and Christophe Schinckus. The institutionalization of econo-
physics in the shadow of physics. Journal of the History of Economic
Thought, 34(01):109–130, 2012.
Dhananjay K Gode and Shyam Sunder. Allocative efficiency of markets with
zero-intelligence traders: Market as a partial substitute for individual ra-
tionality. Journal of political economy, pages 119–137, 1993.
Sanjeev Goyal. Connections: an introduction to the economics of networks.
Princeton University Press, 2012.
David Graeber. Debt: The first 5,000 years. Melville House, 2011.
A.G. Haldane and R.M. May. Systemic risk in banking ecosystems. Nature,
469(7330):351–355, 2011a. ISSN 0028-0836.
Andrew G Haldane and Robert M May. Systemic risk in banking ecosystems.
Nature, 469(7330):351–355, 2011b.
H. Hassanzadeh and M. Pooladi-Darvish. Comparison of different numerical
Laplace inversion methods for engineering applications. Applied Mathemat-
ics and Computation, 189(2):1966–1981, 2007.
Espen Gaarder Haug and Nassim Nicholas Taleb. Option traders use (very)
sophisticated heuristics, never the black–scholes–merton formula. Journal
of Economic Behavior & Organization, 77(2):97–106, 2011.
B. Hayes. Follow the money. American Scientist, 90(5):400–405, 2002.
126
Bibliography
Cars H Hommes. Modeling the stylized facts in finance through simple non-
linear adaptive systems. Proceedings of the National Academy of Sciences
of the United States of America, 99(Suppl 3):7221–7228, 2002.
A. Ilinskaia and K. Ilinski. How to reconcile market efficiency and technical
analysis. Arxiv preprint cond-mat/9902044, 1999.
K. Ilinski. Physics of Finance. Arxiv preprint hep-th/9710148, 1997.
Kirill Ilinski. Physics of finance: gauge modelling in non-equilibrium pricing.
Wiley, 2001.
K. Imakubo and Y. Soejima. The transaction network in Japanese interbank
money markets. Monetary and Economic Studies, 28:107–150, 2010.
H. Inaoka, T. Ninomiya, K. Taniguchi, T. Shimizu, and H. Takayasu. Fractal
network derived from banking transaction–an analysis of network structures
formed by financial institutions. Bank of Japan Working Papers, 4, 2004.
G. Iori, R. Reno, G. De Masi, and G. Caldarelli. Trading strategies in the Ital-
ian interbank market. Physica A: Statistical Mechanics and its Applications,
376:467–479, 2007.
G. Iori, G. De Masi, O.V. Precup, G. Gabbi, and G. Caldarelli. A network
analysis of the Italian overnight money market. Journal of Economic Dy-
namics and Control, 32(1):259–278, 2008.
S Ispolatov, PL Krapivsky, and Sidney Redner. Wealth distributions in asset
exchange models. The European Physical Journal B-Condensed Matter and
Complex Systems, 2(2):267–276, 1998.
EV Ivashkevich, DV Ktitarev, and VB Priezzhev. Waves of topplings in an
abelian sandpile. Physica A: Statistical Mechanics and its Applications, 209
(3):347–360, 1994.
Matthew Jackson and Yves Zenou. Economic Analyses of Social Networks
Volume I: Theory. Edward Elgar Publishing, 2013.
Matthew O Jackson. Social and economic networks. Princeton University
Press, 2008a.
M.O. Jackson. Social and economic networks. Princeton Univ Pr, 2008b.
127
Bibliography
Henrik Jeldtoft Jensen. Self-organized criticality: emergent complex behavior
in physical and biological systems, volume 10. Cambridge university press,
1998.
Neil Johnson, Guannan Zhao, Eric Hunsader, Jing Meng, Amith Ravindar,
Spencer Carran, and Brian Tivnan. Financial black swans driven by ultra-
fast machine ecology. arXiv preprint arXiv:1202.1448, 2012.
Mark Suresh Joshi. The concepts and practice of mathematical finance, vol-
ume 1. Cambridge University Press, 2003.
J.N. Kapur. Maximum-entropy models in science and engineering. John Wiley
& Sons, 1989.
Steve Keen. Debunking Economics-Revised and Expanded Edition: The Naked
Emperor Dethroned? Zed Books, 2011.
Richard W Kenyon and David B Wilson. Spanning trees of graphs on sur-
faces and the intensity of loop-erased random walk on Z2. arXiv preprint
arXiv:1107.3377, 2011.
Gew-rae Kim and Harry M Markowitz. Investment rules, margin, and market
volatility. The Journal of Portfolio Management, 16(1):45–52, 1989.
Stephan Koehler. Fitting sum of two decaying exponentials, 2012.
E.D. Kolaczyk. Statistical analysis of network data: methods and models.
Springer Verlag, 2009.
Zvonko Kostanjcar, Kristian Hengster-Movric, and Branko Jeren. Model of
discrete dynamics of asset price relations based on the minimal arbitrage
principle. Central European Journal of Physics, 9(3):865–873, 2011.
F. Kyriakopoulos, S. Thurner, C. Puhr, and SW Schmitz. Network and eigen-
value analysis of financial transaction networks. The European Physical
Journal B-Condensed Matter and Complex Systems, 71(4):523–531, 2009.
Laurent Laloux, Pierre Cizeau, Jean-Philippe Bouchaud, and Marc Potters.
Noise dressing of financial correlation matrices. Physical review letters, 83
(7):1467, 1999.
Blake LeBaron. Building the santa fe artificial stock market. Physica A, 2002.
128
Bibliography
S. Li, J. He, and Y. Zhuang. A network model of the interbank market. Physica
A: Statistical Mechanics and its Applications, 2010. ISSN 0378-4371.
Xiaolin Li, Marc Parizeau, and Rejean Plamondon. Training hidden markov
models with multiple observations-a combinatorial method. Pattern Anal-
ysis and Machine Intelligence, IEEE Transactions on, 22(4):371–377, 2000.
Thomas Lux and Michele Marchesi. Scaling and criticality in a stochastic
multi-agent model of a financial market. Nature, 397(6719):498–500, 1999.
Benoit Mandelbrot. The variation of some other speculative prices. Journal
of Business, pages 393–413, 1967.
SS Manna. Two-state model of self-organized criticality. Journal of Physics
A: Mathematical and General, 24(7):L363, 1991.
Sergei Maslov, Chao Tang, and Yi-Cheng Zhang. 1/f noise in bak-tang-
wiesenfeld models on narrow stripes. Physical Review Letters, 83(12):2449,
1999.
D. Matthes and G. Toscani. On steady distributions of kinetic models of
conservative economies. Journal of Statistical Physics, 130(6):1087–1117,
2008.
R.M. May, S.A. Levin, and G. Sugihara. Ecology for bankers. Nature, 451
(21):893–895, 2008.
Joseph L McCauley. Response to“worrying trends in econophysics”. Physica
A: Statistical Mechanics and its Applications, 371(2):601–609, 2006.
Joseph L McCauley. Nonstationarity of efficient finance markets: FX market
evolution from stability to instability. International Review of Financial
Analysis, 17(5):820–837, 2008.
G.J. McLachlan and D. Peel. Finite mixture models. Wiley-Interscience, 2000.
Michael McLeay and Amar Radia. Money creation in the modern economy.
Bank of England Quarterly Bulletin, 1:14–27, 2014.
Esteban Moro. The minority game: an introductory guide. arXiv preprint
cond-mat/0402651, 2004.
129
Bibliography
CF Moukarzel, S. Goncalves, JR Iglesias, M. Rodrıguez-Achach, and
R. Huerta-Quintanilla. Wealth condensation in a multiplicative random
asset exchange model. The European Physical Journal-Special Topics, 143
(1):75–79, 2007.
Taleb Nassim. The black swan: the impact of the highly improbable. New York
Random House, 2007.
M.E.J. Newman. Mixing patterns in networks. Physical Review E, 67(2):
26126, 2003.
M. Patriarca, A. Chakraborti, E. Heinsalu, and G. Germano. Relaxation in
statistical many-agent economy models. The European Physical Journal B
- Condensed Matter and Complex Systems, 57(2):219–224, 2007.
Roberto Pieraccini. The voice in the machine: building computers that under-
stand speech. The MIT Press, 2012.
M. Piraveenan, M. Prokopenko, and A. Zomaya. Assortative Mixing in Di-
rected Biological Networks. IEEE IEEE/ACM Transactions on Computa-
tional Biology and Bioinformatics, 2010. ISSN 1545-5963.
Vahagn S Poghosyan, Vyatcheslav B Priezzhev, and Philippe Ruelle. Return
probability for the loop-erased random walk and mean height in the abelian
sandpile model: a proof. Journal of Statistical Mechanics: Theory and
Experiment, 2011(10):P10004, 2011.
VB Priezzhev. Structure of two-dimensional sandpile. i. height probabilities.
Journal of statistical physics, 74(5-6):955–979, 1994.
Gunnar Pruessner. Self-organised criticality: theory, models and characteri-
sation. Cambridge University Press, 2012.
Lawrence R Rabiner. A tutorial on hidden markov models and selected ap-
plications in speech recognition. Proceedings of the IEEE, 77(2):257–286,
1989.
Craig W Reynolds. Flocks, herds and schools: A distributed behavioral model.
In ACM SIGGRAPH Computer Graphics, volume 21, pages 25–34. ACM,
1987.
130
Bibliography
Peter Richmond, Stefan Hutzler, Ricardo Coelho, and Przemek Repetow-
icz. A Review of Empirical Studies and Models of Income Distributions
in Society, pages 131–159. Wiley-VCH Verlag GmbH & Co. KGaA,
2006. ISBN 9783527610006. doi: 10.1002/9783527610006.ch5. URL
http://dx.doi.org/10.1002/9783527610006.ch5.
Bertrand M Roehner. Fifteen years of econophysics: worries, hopes and
prospects. Science and Culture, 76:305–314, 2010.
K.B. Rørdam and M.L. Bech. The topology of Danish interbank money flows.
FRU Working Papers, 2008.
J Barkley Rosser Jr. The nature and future of econophysics. In Econophysics
of Stock and other Markets, pages 225–234. Springer, 2006.
Seref Sagiroglu and Duygu Sinanc. Big data: A review. In Collaboration
Technologies and Systems (CTS), 2013 International Conference on, pages
42–47. IEEE, 2013.
M. Ali Saif and Prashant M. Gade. Emergence of power-law in a market with
mixed models. Physica A: Statistical Mechanics and its Applications, 384
(2):448–456, 2007.
E Samanidou, E Zschischang, D Stauffer, and Thomas Lux. Agent-based
models of financial markets. Reports on Progress in Physics, 70(3):409,
2007.
Christophe Schinckus. Is econophysics a new discipline? the neopositivist
argument. Physica A: Statistical Mechanics and its Applications, 389(18):
3814–3821, 2010.
Joseph Alois Schumpeter. Theory of economic development: an inquiry into
profits, capital, credit, interest, and the business cycle. Harvard University
Press. Cambridge, Mass., 1934.
F. Schweitzer, G. Fagiolo, D. Sornette, F. Vega-Redondo, and D.R. White.
Economic networks: What do we know and what do we need to know?
Advances in Complex Systems, 12(4-5):407–422, 2009.
William F Sharpe. Capital asset prices: A theory of market equilibrium under
conditions of risk*. The journal of finance, 19(3):425–442, 1964.
131
Bibliography
F. Slanina. Inelastically scattering particles and wealth distribution in an open
economy. Physical Review E, 69(4):46102, 2004.
A.A. Slavnov and LD Faddeev. Gauge Fields, Introduction to Quantum The-
ory (Frontiers in Physics, Vol. 50). Benjamin-Cummings Publishing Co.,
1980.
Andrey Sokolov, Tien Kieu, and Andrew Melatos. A note on the theory of
fast money flow dynamics. The European Physical Journal B-Condensed
Matter and Complex Systems, 76(4):637–642, 2010a.
Andrey Sokolov, Andrew Melatos, and Tien Kieu. Laplace transform analysis
of a multiplicative asset transfer model. Physica A: Statistical mechanics
and its applications, 389(14):2782–2792, 2010b.
Andrey Sokolov, Rachel Webster, Andrew Melatos, and Tien Kieu. Loan and
nonloan flows in the australian interbank network. Physica A: Statistical
Mechanics and its Applications, 391(9):2867–2882, 2012.
Kimmo Soramaki, Morten L. Bech, Jeffrey Arnold, Robert J. Glass, and Wal-
ter Beyeler. The topology of interbank payment flows. Physica A: Statistical
Mechanics and its Applications, 379(1):317–333, 2007.
D. Sornette. Gauge theory of Finance? International Journal of Modern
Physics C, 9(3):505–508, 1998.
Mark Stamp. A revealing introduction to hidden Markov models, 2004. URL
www.cs.sjsu.edu/$\sim$stamp/RUA/HMM.pdf.
Matthew Stapleton, Martin Dingler, and Kim Christensen. Sensitivity to
initial conditions in self-organized critical systems. Journal of statistical
physics, 117(5-6):891–900, 2004.
Mieko Tanaka-Yamawaki and Seiji Tokuoka. Adaptive use of technical indi-
cators for the prediction of intra-day stock prices. Physica A: Statistical
Mechanics and its Applications, 383(1):125–133, 2007.
Mark P Taylor and Helen Allen. The use of technical analysis in the foreign
exchange market. Journal of international Money and Finance, 11(3):304–
314, 1992.
132
Bibliography
Stefan Thurner and Sebastian Poledna. Debtrank-transparency: Controlling
systemic risk in financial networks. Scientific reports, 3, 2013.
Stefan Thurner, J Doyne Farmer, and John Geanakoplos. Leverage causes fat
tails and clustered volatility. Quantitative Finance, 12(5):695–707, 2012.
Johannes Voit. The statistical mechanics of financial markets. Springer Berlin,
2003.
Philipp Weber, Fengzhong Wang, Irena Vodenska-Chitkushev, Shlomo Havlin,
and H Eugene Stanley. Relation between volatility correlations in financial
markets and omori processes occurring on all scales. Physical Review E, 76
(1):016109, 2007.
Avinash Chand Yadav, Ramakrishna Ramaswamy, and Deepak Dhar. Power
spectrum of mass and activity fluctuations in a sandpile. Physical Review
E, 85(6):061114, 2012.
Victor M. Yakovenko and J. Barkley Rosser Jr. Colloquium: Statistical me-
chanics of money, wealth, and income. Reviews of Modern Physics, 81(4):
1703–1725, 2009.
K Young. Foreign exchange market as a lattice gauge theory. American
Journal of Physics, 67(10):862–868, 1999.
Shipeng Zhou and Liuqing Xiao. An application of symmetry approach to
finance: Gauge symmetry in finance. Symmetry, 2(4):1763–1775, 2010.
133
Minerva Access is the Institutional Repository of The University of Melbourne
Author/s:
SOKOLOV, ANDREY
Title:
Application of non-equilibrium statistical mechanics to the analysis of problems in financial
markets and economy
Date:
2014
Persistent Link:
http://hdl.handle.net/11343/44218
File Description:
Application of non-equilibrium statistical mechanics to the analysis of problems in financial
markets and economy
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