Numerical a nalysis of proximity oscillator networks · Pepito Rossi. Acknowledgements First and foremost I have to thank my beloved family, for their invaluable lessons and for the

SCUOLA POLITECNICA E DELLE SCIENZE DI BASE

TESI DI LAUREA MAGISTRALE IN

INGEGNERIA DELL’AUTOMAZIONE

Numerical analysis of proximity oscillator networks

Relatore Candidato Chiar.mo Prof. Francesco Garofalo Correlatori Ing. Pietro De Lellis Ing. Francesco Lo Iudice

Giovanni Pugliese Carratelli

M58/30

Anno Accademico 2013/2014

[...] io nun’aggio mai vist a’ coscienza affianco a piglia e’ mazzate pure per me.

Ci song sempe stat’ sul dignıta e coraggio.

Pepito Rossi

Acknowledgements

First and foremost I have to thank my beloved family, for their invaluable lessons and

for the exceptionally interesting and stimulating environment I have had the good

fortune to live. Home has given me an impulse to shape my character to open my doors

towards curiosity, sense of adventure and competitiveness. Papa, thank you for your

endless patience, love, and the continuous model of life you are for me. Mum thank

you for your care, support, love and for having believed in my choices. Giacumı, thank

you for your complicity, laughes and chats we enjoyed and enjoy together.

As a proud student of this university I have to thank all the professors of the

Automation group at DIETI. From all of them I have had an occasion to learn a

lot both professionally and under a human point of view. Professor Garofalo (a.

jokingly k.a. as The Oracle among my study group), Professor Bruno Siciliano (The

Master), Professor Mario di Bernardo (The Messia), Professor Ambrosino (The Captain)

and Professor Giovanni Celentano (The Pope) have all contributed to give me great

engineering lessons, but they overall have remarkably enriched in me the passion for

what I do. While skills and knowledge I may not need or forget, the love for what I do

will never leave me. Thank you.

Thanks to all the SINCRO group, and first of all to Professor Francesco Garofalo,

for having given me the opportunity to spend beautiful months learning, growing and

5

enjoying my time in a marvellous research group. It was thanks to his knowledge

and enthusiasm for control, science and engineering that gave me the confidence and

persistence to tackle my problems.

A very special thanks goes to Piero De Lellis who ignited in me interest for optimal

control and networks during classes. He has been of great support, and a very nice

person from the day I held the exam with him, until today. He showed me the difference

between the student world and professionals world, putting me constantly in difficult

situations and showing me I could always get out of them. Piero, your knowledge

constant support, passion and vision have been a great guide for me.

Many thanks to Franceco Lo Iudice. The chats we have had and your experience

have showed me how to look at things under many different points of view and opened

new roads towards solutions. France, I hope we will get to work again together soon or

later!

I would like to thank all my colleagues and friends. Great days I have passed side

by side with Angelo, Cesare and Mirco. Your passion and determination have been a

source for persistence and hard work in our common goal. Angelo, some day we will

get to catch up and live some great experiences together; I know our friendship will

fly even higher. Cesare, your guidance in practical aspects has been great for me in

numerous occasions as well as the laughes with have had together. Mirco, your strength

and perseverance, have been of great support in a lot of occasions; walking with you

towards a common goal has often kindled in me persistance I did not think I had.

To my Friends Andrea, Giuseppe, Luigi, Jordan, Marcello, and Stelvio thank you

for the great memories, experiences, and overall joyful moments on soccer fields and

water. Andrea, Jordan and Marcello, you are the proof that true friendship can live

anything; Goodfella once, Goodfella forever. Giuseppe, your courage and relentless way

of living life have been a model I have admired since I was a kid, some things will be

with me forever. Luigi your constant presence and example of analytical analysis of life

have allowed me to as objective as possible in a lot of situations. We have built a great

friendship, that I admire every day more. Stelvio, thoughts, opinions, chats and such a

similar way of living life have guided us very close and will even further, IF we hold on.

Friends, the best is yet come.

Abstract

The problem of a large ensemble of interacting units ha attracted the attention of

diverse field of science and engineering. This problem was tackled usign the complex

network paradigm which derives the emerging feature of the whole system from the

interaction among the single units. For example, the problem of goods transportation

may be regarded as a network problem where a possible goal is to find a minimum

cost path towards the destination of the goods. Interconnection of power systems, or

interconnection of computers (e.g. the World Wide Web) and even of social interactions

can be suitably modeled and analyzed as networks. In biology, ensembles of single cells

to perform a given function can be explained in terms of networks.

In classical Complex network models the connections among the entities are often

static, whereas in real-world applications such interconnections might be time-dependant.

For instance, in engineering, and specifically in sensing, recent advances have made

it possible to build networks of sensors that rely on wireless communication and

autonomous power supply. Such sensing networks, may loose the original designed web

of connections if faults occur to power supply or communication. Thus, their design

accounts for possible variations of the connections, abandoning the paradigm of a rigid

network structure. Moreover, in proximity networks of mobile agent the exchange of

information may be possible only if the distance is below a given threshold. For instance,

the formation control of Unmanned Air Vehicles or the control of satellites such as GPS

may be regarded as proximity networks. Social networks, or the circadian function of

some cells may as well be regarded as proximity networks.

In this work we focus our attention on state-dependant proximity networks, which

are an open research area. In particular, we investigate the effect of proximity rule on the

proprieties of interconnection topology. Specifically we focus on networks of Kuramoto

oscillators coupled trough proximity rules. The Kuramoto model is a model for the

behavior of a large set of coupled oscillators. Its formulation was motivated by the

behavior of systems of chemical and biological oscillators, and it has found widespread

applications such as in neuroscience and biology. The outstanding adaptability of the

model have made it suitable to be studied in many different contexts ranging from

physics to chemistry.

The aim of this thesis is to numerically investigate the proprieties of proximity

Kuramoto oscillators. In particular we at describing the topological bifurcation phe-

nomenon, that may take place for coupling strength in a given interval.

The outline of this work is as follows

• In chapter 1 we review and introduce some basics concepts and tool of the complex

network theory. Specifically an insight is given into some graph theory useful for

in the following

• In chapter 2 a literature review is carried out showing the state of the art with

regard to complex network theory. Articles, and internal reports of the SINCRO

group have been the primary research source. Specifically the analysis focuses on

proximity oscillator networks, and since the subject is relatively new, few articles

have been found about this subject.

• Chapter 3 introduces the model and the notation for proximity networks and

clarifies the objectives of the numerical analysis. Successively we give examples of

some phenomena for a 5 node network thus highlighting some interesting aspects.

Successively a full characterization of topological bifurcation is performed on a 5

node network.

Contents

Abstract i

Contents ii

List of Figures vi

1 Background on Complex Network theory 1

1.1 Modeling a complex network . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Agent dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 Coupling protocol . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.3 Network topology . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.4 A brief history on the evolution of complex networks . . . . . . 8

1.2 Emerging behaviors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.1 Consensus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2.2 Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3.1 Biological systems and neuroscience . . . . . . . . . . . . . . . . 18

1.3.2 Computer science and engineering . . . . . . . . . . . . . . . . . 22

1.3.3 Social sciences and economy . . . . . . . . . . . . . . . . . . . . 26

i

1.4 Proximity networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.5 Switching and state dependent networks literature review . . . . . . . . 31

2 The Kuramoto model 33

2.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.2 Phase synchronization and phase locking . . . . . . . . . . . . . . . . . 37

2.2.1 Current stability results: an overview . . . . . . . . . . . . . . . 40

3 Proximity Kuramoto oscillators 41

3.1 The restricted visibility Kuramoto model . . . . . . . . . . . . . . . . . 42

3.1.1 Equilibria and emerging topologies . . . . . . . . . . . . . . . . 46

3.2 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2.1 A simple example: the chain topology . . . . . . . . . . . . . . . 49

3.2.2 Simulations for a 5 node network . . . . . . . . . . . . . . . . . 50

Conclusions and future work

References

List of Figures

1.1 Single system-controller scheme typical in control theory . . . . . . . . 2

1.2 Multiple system interaction scheme . . . . . . . . . . . . . . . . . . . . 2

1.3 Example of a un directed network of N = 5 nodes . . . . . . . . . . . . 4

1.4 Example of a scale free network compared to an ER network . . . . . . 10

1.5 A network of integrator agents in which agent i receives the state xj of

its neighbor, agent j, if there is a link (i; j) connecting the two nodes . 12

1.6 Block diagram for a network of interconnected dynamic systems all with

identical transfer functions P (s) = 1/s . . . . . . . . . . . . . . . . . . 12

1.7 MSF examples where it may be seen that in the fist case no synchro-

nization is achievable, in the second synchronization is achievable il λ2 is

above a threshold, and in the third case synchronization is achievable

within a certain range . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.8 The all-to-all configuration is depicted in the top part of the figure and

it’s eigenratio is independent from N . Due its high number of connection

synchronization is easily achieved at fixed σ. The second part of the

figure has a lower values of the eigen ration and show a that increasing

N reduces ostaculates synchronization . . . . . . . . . . . . . . . . . . 18

iii

2.1 Mechanical analog of a coupled oscillator network . . . . . . . . . . . . 35

3.1 The depicted oscillator has limited visibility over θ . . . . . . . . . . . 42

3.2 Two un coupled oscillators, say oscillator i and j, for which aij = 0 . . 44

3.3 Two coupled oscillators, say oscillator i and j, for which aij = 1 . . . . 44

3.4 Corrective term to the natural frequency ωi for oscillator i when inter-

acting with oscillator j for θvis = π6

. . . . . . . . . . . . . . . . . . . . 44

3.5 Synchronisation is not achieved for g = 1.0, N = 5, θvis = π10

. . . . . . 45

3.6 Synchronisation is achieved for g = 5.0, N = 5, θvis = π10

. . . . . . . . 46

3.7 Here we qualitatively represent the regions of interest for our numerical

analysis of topology bifurcation. On the horizontal axis we consider the

torque parameter g and on the vertical axis we consider a measure of the

topology. The size of the regions are exemplified to be of the same. In

red we denote the the region for which synchronization is not archived

since g is below the critical value. The two blue ares show the all-to-all

areas and in green the we denote the area of different equilibrium topologies 48

3.8 Qualitative diagram of a chain topology for a 5 node network . . . . . . 49

3.9 In the top part of the diagram the N velocities of the oscillators are

diagramed showing that frequency entrainment has been reached. The

lower part of the diagram shows a plot with respect to time of the

difference of the relative position of the N − 1 oscillators respect to

oscillator 1. Approximately at Time=3.5[s], the oscillators are in the

All-to-All topology. The value of g for this simulation is set to g = 3.0 . 50





oscillator 1. Notice how phase locking is archived for the system with

the chain equilibria topology. The value of g for this simulation is set to

g = 6.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51





oscillator 1. Approximately at Time=3.5[s], the oscillators are in the

All-to-All topology. The value of g for this simulation is set to g = 11.0 51

3.12 Topology equilibria diagram for fixed initial condition: the chain topology.

The network parameters have been set as follows: N = 5, ∇1,i[ωi] =

0.1 · 2π, θvis = π10

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.13 λ2 occurrences function for the initial conditions of the N = 5 network 54

3.14 < ki > occurrences function for the initial conditions of the N = 5 network 54

3.15 N = 5 network topology bifurcation diagram . . . . . . . . . . . . . . . 56

3.16 3D bar diagram showing the number of occurrences of equilibrium topolo-

gies with respect to the gain grid and the algebraic connectivity of the

initial condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.17 Second view of the 3D bar diagram, showing the number of occurrences

of equilibrium topologies with respect to the gain grid and the algebraic

connectivity of the initial condition . . . . . . . . . . . . . . . . . . . . 57

3.18 3D bar diagram showing the number of occurrences of equilibrium topolo-

gies with respect to the gain grid and the average degree of the initial

condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.19 Second view of the 3D bar diagram, showing the number of occurrences of

equilibrium topologies with respect to the gain grid and the the average

degree of the initial condition . . . . . . . . . . . . . . . . . . . . . . . 59

1Background on Complex Network theory

A network approach to study the phenomenon of interest implies a change of perspective

towards the subject of study; specifically moving interest from the study of proprieties

of a single entity, e.g. a generic system of interest, towards the behaviour of a collection

of single systems interacting between each other. An example can be found in control

theory, which has traditionally focused on a single system or at most on the interaction

of two systems, the plant and the controller; the former being for example a car and the

latter for instance a driver as shown in Fig. 1.1. Recent work has shifted the interest in

this field towards the interaction of a growing number of systems, changing the paradigm

from the low cardinality of a system-controller scheme to high cardinality scheme. When

1

Background on Complex Network theory

Figure 1.1: Single system-controller scheme typical in control theory

the number of agents1 increases, the collective behavior of the network strictly depends

upon the interconnection among the nodes. The interconnection of the systems shows

Figure 1.2: Multiple system interaction scheme

the emergence of a collective behavior and proprieties that an analysis of the single

individual agents alone does not explain. The web of connections between nodes can

assume various forms; the connections among the nodes, referred in literature as the

topology, is encoded in the network. Another two important aspects that account for the

behaviour of a network are the way the nodes couple, often referred as communication

protocol, and the dynamics of each node.

We now introduce notation for complex network and give an insight to some

important concepts.

1note that further in the text the terms agent, node, oscillator, system will be used indifferentlyunless differently stated

2

1.1 Modeling a complex network


A classic way of modeling a network, is shown in Eq. (1.1)

xi = fi(xi) + gi(xi)ui,∀i = 1, . . . , N (1.1)

where N is the number of nodes in the network, xi ∈ Rn is the state of the single node,

ui ∈ Rm,m < N is the coupling model between nodes. fi and gi are vectorial functions

describing the dynamics of each node.

1.1.1 Agent dynamics

For each agent note that some modeling assumption are held in Eq. (1.1): the first

concernes the dependance of fi and gi with respect to time; these functions are assumed

independent from time and such systems are called autonomous systems. A second

assumption regards the mathematical structure of such equation, we are assuming that

the relationship between the control input ui and gi is linear and that function gi is

added to fi.

Some stronger assumption are some times held, for instance in control networks the

dynamics of the single agent is modeled as a single or double integrator. Notice that

this assumption is excessively restrictive, a practical example can be given by assuming

each nodes as subject a feedback linearization. While on the one hand this assumption

simplifies the analysis of the collective behaviour on the other hand it is restrictive do

to the limitations of the feedback linearzation theorem as may be read from [1].

3


1.1.2 Coupling protocol

A typical coupling model has the following form

ui = g

N∑j=0

aij[h(xj)− h(xi)] (1.2)

where g is the coupling gain, aij is element of i-th row and j-th column of the adjacency

matrix that will be defined in Sec. 1.1.3. h(xj) is function describing the way in which

the agents exchange their state, if h is a linear function the coupling between nodes is

a linear diffusive model. Notice that Eq.(1.1.2) holds some assumptions. Specifically,

note that function h has no direct dependance on time, g ∈ R is equal for all nodes,

and elements aij show no dependance on time. Other models are possible by removing

such assumptions, and by obtained by considering delays, adaptive gains or for example

a differential operator for function h. This last aspect has a interesting mechanical

interpretation for a first order differential coupling: the interaction is described as

viscous-elastic interaction, such as a spring with a viscous damper.

1.1.3 Network topology

As previously introduced, the topology of the network is of primary importance with

respect to the behaviour of the network. A network is formally defined in graph theory

as a pair of sets G = n,m such that |n| = N is the number of nodes and |m| = M is the

number of edges. In Fig.1.3 a network is depicted. At first let us make clear that the

1

2

4

3

5

Figure 1.3: Example of a un directed network of N = 5 nodes

4


arcs between nodes are referred as edges2 and may be directed or indirected. A directed

graph represents a coupling that yields only from node i to node j but not viceversa. A

undirected graph allows a symmetric interaction between the generic node i and node j.

In such scenario two main tools are to be recalled, the first is the adjacency matrix, and

the second is the Laplacian of the network. Further we introduce some basic properties

and tools of a network.

Adjacency matrix

The adjacency matrix accounts for the network network connections

A =

a11 . . . a1N... . . .

...

aN1 . . . aNN

(1.3)

where in the case of a undirected topology aij = 1 if a connection is held between

nodes i and node j, aij = 0 otherwise. Notice that in case of undirected graphs matrix

A is also symmetric. In case the network connection are oriented aij = 1 if there is

a connection from node i to node j, aij = −1 if there is a connection from node j to

node i; if there is no connection between node i and node j then aij = 0. In the case of

Fig.1.3 the adjacency matrix is the following

A =

1 1 1 1 0

1 1 0 1 1

1 0 0 1 1

0 1 1 0 0

0 1 0 0 0

(1.4)

2further referred as connections

5


Notice that on the diagonal of the adjacency ring ones have been place, but in literature

these may assume either 0 or 1; specifically the if aij = 1, i = j then a self connection is

established.

Average path length

The average path length Le of a network is defined as the mean distance between two

nodes, averaged over all pairs of nodes. Hence, Le determines the effective ’size’ of

a network, the most typical separation of one pair of nodes therein. In a friendship

network, i.e. a network of relationship, for instance, Le is the average number of friends

existing in the shortest path connecting two persons in the network.

Degree Distribution

The simplest and perhaps also the most important characteristic of a single node is

its degree. The degree ki of node i is defined as the total number of its connections.

Thus, the larger the degree, the ’more important’ the node is in a network. The average

of ki over all i is called the average degree of the network, and is denoted by < k >.

The spread of node degrees over a network is characterized by a distribution function

P (k), which is the probability that a randomly selected node has exactly k edges. A

regular lattice has a simple degree sequence because all the nodes have the same number

of edges; and so a plot of the degree distribution contains a single sharp spike (a.k.a

delta distribution). Any randomness in the network will broaden the shape of this

peak distributing the degree. In the limiting case of a completely random network, the

degree sequence obeys the familiar Poisson distribution; and the shape of the Poisson

distribution falls off exponentially away from the peak value < k >. Because of this

exponential decline, the probability of finding a node with m edges becomes very small

for m >>< k > [2].

6


Betweenness distribution

Betweenness is an important measure to assess how a node is central in a network. This

metric in fact computes how many shortest paths traverse a node, therefore giving an

information of the importance of the node in the path management.

Clustering coefficient

Let us consider a network of friendship. In such a network one may be interest to

evaluate the possibility of friend’s friend being a direct friend. In other words the

possibility that two of someone’s friends are friends of each other. This property refers

to the clustering of the network. More precisely, one can define a clustering coefficient

C as the average fraction of pairs of neighbors of a node that are also neighbors of each

other. Suppose that a node i in the network has ki edges which connect this node to ki

other nodes. These nodes are all neighbors of node i. It can be shown that, at most

ki(ki − 1)/2 edges can exist among them, and this occurs when every neighbor of node

i connected to every other neighbor of node i. The clustering coefficient Ci of node i is

then defined as the ratio between the number Ei of edges that actually exist among

these ki nodes and the total possible number ki(ki−1)/2, namely, Ci = 2Ei/(ki(ki−1)).

The clustering coefficient C of the whole network is the average of Ci over all i. Clearly,

C ≤ 1; and C = 1 if and only if the network is globally coupled, which means that every

node in the network connects to every other node. In a completely random network

consisting of N nodes, C ∼ 1/N , which is very small as compared to most real networks.

Laplacian matrix

Consider a graph G, the graph Laplacian is defined as:

L = D − A

7


where D = diag(k1, ..., kN) is the degree matrix of G with elements ki =∑

j 6=i aij and

zero off-diagonal elements. By definition, L has a right eigenvector of 1 associated with

the zero eigenvalue because of the identity L1 = 0. For the case of undirected graphs,

the Laplacian graph satisfies the following Sum-Of-Squares (SOS) property:

xTLx =1

2

∑(i,j)∈E

aij(xj − xi)2

Spectral proprieties of the Laplacian matrix are instrumental in analysis of con-

vergence of the network dynamics. According to Gershgorin circle theorem [3], all

eigenvalues of L in the complex plane are located in a closed disk centered at ∆+0j with

a radius of ∆ = maxi(ki), i.e., the maximum degree of a graph. For undirected graphs,

L is a symmetric matrix with real eigenvalues and, therefore, the set of eigenvalues of L

can be ordered sequentially in an ascending order as

0 = λ1 ≤ λ2 ≤ ... ≤ λn ≤ 2∆

The zero eigenvalue is known as the trivial eigenvalue of L. For a connected graph

G, λ2 > 0 (i.e., the zero eigenvalue is isolated). The second smallest eigenvalue of

Laplacian λ2 is called algebraic connectivity of a graph. Algebraic connectivity of the

network topology is a measure of how connected the network is.

1.1.4 A brief history on the evolution of complex networks

A brief review is here given, for the main results on complex networks on the vast

literature at this regard. One of the early papers in the field of complex networks was

the article by Erdos and Renyi in 1959 [4](ER). They introduced the random graph

model knows as the ER model, generated from an initial set of disconnected nodes

which are linked by arcs between randomly chosen pairs of nodes, which gives random

8


graphs a Poisson-distributed degree distribution. ER graphs have been extensively

studied and many modifications have been proposed in order to fit this model to real

networks [5].

In 1998 Watts and Strogatz introduced the small-world network model (WS). The

WS model aims to generate graphs that have a high node clustering and present the

small-world property, i.e. the ability to reach any given point within the network in a

fairly small number of steps relative to the network size. These graphs are generated

by rewiring an initial ring of nodes that are symmetrically connected to their nearest

neighbors. With a small number of these shortcuts, the network acquires small-world

properties.

The growth of the World Wide Web inspired Barabasi and Albert in 1999 to create

one of the most comprehensively studied network models (BA). The BA model is a

scale-free network, which is a connected graph or network with the property that the

number of links k originating from a given node exhibits a power law distribution

P (k) ∼ k−γ . A scale-free network can be constructed by progressively adding nodes to

an existing network and introducing links to existing nodes with preferential attachment

so that the probability of linking to a given node i is proportional to the number of

existing links ki that node has. Scale-free networks occur in many areas of science and

engineering, including the topology of web pages (where the nodes are individual web

pages and the links are hyper-links), the collaborative network of Hollywood actors

(where the nodes are actors and the links are co-stars in the same movie), the power

grid of the western United States (where the nodes are generators, transformers, and

substations and the links are power transmission lines), and the peer-reviewed scientific

literature (where the nodes are publications and the links are citations).

9


Figure 1.4: Example of a scale free network compared to an ER network

1.2 Emerging behaviors

A lot of attention has been put towards two behaviours of the collective dynamics of a

complex network.

• Consensus

• Synchronization

In both cases the evolution of the dynamics of the agents converges towards a common

evolution.

Consensus has a long history in computer science and form the foundation of the

field of distributed computing. In networks of agents (or dynamic systems), ”consensus”

means to reach an agreement regarding a certain quantity of interest that depends

on the state of all agents. A consensus algorithm (or protocol) is an interaction rule

that specifies the information exchange between an agent and all of its neighbors on

the network. The theoretical framework for posing and solving consensus problems for

networked dynamic systems was introduced by Olfati-Saber and Murray in [3].

Synchronisation is to be regarded as the correlated and coordinate in time behaviour

of heterogenous processes towards an objective. In the following we introduce both

consensus and synchronization, where the latter is essential for the objectives of this

thesis.

10


1.2.1 Consensus

The topology of a network of agents is represented using a directed graph G = (V,E)

with the set of nodes V = (1, 2, ..., n) ; and edges E ⊆ V × V . The neighbors of agent

i are denoted by Ni= j ∈ V : (i, j) ∈ E. A simple consensus algorithm to reach an

agreement, regarding the state of n integrator agents with dynamics

xi = ui

can be expressed as an nth-order linear system on a graph

xi(t) =∑j∈N

(xj(t)− xi(t)) + bi(t), xi(0) = zi

The collective dynamics of the group of agents that following this protocol can be

written as:

x = −Lx

where L = lij is the Laplacian (introduced in 1.1.3) graph of the network and its

elements are defined as follows:

lij =

1, j ∈ Ni

‖ki|, j = 1.

Here, |ki| denotes the number of neighbors of node i (or degree of node i).

Note that according to the definition of graph Laplacian (introduced in 1.1.3), all

row-sums of L are zero because of∑

j lij = 0. Therefore, L always has a zero eigenvalue

λ1 = 0. This zero eigenvalues corresponds to the eigenvector 1 = (1, ..., 1) because 1

belongs to the null-space of L(L1) = 0.

Consider a network of decision-making agents with dynamics xi = ui, as in Fig.1.6

interested in reaching a consensus via local communication with their neighbors on a

graph G = (V,E). By reaching a consensus, we mean asymptotically converging to a

11


Figure 1.5: A network of integrator agents in which agent i receives the state xj of its neighbor,agent j, if there is a link (i; j) connecting the two nodes

Figure 1.6: Block diagram for a network of interconnected dynamic systems all with identicaltransfer functions P (s) = 1/s

12


one-dimensional agreement space characterized by the following equation:

x1 = x2 = ... = xn.

This agreement space can be expressed as x = α1 where 1 = (1, ..., 1) and α ∈ R is the

collective decision of the group of agents. Let A be the adjacency matrix of graph G.

The set of neighbors of a agent i is Ni and defined by

Ni = j ∈ V : aij 6= 0 ; V = 1, ..., n.

Agent i communicates with agent j if j is a neighbor of i (or aij 6= 0). The set of all

nodes and their neighbors defines the edge set of the graph as

E = (i, j) ∈;V × V : aij 6= 0

It is shown in [3] that the linear system

xi(t) =∑j∈Ni

aij(xj(t)− xi(t))

is a distributed consensus algorithm, i.e., guarantees convergence to a collective decision

via local interagent interactions. Assuming that the graph is undirected, i.e. aij = aji,

it follows that the sum of the state of all nodes is an invariant quantity, or∑

i xi = 0.

In particular, applying this condition twice at times t = 0 and t = ∞ gives the

following result

α =1

n

∑i

xi(0)

In other words, if a consensus is asymptotically reached, then necessarily the collective

decision is equal to the average of the initial state of all nodes. A consensus algorithm

with this specific invariance property is called an average-consensus algorithm [3]. The

13


dynamics of system can be expressed in a compact form as:

x = −Lx

where L is again the Laplacian of graph G. As stated earlier y definition, L has a right

eigenvector of 1 associated with the zero eigenvalue because of the identity L1 = 0. For

the case of undirected graphs, the Laplacian graph satisfies the following Sum-Of-Squares

(SOS) property:

xTLx =1

2

∑(i,j)∈E

aij(xj − xi)2

By defining a quadratic disagreement function as

φ(x) =1

2xTLx

it becomes apparent that algorithm is the same as

x = −∇φ(x)

or the gradient-descent algorithm. This algorithm globally asymptotically converges to

the agreement space provided that two conditions hold: 1) L is a positive semidefinite

matrix; 2) the only equilibrium is α1 for some α. Both of these conditions hold for a

connected graph and follow from the SOS property of the Laplacian graph.

1.2.2 Synchronization

The question of interest is that, under what conditions can the highly complicated

coupled system exhibit coherent behavior, i.e., they completely synchronize, or more

precisely,

limt→+∞

||xi − xj|| = 0,∀i, j

14


A famous approach named master stability function (MSF) analysis was introduced

by Pecora and Carroll in 1998[6], which allows one to answer the question of how

synchronization stability depends on the dynamics, coupling form, and network topology.

Here we review the MSF approach.

At first let us consider the system in Eq.1.1, it may be rewritten as

xi = f(xi) + σ

N∑j=1

Lijh(xj)

where h is a copulating function. By considering such system at a certain time t∗ to

guarantee existence and stability of a synchronous evolution we suppose a synchronous

condition to be achieved thus yielding

x1 = x2 = · · · = xn = xs

. The key idea is to find synchronization conditions for the local transversal stability

of the synchronization manifold. Thus the system is linearized at xs defining the i-th

evolution variable around xs as ξi

ξi = fx(xs)ξ + σ

N∑j=1

[lijh(xs) + hx(xs)ξ]

that may be re written as

ξi = fx(xs)ξ + σN∑j=1

Lijhxxsξ

Now since the previous equation depends on the Jacobian of the nodes and from the

topology thanks to the presences to the la laplacian we may re write the equation in a

15


un directed network by diagonalizing the Laplacian

ζi = [fx(xs) + σλihx(xs)]ζi

such equation expresses how the transversal dynamics to the manifold are function of

coupling strength and dynamics of the single nodes. Thus in one equation we have

the dynamics associated parallel to the synchronization manifold ( by recalling the

proprieties of the Laplacian,i.e. λ1 = 0 ) and the the transversal modes λi,∀i = 2, . . . , N .

Thus, if all the other ζ → 0, then all the transverse perturbation dies out, and a slightly

perturbed state will come back to the synchronous state xs. This requires that the

solution of the previous equations goes to zero for all i = 2, . . . , N . If the dynamics fx

and hx have variable coefficients the evaluation of stable dynamics is to be carried out

with Lyapunov exponents. Thus the requirement that ζ → 0 is equivalent as requiring

the largest Lyapunov exponent of the flow of the previous Equations associated with

the motion xs = f(xs) being negative. Thus, the local synchronization stability can be

determined by checking the largest Lyapunov exponent for all i ≥ 2. Synchronization is

locally stable if and only if for each i ≥ 2, the corresponding largest Lyapunov exponent

is negative.

The last step in the analysis is the plot of the maximum Lyapunov exponent (MSF)

for a generic coupling, given f and h. Pecora and Carroll introduced a stability function

called master stability function Θ as a function of α, where Θ(α) equals the largest

Lyapunov exponent of the parametric equation

ζi = [fx(xs)− αhx(xs)]

Synchronization stability can be determined by simply checking whether Θ(σλi) < 0,i.e.

it requires σλi to be in the range where the MSF is negative, for all i ≥ 2, if it

is, then synchronization is locally stable; and not if not. Thus the network is more

16


synchronizable, the large is the eigen ratio λ2/λN The introduction a the MSF allows

Figure 1.7: MSF examples where it may be seen that in the fist case no synchronization isachievable, in the second synchronization is achievable il λ2 is above a threshold, and in the third

case synchronization is achievable within a certain range

some interesting considerations especially in terms of control. Indeed by setting σ it may

be possible to find the topology that allows synchronization, or equivalently, by setting

the topology it may be possible to find a value σ at which the system synchronizes. Two

examples are reported in Fig.1.8, and may be useful for some intuitive interpretation of

the numerical analysis of the proximity networks, subject of this thesis.

17


Figure 1.8: The all-to-all configuration is depicted in the top part of the figure and it’s eigenratiois independent from N . Due its high number of connection synchronization is easily achievedat fixed σ. The second part of the figure has a lower values of the eigen ration and show a that

increasing N reduces ostaculates synchronization

1.3 Applications

In this section we will overview the applications to specific problems in such different

scientific fields as biology and neuroscience, engineering and computer science, and

economy and social sciences. There are several problems where the application of the

ideas and techniques developed in relation to synchronization in complex networks

are very clear and the results help to understand the interplay between topology and

dynamics a lot of scenarios. There are other cases, for which most of the applications

so far have been developed in simple patterns of interaction, but extension to complex

topologies is necessary because it is its natural description.

1.3.1 Biological systems and neuroscience

In biology, complex networks are found at different scales: from the molecular level

up to the population level, passing through many intermediate scales of biological

systems. In some of these networks, dynamical interactions between units, which

are crucial for our current understanding of living systems, can be analyzed in the

18

1.3 Applications

framework of synchronization phenomena. We review some of these application scenarios

where synchronization in networks has been shown to play an essential role. Thus,

at the molecular level we can analyze the evolution of genetic networks and at the

population level the dynamics of populations of species coupled through diffusion along

spatial coordinates and through trophic interactions. Amongst these two extremes we

find a clear application in the analysis of circadian rhythms. On a different context,

neuroscience offers an application level for the synchronization of individual spiking

neurons.

Circadian rhythms

A circadian rhythm is a roughly 24-hour cycle in the physiological processes of living

systems; usually endogenous, or when it is exogenous it is mainly driven by daylight.

Understanding circadian rhythms is crucial for some physiological and psychological

disorders. Circadian rhythms are known to be dependent on the network of interactions

between different subsystems. For example, daylight sensed by eyes and processed

by the brain develops a chain of interactions that affects even the behavior of certain

groups of cells. On a different scenario, non-oscillatory cardiac conducting tissues, when

driven rhythmically by repetitive stimuli from their surroundings, produce temporal

patterns including phase locking, period-doubling bifurcation and irregular activity.

Synchronization phenomena in complex networks of coupled circadian oscillators

has been recently investigated experimentally on plant leaves. The vein system is in

this case the complex network substrate of the synchronization process. Plant cells

are coupled via the diffusion of materials along two types of connections: one type

that directly connects nearest-neighboring cells and the other type that spreads over

the whole plant to transport material among all tissues quickly. Analyzing the phase

of circadian oscillations, the phase-wave propagations and the phase delay caused by

the vein network, synchronization of circadian oscillators in the leaf can be attained.

19


The role of the topology of interactions is again fundamental in the development of

synchronization. This work is representative of the new type of applications we can find

in the very recent literature about synchronization in complex networks. This particular

case of circadian rhythms in plants might be extended to other living systems, including

humans.

Ecology

In nature fluctuations in animal and plant populations display complex dynamics.

Mainly irregular, but some of them can show a remarkably cyclical behavior and take

place over vast geographical areas in a synchronized manner. One of the best documented

cases of such situation are the population fluctuations in the Canadian lynx, obtained

from the records of the fur trade between 1821 and 1939 in Canada. Fluctuations

in lynx populations show a 10-year periodic behavior from three different regions in

Canada. On the other hand, there are some evidences that the existence of conservation

corridors favoring the dispersal of species and enhancing the synchronization over time

increases the danger of global extinctions.

One of the first explanations for such types of behavior was that of synchronous

environmental forcing, this is the so-called Moran effect. There are, however, other

explanations for this phenomenon, but in any case the problem highlights the importance

of integrating explicitly spatial and trophic couplings into current metacommunity

theories. Some efforts along these lines have already been made by considering very

simple trophic interaction in spatially extended systems. For example, a three-level

system (vegetation, herbivores, and predators), where diffusive migration between

neighboring patches is taken into account. They find that small amounts of migration are

required to induce broad-scale synchronization. Another interesting study is performed

with an extremely simple model, it is found that changing the patterns of interaction

between consumers and resources can lead to either in-phase synchrony or antiphase

20

1.3 Applications

synchrony.

Nowadays we know, however, about the inherent complexity of food-webs. Food

webs have been studied as paradigmatic examples of complex networks, because they

show many of their non-trivial topological features. Furthermore, the existence of

conservation corridors affecting the migration between regions adds another ingredient

to the structure of the spatial pattern. It is precisely this complexity in the trophic

interactions coupled to the spatial dependence that must to be considered in detail in

the future to get a deeper understanding of ecological evolution.

Neuronal networks

Synchronization has been shown to be of special relevance in neural systems. The

brain is composed of billions of neurons coupled in a hierarchy of complex network

connectivity. The first issue concerns neural networks at the cellular level. In the last

years, significant progress has been made in the studies about the detailed intercon-

nections of different types of neurons at the level of cellular circuits. At this level,

the neuronal networks possess complex structure, sharing SW and SF features. Here

are two basic neuron types: excitatory principal cells and inhibitory interneurons. In

contrast to the more homogeneous principal cell population, interneurons are very

diverse in terms of morphology and function. There is inverse relationship between the

number of neurons in various interneuron classes and the spatial extent of their axon

trees–most of the neurons have only local connections, while a small number of neurons

have long-range axons. These properties of neuronal networks reflects a compromise

between computational needs and wiring economy.

On the one hand, the establishment and maintenance of neuronal connections

require a significant metabolic cost that should be reduced, and consequently the wiring

length should be globally minimized. Indeed, the wiring economy is apparent in the

distributions of projection length in neural systems, which show that most neuronal

21


projections are short. However, there also exists a significant number of long-distance

projections.

Large-scale synchronization of oscillatory neural activity has been believed to play

a crucial role in the information and cognitive processing. At the level of cellular

circuits, oscillatory timing can transform unconnected principal cell groups into temporal

coalitions, providing maximal flexibility and economic use of their spikes. Brains have

developed mechanisms for keeping time by inhibitory interneuron networks. The wiring

will be the most economic if the connections were all local. However, in this case

physically distant neurons are not connected, and synaptic path length and synaptic

delays become exceedingly long for synchronization in large networks. It was shown

with a model of interneuronal networks containing local neurons (Gaussian distribution

of projection length) and a fraction of long-range neurons (power law distribution of

projection length), that the ratio of synchrony to wiring length is optimized in the SW

regime with a small fraction of long-range neurons. Thus, most wiring is local and

neurons with long-range connectivity and large global impact are rare, as consistent

with observations. It was argued that the complex wiring of diverse interneuron classes

could represent an economic solution for supporting global synchrony and oscillations at

multiple time scales with minimum axon length. While such mathematical consideration

can predict the scaling relationship among the interneuron classes in brain structures

of varying sizes, understanding the role of complex neuronal connectivity, most likely

mediated by synchronization, is still one of the main challenges in neuroscience.

1.3.2 Computer science and engineering

Complex networks and synchronization dynamics are relevant in many computer science

and engineering problems. For example, in computer science, synchronization is desirable

for an efficient performance of distributed systems. Sometimes, the goal of the distributed

system is to achieve a global common state i.e. consensus. Nowadays these systems

22

1.3 Applications

are becoming larger and larger and their topologies more and more complex. On the

other hand, some engineering problems also face the need of maintaining coordination

at the level of large scale complex networks, for example in problems of distribution of

information, energy or materials.

Parallel computation

The simulation of large systems are, nowadays, mainly implemented as parallel dis-

tributed simulations where parts of the system are allocated and simulated on different

processors, as in the case of the calculations for this thesis. A large class of interacting

systems including financial markets, epidemic spreading, traffic, and dynamics of physi-

cal systems in general, can be described by a set of local state variables allowing a finite

number of possible values. As the system evolves in time, the values of the local state

variables change at discrete instants, either synchronously or asynchronously, depending

on the dynamics of the system. The instantaneous changes in the local configuration

are called discrete events, forming what has been coined as a parallel discrete-event

simulation (PDES). The main difficulty of PDES is the absence of a global pacemaker

when dealing with asynchronous updates. This imposes serious problems because

causality and reproducibility of experimental results are desired. In a conservative

scheme, processes modeling physical entities are connected via channels that represent

physical links in the target system. Since PDES events are not synchronized via a

global clock, they must synchronize by communication between nodes.

Consensus problems

Consensus problems, understood as the ability of an ensemble of dynamic agents to

reach a unique and common value in an asymptotically stable stationary state, have

a long history in the field of computer science, particularly in automata theory and

distributed computation. In many applications, like for instance cooperative control

23


on unmanned air vehicles, formation control or distributed sensor networks, groups of

agents need to agree upon certain quantities of interest. As a result, it is important to

address these problems of agreement within the assumption that agents form a complex

pattern of interactions. These interactions can be directed or undirected, fixed or mobile,

constant or weighted, keeping then many of the ingredients we have been discussing in

this thesis. Another interesting fact in this sort of problems is the existence of time

delays in the communication process. Let us consider a dynamic graph in which the

connectivity pattern of the nodes can change in time. At each node, a dynamical agent

evolves in time according to the dynamics

xi = f(xi, ui)

similarly as in Eq. 1.1, where f(xi, ui) is a function that depends on the state of the

unit xi , and on ui that describes the influence from the neighbors. The χ-consensus

problem in a dynamical graph is a distributed way to reach an asymptotically stable

equilibrium x∗ satisfying x∗i = χ(x(0)), ∀i where χ(x(0)) is a function of the initial

values (e.g. the average or the minimum values).

The authors in [7] present two protocols that solve consensus problems in a network

of agents:

• fixed or switching topology and zero communication time-delay:

xi =N∑

i,j=1

aij(t)(xj(t)− xi(t))

• fixed topology and non-zero communication time-delay τij > 0

xi =N∑

i,j=1

aij(xj(t− τij)− xi(t− τij))

24

1.3 Applications

We note that the analysis of the asymptotic behavior of such linear system is similar to

the stability analysis performed in the framework of the MSF(see par. 1.2.2).

For a switching topology, they find that if the dynamics of the network is such

that any graph along the time evolution is strongly connected and balanced then the

switching system asymptotically converges to an average consensus. Concerning time

communication-delays, the important result is that if all links have the same time-delay

τ > 0, and the network is fixed, undirected and connected, the system solves the

average consensus if τ ∈ ( π/2λN ). In this case, in a similar way as discussed in previous

applications, there are two tradeoff issues that can be related to problems of network

design; one concerns the robustness of the protocol with respect to time-delays, and

the other to communication cost.

Power-Grids

Power grids are physical networks of electrical power distribution lines of generators and

consumers. In the pioneering paper by Watts and Strogatz [8](and therein references)

it was already reported that the power-grid constitutes one of the examples of a self-

organized topology that has grown without a clear central controller. This topology is

indeed very sensitive to attacks and failures. From its topological point of view there

are several analyses on power-grids in different areas of the world and some models

have been proposed to deal with the cascading process of failures.

The principles of electricity generation and distribution are well known. Synchro-

nization of the system is understood as every station and every piece of equipment

running on the same clock, which is crucial for its proper operation. Cascading failures

related to de-synchronization can lead to massive power blackouts.

Consider the power produced at a generator. It can then be dissipated, accumulated,

or transmitted along the electric line. The first two terms (dissipation and accumula-

tion) depend on the frequency of the generator whereas the last one (transmission) is

25


proportional to the sinus of the phase difference between the generator and the machine

at the other extreme of the line. Then, a simple energy balance equation relates the

evolution of the phase (first and second time derivatives) with sinus of phase differences.

Applying this simplified approach to a networked system of generators and machines,

they arrive to a set of second order Kuramoto-like differential equations

θi + αθi = ωi +K∑j

sin(θj − θi)

where ωi is related to the power generated at the element and to the dissipated

power, and K, representing the stength of the coupling, is related to the maximum

transmitted power.

Within this framework,they analyze, as an application, under which conditions the

system is able to restore to a stable operation after a perturbation in simple networks

of machines and generators. To the best of our knowledge this is a first approximation

to the real applicability of the knowledge about synchronization in complex networks

to power grids.

1.3.3 Social sciences and economy

In the last decades, social sciences and economy have become one of the favorite applica-

tions for physicists and engineers. In particular, tools and models from statistical physics

can be implemented on what some people has called social atoms, i.e. unanimated

particles are replaced by agents that take decisions, trade stocks or play games. Simple

rules lead to interesting collective behaviors and synchronization is one of them, because

some of the activities that individual agents do can become correlated in time due

to its interaction pattern, which, in turn, is clearly another example of the complex

topologies considered along the review. In social systems, however, it is not an easy

task to identify the relationship between agents (being humans or collectives in social

26

1.3 Applications

interactions, stock prices in finances, or countries in the World Trade Web).

Finance

When reading the economic news, it is not difficult to identify the existence of economic

cycles in which Gross Domestic Products (GDP’s), economic sectors, or stock options

raise and fall. Most of the time this does not happen for isolated countries, sectors or

options but it occurs in quite a synchronized way, although some delays are noticeable.

Wwe are focusing on synchronization in complex networks, and this is what we can

identify in many economical sectors: there exists a complicated pattern of interactions

among companies or countries and the dynamics of each one is quite complex. But, in

contrast to many networks with a physical background, here we neither know in detail

the node dynamics nor its connectivity pattern. In this situation it is useful to look at

the problem from a different angle. By analyzing some macroscopic outcomes, we get

some insight into the agents’ interactions.

In the economic literature, synchronization is measured by a correlation coefficient,

based on the idea that correlated (synchronized) business cycles should generate cor-

related returns. The point is to identify what types of interactions lie behind market

co-movements. Synchronization is the result from two different effects. On the one

hand, there are different types of common disturbances (world interest rates, oil price,

or political uncertainty). On the other hand, there exist strong interactions between the

agents (financial relationships, sector dependencies, co-participation in director boards,

etc.). It is precisely, these interactions that play a crucial role in the synchronized

behavior along economic cycles of tightly connected agents and the analysis of the

correlations can help in shedding light on the strength of the different factors.

27


1.4 Proximity networks

In this paragraph we introduce the concept of proximity based upon [9] and therein

references. At this regard it we highlight that proximity is a subtle notion, whose

definition can depend on a specific application. Usually the notion of proximity is

strongly tied to the definition of an edge in the network. In a network where links

represent phone or email communication, proximity measures potential information

exchange between two non-linked objects through intermediaries. Where edges represent

physical connections between machines, proximity can represent latency or speed of

information exchange. Alternatively, proximity can measure the extent to which the two

nodes belong to the same cluster, as in a co-authorship network where authors might

publish in the same field and in the same journals. In other cases, proximity estimates

the likelihood that a link will exist in the future, or is missing in the data for some

reason. For instance, if two people speak on the phone to many common friends, the

probability is high that they will talk to each other in the future, or perhaps that they

already communicate through some other medium such as email. There are many uses

for good proximity measures. In a social network setting, proximities can be used to

track or predict the propagation of a product, an idea, or a disease. Proximities can help

discover unexpected communities in any network. A product marketing strategist could

target individuals who are in close proximity to previous purchasers of the product, or

target individuals who have many people in close proximity for viral marketing.

For the definition of proximity we sequentially refine a series of candidate definitions,

starting with the simplest one: shortest path. Notationally, we assume we have a graph

G(V,E) where the “network objects” are nodes (V)and the links between them are

edges(E). The weight of edge(i, j) is denoted by wij > 0 and reflects the similarity of i

and j.

Graph-theoretic distance The basic definitions we need for network proximity

28

1.4 Proximity networks

are taken from graph theory. The most basic one is the graph theoretic distance, which

is the length of the shortest path connecting two nodes, measured either as the number

of hops between the two nodes, or the sum of the edge weights along the shortest path.

The main rationale for considering graph theoretic distance is that proximity decays as

nodes become farther apart. Intuitively, information following a path can be lost at any

link due to the existence of noise or friction. Therefore two nodes that are not connected

by a short path are unlikely to be related. Distance in graphs can be computed very

efficiently. However, this measure does not account for the fact that relationships

between network entities might be realized by many different paths. In some instances,

such as managed networks, it may be reasonable to assume that information between

nodes is propagated only along the most “efficient” routes. However, this assumption is

dubious in real world social networks, where information can be propagated randomly

through all possible paths. Ideally, proximity should be more sensitive to edges between

low-degree nodes that show meaningful relationships, and take into account multiple

paths between the nodes

Network flow Consider another concept from graph theory– maximal network flow.

We assign a limited capacity to each edge (e.g.one proportional to its weight) and then

compute the maximal number of units that can be simultaneously delivered from node

s to node t. This maximal flow can be taken as a measure of s-t proximity. It favors

high weight (thus, high capacity) edges and captures the premise that an increasing

number of alternative paths between s and t increase their proximity problematic with

this definition is that the maximal s− t flow in a graph equals the minimal s− t cut–

that is, the minimal edge capacity we need to remove to disconnect s from t. In other

words, the maximal flow equals the capacity of the bottleneck, making such a measure

less robust.

Effective conductance (EC) A more suitable candidate comes from outside the

classical graph-theory concepts - modeling the network as an electric circuit by treating

29


the edges as resistors whose conductance is the given edge weights. This way, higher

weight edges will conduct more electricity. Descriptions are found in standard references.

When dealing with electric networks, a natural s − t proximity measure is found by

setting the voltage of s to 1, while grounding t (so its voltage is 0) and solving a system

of linear equations to estimate voltages and currents of the network. The computed

delivered current from s to t , is also called the effective conductance, or EC. Effective

conductance appears to be a good candidate for measuring proximity. It was used for

such purposes in different occasions, like in the graph-layout algorithm of Cohen [10], or

for computing centrality measures in social networks. Faloutsos et al also considered EC

in their study of connection subgraphs. An important advantage is that it accounts for

both path length (favoring short paths, like graph-theoretic distance) and the number

of alternative paths (more is better, like maximal flow), while avoiding dependence on

a single shortest path or a single bottleneck.

Specifically, in the study of multi-agent systems, most of the biologically-inspired

models proposed in the literature are based on the interaction rules introduced by

Reynolds in 1986(see [11] and there in references): two agents mutually interact only

if their distance is below a given threshold, and the kind of interaction (attraction,

alignment, repulsion) depends on the distance. In the context of this thesis we will refer

to proximity networks that base the interaction between agents rule on the distance

amongst agents. In 3 we will introduce some equations that describe the network

dynamics where the position of the agents is the solution to a set of interacting Ordinary

Differential Equations (ODE). So the proximity measure we shall use is a state dependant

rule, that will modify with respect to the state evolution the topology of the network,i.e.

the topology is time varying. Thus in the following, and specifically in Ch. 3, we shall

consider proximity state dependant networks.

30

1.5 Switching and state dependent networks literature review

1.5 Switching and state dependent networks

literature review

Switching topologies, as well as state dependant topologies have been subject of literature

in past years. In [12] dynamically changing state dependant topologies are considered

for a velocity consensus problem for two satellites. For coordination of mobile agents

that considers switching topologies authors in [13] show results which demonstrate that

the nearest a neighbor rule can cause all agents to eventually move in the same direction

despite the absence of centralized coordination and despite the fact that each agent’s

set of nearest neighbors change with time as the system evolves.

State dependant consensus models have been take into account in [14] that take into

account social opinion dynamics. In their model each agent has an opinion represented

by a real number, and updates its opinion by averaging all agent opinions that differ

from its own by less than 1. They then prove the convergence into clusters of agents,

with all agents in the same cluster holding the same opinion.

An optimization approach to handle proximity networks, specifically of sensor

networks, has also been proposed in literature in [15]. They address the issues associ-

ated with the steady connectivity which reduces the overall power consumption, and

successively a comparison study is made on these issues.

31


32

2The Kuramoto model

Synchronization in networks of coupled oscillators is a prevalent topic in various

scientific disciplines ranging from biology, physics, and chemistry to social networks

and technological applications. A coupled oscillator network is characterized by a

population of heterogeneous oscillators and a graph describing the interaction among

the oscillators. These two ingredients rise rich dynamics that keeps on fascinating the

scientific community [16] and therein references.

Consider a system of N oscillators, each characterized by a phase angle θi ∈ R

and a natural rotation frequency ωi ∈ R. The dynamics of each isolated oscillator

are thus θi = ωi for i ∈ 1, 2, ..., N . The interaction topology and coupling strength

33

The Kuramoto model

among the oscillators are modeled by a connected, undirected, and weighted graph

G = (V,E,A) with nodes V = (1, 2, ..., N), edges E ⊂ V × V , and positive weights

aij = aji > 0 for each undirected edge i, j ∈ E. The interaction between neighboring

oscillators is assumed to be additive, anti-symmetric, diffusive, and proportional to

the coupling strengths aij. In this case, the simplest 2π-periodic interaction function

between neighboring oscillators i, j ∈ E is sin(θi− θj), and the overall model of coupled

oscillators reads

θi = ωi −N∑j=1

aij sin(θi − θj) (2.1)

Despite its apparent simplicity, this coupled oscillator model gives rise to rich dynamic

behavior, and it is encountered in ubiquitous scientific disciplines ranging from natural

and life sciences to engineering.

2.1 The model

A variation of the considered coupled oscillator model 2 was first proposed by Winfree

in 1967 [17]. Winfree considered general (not necessarily sinusoidal) interactions among

the oscillators. He discovered a phase transition from incoherent behavior with dispersed

phases to synchrony with aligned frequencies and coherent (i.e., nearby) phases. Winfree

found that this phase transition depends on the trade-off between the heterogeneity

of the oscillator population and the strength of the mutual coupling, which he could

formulate by parametric thresholds. However, Winfree’s model was too general to

be analytically tractable. Inspired by these works, Kuramoto in 1975 [18] simplified

Winfree’s model and arrived at the coupled oscillator dynamics 2 with a complete

interaction graph and uniform weights aij = K/N .

θi = ωi −K

N

N∑j=1

sin(θi − θj) (2.2)

34

2.1 The model

Kuramoto showed that synchronization occurs in the model 2.1 if the coupling gain K

exceeds a certain threshold Kcritical function of the distribution of the natural frequencies

ωi. The dynamics 2.1 are nowadays known as the Kuramoto model of coupled oscillators,

and Kuramoto’s original work initiated a broad stream of research.

A mechanical analog of a coupled oscillator network is the spring network shown in

Fig. 2.1. This network consists of a group of kinematic particles constrained to move

on a unit circle and assumed to move without colliding. Each particle is characterized

by the phase angle θi ∈ R and is subject to an external driving torque ωi ∈ R.

Figure 2.1: Mechanical analog of a coupled oscillator network

Pairs of interacting particles i and j are coupled through a linear-elastic spring with

stiffness aij > 0. The overall spring network is modeled by a graph, whose nodes are the

kinematic particles, whose edges are the linear elastic springs, and whose edge weights

are the positive stiffness coefficients aij = aji. Under these assumptions and by writing

the mechanical system as a first-order vector field, it can be shown [19] that the system

of spring-interconnected kinematic particles obeys the coupled oscillator dynamics 2.

The population of oscillators exhibits the dynamic analog to an equilibrium phase

transition. When the natural frequencies of the oscillators are too diverse compared to

the strength of the coupling, they are unable to synchronize and the system behaves

incoherently. However, if the coupling is strong enough, all oscillators freeze into

35

The Kuramoto model

synchrony. The transition from one regime to the other takes place at a certain

threshold. At this point some elements lock their relative phase and a cluster of

synchronized nodes develops. This constitutes the onset of synchronization. Beyond

this value, the population of oscillators is split into a partially synchronized state made

up of oscillators locked in phase and a group of nodes whose natural frequencies are

too different as to be part of the coherent cluster. Finally, after further increasing the

coupling, more and more elements get entrained around the mean phase of the collective

rhythm generated by the whole population and the system settles in the completely

synchronized state. Kuramoto worked out a mathematically tractable model to describe

this phenomenology, namely he proposed the all-to-all sinusoidal coupling where the

governing equations is 2.

The collective dynamics of the whole population is measured by the macroscopic

complex order parameter

r(t)eiφ(t) =1

N

N∑j=1

eiθj(t) (2.3)

where the modulus 0 ≤ r(t) ≥ 1 measures the phase coherence of the population and

φ(t) is the average phase. The values r 6= 1 and r 6= 0 describe the limits in which

all oscillators are either phase locked or move incoherently, respectively. With some

algebra we obtain:

θi = ωi +Kr sin(φ− θi),∀i = 1, . . . , N

The first quantity provides a positive feedback loop to the system’s collective rhythm:

as r increases because the population becomes more coherent, the coupling between the

oscillators is further strengthened and more of them can be recruited to take part in the

coherent pack. Moreover, the previous equation allows to calculate the critical coupling

Kcritical and to characterize the order parameter limt→∞rt(K) = r(K). Looking for

steady state solutions, one assumes that r(t) and φ(t) are constant. Now by following

36

2.2 Phase synchronization and phase locking

[8], by integrating r(t) it is found Kcritical = 2πg(0)

where g(ω) is the distribution of

speeds, usually assumed to be unimodal and symmetric about its mean frequency Ω.


To deal with the KM on complex topologies, it is necessary to reformulate Eq. 2.1 to

include the connectivity following waht was introduced in 1.1.3, the KM is re written to

θi = ωi +N∑j=1

σijaij sin (θj − θi) (2.4)

where σij is the coupling strength between pairs of connected oscillators and aij are

the elements of the connectivity matrix. The original Kuramoto model is recovered by

letting aij = 1,∀i 6= j(all − to− all) and σij = K/N, ∀i, j.

The first problem when defining the KM in complex networks is how to state the

interaction dynamics. In contrast with the mean field model, there are several ways to

define how the connection topology enters in the governing equations of the dynamics. A

good theory for Kuramoto oscillators in complex networks should be phenomenologically

relevant and provide formulas amenable to rigorous mathematical treatment.

For the original model 2.1, the coupling term on the right hand side is an inten-

sive magnitude because the dependence on the size of the system cancels out. This

independence on the number of oscillators N is achieved by choosing σij = K/N . This

prescription turns out to be essential for the analysis of the system in the case of

limN →∞ in the all-to-all case. However, choosing σij = K/N for the governing

equations of the KM in a complex network makes them to become dependent on N .

Therefore, the coupling term tends to zero except for those nodes with a degree that

scales with N . Note that the existence of such nodes is only possible in networks with

power-law degree distributions, but this happens with a very small probability as k−γ ,

37

The Kuramoto model

with γ > 2. In these cases, mean field solutions independent of N are recovered, with

slight differences in the onset of synchronization of all-to-all.

A second prescription consists in taking σij = K/ki (where ki is the degree of node

i) so that σij is a weighted interaction factor that also makes the right hand side of

Eq. 2.1. This form has been used to solve the paradox of heterogeneity [20] that

states that the heterogeneity in the degree distribution, which often reduces the average

distance between nodes, may suppress synchronization in networks of oscillators coupled

symmetrically with uniform coupling strength. This result refers to the stability of

the fully synchronized state, but not to the dependence of the order parameter on

the coupling strength (where partially synchronized and unsynchronized states exist).

Besides, the inclusion of weights in the interaction strongly affects the original KM

dynamics in complex networks because it can impose a dynamic homogeneity that

masks the real topological heterogeneity of the network.

The prescription σij = K/const, which may seem more appropriate, also causes

some conceptual problems because the sum in the right hand side of Eq. 2.1 could

eventually diverge. The constant in the denominator could in principle be any quantity

related to the topology, such as the average connectivity of the graph, or the maximum

degree kmax . Its physical meaning is a re-scaling of the temporal scales involved in the

dynamics. However, except for the case of σij = K/kmax , the other possible settings

do not avoid the problems when N →∞ On the other hand, for a proper comparison

of the results obtained for different complex topologies, the global and local measures

of coherence should be represented according to their respective time scales. Therefore,

given two complex networks A and B with kmax = kA and kmax = kB respectively, it

follows that to make meaningful comparisons between observables, the equations of

motion Eq. 2.2 should refer to the same time scales, i.e., σij = KA/kA = KB/kB = σ.

38


With this formulation in mind, Eq. 2.2 reduces to ˙

θi = ωi + σ

N∑j=1

aij sin(θj − θi),∀i = 1, . . . N (2.5)

independently of the specific topology of the network. This allows us to study the

dynamics of Eq. 2.2 on different topologies, compare the results, and properly inspect

the interplay between topology and dynamics in what concerns synchronization. There

are also several ways to define the order parameter that characterizes the global dynamics

of the system, some of which were introduced to allow for analytical treatments at the

onset of synchronization.

The KM so far we have referred to populations where the oscillators are nearly

identical in the sense that they can have slightly different frequencies. Whenever there

is a subset of units that play a special role, in the sense that they have substantially

different frequencies than the rest in the population or they affect some units but are

not affected by any of them, one usually refers to them as pacemakers. The effect of

pacemakers has been studied in regular networks, as for instance in one-dimensional

rings, two-dimensional tori and Cayley trees. So far, the only approach in a complex

topology has been performed in [21]. There, the authors considered a system of identical

units (same frequency) and a singular pacemaker. For an ER network they found that

for a large coupling the pacemaker entrains the whole system (all units with the same

effective frequency, that of the pacemaker), but the phase distribution is hierarchically

organized. Units at the same downward distance from the pacemaker form shells of

common phases. As the coupling strength is decreased the entrainment breaks down

at a value that depends exponentially on the depth of the network. This result also

holds for complex networks, although the analytical explanation is only valid for ER

networks.

39

The Kuramoto model

2.2.1 Current stability results: an overview

Different levels of synchronization are typically distinguished for the Kuramoto model

in 2.1. The case when all angles θi(t) converge exponentially to a common angle θ∞ ∈ R

for t → inf is referred to as exponential phase synchronization and can only occur if

all natural frequencies are identical. If the natural frequencies are non-identical, then

each pairwise distance |θi(t)− θj(t)| can converge to a constant value, but this value

is not necessarily zero. The following concept of phase cohesiveness addresses exactly

this point. A solution θ for Eq.2.1 is phase cohesive if there exists a length γ ∈ [0, π[

such that θ(t) ∈ ∆(γ) for all t→∞, i.e., at each time t there exists an arc of length γ

containing all angles θi(t). A solution θ achieves exponential frequency synchronization

if all frequencies θi converge exponentially fast to a common frequency θ∞ ∈ R. Finally,

a solution θ achieves exponential synchronization if it is phase cohesive and it achieves

exponential frequency synchronization.

If a solution θ(t) achieves exponential frequency synchronization, all phases asymptot-

ically become constant in a rotating coordinate frame with frequency θ∞ , or equivalently,

all phase distances |θi(t)− θj(t)| asymptotically become constant. Hence, the terminol-

ogy phase locking is sometimes also used in the literature to define a solution θ that

satisfies θi(t) = θinf ,∀i ∈ 1, ..., N and ∀t ≥ 0. In the networked control community,

boundness of angular distances and consensus arguments are typically combined to

establish frequency synchronization.

40

3Proximity Kuramoto oscillators

In this chapter we numerically investigate some phenomena of proximity a network

of Kuramoto oscillators. Proximity networks here considered, introduced in 1.4, are

a class of networks in which the topology is time varying and, more specifically we

consider the topology to depend on the state of the network. Some concepts introduced

in ch. 1 and ch.2 will be used, at first to introduce a mathematical model for proximity

state dependant networks, and further to analyze the result of the numerical analysis.

A key aspect of state dependant networks, on which this analysis is focused is on

the evolution of the topology of the network. After showing some interesting aspects of

proximity networks, at first we will evaluate if frequency synchronization is possible

41

Proximity Kuramoto oscillators

in proximity networks, successively we introduce the concepts of equilibrium for a

topology, and lastly after showing some examples we investigate a phnomenon of

topology bifurcation in a 5 node network.

3.1 The restricted visibility Kuramoto model

The main difference between proximity Kuramoto networks and Kuramoto oscillator

networks resides in the fact that the corrective term of Eq.2.2, reported in Eq.3.1 for

ease, may be present or zero depending on the state of the network.

θvis

Figure 3.1: The depicted oscillator has limited visibility over θ

Therefore to account for this we introduce the restricted visibility Kuramoto model

in Eq. 3.2,

θi = ωi + g

N−1∑j=1

aij sin (θj − θi) (3.1)

θi = ωi + g

N−1∑j=1

aij(t)sin(θij) (3.2)

where θi indicates the angular position of the the i-th oscillator, θij = θi−θj indicates

the relative angular position between oscillator i and j, g is the coupling gain between

the oscillators, N is the total number of oscillators of the network and ωi is the natural

frequency of each oscillator.

The terms aij(t), are the elements of the adjacency matrix A of the network and

unequivocally define the topology of the network. Such time varying elements take into

42


account for the restricted visibility conditions as follows

aij(t) =

1, if min mod (θij), mod (−θij) ≤ θvis ≤ π

2

0, otherwise

(3.3)

In Eq. 3.3 we indicate as mod the modulus operator as follows: given a ∈ R then

mod (a) := b and is the remainder of a modulo 2π, with b being the unique solution

of b = a− 2πq, q ∈ Z and b ∈ [0, 2π[ and takes into account for the periodicity of the

route of the oscillators over 2π radiants. The min operator calculates the minimum

value between the relative angular position between oscillator i and j and oscillator j

and i, in other words it defines the phase distance between oscillator i and j. Let us so

denote the phase distance as αij(t) = min mod (θij), mod (−θij). Each oscillator

has a limited field of visibility over the route: in other words if the phase distance, is

below a threshold then the corrective term is present otherwise the corrective term is

zero. Thus a connection is established between oscillator i and oscillator j, i.e. two

oscillators are coupled, if αij < θvis. Note that Eq. 3.3, due to its definition, yields

a symmetric adjacency matrix A and thus the network we consider is a undirected

network. Moreover note that if θvis > π Eq. 3.2 reduces to the classic Kuramoto model

of with heterogenous frequencies.

A graphic representation to better understand how the coupling is established

is given in Fig. 3.2 and Fig. 3.3. Fig. 3.3 shows two uncoupled oscillators while

Fig. 3.3 shows two coupled oscillators with the overlapped visibility fields and thus

interacting. If the coupling is established, the corrective term to the natural frequency

of the oscillator, is g sin(θij), with aij 6= 0, that by using a mechanical analog, may be

interpreted as non linear elastic torque with force parameter g. Fig. 3.4 shows the torque

corrective coupling term. An important remark is to be done with respect to frequency

synchronization, indeed results in [22] hold even under the assumption of a proximity

network. With this regards, independently form the topology, entrainment frequency is

43


Figure 3.2: Two un coupled oscillators, say oscillator i and j, for which aij = 0

Figure 3.3: Two coupled oscillators, say oscillator i and j, for which aij = 1

0 θvisπ2

π0

0.2

0.4

0.6

0.8

1

θij[rad]

Cou

pling

term

Coupling term

Figure 3.4: Corrective term to the natural frequency ωi for oscillator i when interacting withoscillator j for θvis = π

6

44


achieved for sufficient values of parameter g. Entrainment frequency, denoted with ω is

obtained by supposing Eq. 3.2 in steady state and summing over i yielding

N

N∑i=1

θi =N∑i=1

ωi +N∑i=1

N∑j=1

aij(t) sin (θji)

and thus the second term on the right of the equations is zero due to the symmetry of

the adjacency matrix, and the anti-symmetry of the sine function, hence the entrainment

frequency is independent from the topology.

ω =

∑Ni=1 ωiN

(3.4)

If phase locking is reached, i.e. θij = 0 then the frequency entrainment in Eq. 3.4 result

hold, and the topology is steady with respect to time. At this regards we point out

that Fig. 3.6 and Fig. 3.5, show that also for a proximity network a critical values of g

is to be reached to allow frequency synchronization [19].

1

2

3

ωi[rads

]

0 5 10 15 20 25 30 35 40 45 500

π2

π

32π

2π

T ime[s]

θ i[rad]

Figure 3.5: Synchronisation is not achieved for g = 1.0, N = 5, θvis = π10

45


−2

0

2

4

ωi[rads

]

0 0.5 1 1.5 2 2.5 3 3.50

π2

π

32π

2π

T ime[s]

θ i[rad]

Figure 3.6: Synchronisation is achieved for g = 5.0, N = 5, θvis = π10

3.1.1 Equilibria and emerging topologies

Let us assume that entrainment frequency has been reached. As described in 3.1, the

phase-locking phenomenon ensures that the network settles in a steady-state topology,

thus the topology is now independent from time. In this scenario all the oscillators are

entrained at the same frequency, settle at a certain phase distance, and the topology

does not vary. Such a topology is an equilibrium topology for the network.

Hence by imposing θi = ω, and assuming phase-lock is achieved we obtain

ω = ωi + gN∑j=1

aij sin (θji),∀i = 1, . . . , N

hence we can recast to

ω − ωig

=N∑j=1

aij sin (θji),∀i = 1, . . . , N

. However, this equality does not admit a solution for low values of g, contradicting

46

3.2 Numerical Analysis

the assumption of phase-locking. In fact, a necessary condition for the existence of a

solution is that

g > gmin =maxi |ω − ωi|

N − 1

as

|N∑j=1

aij sin(θij)| ≤ N − 1,∀j = 1, . . . , N

as multiple equilibria may exist and give rise to different emerging topologies.

Amongst the possible equilibrium topologies for system 3.2, let us point out that

highlight that for high1 values of torque parameter g the emerging topology is the all-

to-all topology. A intuitive physical interpretation can given by following the mechanics

analog of system 3.2 introduced in Sec. 3.1, i.e. elastic torque coupling with high force

parameters tend to attract the oscillators.


Based on the considerations in section 3.1.1 we perform a numerical analysis to evaluate

the dependency of equilibria topologies with respect to parameter g. Moreover once

frequency entrainment has been achieved, for low values of g the network in Eq.

3.2 reaches the all-to-all equilibrium topology. For increasing values of g due the

considerations in Par 3.1.1, we expect the equilibrium topologies to be feasible. Thus

numerically we analyze for different values of g which topologies may admit a non

all-to-all equilibrium; since multiple topologies may admit an equilibrium for a fixed

value of g we expect a dependance from the initial condition set leading to a bifurcation

in the topologies of the system in dependance of g and the initial condition set.

Fig. 3.7 shows a possible example of the values of g interested to equilibrium

topologies when plotting a measure of the topology on the vertical axis with respect to

1notice that high values of parameter g is to be contextualized with reference to the parameters ofthe network: N , ωi,. . .

47


increasing values of g on the horizontal axis.

Figure 3.7: Here we qualitatively represent the regions of interest for our numerical analysisof topology bifurcation. On the horizontal axis we consider the torque parameter g and on thevertical axis we consider a measure of the topology. The size of the regions are exemplified to beof the same. In red we denote the the region for which synchronization is not archived since g isbelow the critical value. The two blue ares show the all-to-all areas and in green the we denote

the area of different equilibrium topologies

Let us define before showing the results of the numerical simulations, what we shall

consider as a measure of topologies. Specifically we introduce an equivalence class index

for the topologies. The index we consider is the number of arcs(the sum of the degree

of every node ki) of the network. The index in Eq.3.5, defined on the undirect graph,

accounts for both the outer edges and inner edges for the i-th node.

Narcs =N∑i=1

ki (3.5)

For example consider a 5 node network, and note that:

• it is easy to calculate and it gives a measure of how many connections there are

between oscillators

• a fully connected network, with N = 5 has a maximum of 20 arcs considering the

link orientation

• some topologies may have the same number of arcs but be different

48


3.2.1 A simple example: the chain topology

A simple numerical example to show the existence of equilibria topologies is the chain

topology. A chain topology consists of N oscillator with N − 2 oscillator coupled with

both the leading and the following oscillators with exception for the first and last

oscillator that hold coupling respectively only over the follower and leader. Fig. 3.8

shows a diagram of a chain topology for a 5 node network.

Figure 3.8: Qualitative diagram of a chain topology for a 5 node network

By integrating the system in (3.2) with an ODE5 solver and by setting the initial

conditions of the network in 3.2 to a chain topology, an example of the feasibility of

the equilibrium topologies can be seen in Fig.3.9, Fig.3.10 and Fig.3.11. Specifically by

setting the parameters as following:

• N = 5

• a gradient of frequencies with respect to nodes is set as ∇1,i[ωi] = 0.1 · 2π

• θvis = π10

• the fastest oscillator is the leader of the chain and the following are ordered by

frequency

• g is build on a grid from 1 to 12 with pace 0.1

49


we see in Fig.3.9, the system steady-state topology is the all-to-all topology for g = 3.0

By increasing g another equilibrium topology is held as shown in Fig. 3.10, the reached

0

1

2

3

4

ωi[rads

]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

π2

π

32π

2π

T ime[s]

θ 1−θ i

[rad]

Figure 3.9: In the top part of the diagram the N velocities of the oscillators are diagramedshowing that frequency entrainment has been reached. The lower part of the diagram shows aplot with respect to time of the difference of the relative position of the N − 1 oscillators respectto oscillator 1. Approximately at Time=3.5[s], the oscillators are in the All-to-All topology. The

value of g for this simulation is set to g = 3.0

equilibria is the original chain. For higher values of g the network reenters to the all to

all equilibria topology as shown in Fig. 3.11. By plotting the equivalence class index

with respect to the grid of g for the chain topology simulation in Fig. 3.12, we notice

how for some values of g the topology changes. Specifically moving from the all to all

configuration with Narcs = 20 to the chain topology with Narcs = 8

3.2.2 Simulations for a 5 node network

Due to the dependance of the equilibrium topologies with respect to the initial conditions,

we carry out a full investigation of a 5 node network for a fixed set of natural velocities,

ωi = ωi,∀i = 1, . . . N . The investigation is performed by integrating the system in Eq.

3.2 for a grid of values of the coupling gain g and for all the possible topologies of the

50


1.5

2

2.5

ωi[rads

]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

Time[s]

θ 1−θ i

[rad]

Figure 3.10: In the top part of the diagram the N velocities of the oscillators are diagramedshowing that frequency entrainment has been reached. The lower part of the diagram shows aplot with respect to time of the difference of the relative position of the N − 1 oscillators respectto oscillator 1. Notice how phase locking is archived for the system with the chain equilibria

topology. The value of g for this simulation is set to g = 6.6

0

2

4

6

8

ωi[rads

]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

Time[s]

θ 1−θ i

[rad]

Figure 3.11: In the top part of the diagram the N velocities of the oscillators are diagramedshowing that frequency entrainment has been reached. The lower part of the diagram shows aplot with respect to time of the difference of the relative position of the N − 1 oscillators respectto oscillator 1. Approximately at Time=3.5[s], the oscillators are in the All-to-All topology. The

value of g for this simulation is set to g = 11.0

51


0 2 4 6 8 10 12 14 16

0

5

10

15

20

g[1t]

Narcs

Figure 3.12: Topology equilibria diagram for fixed initial condition: the chain topology. Thenetwork parameters have been set as follows: N = 5, ∇1,i[ωi] = 0.1 · 2π, θvis = π

10

system. Specifically note that with a N = 5 node network, due to the symmetry of the

adjacency matrix 2N2−N

2 = 210 topologies could be possible.

Let us point out at this regard a consideration about the possible set of feasible

topologies for a given number of nodes. Indeed by setting N > 3, due to Eq. 3.3

topologies where more than 2 nodes have exclusive visibility over a third are not possible.

In other words, some topologies that may be classified by an adjacency matrix are not

possible for the system in Eq. 3.3.

For example:

1 0 0 0 1

0 1 0 0 1

0 0 1 0 1

0 0 0 1 1

1 1 1 1 1

(3.6)

Note that:

• node 5 has visibility towards all the nodes

• nodes for i = 1, 2, 3, 4 have exclusively visibility on node 5

• a similar situation on a clusters of 3 nodes is of no relevance

52


In order to account for every topology of the network a set of 20 initial conditions is

generated via Monte Carlo techniques described in the following paraph.

Monte Carlo techniques

The steps for the Montecarlo generation of the initial conditions for the 5 node network

are the following:

• all 210 permutations of the adjacency matrix are generated

• a random vector RN is generated with values within [10−7, 2π]

• the adjacency matrix is calculated

• a search is performed using the calculated matrix as the research key

• every found condition associated with such matrix is saved in a structure

20 conditions where to be reached for each topology and due to the considerations in

Par.3.2.2, 637 topologies where found to be possible and 387 where not. Which result

in 20 · 637 = 12740 conditions to be integrated for system 3.2.

The distributions of occurrences of the initial condition set with respect to the

algebraic connectivity and average degree < ki > are respectively reported in Fig.3.13

and Fig.3.14 and will be useful to further understand the results of the simulation over

the grid of the coupling term.

Results and considerations

The network in 3.2 has been integrated with an ODE45 solver for 120 points of the

coupling parameters g with values ranging from g = 0.1 to g = 12. Each of the

12740 generated initial conditions has been set for the system 3.2, and the network

has integrated for all the values of the gain grid g, accounting for a total of 1528800

53


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

100

200

300

400

λ2

λ2

occ

urr

ence

s

Figure 3.13: λ2 occurrences function for the initial conditions of the N = 5 network

0 0.5 1 1.5 2 2.5 3 3.5 40

50

100

150

200

< k >

<k>

occ

urr

ence

s

Figure 3.14: < ki > occurrences function for the initial conditions of the N = 5 network

54


simulations. The single simulation time has been set to 1000[s], and a routine has

been developed to stop the simulation after that the system has reached frequency

synchronization. Such routine, after calculating the entrainment frequency, compares

the speed of each oscillator with respect to such frequency. If the difference of all speeds

is below a given threshold then the simulation is interrupted, saving the needed data.

The data from the simulations shows a topology bifurcation diagram for the system

with the following parameters:

• N = 5

• a gradient of frequencies with respect to nodes is set as ∇1,i[ωi] = 0.1 · 2π

• θvis = π10

• the fastest oscillator is the leader of the chain and the following are ordered by

frequency

• g is build on a grid from 1 to 12 with pace 0.1

The diagram is given in Fig. 3.15 and shows that for a set of the g, the network has a

topology bifurcation for which equilibrium topologies, other that the all to all topology,

are feasible depending on the initial condition. Approximately 5% of the generated

initial condition topologies have shown the presence of feasible equilibrium topologies.

Specifically the topologies bifurcate from the all-to-all topology for an interval of g,

indicated by gmin ' 1.7, gmax ' 10 and in the region delimited by the black line, by

varying the initial conditions numerous equilibrium are feasible. For g < gmin and

g > 1.5 the all-to-all topology is achieved for all initial conditions.

To further investigate a relationship between the equilibrium topologies and the

initial conditions in the system we report in Fig. 3.16a bar diagram relating: the

algebraic connectivity of the initial condition λ2, the gain grid, and the number of not

all-to-all equilibrium topologies reached for the system 3.2, at the end of simulation.

55


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

0

2

4

6

8

10

12

14

16

18

20

g[1t]

Narcs

Figure 3.15: N = 5 network topology bifurcation diagram

56


The diagram then is rotated and reported in Fig. Such diagram shows that for initial

Figure 3.16: 3D bar diagram showing the number of occurrences of equilibrium topologies withrespect to the gain grid and the algebraic connectivity of the initial condition

Figure 3.17: Second view of the 3D bar diagram, showing the number of occurrences ofequilibrium topologies with respect to the gain grid and the algebraic connectivity of the initial

condition

conditions with λ2 = 0 no equilibria topologies are feasible, for increasing values of the

algebraic connectivity2 of the initial condition set not all-to-all equilibrium topologies

2note that λ2 is a measure of well a graph is connected

57


appear. Specifically for low values of λ2, low values of the gain g are needed, and for

increasing values of λ2 high values are need to hold the equilibrium. Such dependance

is in accordance with the fact that to hold in equilibrium topologies with a low number

of connections(or nodes), i.e. low values of λ2, low values of g are needed. Analogously

for initial topologies that are al ready well connected, i.e. high values of λ2, higher

values of g are need to maintain equilibria.

In Fig. 3.18 and Fig. 3.19, a bar diagram relating: the average degree < ki >

of the initial condition, the gain grid, and the number of not all-to-all topologies, i.e.

equilibrium topology, reached for the system 3.2, at the end of simulation.

Figure 3.18: 3D bar diagram showing the number of occurrences of equilibrium topologies withrespect to the gain grid and the average degree of the initial condition

Anova Analysis

In order to highlight the dependance between the initial condition algebraic connectivity

and the results obtained we perform a 1-way ANOVA analysis [23]. Such analysis will

establish if the the main source of variation for the presence of equilibria topologies

obtained is to be attributed to the variation of initial conditions or is of other nature.

In other words we statistically show that there is a dependance between the initial

58


Figure 3.19: Second view of the 3D bar diagram, showing the number of occurrences ofequilibrium topologies with respect to the gain grid and the the average degree of the initial

condition

conditions topology and the possible equilibria topology.

We consider for a first ANOVA analysis as a factor, the algebraic connectivity λ2

and its values as levels of such factor. In Tab.3.1 we report the ANOVA table for the

distribution of not all-to-all equilibria topologies with respect to the initial condition

λ2. Such table shows, under the column MS that the main source of variation is to be

Source SS df MS F Prob>F

Between 57375.8 8 7171.97 32.1 1.76113e-43Within 166883 747 223.4 - -

Total 224258.8 755 - - -

Table 3.1: Anova table for the distribution of not-all-to-all- equilibrium topologies with respectto λ2

attributed to between(levels) variance. In other words for all the simulated values of g

the main contribution to the variations of the presence of equilibrium topologies is to

be attributed to the initial condition set, and thus to the algebraic connectivity of the

initial condition topology.

A second analysis we carry out is by considering as a factor the average degree

59


< ki > of the initial conditions and as levels its values. The resulting ANOVA table is

reported in Tab. 3.2 That similarly as Tab.3.1 shows a primary source of variation of

Source SS df MS F Prob>F

Between 42473.3 5 8494.67 22.36 3.01614e-20Within 189198.9 498 379.92 - -

Total 231672.2 503 - - -

Table 3.2: Anova table for the distribution of not-all-to-all- equilibrium topologies with respectto < ki >

the presence of equilibrium topologies between the levels of the factor.

60

Conclusions and future works

In this thesis, a numerical analysis of proximity Kuramoto oscillators networks was

carried out. Recalling some basic concepts of complex network theory, we have intro-

duced a proximity rule in a network of heterogenous Kuramoto oscillators, developing a

restricted visibility model. We have illustrated how multiple equilibria and equilibrium

topologies may exist.

Trough an intensive set of numerical simulations, we have shown the emergence of

an interesting phenomenon that we called topological bifurcation: when the coupling

strength belongs to an interval, the topology may converge to a different steady-state

configuration depending on the initial conditions.

Future works will be devoted to investigate the possible emergence of these phe-

nomenon for different individual dynamics and coupling rules. Also we envision that

control strategies may be developed to control both the individual dynamics and the

emerging topologies.

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Numerical a nalysis of proximity oscillator networks · Pepito Rossi. Acknowledgements First and foremost I have to thank my beloved family, for their invaluable lessons and for the