Utopias of Perfection and their Dystopias

Proceeding of the International Scientifical Conference.

Volume III.

189

UTOPIAS OF PERFECTION AND THEIR DYSTOPIAS

Livio C. Piccinini

Maria Antonietta Lepellere

Ting Fa Margherita Chang

University of Udine

Abstract. The search for optimality is a must in the study of biological evolution and many

models have been proposed in order to grasp its main operating gears. These models can also

give a deep comprehension of many social and economical phenomena and are studied in the

so called econophysics. Bak Sneppen is one of the most significant models because it balances

at best explication power and simplicity. Unlike cellular automata models, where phenomena

happen in a local framework without global supervision, Bak Sneppen models join locality

and globality, as many contemporary economical models require. The authors try to re-read

these models between utopic and dystopic frames, showing with simplified models some

behaviors that had not been considered in the previous literature. A search for uniformity

rather than optimality is also studied, since it seems to rule fashion evolution, as well as

political behavior of people (Band wagon).

Keywords: Optimality Utopias, Totality Utopias, Bak Sneppen Model, Modal Evolution,

Partitioned Frames and Evolution,

1. Introduction

The Bak-Sneppen model was originally introduced as a simple model of

evolution by Per Bak and Kim Sneppen [3] (compare also [2], [19]). Their

original model can be defined as follows. There is a circle of N nodes and they

are occupied by N different species, each of which has been assigned a random

fitness. The fitness values are independent and uniformly distributed on (0; 1).

At each discrete time step the system is updated by locating the lowest fitness

and replacing this fitness, and those of its two neighbors, by independent and

uniform (0; 1) random variables. Bak-Sneppen models can be defined on a wide

range of graphs using the same update rule as above. What the Bak Sneppen

model illustrates is that even random processes can result in self-organization to

a critical state (see [14] for a discussion). Threshold fitness rises, rapidly at first,

then exponentially slows until it reaches about 0.66, the critical state from which

level extinction sweep back and forth through the ecosystem. None of the

changes observed in the system are designed, however, to increase the critical

threshold or lead to extinction, but the dynamics of the model lead inevitably to

self organized criticality.

Brunk in [6] suggests that self-organized criticality is the sort of process that

should have great intuitive appeal to social scientists. Like earthquakes, the

precise prediction of riots; wars; and the collapse of economics, governments,

and societies is not possible in human systems in which catastrophic events can

be triggered by minuscule causes. A large bibliography of applications of self

organized criticality can be found in Turcotte [22] (see also [1] for an economic

https://www.researchgate.net/publication/13235534_Bak_P_Snepen_K_Punctuated_equilibrium_and_criticality_in_a_simple_model_of_evolution_Phys_Rev_Lett_59_381-384?el=1_x_8&enrichId=rgreq-179e26ac-41f2-4340-a719-b3327a489bc3&enrichSource=Y292ZXJQYWdlOzI2ODIyMTE2NDtBUzoxNjM0MTM4NTQ1MzE1ODVAMTQxNTk3MjMwNzYyMw==


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application). The authors anyhow think that a sound basis for applying Bak-

Sneppen model of contact with neighbors is given by Duesenberry’s

demonstration effect. Its first presentation can be found in Duesenberry [9],

while many application in consumer’s economy and sociology can be found for

example in Cavalli [7].

Although the Bak-Sneppen model is extremely simple, it has not yet been

completely solved in spite of numerous analytical and numerical investigations.

Motivated by the difficulty of analyzing rigorously even the one-dimensional

version of the Bak-Sneppen model, J. Barbay and C. Kenyon [4] proposed a still

simpler model with discrete fitness values1. In their model each species has

fitness 0 or 1, and each new fitness is drawn from the Bernoulli distribution with

parameter p. Since there are typically several least fit species, the process then

repeatedly chooses a species at random for mutation among the least fit species.

They prove bounds on the average numbers of ones in the stationary distribution

and present experimental results. Parameter p can substitute up to a certain

level a plurality of values, but it cannot explain the staircase phenomenon found

by the authors in [18] section 5. Hence binary structure, though simple and

appealing is not sufficient for a thorough description of what may happen.

An important feature of discrete-value model is that it is possible to analyze the

mode, and that an ordered fitness function is no longer required. Discrete fitness

can be replaced by names of colors, a probability can be attached to each color

(maybe the same to all of them). As usual the modal color is the color that has

most items; we call antimodal the color that has at least one item, and whose

total number of items is the least. In case of ties one of the tied colors is chosen

at random. The process is as follows: at each discrete time step one of the

antimodal items and its two neighbors are replaced at random by a color chosen

according to the law of probability. We call it Modal Bak-Sneppen Process.

Some particular cases will be discussed in sections 2 and 3.

2. Globality and Uniformity Utopias

Utopias could work at their best when no competition arises. A global model of

perfection should foresee that all its members reach the top level. Similarly in a

modal process unanimity should be attained. At first it seems that utopia cannot

coexist with randomness. Strangely enough that is not true; simply, when

randomness is allowed, more time is required. A fundamental feature is

globality, so that the choice of the evolving node will be no longer sequential or

random, but will be connected to some global evaluation of all the nodes. This

happens for example in DNA analysis, where longest matching is chosen for

improvement.

1 Also this case is by no means trivial, as it was shown by Meester and Znamenski ([16]). The

reduction of number of species was recently considered by Schlemm in [20].

https://www.researchgate.net/publication/2240101_On_the_discrete_Bak-Sneppen_model_of_self-organized_criticality?el=1_x_8&enrichId=rgreq-179e26ac-41f2-4340-a719-b3327a489bc3&enrichSource=Y292ZXJQYWdlOzI2ODIyMTE2NDtBUzoxNjM0MTM4NTQ1MzE1ODVAMTQxNTk3MjMwNzYyMw==


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The simplest case is the random optimization: in a totally random process a

random node changes and assumes a random value, so that after a transitory

phase in which some memory of the original values still survives, it reaches a

purely random distribution of values, that coincides in probability with a casual

extraction with replacement. The process becomes much more interesting if we

add some rule for the choice of the evolution node. In ordered sets a good idea is

to choose the node (or one of the nodes) that attains the minimum value, and

replace its value by a random one. This process leads to optimality, since all

nodes that reach a value greater than the present minimum are left unchanged.

Those that already attain the maximum will undergo no change until all nodes

reach the maximum. At this utopian stage one of the optima, at random, will be

chosen for changing, and then it (and only it) will keep changing until it

resumes again the optimal value. It must be remarked that in order to reach

optimality it is not enough to state a fixed objective and to require that what lies

below changes. The final result would be only a distribution lying between the

objective and the potential maximum, hence the objective should be made

dynamical.

An interesting process can arise also without ordered structures of values. In

this case we deal with a finite (and small) set of values (colors). It is enough to

choose an antimodal node and replace it at random. We call this process simple

modal uniformization. The structure will converge anyhow to a mode involving

all nodes but one; in particular if at the initial time more than half of the nodes

have the same color this will also be the final modal color. For more detail the

authors have proved the following property:

“Suppose there are K colors, ordered in a sequence increasing according to their

cardinalities ci. Consider the inequalities

E1 c1+ c2 + c3 +...+ cK-1 cK

E2 c1+ c2 + c3 +...+ cK-2 cK-1

…

EK-3 c1+ c2 + c3 c4

EK-2 c1+ c2 c3

EK-1 c1 = c2

Suppose that En is the first inequality that does not hold. Then only the colors

labelled by K, K-1,…, K-n+1 may become final modal color. If all the

inequalities hold, then each color may become final”.

If the first inequality is false, color K is the only final color (as stated before)

since even if all non modal colors are changed into one and the same color (what

has a non zero probability) this color would still remain antimodal. Repeated use

of this argument achieves the proof. The computation of the probability of each


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outcome uses Bellman’s technique of dynamic programming,2 but is somewhat

technical and cannot be reported here.

A somewhat opposite process arises when one of the modal nodes is chosen for

replacement; in this case the evolution tends to a division into groups that have

all the same cardinality (plus/minus 1). Both schemes arise in political and

social analysis: Band Wagon in one case, Biocultural Diversity in the other case.

A balanced division of items is also useful in the creation of optimal codes in

information theory, according to Shannon entropy principle3. This is a technical

utopia that scholars devoted to data mining must efficiently pursue, as Mc

Cormick recalls in [13] when he explains the reasons that made Google so

successful.

3. Antagonistic Forces

Random collateral actions disturb the utopian search of optimality or totality.

What is relevant is that they do not succeed in a complete destruction of the

optimizing process, but rather lead to a self organized system of critical states.

Many different ways of collateral disturbance have been proposed; also cellular

automata or heat equation may enter in this scheme. Bak-Sneppen model was

proposed as a conceptual kernel of many description efforts; it supposes that the

two adjoining nodes (on the circle) are affected independently from their values,

and undergo a random change. Many other schemes have been proposed: the

graph setting can be changed, inasmuch one or more nodes may be involved, or

connected nodes may depend probabilistically according to an influence graph.

A limit, non local case, is given by a random choice of one or more nodes.

Incidentally we remark that this randomized case is much easier to be

analytically dealt with.

For any finite number of nodes and for a discrete number of fitness levels (or

colors, in the modal case) it is possible to describe explicitly the system using

Markov chains4. For each given set of probabilities the transition matrix can be

constructed by a computer program developed by the authors, but if the

dimension becomes too large too many states of the system must be looked for

and the search for average frequencies becomes very slow and numerically

unstable.

The examples are given for the simple case consisting of 5 nodes, a single

collateral disturbance and two possible fitness values, namely 0 and 1, each of

them with probability ½. A reduced state representation for the random neighbor

case is simply the number of 1’s present in the system. The transition matrix in

case of optimization is given in table 1.

2 We refer to [5]

3 A classical work is ([21]), but also [14] can be referred to.

4 Compare for example [23]


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Table 1.

Transition matrix for random optimization with one collateral disturbance

0 1 2 3 4 5

0 1/4 1/16 0 0 0 0

1 2/4 5/16 1/8 0 0 0

2 1/4 7/16 3/8 3/16 0 0

3 0 3/16 3/8 7/16 1/4 1/4

4 0 0 1/8 5/16 2/4 2/4

5 0 0 0 1/16 1/4 1/4

A suitable rearrangement of the columns supplies the corresponding matrixes for

the uniformity processes. The average frequencies of the 6 states are

summarized in table 2, where F indicates the column where fitness is present,

BW and BD indicate the columns of modal processes, BW for the Band Wagon

and BD for the Biocultural Diversity. Column R represents the frequencies of

the states in a purely random process (binomial coefficients).

Table 2.

Averages frequencies of states of random optimization (one collateral disturbance)

State F BW BD R

0 0.002 0.0875 0.011 0.031

1 0.022 0.2125 0.137 0.156

2 0.113 0.2 0.352 0.313

3 0.323 0.2 0.352 0.313

4 0.385 0.2125 0.137 0.156

5 0.155 0.0875 0.011 0.031

Of course the last three columns are symmetrical, since they are modal, while

the first column is concentrated on the high value states since it corresponds to a

phenomenon oriented towards optimization. In the first column the average

frequency of 1-fitness can be computed, and is equal to 0.707 (to be compared

with the pure random frequency of 0.5). As for modal processes, Band Wagon

concentrates the frequencies upon the presence of a majority (either 1 or 4),

while on the opposite Biocultural Diversity concentrates in the middle of the

scale (a comparison may be made with column R)

In linear phenomena random disturbance is the most effective agent in

preventing optimization. Neighborhood contagion, as is well known in

epidemiology, slackens the diffusion rate, hence sharper results are to be

expected when Bak Sneppen processes are involved. The same case as before is

now considered, choosing as neighbor one node (say the left node). Even the

simplest representations of states must take into account the geometry of the

structure, but they can be invariant for rotation. Hence in the case of 5 nodes

and 2 fitness levels, the states with 2 and 3 nodes at level 1 must be split into

the case where these nodes are adjoining (states 2A and 3A) and the case where

they are split (states 2S and 3S). The transition matrix for the Markov chain is

given in table 3.


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Table 3.

Transition matrix for unilateral Bak-Sneppen process with 5 nodes

0 1 2A 2S 3A 3S 4 5

0 2/8 1/16 0 0 0 0 0 0

1 4/8 5/16 1/12 2/12 0 0 0 0

2A 1/8 3/16 3/12 1/12 1/8 1/8 0 0

2S 1/8 4/16 1/12 4/12 0 1/8 0 0

3A 0 2/16 3/12 1/12 2/8 1/8 1/4 1/4

3S 0 1/16 2/12 3/12 1/8 3/8 0 0

4 0 0 2/12 1/12 3/8 2/8 2/4 2/4

5 0 0 0 0 1/8 0 1/4 1/4

The table of results (table 4) is built in the same way as the previous one, but in

order to simplify comparison the frequencies corresponding to the states 2A - 2S

and 3A – 3S have been summed together

Table 4.

Results of Bak Sneppen unilateral process and related problems. Case of 5 nodes.

State F BW BD R

0 0.001 0.094 0.011 0.031

1 0.014 0.219 0.136 0.156

2 = 2A+2S 0.085 0.187 0.353 0.313

3 = 3A+3S 0.308 0.187 0.353 0.313

4 0.414 0.219 0.136 0.156

5 0.177 0.094 0.011 0.031

The frequency of 1-fitness now has increased to 0.721. Also the majority effect

in Band Wagon modal optimization has been increased, while in the case of

Biocultural Diversity the neighborhood effect is less noticeable.

4. Partitions Defend Utopian Optimization

In this section the case of actual discrete two-sided Bak-Sneppen process will be

investigated. For a richer choice of cases a non prime number of nodes is

required; the choice is 6, so that it is easy to enumerate all the possible structures

(apart from rotations they are 14) and to calculate the corresponding transition

matrix. The generalization consists in changing the two connections of Bak-

Sneppen model in all possible (different) combinations, so that, in view of

symmetries, the number of different cases sums up to 6. In table 5 the two links

(according to positive rotation) are listed. Bak-Sneppen classic case lies in the

first row. In the third column, as a synthesis of the process, the average

percentage of 1’s is given.


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Table 5.

Binary Bak-Sneppen on a frame of 6 nodes

1st

link 2nd

link Average

1 5 (=-1) 0.637389

1 2 0.620732

1 3 0.620732

1 4 0.607729

2 3 0.620732

2 4 0.75

random random 0.61229

In view of the increased collateral disturbance both the average of Bak-Sneppen

process (1st row) and of the random process (last row) are much lower than what

arose from tables 4 and 2, that amounted respectively to 0.721 and 0.707.

Usually the average of structured processes is higher than in random processes,

because long chains of maxima are structurally protected, but an exception is

shown for the case 1- 4 in which the two operating nodes are just opposite,

hence tend to break long chains of maximums wherever they can be located.

The table highlights an interesting phenomenon, that may arise only in the case

of a non prime number of nodes (hence the reason for choosing 6 nodes). The

average of the last case (2-4) is much greater than all the remaining averages. In

the transition matrix of the associated Markov chain there exists a persistent

group5 that is smaller than the whole set of 14 structures. Namely it contains

only the 4 structures 01 01 01, 01 01 11, 01 11 11, 11 11 11, while the remaining

10 form a transient group: this means that there exist sets of transformations that

allow to pass from any of them to any other, but there is the possibility of falling

outside into the persistent group without possibility of returning back. Nodes 2

and 4, together with the minimum conventionally placed at 0, form a subset that

has no interference with nodes 1-3-5, that on the contrary are activated when

minimum is attained in one of those nodes. Whenever an operation is performed,

one 3-element subset is left unchanged, while the other 3-element subset is

totally changed at random. The two subsets do not change between different

operations, what on the contrary happens in all the remaining cases. The process

cannot anyhow be divided into two independent subprocesses, since the

minimum must be looked for among both the subsets (for example a subset

containing 011 enters in this search, while a subset 111 enters in the search only

if also the other one is a 111 subset, in which case there are 6 minima of value

1). A further interaction is given by the number of minima; for example in the

case 000 011 the first subset has probability ¾ of being chosen, while the other

one only ¼.

When one set reaches the configuration 111 (probability 1/8) it becomes stable,

in the sense that it can be changed only if the global minimum is 1, that is the

configuration is 111 111. In this case anyhow at least one of the two subsets will

5 We follow Hsu’s nomenclature [12], compare also Chang-Piccinini [8]


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save the configuration 111, that can no longer be destroyed. The four structures

above in fact can be read, keeping the two subsets divided, as 000 111, 001

111, 011 111, 111 111. All computations become very simple since there is no

longer a Bak Sneppen structure. In particular the average delay time for reaching

the persistent group starting from any configuration of the transient group is 8,

and does not depend on the initial structure. The average time for reaching the

top configuration 111 111 starting from 000 000 (or any other transient) is 16,

while in the standard Bak Sneppen process is 23.07572. The increase in time is

due to the impossibility of protecting the structure 111 from decay.

5. Staircases and Sudden Balance Break

A richer example of Bak-Sneppen process will be shown. The simplest case that

shows the main features requires 3 subsets of 3 nodes each, and a ternary set of

values should be considered, say 0,1,2. This corresponds to 9 nodes and

displacements of 3-6. A simplified analysis is the following: label 0 is attached

to any subset in which the minimum is 0 (not regarding their number), label 1 to

any subset in which the minimum is 1 and label 2 to the set 222. Track of the

single subsets will not be kept, but they will be enumerated, getting the states

000, 100, 110, 111, 200, 210, 211, 220, 221, 222. Table 6 shows the transition

matrix.

Table 6.

Transitions in a 9 node tripartite Bak Sneppen set with 3 values.

A \DA 000 100 110 111 200 210 211 220 221 222

000 17/27

100 9/27 17/27

110 9/27 17/27 17/27

111 9/27 9/27

200 1/27 17/27

210 1/27 9/27 17/27 17/27

211 1/27 1/27 9/27 9/27

220 1/27 17/27 17/27 17/27

221 1/27 1/27 9/27 9/27 9/27

222 1/27 1/27 1/27

There are two transient groups (110, 111) and (210, 211) and one persistent

group (220, 221, 222). The final average is thus 1.667, since two subsets have

the form 222, and the third one is random on the three values 0, 1, 2. The

average times are 54 from 000, 100, 110,111 to the persistent group, 27 from the

groups 200, 210, 211 to the persistent group. The scheme of transitions is

represented in figure 1.


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Fig. 1. Transition graph in a 9 node tripartite Bak Sneppen set with 3 values.

The transitions from 100 to 210 and from 110 to 211 show an interesting fact:

the subset that is changed into the optimal label 2 is not one labelled with 1, but

one that is labelled with 0, that is, its minimum must be lower than that of the

best subset. In Bak-Sneppen models, as sometimes also in real life, the principle

is “quieta non movere” (Let quiet things stay). For example in a situation

A=122, B=112, C=011, it is impossible that the top ones (A and B) reach the

state 222, while it is possible, even if unlikely, that this state is reached by C.

The idea is that movement requires dissatisfaction, while further movements are

caused by some form of nested neighborhood with unsatisfied people that can

in turn generate new dissatisfaction: We can remind the “happiness paradox” of

Easterlin ([10], [11]).

Examples more connected with economics and theory of decisions are given by

the evolution of road nets and urban systems. Here the influence of a single

intervention is somehow localized (hence “partitioned nets”), but usually the

decision is taken at a global level, choosing the worst section of a road system or

the most impoverished part of a city. An intervention on the set of roads perturbs

the neighbors in an unpredictable way, allowing a change in the overall structure

of the connexion system. The new road usually satisfies up-to-date standards

that are not satisfied by the unchanged sections. In cases of good planning also

the neighbor roads are improved, so that the overtaking phenomenon can arise.6

In urban systems an overtaking case is encountered when a large and central or

semicentral area is sold out at a very low price because of its decay, and is used

for a top level reconstruction. A classical case can be found in London’s East

End, especially around the Tower Bridge, but also Liverpool and Valencia

present interesting evolutions.

In numerical simulations as soon as the dimension of the subsets is increased it

becomes more and more difficult to reach the persistent group; for example

6 Compare also the paper by two of the authors [17]

000 100 110 111

210 211200

220 221

222

Transient

group

Transient

group

Persistent

group

https://www.researchgate.net/publication/4890476_Income_and_Happiness_A_Unified_Theory?el=1_x_8&enrichId=rgreq-179e26ac-41f2-4340-a719-b3327a489bc3&enrichSource=Y292ZXJQYWdlOzI2ODIyMTE2NDtBUzoxNjM0MTM4NTQ1MzE1ODVAMTQxNTk3MjMwNzYyMw==


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already subsets of 18 nodes (for a total of 36) very often require more than

100,000 iterations in order to reach the persistent group. The transition, when it

happens, is very similar to an “avalanche”, a typical structure of Bak Sneppen

processes and in general of punctuated equilibrium. In some cases the process is

not complete, hence it can be reverted, but finally it happens that the threshold is

reached. Increasing the number of values multiple thresholds arise; the lower

levels can be easily overcome, while the top levels can prove to be almost

unreachable. This is for example the case of ten values and forty nodes, in which

the top level has never been reached in ten simulations of 1,000,000 iterations.

Conclusions

While aiming at utopia one of the main indicators is the expected time to reach

it, should success be possible. Similarly the expected time of destruction by

dystopias is meaningful. Using the transition matrix one can compute the

average waiting time. In section 5 it has been shown that in partitioned frames a

final state can be reached which is not reversible, that is isolation pays, as all

authoritarian governments know, even if evolution can be hindered. On the

contrary when isolation is not achieved and random disturbances arise, all states

are again reachable, but the most frequent is an equilibrium distribution higher

than the average. On the contrary in modal uniformity search sudden changes of

mode can occur, as it happens with political majorities. Taking again into

account the 5-nodes model with one collateral disturbance it is possible to

compare the effects of protection against changes (table 7). The benchmark is

the purely random replacement of two nodes (R); the comparisons are with

antimodal choice plus random disturbance (RD) and antimodal choice plus next

neighbor disturbance (unilateral Bak-Sneppen, ND). The first columns list the

starting state, while table 7 shows the expected time to reach a new consistent

majority (namely 4 out of 5).

Table 7.

Expected time for the majority inversion in 5-nodes model

R RD ND

0 12.4 26.7 35.6

1 11.6 26.7 35.6

2A 10.1 22.7 28.3

2S 10.1 22.7 31.6

The greater protection given by Bak Sneppen model confirms that in linear

phenomena where no critical threshold arises a localized dissatisfaction is less

effective. As soon as the number of nodes increases the expected waiting times

grow faster and faster. For example for 13 nodes to pass from 3 to 10 in case RD

it amounts already to 944.0 (authors’ computation).


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Summary

The paper considers some simple models of evolution tending to utopian development

towards perfection or uniformity, and shows how dystopic antagonistic forces prevent or

hinder the achievement of the goal. It is also shown that in linear phenomena, where no

threshold is available, locality of evolution hinders the destruction of achieved objectives, but

can also prevent further improvements (turris eburnea, ivory tower effect). The interesting

phenomenon of overtaking is considered in the last section, where a theoretical three level

model and real world examples are shown that satisfy the Bible sentence “the last will be the

first”. The examples show that this fact not necessarily happens, but show also that its

probability is not negligible. In figure 1, containing the theoretical scheme, the Bible is

satisfied along the diagonals, while vertical lines correspond to everyday life.

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Livio C.Piccinini Department of Civil

Engineering and Architecture,

University of Udine

E-mail: [email protected]

Maria Antonietta

Lepellere

Department of Civil


University of Udine


Ting Fa Margherita

Chang

Department of Civil


University of Udine


Documents

Utopias of Perfection and their Dystopias