Effective Games in Multilayer Complex Networks · Effective Games in Multilayer Complex Networks Miguel Dziergwa de Carvalho Thesis to obtain the Master of Science Degree in Information

Effective Games in Multilayer Complex Networks

Miguel Dziergwa de Carvalho

Thesis to obtain the Master of Science Degree in

Information Systems and Computer Engineering

Supervisors: Prof. Francisco Joao Duarte Cordeiro Correia dos SantosProf. Flavio Luıs Portas Pinheiro

Examination Committee

Chairperson: Prof. Antonio Manuel Ferreira Rito da SilvaSupervisor: Prof. Francisco Joao Duarte Cordeiro Correia dos Santos

Members of the Committee: Prof. Carlos Antonio Roque Martinho

October 2018

Acknowledgments

This work represents a hallmark in my life. With it came times of desperation and insecurity, but also

moments of fulfillment and enlightenment. To my family, friends and teammates who supported me in

the former, to professor Francisco Santos whose blissful spirit inspired mine and helped me reach the

latter, and to professor Flavio Pinheiro and Fernando Santos for going out of their way and providing me

numerous explanations. I would like to express my gratitude to you all with the hope that the path I will

take, and all its endeavors, will be accompanied by such wonderful people.

Abstract

Understanding the mechanisms behind the origin and maintenance of cooperation has been the focus

of much research during the last decades. In that context, the underlying structure of social interactions

has been shown to greatly influence the chances of reaching high levels of cooperation. However, real

networked systems are often shaped by multiple interdependencies that are not conveniently captured

by a single network. For instance, in social networks, individuals can be connected through different

types of relationships — originating from collaboration, professional, friendship, or family ties — which

can be conveniently described by a multilayer network. In this thesis, we analyze how a population struc-

tured as a multilayer network of interactions alters the chances of reaching cooperation. In particular,

we aim to understand under which conditions networked interactions effectively transform, globally, the

social dilemmas of cooperation that individuals locally face. To that end, we implement a novel numerical

tool that allows us to track the self-organization of cooperation in networked populations with an arbitrary

number of layers. Our results show that interactions in multiple layers can transform the original dilem-

mas, creating new basins of attraction and stable equilibria, absent in a single type of network. Finally,

we show that these game transformations are not trivial. Cooperation may either increase or decrease,

depending on factors such as the number of layers, the strength of the dilemma, the topology of the

network, or the level of degree overlap among layers.

Keywords

Cooperation; Multilayer; Network; Transformations; Game Theory.

iii

Resumo

As ultimas decadas testemunharam a descoberta de varios mecanismos responsaveis pela evolucao e

emergencia da cooperacao na natureza e nas sociedades. Nesse contexto, foi demonstrado teoricamente

e empiricamente que o facto de os indivıduos interagirem atraves de complexas redes de interaccao

aumenta a probabilidade de alcancar comunidades mais cooperantes e altruıstas. No entanto, estes sis-

temas sao geralmente caracterizados por multiplas interdependencias que nao sao correctamente cap-

turadas por uma unica rede. Por exemplo, nas redes sociais, os indivıduos podem ser ligados por meio

de diferentes tipos de relacionamentos - de colaboracao, amizade ou lacos familiares - que podem ser

convenientemente descritos por uma rede multi-camada. Nesta tese, analisamos como uma populacao

estruturada por uma rede de interacoes multi-camada altera a propensao para a cooperacao. Em par-

ticular, estudamos em que condicoes as interacoes em rede transformam os dilemas de cooperacao em

que os indivıduos participam. Para fazer isso, implementamos uma ferramenta numerica - chamada de

Gradiente Medio de Seleccao - que nos permite compreender a dinamica colectiva e a auto-organizacao

da cooperacao em populacoes em rede com um numero arbitrario de camadas. Os resultados sugeren

que interacoes em multiplos domınios podem transformar os dilemas originais de uma forma pouco triv-

ial, criando novas bacias de atraccao e equilıbrios estaveis ausentes quando se considera apenas uma

unica rede (ou domınio de interaccao), promovendo (ou nao) a cooperacao, dependendo do numero

de camadas, da forca do dilema, da topologia da rede, ou o nıvel de sobreposicao de graus entre

camadas.

Palavras Chave

Cooperacao; Multi-camada; Rede; Transformacoes; Teoria de Jogos.

v

Contents

1 Introduction 1

1.1 What are game transformations? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 A multilayer approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Work goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Background theory 7

2.1 Evolutionary game theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Evolutionary stable strategies and effective games . . . . . . . . . . . . . . . . . . 9

2.1.2 Replicator equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.3 2-person games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.4 N-person games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Multilayer classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Graph topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 The Average Gradient of Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Related work 19

3.1 Mechanisms for the emergence of cooperation . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 How structure affects cooperation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.1 Heterogeneous vs Homogeneous . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.2 Degree correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Connecting local and global behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Model 27

4.1 Creating and validating our framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Understand the FFC as a function of the game incentives and the number of layers . . . . 30

4.3 Extending the AGoS to multilayer networks . . . . . . . . . . . . . . . . . . . . . . . . . . 31

vii

5 Evaluation 33

5.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.1.1 Regarding framework validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.1.1.A FFC for different types of networks and for different game incentives . . . 35

5.1.1.B AGoS for different networks and for different game incentives . . . . . . . 36

5.1.1.C FFC in degree varying multilayers . . . . . . . . . . . . . . . . . . . . . . 38

5.1.2 Regarding our own objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.1.2.A Final fraction of cooperators (FFC) . . . . . . . . . . . . . . . . . . . . . 40

5.1.2.B Multilayer AGoS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.2.1 Regarding framework validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.2.2 How multilayer networks impact cooperation . . . . . . . . . . . . . . . . . . . . . 42

5.2.3 Understanding the changes by analyzing the AGoS . . . . . . . . . . . . . . . . . 49

6 Conclusion 55

6.1 Summary of contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.3 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

A Example runs 63

B The expected results, according to previous papers 65

viii

List of Figures

1.1 Comparing obtained and expected gradients of selection for a PD, in different network

topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Evolution process in finite well-mixed populations. In networked populations, Y is ran-

domly selected from X ’s first neighbors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Payoff matrix of 2-person and two strategy games. R,S,T and P are the payoffs; C and D

are the possible strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 The gradient of selection in Snowdrift games in well-mixed populations. . . . . . . . . . . 12

2.4 The gradient of selection in Stag-Hunt games in well-mixed populations. . . . . . . . . . . 12

2.5 The gradient of selection in the Prisoner’s dilemma in well-mixed populations. . . . . . . . 13

2.6 The orange node would participate in the PGG with center on himself and with center on

its colored neighbors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.7 3 layers that share the same nodes but are connected through different links, from [1]. . . 14

2.8 A complete graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.9 A HR graph with average degree k=2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.10 A degree distribution for a Single-scale graph. . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.11 An exponential decaying tail (b) meaning that there is a single scale for the connectivity k,

from [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.12 Cumulative distribution following a power law, meaning we are in the presence of a scale

free network,from [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.13 The AGoS applied to different networks, from [4]. . . . . . . . . . . . . . . . . . . . . . . 18

3.1 Graphic representation of the 5 mechanisms described, from [5]. . . . . . . . . . . . . . . 22

3.2 In a) there are strong degree correlations and in b) there are not. . . . . . . . . . . . . . . 23

3.3 Distribution showing the frequency spent in each state, for two different values of B (ben-

efit), from [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4 Three instances of the AGoS captured at different evolutionary times, from [6]. . . . . . . 24

ix

5.1 After choosing a model for the population and a pair (S,T), this is how the simulation

performs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.2 The gradients calculated during the 1000 simulations that started at fi = 0.45 in different

SFBA networks (β = 0.1, B = 1.15) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.3 Implementations of the degree correlations . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.4 Obtained FFC for different networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.5 Obtained AGoS over 30 generations, for different values of B. . . . . . . . . . . . . . . . . 43

5.6 Obtained FFC as a function of the number of layers (xx) and the degree correlations (yy),

for a) Harmony game and b) Prisoner’s dilemma. . . . . . . . . . . . . . . . . . . . . . . . 44

5.7 Obtained FFC for L = 16 as a function of the initial fraction of cooperators (xx) and the

degree correlations (yy), for a) Harmony game and b) Prisoner’s dilemma. . . . . . . . . . 45

5.8 The FFC as a function of B in networks with different layers (B = benefit, ffc = average

final fraction of cooperators, L = total number of layers) . . . . . . . . . . . . . . . . . . . 46

5.9 The impact of multilayers in cooperation, for L = 2, when in comparison to L = 1 . . . . . 47

5.10 AGoS for HR networks, during 30 generations (B = benefit, L = total number of layers,

β = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.11 AGoS for SFBA networks with correlations, during 30 generations (B = benefit, L = total

number of layers, β = 0.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.12 AGoS for SFBA networks without correlations, during 30 generations (B = benefit, L =

total number of layers, β = 0.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

A.1 Fraction of cooperators per round, in a HR network, for B=1.005. . . . . . . . . . . . . . . 64

A.2 Fraction of cooperators per round, in a SFBA network ,for B=1.25. . . . . . . . . . . . . . 64

B.1 Expected FFC, according to [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

B.2 Expected AGoS over 100 generations, for different values of B from [6]. . . . . . . . . . . 67

B.3 Expected FFC as a function of the number of layers and the degree correlations, for a)

Harmony game and b) Prisoner’s dilemma, according to [8]. . . . . . . . . . . . . . . . . . 68

B.4 Expected FFC as a function of the initial fraction of cooperators and the degree correla-

tions, for a) Harmony game and b) Prisoner’s dilemma, according to [8]. . . . . . . . . . . 68

x

1Introduction

Contents

1.1 What are game transformations? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 A multilayer approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Work goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1

2

Although Network Science is a relatively new field of study – which combines tools and principles

from physics, applied mathematics and computer science – it has proven helpful to several other areas

of research. The study of interactions and the topology of networks presents itself as a map to interpret

phenomena in Economic [9, 10], Ecological [11], and Sociological [12] Systems, as well as in Cognitive

Sciences [13].

In the study of social systems, the main goal is trying to understand how an individual can influ-

ence (or be influenced by) others’ opinions and/or ideas. This can be useful for designing marketing

strategies that use prominent figures, or manipulate electoral outcomes by influencing the right set of

people [14] [15]. From a theoretical perspective, individuals can be abstracted as nodes and the relations

between them as links in the network. The links can represent, for instance, an acquaintance relation,

a co-working connection or simply that individuals share a common space such as a gym. After having

a suitable representation, one can describe any particular rules (or social norms) that dictate how indi-

viduals interact and learn new traits or strategies through those links. With these elements we can start

a simulation to study the emergent behavior on a macro-, population-wide level, stemming from simple

rules governing how individuals interact, at micro-level, [16]. We have already argued that networks can

model a plethora of different systems. With the goal to make our study as general as possible, to cover

a wider range of applications, we will introduce variability by manipulating the characteristics that our

networks will have. This is done by following methods already defined for the creation of networks, like

the Barabasi-Albert method [17], or by other methods that manipulate a network after its creation, like

the Xulvi method [18]. Another area where the methods and tools of network science can be applied

are biological systems. Networks are a natural underlying structure to a lot of processes occurring in

biology, from protein-protein [19] to predator-pray [20] interactions. By performing a similar reasoning

as the one described above, we can simulate behavior and discover how a network would evolve. By

knowing this we can act on our information, anticipate (prevent, catalyze, augment) the outcome and,

theoretically, change the world. But this change is hardly performed if it is not endorsed by large scale

and/or powerful organizations, like companies or governmental entities. These are usually the motors

behind implementing economic incentives and social rewards, like bonuses at work for being a ”friendly”

colleague or reduced taxing for people that do recycling. We will focus on person-to-person interactions,

but one possible extension to the problem would be to understand how can we strive towards reaching a

cooperation goal that can benefit many people but that requires an effort from a large group, when there

can be conflicting interests or costly sacrifices that could stray the group away from said goal?

In this chapter we give a brief introduction that provides some context about the scope and the main

topics we will approach. We show the importance of networks and the motivation behind multilayer ones,

we explain what are game transformations and provide the goals and outline of this work.

3

1.1 What are game transformations?

A popular approach to model interactions between agents, and the payoffs herein accrued, is to

use Game Theory [21]. In this context, Game Theory allows the modeling of real-life situations while

providing an intuitive setting. In particular, we shall use Game Theory to model cooperation in social

dilemmas. These are scenarios in which interactions may represent a contest over a shared resource

or a conflict between individual and social interests. Different cooperation dilemmas can be obtained by

conveniently tuning a few game parameters (see figure 2.2).

Empirical observations often suggest that the observed behavior of the population does not match

the expected outcome derived from a Game Theoretical rational. Cooperation represents an obvious

example; it is widespread in nature and societies, from simple organisms to complex Human interactions,

even if purely rational analysis would suggest the opposite [22].

For example, a Prisoner’s Dilemma (PD) taking place in a well-mixed population ( where all individ-

uals can interact with each other) is expected to lead to the tragedy of the commons where everybody

refuses to cooperate, yet this is not necessarily true when considering a different type of network, as

you can see in figure 1.1. This change happens because the population is no longer characterized as

well-mixed, but now has a heterogeneous character.

We hypothesize that the differences in the theoretically predicted outcomes in cooperation games,

and the observed behaviors in reality, differ as a byproduct of different social networks of interactions.

Decision-making in a network may thus alter in a fundamental way the global dynamics of coopera-

tion in a population, an hypothesis supported by both experimental [23, 24] and theoretical [7, 25, 26]

frameworks. Ultimately, networks lead to a game transformation [6, 27, 28], a break in the symme-

try between the local and global dynamics of the population, which one can capture by studying the

temporal behavioral evolution of the networked population.

In this figure 1.1, from a defection dominance dilemma, where defectors (Ds) were expected to

dominate in the population, to a coordination problem, where both cooperators (Cs) and defectors can

overtake the other and establish themselves as the dominant behavior, represented by the two stable

roots at x = 0 and x = 1 of figure 1.1 a). Another example consists in the change in ability that epidemic

outbreaks have to overtake networks. These can spread across some classes of network while being

very limited to the infectiousness of the disease, in well-mixed populations.

These transformations are observable by analyzing the evolution of the behavioral dynamics inside

a network, e.g by examining how the stable roots shift, thus allowing us to establish the links between

individual and collective behavior.

4

(a) AGoS in a BA network from [6] (b) The gradient of selection in the Prisoner’s dilemmain well-mixed populations.

Figure 1.1: Comparing obtained and expected gradients of selection for a PD, in different network topologies

1.2 A multilayer approach

When scientists or engineers are making an attempt at building a model, they always have as a goal

to do the best possible approximation of reality. Looking at previous works we thought that single-layer

studies were actually using an over-simplified representation, so, with the aforementioned goal in mind,

we decided that we would portrait our networks of interactions as having a multilayer aspect. This allows

for individuals to have connections across different levels with others, representing the different networks

that an individual can be a part of. For example, when considering the flows of information inside a social

network, if one were to study only the face-to-face interactions he would have a poor reflection of reality.

While, if considering the interactions that could occur between mobile phones, or through Facebook,

or through Skype one would have more trustworthy data about the way people communicate with one

another.

In section 2.2 we give some insight on the different terminology regarding the family of words of

”multi-”, that we will be using.

1.3 Work goals

We want to understand how the global phenomena is linked to individual mechanisms, and identify

the effective game transformations that the network of contacts brings about. We will analyze how

simultaneous strategic interactions in multiple layers — here modeled through multiplex networks — can

transform the original dilemmas, fostering (or not) cooperation when compared with a single layer or

network. To do so, we will focus on extending the AGoS, a tool proposed in a previous paper [5,6], to a

multilayer architecture where we vary the structure of the networks, the degree that nodes share across

5

layers, and the importance given to the relative fitness (intensity of selection) of others. This work is an

attempt at answering the following question: What game transformations occur when considering

different types of networks in a multilayer structure?

By answering this question we are able to establish a link between individual behavior at a local

level and collective behaviors for evolutionary processes in multilayer networks. Finally, the abstract

nature of the new computational tool we develop renders this framework readily applicable to other time-

dependent processes that may also occur in multilayer networks, such as N-person collective dilemmas,

opinion dynamics, ecological processes, or disease spreading, among others.

1.4 Outline

In Chapter 1 we provide some context and motivation of our work. In Chapter 2 we will present the

tools and theory required to understand the topics we approach. In Chapter 3 we show some related

work already performed in the area and how it influences ours. In Chapter 4 we describe the model and

methodology that we will be following. In Chapter 5 we explain in detail our methods and results. In

Chapter 6 we provide a final analysis of the work and remarks regarding future work.

6

2Background theory

Contents

2.1 Evolutionary game theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Multilayer classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Graph topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 The Average Gradient of Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

7

8

We will now present the knowledge and basic concepts one has to have to understand the topics

discussed in this work.

2.1 Evolutionary game theory

Evolutionary game theory applies principles from evolutionary theory to populations of individuals

whose interactions follow a game theoretical setting. Individuals (assumed to be rational decision-

makers that look to maximize their rewards) change or adapt their strategies, based on the returns

obtained from interacting with others. It is, thus, a mathematical framework that allows us to formal-

ize these interactions, through the definition of the range of possible strategies and the payoffs each

strategy has (as a function of the strategies others are using). By letting individuals interact over a suf-

ficient amount of rounds, the best strategies (the ones that return a higher payoff) start to be used by a

greater amount of individuals, similar to the way a favorable trait would gain prevalence in the context of

a Darwinian natural selection process, while a less fit strategy (with a lower payoff) would disappear.

In this chapter we first present the framework given by evolutionary game theory, then we explain the

multilayer architecture and the types of graph that are useful. Lastly we define and motivate the use of

the AGoS.

2.1.1 Evolutionary stable strategies and effective games

A strategy is said to be an evolutionary stable strategy (ESS) if an infinitesimally small amount of

individuals using a different strategy are not able to spread over the population. This definition will be

useful because particular dilemmas have known points of stability, therefore, if we can show that these

stability points changed, then this can be evidence of the actual dilemma also changing. The concept of

effective game, that gives name to this work, thus becomes of central importance since the real games,

the ones that are actually being played, can differ from the theoretical games that evolutionary game

theory would predict to occur.

2.1.2 Replicator equation

The replicator equation describes the evolutionary dynamics of infinite and well-mixed populations.

Thus, we describe how the frequency of a strategy i would evolve in a population:

xi = xi[fi(x)− φ(x)], φ(x) =

n∑j=1

xjfj(x) (2.1)

9

where xi is the fraction of type i in the population, fi(x) is the fitness of type i and φ(x) is the aver-

age population fitness. In the particular case when we only consider 2 strategies the equation above

simplifies to:

x = x(1− x)(fC − fD) (2.2)

where x represents the fraction of cooperators ( (1− x) the fraction of defectors) and fC represents the

fitness of cooperators (fD the fitness of defectors). This derivative x can also be seen as the gradient

of selection g(x). If g(x) > 0 then the number of cooperators increases, if g(x) < 0 then it decreases.

In general, strategies which exceed the average fitness φ(x) tend to spread while others tend to die

out. The replicator equation describes evolution for infinite well-mixed populations, and only constitutes

a good approximation when modelling finite large ones. In reality, populations are finite and potentially

small; also, individuals tend to interact following an underlying network of contacts and deviating from

the well-mixed assumption. This way, we will be working with numerical simulations that assume finite

populations, which means we should no longer use the replicator equation and have to start considering

that agents change strategies stochastically as depicted in figure 2.1, i.e, agents choose to imitate

behavior probabilistically, based on the success of the other strategy.

Figure 2.1: Evolution process in finite well-mixed populations. In networked populations, Y is randomly selectedfrom X ’s first neighbors.

Furthermore, in well-mixed populations, individuals with the same strategy are considered equivalent,

this is not true in structured populations where the context surrounding an individual has to be taken in

10

consideration as well. Individuals with the same strategy are not necessarily equivalent since they also

depend on their position in the network and on their neighborhood, at each time step. This motivates

the use of the AGoS described in section 2.4.

2.1.3 2-person games

Pairwise interactions between individuals are called 2-person games. Each type of game (Prisoner’s

dilemma, Stag-Hunt game, Snowdrift game) has a payoff matrix (figure 2.2) associated (the names

come from famous examples representing the dynamics the matrix originates) and individuals can either

behave as Cooperators (C) or Defectors (D). When both individuals cooperate they get a reward R,

when both defect they get a punishment P and when one cooperates and the other defects, the former

gets S (also known as the sucker’s payoff) and the latter gets T (also know as temptation to defect).

The dilemmas occur when individual choices result in defecting instead of, the more beneficial, mutual

cooperation. By normalizing R to 1 and P to 0 we only have two variables remaining, T and S (instead

of four originally) which will be in the range of 0 ≤ T ≤ 2 and −1 ≤ S ≤ 1. This simplification has no loss

of generality [7].

Figure 2.2: Payoff matrix of 2-person and two strategy games. R,S,T and P are the payoffs; C and D are thepossible strategies.

The different dilemmas occur for different relative orders of the values of R,T,S and P:

• Snowdrift: The scenario for this dilemma can be described as a snowy road where there are 2

people stuck. Just one person is enough to clear the path for both to pass. If one goes to clear

the road, the other can drive away, and vice-versa. This dilemma can be formulated with the order

T > R > S > P , which means players prefer to defect unilaterally than to mutual cooperate. The

relation T > R can be associated with greed. In this type of dilemmas there is no ESS, since the

population tends to remain stable at a fraction of cooperators near 0.5 as you can see in figure 2.3.

This represents an example of co-existence between cooperators and defectors.

• Stag-Hunt Dilemma: The scenario for this dilemma can be described as hunters trying to catch a

big stag, representing a high reward, but requires mutual cooperation in order to achieve it. This

can be formulated as R > T > P > S, which means players prefer to mutually defect instead of

11

Figure 2.3: The gradient of selection in Snowdrift games in well-mixed populations.

unilateral cooperation. The relation P > S can be associated with fear. In the example, facing

a big pray alone (unilateral cooperation) would be more dangerous than to do nothing (mutual

defection). In this type of dilemmas both C and D are ESS. Each strategy leads to a stable fraction

of cooperators equal to 1 or 0 respectively, as you can see in figure 2.4. This represents an

example of bi-stability.

Figure 2.4: The gradient of selection in Stag-Hunt games in well-mixed populations.

• Prisoner’s Dilemma: The scenario for this dilemma can be described as 2 criminals being impris-

oned and each is being offered the same deal: To either testify against the other or to cooperate

with the other by remaining silent. This can be formulated as T > R > P > S, which means both

fear and greed are present, and the individual rational choice is to defect, despite mutual coopera-

tion being the best option. In this type of dilemmas, D is an ESS. Regardless of the initial fraction

of cooperators, individual rational choice will lead to the tragedy of the commons, where there are

no cooperators remaining in the population, as you can see in figure 2.5.

We defined the 2-person games because they allow us to introduce some basics on evolutionary

theory and they are the model of interaction we will mainly use, both for debugging our framework and

for our studies.

12

Figure 2.5: The gradient of selection in the Prisoner’s dilemma in well-mixed populations.

2.1.4 N-person games

In this setting the game involves groups of people, instead of just two persons. Individuals interact

with their k > 2 neighbors, represented as nodes in a network, forming a group of size N = k + 1,

(figure 2.6 shows the nodes and a 5-person game), known as an N-person game. If the game is played

in a well-mixed population the replicator equation holds true, accurate and simplifies the calculus of the

gradient of selection. If the population is structured, we should recur to the use of the AGoS.

The most common N-person game is a generalization of the Prisoner’s dilemma, applied to groups,

also known as a Public goods game, or PGG. An individual can be a part of several PGGs, the number

of which being equal to a node’s degree. Cs contribute with an amount c to the public good and Ds do

not contribute. In a conventional PGG the total pot is multiplied by an enhancement factor r ≥ 1 and

equally split across all N members of the group [26]. When considering non-linear rewards, the total

of contributions has to reach a certain amount (or threshold) M (below which failure happens) to reap

the rewards, e.g. ecological tragedies [29, 30], in this case also known as free-riding. The probability

with which the tragedy happens, also known as risk perception, is often a variable considered in these

experiments. One difference that separates the 2-person from the N-person games is that, in the PGGs,

targeted interaction is not possible, i.e, individual behavior cannot be selectively rewarded or punished.

N-person dilemmas often involve a much larger complexity than the one addressed in this thesis,

which focuses on 2-player symmetric games.

Figure 2.6: The orange node would participate in the PGG with center on himself and with center on its coloredneighbors.

13

2.2 Multilayer classification

Usually, the simpler solution is the best one. For this reason we will represent networks as graphs

(concretely defined in section 2.3), since they are intuitively the best structure to do so. A multilayer

network is thus represented in a multigraph.

In single-layer interactions individuals (or nodes) only interact in a single dimension, they are either

connected through a link or they are not. In reality, individuals can have different groups of interactions,

and can be seen as connected or not through more than a single plane, e.g, they can be co-workers and

they can attend the same gym, making them connected at two distinct levels. This motivates the use

of multilayer networks. In these networks, nodes are represented in different layers and their strategies

can be represented in vector form.

Figure 2.7: 3 layers that share the same nodes but are connected through different links, from [1].

We will describe the 3 main types of multilayer networks, following the definition of [31], and argue

which is the most convenient for this work. The types are:

• Multiplex networks All the layers have the same set of nodes (or share a fraction). The only

difference among layers is the way that the nodes are connected.

• Interdependent networks Different layers have all (or nearly) different nodes and there can be

links that establish connections between some nodes, across different layers, also known as de-

pendency links. These links are not necessarily physical, they merely portrait a dependency.

• Interconnected networks Very similar to interdependent networks but the links that may exist

between layers are physical ones.

14

The idea is that each node would take part in several interaction networks, depicting various areas

it can be inserted in, like family ties or work-related. Each one of these networks can have distinct

dilemmas or interests affecting them. Since having a somewhat constant set of nodes would be of use

to us,because it is the most approximate way of representing human interactions among these three,

the option that best suited our case were the Multiplex networks.

In the upcoming sections we will relax the language and use the term multilayer often to refer to

multiplex structures.

2.3 Graph topologies

In this work we will refer to a variety of network types, thus we dedicate a section to introduce their

characteristics.

Previous work has observed that real-world networks tend to be heterogeneous, i.e, there are people

with a lot of contacts and others with only a few [32]. To recreate this reality and the different levels of

heterogeneity observed, several graph models are used, to largely cover the spectre of this network

feature. They are presented here by increasing heterogeneity:

• Complete graphs: where every individual is connected with everybody else (figure 2.8). This kind

is the most homogeneous graph type that exists. It is an example of a finite well-mixed population

where everyone can interact with anyone else, usually used as a reference or benchmark since it

has 0 heterogeneity.

Figure 2.8: A complete graph.

• Homogeneous random (HR) graphs [33] : where every individual has the same number of neigh-

bors, but is not necessarily attached to all the nodes in the graph (as in the complete graph), and

the neighbors are random (figure 2.9). In these networks, individuals with the same strategy are

not necessarily equally fit because they now have a limited amount of connections, thus they un-

able to contact with everyone, contrary to what happens in well-mixed populations. This makes

the fitness of an individual context-dependent. An example would be a cooperator surrounded by

cooperators, or surrounded by defectors, obviously he would survive better in the former scenario.

This network is created by using a regular graph and then randomizing the links.

15

Figure 2.9: A HR graph with average degree k=2.

• Single-scale graphs: where individual degree (number of contacts) does not deviate appreciably

from the average degree of the graph (figure 2.10). To generate a population like this, one uses the

configuration model which leads to random graphs that follow a certain distribution. In this method,

the degree of each vertex is pre-defined, instead of having a probability distribution from which the

given degree is chosen.

Figure 2.10: A degree distribution for a Single-scale graph.

Figure 2.11: An exponential decaying tail (b) meaning that there is a single scale for the connectivity k, from [2]

• Scale-free (SF) graphs [34]: where the degree distributions decay with a power-law, showed in

Fig. 2.12. The best way to create a graph of this type is to use the method proposed by Barabasi

and Albert [32, 34] which combines growth and preferential attachment (we will refer to these

16

as SFBA). This method consists in continuously adding nodes to a network, that starts with m0

nodes, in a way that the probability of linking to an existing node is proportional to a node’s degree.

This way, nodes with higher degree or hubs, are more likely to attach to the new nodes therefore

allowing for the preferential attachment property to emerge in the network.

Figure 2.12: Cumulative distribution following a power law, meaning we are in the presence of a scale free net-work,from [3]

Using the above referred types is also useful because they allow us to cross our findings with previous

works that use similar networks.

2.4 The Average Gradient of Selection

The average gradient of selection - AGoS - is a time-dependent variable that is used as a tool to study

games on graphs. It can provide the same information that the replicator equation does in well-mixed

populations, but in structured ones. The AGoS is defined as the average over all possible transitions

taking place in every node of the network throughout evolution and the average over many generations.

It takes in consideration the influence each node can have on its neighbors. The AGoS is useful since it

captures the population-wide dynamics due to its mean-field character. It can be computed for arbitrary

intensity of selection β (present in the Fermi function, which conditions when to imitate other behavior),

for arbitrary finite populations and for any kind of network structure. We will use it to track the self-

organization of cooperators when interacting with defectors [6].

To describe the AGoS we first calculate, for every node i, the probability of adopting a different

strategy at time t,

Ti(t) =1

ki

ni∑m=1

[1 + e−β(Pm(t)−Pi(t))]−1 (2.3)

where ki is the degree (the number of connections) of node i and ni is the number of neighbors of i

17

that are using a different strategy. The AGoS for a given time t, a simulation p and for j cooperators is:

Gp(j, t) = T+A (j)− T−

A (j) (2.4)

where

T±A (j, t) =

1

N

AllDs/AllCs∑i=1

Ti(t) (2.5)

or separately:

T+A (j, t) =

1

N

Ds∑i=1

Ti(t) (2.6)

T−A (j, t) =

1

N

Cs∑i=1

Ti(t) (2.7)

Equation 2.6 and 2.7 represent the increase that can happen in the number of cooperators and

defectors at time t, respectively. Notice that the sum over the defectors contributes to the increase in

cooperators and vice-versa.

The final and time-independent AGoS is averaged over all Ω simulations and Λ time-steps (or

rounds):

G±A(j) =

1

ΩΛ

Λ∑t=1

Ω∑p=1

Gp(j, t) (2.8)

It has been shown through the use of the AGoS (to calculate the gradients in figure 2.13) that, for

both heterogeneous and homogeneous networks, even though locally the interactions are, in theory, the

ones of a Prisoner’s dilemma, globally the dynamics resembles that of a coordination and co-existence

problems, respectively for each network [4]. This is explained in further detail in section 3.2.1.

Figure 2.13: The AGoS applied to different networks, from [4].

18

3Related work

Contents

3.1 Mechanisms for the emergence of cooperation . . . . . . . . . . . . . . . . . . . . . . 21

3.2 How structure affects cooperation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Connecting local and global behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

19

20

Human beings are competitive by nature, with other species and, more importantly to this study,

among ourselves. Every living thing is designed with the intent to propagate its genetic material and to

ensure its species survival, regardless, cooperation can be observed, from cells to sheep herds. This

has been motivation for several studies that try to understand why this cooperation exists, under which

conditions, or what factors can make it increase or decrease. In this section we present some of the

existing work and what others have done to study and discuss similar problems.

3.1 Mechanisms for the emergence of cooperation

Martin A. Nowak was the first to try and explain this apparent contradiction between cooperative and

selfish behavior. He proposed the existence of 5 mechanisms (figure 3.1) that enable said cooperation

[35], in detriment of natural selection principles:

• Kin selection: This occurs when behavior is conditioned by kin recognition. If I find my interacting

partner to be somewhat related to me, I will more likely cooperate with him than I would with a

complete stranger.

• Direct reciprocity: This mechanism is only relevant when several interactions with the same

partner occur. If I cooperate in the present interaction then I may be repaid with cooperation

in some future interaction, which means my actions can have a direct impact on future actions

towards me.

• Indirect reciprocity: This mechanism is only relevant when several interactions occur and third-

parties observe or have knowledge about them. This attributes a reputation to each person. When

choosing how to act, I now have in consideration how my partner has interacted with me, how he

interacted with others, and how my decision will affect my own reputation.

• Structured populations: When populations are structured, individuals tend to interact with others

closer to them. This can form clusters of cooperators which prevail, since pairs (or groups) of

cooperators have a higher payoff then pairs of defectors.

• Group selection: This mechanism occurs when, besides having competition within a group of

individuals, there is also competition between different groups. In this case groups of cooperators

are likely to outperform groups of defectors, even though defectors can win within groups.

In 2013, 7 years after these mechanisms were proposed, Rand and Nowak [36] showed experimental

evidence to corroborate the earlier theoretical work.

21

Figure 3.1: Graphic representation of the 5 mechanisms described, from [5].

3.2 How structure affects cooperation

3.2.1 Heterogeneous vs Homogeneous

One mechanism that received particular attention was related to how the populations are struc-

tured. Clusters of cooperators or defectors can be formed in complex networks of interaction, which are

ubiquitous in social and living systems. These networks are known to profoundly affect the dynamical

processes that take place on them, from opinion dynamics to disease spreading [37], [17], [38], [39]. The

fact that individuals interact with others along the links of complex networks has been shown to greatly

influence the chances of reaching to high levels of cooperation, [30], [40], [26], [41], [42]. Heterogeneity

is one of the motifs leading to changes in structure. Santos, Pacheco and Lenaerts [7] have proven

that, when high levels of heterogeneity occur, the chances for cooperation also increase. According to

them, this stems from the interplay between two mechanisms. The first is the fact that heterogeneous

networks, by means of the randomness with which they are created, can have links that connect nodes

that are far away from each other (also known as shortcuts). This makes the formation of clusters of

cooperators harder to occur, decreasing the overall prevalence of cooperators. The second is that in-

dividuals can now interact in different quantities. A node with a higher degree will play in more games

than one with fewer, per generation. This makes for a new way for cooperators to survive. Notice that

one inhibits while the other enhances cooperation. The overall result is that cooperators are now more

likely to prevail.

3.2.2 Degree correlations

In a multilayer structure, that can represent different domains, it is interesting to question whether

the relevance of the individuals across layers can have an impact in the resulting cooperation. This

22

was modeled by creating multiplexes where nodes have a variable degree correlation between layers.

Recently, Kleineberg and Helbing showed that, for heterogeneous networks, when a large enough de-

gree correlation is present in association with many layers, the emerging behavior is determined by the

initial conditions of the network [8]. This can be explained by the fact that hubs can accumulate a higher

payoff, which means they are more likely to be imitated by nodes in a certain domain, regardless if

the hub’s high payoff was due to its behavior on this domain or not. This makes for what the authors

call a ”topological enslavement”, where the initial assignment of the hub’s behavior will dictate the out-

come, meaning that the strategic incentives (the game parameters) become irrelevant because they are

subdued by the ability that hubs have to deceive others (figure 3.2).

Figure 3.2: In a) there are strong degree correlations and in b) there are not.

Considering the green edge in layer 1 of a), we can see that node B would likely imitate node A in

this layer, becoming a cooperator, even though A’s success comes from it being a defector in the other

layers. Opposingly, in the green edge of layer 1 of b), node A would likely imitate node B and would

thus be unable to drive the layer towards its initial cooperative behavior. In the presence of degree

correlations, the layer is dependent of the initial setting while on its absence it is not.

3.3 Connecting local and global behavior

In section 1.1 we have explained what a game transformation is. We will pick up on that and now

explain what methods are used to effectively observe and study these same transformations. The main

idea is that, in order to observe a transformation we cannot only observe the initial and final state of an

experience, so we must create ways that allow us to retrieve information in a more continuous manner.

23

With this in mind we can:

• Count the states ( in a population of size N, there are N+1 possible states, one for each possible

fraction of cooperators ) in which the population is more frequently (figure 3.3).

Figure 3.3: Distribution showing the frequency spent in each state, for two different values of B (benefit), from [6].

• Use the AGoS, defined in section 2.4, to capture at each time step whether the number of cooper-

ators tends to increase or decrease.

• See how the AGoS is evolving as the individuals are playing the game and adapting, by stopping

the evolution at a fix generation t, for several t’s (figure 3.4)

Figure 3.4: Three instances of the AGoS captured at different evolutionary times, from [6].

We will use some of the methods referred here to analyze our results and derive conclusions from

them.

3.4 Summary

Previous work has already discovered how to explain cooperation in 2-person interactions of a PD

played in a single layer. It has also shown that cooperation varies with the correlations between degrees

24

(inter-layers) and the degree of heterogeneity (intra-layer).

We now propose to study how these variables, as well as the number of layers and the strength of

the dilemmas, when mixed together, change the outcome of cooperation. We will not, however, try to

explain why the eventual changes occur.

25

26

4Model

Contents

4.1 Creating and validating our framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Understand the FFC as a function of the game incentives and the number of layers 30

4.3 Extending the AGoS to multilayer networks . . . . . . . . . . . . . . . . . . . . . . . . 31

27

28

In this section we will describe the methodology we followed during our work. We can divide our

process in two separate parts: The first part consists in creating and validating our framework, by repli-

cating the results obtained in previous works (detailed in the dissertation); the second part consists in

extending the previous methods to model the evolutionary dynamics of multilayer networks, in particular

the interactions and transformations occurring in them.

4.1 Creating and validating our framework

We will begin by developing a framework that will allow us to extract information that is necessary

to link the global patterns that emerge from the local interactions. These patterns are usually studied

through the use of gradients of selection like the ones in section 2.1.3. When regarding real networks,

these gradients are harder to obtain, usually one can only resort to simulations to see the final levels of

cooperation and not the ”path” that agents took while evolving, thus the usage of the AGoS. It allows us to

see if the network is following the expectations from standard evolutionary game theory. This framework

will be coded in Python and will use the NetworkX [43] as well as the multiNetX [44] packages, which

were designed for the creation, manipulation and study of multilayer complex networks.

Our framework has to be modular so that it becomes easy to change parameters between experi-

ences. These changes can be in the type of game, the type of network, the size of the population, the

number of layers, the number of rounds, etc.

Using our framework we will attempt to recreate the results of three papers, ”Evolutionary dynam-

ics of social dilemmas in structured heterogeneous populations” [7], ”From local to global dilemmas in

social networks” [6] and ”Topological enslavement in evolutionary games on correlated multiplex net-

works” [8]. This is made with two goals in mind, one is to better understand the conclusions that these

works reached, since they approach a topic very similar to the thesis. The second goal is to debug our

framework, i.e, if we reach the same results that both papers did, we can claim with more certainty that

our framework is well developed and can trust its results when we apply it to our work.

The first paper tries to find correlations between heterogeneity and the emergence of cooperation.

It follows an evolutionary dynamic where a random pair of directly connected individuals (x and y ) will

engage in a round of the game with their respective neighbors and store their obtained cumulative

payoffs (Px and Py). After obtaining the payoffs, if Py > Px, x will replace its behavior with y ’s with a

probability given by :

p = (Py − Px)/(k>D>) (4.1)

where k> = max(kx, ky) and D> = max(T, 1)−min(S, 0).

The second paper tries to link the global behavior to the local interactions, using the AGoS. Instead

of considering pairs (S, T ), they simplify by defining one single variable B (benefit) (that can be unfolded

29

as T = B and S = 1 − B), this takes advantage of the previous knowledge regarding variations in

behavior that occur in prisoner’s dilemmas (our main dilemma of interest). These are greater (for a PD)

as T increases and S decreases, which is exactly the change we produce by using this single variable.

It also uses a different function to update one’s behavior, called the Fermi update [45,46], where a node

adopts new behavior with probability :

p = [1 + e−β(Py−Px)]−1 (4.2)

where β is the intensity of selection.

Regarding our simulations, we will use the Fermi function (4.2) because it provides very similar

results and it is a step function, which does not require a normalization constant, unlike the first function

described in 4.1. Both update rules mimic cultural evolution where individuals probabilistically imitate

better performing strategies [41,46].

The third paper calculates the final fractions of cooperators, but now in multilayer networks in which

degree correlations are defined by v ∈ [0, 1], where 0 represents no correlation and 1 maximum correla-

tion.

To see how each strategy survives in the population, we will run several simulations and plot the

resulting fractions of cooperators so that we can compare them with the ones obtained by the first

paper. This allows us to know if we are ”playing the game” correctly. We will also calculate the AGoS of

several populations, along with their probability densities, and compare them to the second paper. This

allows us to know if we are calculating the AGoS correctly. We will calculate the fractions of cooperators

when considering multilayers with different degree correlations and compare them to the third paper.

This allows us to know if we are defining the multilayer and, again, ”playing the game” correctly as well,

and if we are manipulating the degree correlations as intended.

After debugging our framework we will begin implementing new features to our simulations.

4.2 Understand the FFC as a function of the game incentives and

the number of layers

In this section we are interested in studying how cooperation varies as a function of the game in-

dividuals play, reflected by B, as well as a function of the total number of layers, L. This will consist

in starting with populations where there are 50% of cooperators (and 50% defectors), and observe the

mean final fraction of cooperators(FFC) after a number of generations. The FFC is now calculated by

averaging over the fractions of all layers, for each pair (B,L). These results will be interesting because

they will raise the questions that we want the AGoS to answer. If we observe that the number of layers

30

increases the FFC for, e.g (B = 1.45), then this is expected to occur due to the interplay between two

things: either less cooperators are required to reach a stable fraction of cooperators and, thus, the root

of the gradient should be smaller; or the gradient is positive and stronger (in absolute value) and exerts

a greater pressure towards cooperation.

4.3 Extending the AGoS to multilayer networks

Our evolutionary dynamics will consist in 2-person interactions between L layers, where the 2 nodes

are randomly selected nodes and the update is performed, using a Fermi function similar to 4.2, but

now the fitness of an individual is calculated by averaging over the payoffs he obtained considering all L

layers, instead of the previous fitness obtained solely in a single layer.

The number of layers we will be able to study is still ill-defined. Due to the complexity of computation

of the AGoS, the time required to perform the simulations is a constraint we are not able to assess at

this point in time. We can only say that we will study, at least, for multilayers with L = 2.

With the change in the evolutionary dynamics, so must the computation of the AGoS change. Thus,

it is now calculated taking in consideration two differences. The probability of a node changing behavior,

at time t, in layer l, is affected by the accumulated payoffs and is now given by:

Ti(t, l) =1

ki

ni∑m=1

[1 + e−β( 1L (

∑Ll=1 Pm,l(t)−

∑Ll=1 Pi,l(t)))]−1 (4.3)

,where ki is the degree of node i and ni is the number of neighbors of i that are using a different

strategy. The same node can have different strategies in different layers. The second change is that,

now, the total pool of nodes has size L×N , leading to:

T±A (j, t) =

1

L×N

AllDs/AllCs∑i=1

Ti(t) (4.4)

We will study how the gradient of selection changes by increasing the amount of layers of the same

type, e.g a couple of HR networks. We will perform this analysis for different values of B. We will do this

for HR, SFBA with correlations and SFBA without correlations.

31

32

5Evaluation

Contents

5.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

33

34

In this chapter we will explain in detail our methods, show our results and provide their analysis.

5.1 Methods

Computer simulations offer scientists a powerful way to explore and investigate models of environ-

ments as well as changes inside them. They allow to ”speed up” the passing of time through the power

of computation making for a useful tool to exploit, among other areas, evolutionary game theory. Imag-

ine the amount of time and resources that we would have to allocate if we were performing a study in a

real environment with hundreds of real people... But simulations are not only sunshine and roses. They

represent only a small scope of a far more complex reality of interactions that is near impossible to grasp

and to formalize completely. Taking this in consideration, it is the best tool we can use to explore and

develop an intuition for many scenarios. We will now detail how we defined our lot of simulations.

5.1.1 Regarding framework validity

All networks have N = L × 1000 nodes and average degree k = 4 (except for the complete graphs

that have average degree k = N − 1). The SFBA, HR, and complete graphs were generated by directly

using the graph generators of NetworkX [43]. The SF Random was created by swapping pairs of edges

of a SFBA graph, also known as the Xulvi method [18]. The Single Scale was created by generating

a sequence of degrees and then using the configuration model (which is a model that creates a graph

with a predefined degree distribution).

5.1.1.A FFC for different types of networks and for different game incentives

To estimate the expected Final Fraction of Cooperators (FFC) for different types of networks and for

different game incentives, our simulations start with 50% of cooperators assigned randomly. The popu-

lation structure is created according to one of the models described above. The evolutionary dynamics

proceed as follows: In each round, a node x and a neighbor y are randomly selected from the network.

X will interact once with each of its neighbors, obtaining an accumulated payoff Px; Y will interact once

with each one of its neighbors, obtaining an accumulated payoff Py. Then x will adopt y ’s strategy with

a probability given by the Fermi function:

p = [1 + e−β(Py−Px)]−1 (5.1)

where β represents the intensity of selection, which allow us to regulate the level of randomness in

the strategy adoption process.

35

Figure 5.1 illustrates the following description of a set of simulations deployed to estimate the FFC.

We start by creating 10 populations with a predefined structure; we let each play out Λ = 250000 rounds

(250 generations) with a fix pair (S, T ) in the range of [−1, 1] and [0, 2], respectively, and we store the

observed FFC. We then reset the population to a new initial random sample of cooperators, maintaining

the population structure; We repeat these steps 500 times for each set of conditions.

Figure 5.1: After choosing a model for the population and a pair (S,T), this is how the simulation performs.

If the results match those of Santos, Pacheco and Lenaerts [7] we have a good indicator that we

were ”playing the game correctly”, i.e, the players were receiving their correct payoffs, according to the

game parameters, from the interactions they were a part of. Also we would could guarantee that the

networks were created correctly.

5.1.1.B AGoS for different networks and for different game incentives

To reach the AGoS as a function of the fraction of cooperators (j/N ) in SFBA and HR networks

we started our simulations with a random initial fraction of cooperators, fi, ranging between 0 to 1.

We let each simulation run during 30 generations (this amount of generations represents the sufficient

condition for the expected gradient to stabilize, according to [6]), and we perform Ω = 1000 simulations

for the vicinity of each initial fraction fi, exemplified in figure 5.2.

Notice that one generation is the time required so that every individual has the chance, on average, to

imitate a random neighbor once. Therefore, in a population of N nodes, one generation corresponds to

N time-steps. Our goal is to compute the difference between the probabilities to increase and decrease

the number of Cs by one in the population (G(j) = T+A (j) − T−

A (j)). For every node i, the probability of

adopting a different strategy at time t is:

36

Figure 5.2: The gradients calculated during the 1000 simulations that started at fi = 0.45 in different SFBA net-works (β = 0.1, B = 1.15)

Ti(t) =1

ki

ni∑m=1

[1 + e−β(Pm(t)−Pi(t))]−1 (5.2)

,where ki is the degree of node i and ni is the number of neighbors of i that are using a different

strategy. Each node has an accumulated payoff that results from interacting once with each neighbor

per iteration and we implemented this as playing one game in all edges of the graph. The AGoS at a

given time t, a simulation p and for j cooperators is:

Gp(j, t) = T+A (j)− T−

A (j) (5.3)

where

T±A (j, t) =

1

N

AllDs/AllCs∑i=1

Ti(t) (5.4)

The final and time-independent AGoS is averaged over all Ω = 1000× fi simulations and Λ = 30000

time-steps.

GA(j) =1

ΩΛ

Λ∑t=1

Ω∑p=1

Gp(j, t) (5.5)

For SFBA networks we will use β = 0.1 and for HR networks we will use β = 1. This represents the

intensity of selection (as in the Fermi function) and is expected to provide good plots for observing the

internal roots of the various graphics.

If the results match those of Pinheiro, Pacheco and Santos [6] we can assert more confidently that

our implementation of the gradient of selection is correct and we can start expanding towards calculating

the gradient in a multilayer context.

37

5.1.1.C FFC in degree varying multilayers

To estimate the levels of cooperation in multilayers with variable degree correlations we performed

two different types of simulations to calculate the FFC (all of these are in SFBA networks, since it makes

no sense to talk about varying the degree correlations in homogeneous networks).

We will use the values of v in the set of 0, 0.5, 1 because they are the resulting degree correlations

obtained between any fix pair of our layers (theoretically). The following description can be accompanied

by figure 5.3. Respectively, these values of v are obtained by means of a random permutation of the

degrees of the nodes across any two layers, named as ”No degree correlation” networks; by means of

a random permutation of 50% of nodes (the ones with odd id’s) across two layers, named as ”Partial

degree correlation”; and by means of making no permutations (creating copies), named ”Total degree

correlations”.

We will run 25 simulations, starting from 50% cooperators, and the value of FFC, after Λ = 10000

time-steps (corresponding to 10 generations), is given by:

FFC =1

L× P

P∑p=1

L∑l=1

FFCl,p (5.6)

, where FFCl,p is the observed fraction of cooperators, after Λ time-steps, on layer l of simulation p.

The FFC was calculated in two different games, a Prisoner’s Dilemma (T = 1.5 and S = −0.5) and a

Harmony game (T = 0.5 and S = 0.5). For each game we performed two types of simulations:

• As a function of the number of layers, L, and the degree correlations, v: We calculated the

FFC for correlated (v = 1), partially correlated (v = 0.5) and uncorrelated (v = 0) networks, and L

varied between 1 to 12.

• As a function of the initial fraction of cooperators fi and the degree correlations v: We

calculated the FFC in networks that started with different initial fractions of cooperators, fi, that

varied in the interval of [0.1, 0.9], and for correlated (v = 1), partially correlated (v = 0.5) and

uncorrelated (v = 0) networks.

If the results match those of Kleineberg and Helbing [8] we can guarantee that our variations in

degree are trustworthy (even though we generate them in a different way than the authors, through

randomizing the degrees) and the evolutionary dynamics considering aggregated payoffs is well imple-

mented.

38

(a) ”No correlation” (or v = 0) - Layer 2 is arandom permutation of layer 1

(b) ”Partial correlation” (or v = 0.5) - Layer 2 isa random permutation of the odd nodes oflayer 1

(c) ”Total correlation” (or v = 1) - Layer 2 is acopy of layer 1

Figure 5.3: Implementations of the degree correlations

39

5.1.2 Regarding our own objectives

5.1.2.A Final fraction of cooperators (FFC)

We wanted to see how did the FFC varied as a function of the game parameter, B, and the total

number of layers, L. We used B varying in the interval of [1, 1.5], and L ranging from 1 to 3. We start

with 50% of cooperators randomly spread in a population of L× 1000 thousand nodes. Each simulation

p comprises a fixed pair (B,L); we create 100 different multilayer graphs, and run independently for

Λ = L× 1M rounds (1000 generations), by the end of which we calculated the FFC for each simulation

p using equation 5.6.

5.1.2.B Multilayer AGoS

Taking in consideration what we described for the AGoS in a single layer, in section 5.1.1.B, we

will now detail only the differences in calculating them. For L = 1, we used Λ = 30000 time-steps,

corresponding to 30 generations. Now, for the number of generations to remain constant for L > 1, we

have to perform L × Λ time-steps instead. The evolutionary dynamics now considers the differences

between accumulated payoffs, instead of the differences between the payoff of nodes in a single layer,

leading to:

p = [1 + e−β( 1L (

∑Ll=1 Py,l−

∑Ll=1 Px,l))]−1 (5.7)

where β is the intensity of selection and L is the total number of layers.

In order to have the payoffs defined we have to let each node play a game with every neighbor. This

is now done by playing a game on every edge of all layers, instead of just one layer. The probability of a

node i changing behavior, at time t, is affected by the accumulated payoffs and is now given by:

Ti(t) =1

ki

ni∑m=1

[1 + e−β( 1L (

∑Ll=1 Pm,l(t)−

∑Ll=1 Pi,l(t)))]−1 (5.8)

where ki is the degree of node i and ni is the number of neighbors of i that are using a different

strategy. Another change is that now the total pool of nodes has size L×N , leading to:

T±A (j, t) =

1

L×N

AllDs/AllCs∑i=1

Ti(t) (5.9)

We also implemented a transient of 1 generation where we merely let the population evolve and do

not calculate the AGoS, to remove any bias. In the end we are still calculating the average gradient for

every configuration j/N that the population passes by.

40

5.2 Results

In subsection 5.2.1 we will only show the results and analyze them briefly, since they are a recreation

of the results obtained by other papers, deferred to appendix B, and the authors did a much better

job explaining them than we would. In subsections 5.2.2 and 5.2.3 we analyze in more detail our own

results.

5.2.1 Regarding framework validity

A brief analysis of our results, in comparison with the expected ones, ensues.

• In figure 5.4 we show our obtained FFC for the different types of networks.

The expected and observed FFC are identical across the whole spectre of networks and the game

parameters. As we increase heterogeneity, we can observe, that so does the ability to sustain

cooperation. In particular, cooperation is able to spread towards domains where greed and fear

are high, as in the Prisoner’s dilemma, the game we are mainly using to study the impact of

multilayers.

This allows us to conclude that the implementation of the creation of the networks, as well as the

evolutionary dynamics, for a single-layer game, is correct.

• In figure 5.5 we show our obtained AGoS for both HR and SFBA networks.

The behavior is according to the expected, apart from some justifiable differences. When consider-

ing B = 1.005 (same is valid for other Bs), we can observe differences between our obtained stable

root Xobtained ' 0.42 and the expected stable root Xexpected ' 58. This occurs because we are

averaging over only the first 30 generations, instead of the first 100 as in the paper, and because,

as shown in figure 3.4, the stable root Xexpected starts near 0 and is shifting towards 0.5 (for HR

networks) as generations go by. This means that the initial values of the gradient, the ones we are

taking in consideration for the AGoS, are weighing the average down. As we show in appendix A,

if we considered a larger amount of generations the population would stabilize around a fraction

of cooperators of around 0.52, and thus, since we know that the quasi-stationary distribution is

aligned with the stable root, it would also be expected that our root Xobtained would shift towards

Xexpected. In short, the difference in generations accounts for the small shift in our roots.

We also notice some lack of definition for 0.5 < j/N < 0.95 in the plots of HR networks. We

believe this is introduced by some involuntary bias of the initial conditions. To prevent this we will

implement a transient time of one generation, meaning that we will ignore the first generation of

each simulation for the calculus of the AGoS from now on (namely for the multilayer simulations).

41

(a) Complete (b) Single-Scale (c) Scale-free random

(d) Scale-free BA (e) HR

Figure 5.4: Obtained FFC for different networks.

Regardless, we think that the AGoS is correctly calculated, since it varies in concordance with the

expected results for all values of B and for both types of networks.

• In figure 5.6 and 5.7 we show how does the FFC vary in SFBA networks with different degree

correlations, with regard to the number of layers and with regard to the initial setting of cooperators.

Even though [8] shows that there exists a dependency between the outcome of cooperation and

the degree correlations for the GMM (Geometric Multiplex Model), we believe that our images

make a strong enough proof that allows us to claim that there also exists a dependency for SFBA

networks. These also allow us to conclude that we are able to tune the degree of correlations.

5.2.2 How multilayer networks impact cooperation

In the following two subsections we finally show our main results. Revisiting, our goal was to under-

stand how does the multilayer aspect impact the overall behavior, for different networks. In figure 5.8

we show the FFC after 1000 generations obtained for HR, SFBA with correlations and SFBA without

correlations, respectively, and for different values of B, reflecting the strength of the dilemma.

42

(a) HR, with β = 1

(b) SFBA, with β = 0.1

Figure 5.5: Obtained AGoS over 30 generations, for different values of B.

• In HR networks, we can see two regimes, one on each side of B ' 1.0125, which we call the

Binflection. We associate the right side, where B > Binflection, with a (what we call) ”high” B,

and the left side, where B < Binflection, with a ”low” B. When B is high, a single-layer can sustain

cooperation while multilayers cannot. For low B, multilayers can reach higher levels of cooperation.

• In SFBA with correlations, we can also observe an inflection point, as in the HR, but now Binflection '

1.25. This time the behavior is almost the complete inverse. For low B, a single-layer reaches a

higher cooperation than a multilayer can, while for high B the multilayer enables cooperation to

survive and a single-layer does not.

• In SFBA without correlations, we do not observe an inflection point, meaning that the outcome is

correlated linearly with B. In this case, for very high or very low B, cooperation is constant and

independent of the number of layers. For all B in between, a single layer provides slightly better

43

(a) (b)

Figure 5.6: Obtained FFC as a function of the number of layers (xx) and the degree correlations (yy), for a) Harmonygame and b) Prisoner’s dilemma.

conditions for cooperation to emerge than multilayers.

In figure 5.9 we present a summary of the analysis in the three previous paragraphs.

After observing our results we had the confirmation that the value of B was an extremely relevant

variable. In the limit, it would be interesting to apply the multilayer AGoS for many different values of

B. Since we had limited time and computational resources we had to choose some values of B that we

deemed of greater importance. Therefore we chose, for every network, one value of high and low B,

and applied the AGoS for each scenario.

44

(a) (b)

Figure 5.7: Obtained FFC for L = 16 as a function of the initial fraction of cooperators (xx) and the degree correla-tions (yy), for a) Harmony game and b) Prisoner’s dilemma.

45

(a)

(b)

(c)

Figure 5.8: The FFC as a function of B in networks with different layers (B = benefit, ffc = average final fractionof cooperators, L = total number of layers)

46

Figure 5.9: The impact of multilayers in cooperation, for L = 2, when in comparison to L = 1

47

48

5.2.3 Understanding the changes by analyzing the AGoS

In this section we show the results obtained by calculating the AGoS in multilayer networks, during 30

generations (we experimented for a larger number of generations in HR networks and the results were

qualitatively the same). By using this tool we aim to understand the impact that multilayers can have in

the emergence (or not) of cooperation, described in the previous section.

• In figure 5.10 we compare the AGoS in HR networks, for two different values of B. When con-

sidering a low B, we can see that multilayers are able to form a basin (many roots very close to

one another) that goes from 0.5 . j/N . 0.7, while the single-layer has a well-defined stable root

xstable ' 0.5. The other significant difference is that, for j/N & 0.5, |GA(j, L = 1)| > |GA(j, L = 2)|,

and since they are both negative, this means that the single-layer exerts a greater pressure towards

xstable.

When considering a high B, we can observe that multilayers do not evidence a root, meaning that

they cannot withstand cooperation and the game is a defection dominance one. On the other hand,

a single-layer shows a stable root xstable ' 0.15, meaning that cooperators can have a chance to

survive, due to the existence of this co-existence point.

In summary, for low B the game remains qualitatively the same but with a net increase in cooper-

ation, while for high B, it changes from a co-existence to a defection dominance one.

• In SFBA with correlations (figure 5.11), when we have a low B we can see that the roots are sen-

sitively the same, and the main difference between single- and multilayer networks is the intensity

of the gradient. In a single-layer there is a greater pressure to evolve towards full cooperation,

justifying the higher values of cooperation observed.

For a high B we have the only instance of our results on cooperation, which cannot be clearly

justified by changes in the AGoS. It was expected that the AGoS for the multilayer would facilitate

cooperation in some way. Even though we can observe a slightly higher pressure towards full

cooperation in 0.7 . j/N . 0.85, this could arguably be counterbalanced by the inverse occurring

in 0.22 . j/N . 0.45, where the pressure towards full defection is smaller in a multilayer. In this

case, where the AGoS between single and multilayers is entangled, it’s hard to justify the behavior.

In summary, for low B, the game remains the same but with an net increase of cooperators and for

high B the gradient does not allow for any valid conclusions.

• In SFBA without correlations (figure 5.12), when we have a low B we can see that a single-layer has

an unstable root xunstable ' 0.3, which is to the left of all the roots that the multilayer evidences.

This means that a single-layer requires less cooperators to reach the tipping point required to

”jump” towards full cooperation. For j/N > xunstable we can observe that a single-layer exerts

49

a larger pressure towards cooperation than a multilayer does. We could argue that multilayers

show many roots, but we are not fully prepared to make that claim since these can be attributed to

fuzzyness in the simulation.

For high B, we can also observe that the single-layer root is to the left of the multilayer one, as

for low B. This time, the multilayer exerts a greater pressure towards defection than a single-layer

does.

In summary, for low B the game remains a coordination one. For high B, it also remains the same

but we can imagine that for a B slightly greater than the one we used, the game would change

from a coordination to a full defection one.

After observing the results we are pushed to the conclusion that the multilayer aspect of the networks

has a non-trivial impact on the outcome of cooperation. For some topologies, a strong dilemma (char-

acterized by a high B) played in a multilayer increases cooperation, while for others topologies it makes

it decrease. This change in the number of cooperators sometimes is sufficient for the effective game

to change, while other times it is not. This same logic can be observed for weaker dilemmas. In some

topologies, the multilayer enhances cooperation and for other inhibits it.

These differences in cooperation can be justified mainly with two arguments. The first is the shift that

can occur to the roots of the AGoS, and the second is the intensity (or amplitude) of the AGoS.

50

(a)

(b)

Figure 5.10: AGoS for HR networks, during 30 generations (B = benefit, L = total number of layers, β = 1)

51

(a)

(b)

Figure 5.11: AGoS for SFBA networks with correlations, during 30 generations (B = benefit, L = total number oflayers, β = 0.1)

52

(a)

(b)

Figure 5.12: AGoS for SFBA networks without correlations, during 30 generations (B = benefit, L = total numberof layers, β = 0.1)

53

54

6Conclusion

Contents

6.1 Summary of contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.3 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

55

56

This last chapter will serve the purpose of summarizing what this thesis achieved and to shed light

on some ideas worth pursuing in the continuation of this work.

6.1 Summary of contributions

Throughout this document we always wanted to maintain a rigorous scientific and rational perspec-

tive. We did not take anything for certain, and we always questioned what we thought was plainly

obvious. This, in part, was the driving force behind wanting to implement a correct framework, validated

with as much empirical evidence as we possibly could. This makes for our first contribution: by creat-

ing our framework we independently validated numerous previous results, thus solidifying them. The

second contribution would be that we achieved results that allow us to observe how multilayers affected

the emergence of cooperation, in a plethora of network topologies, and for different game intensities (all

inside a Prisoner’s dilemma). The third and final contribution would be that we managed to explain and

justify, by using our multilayer extended version of the AGoS, how these changes in behavior occurred

inside the networks.

This allows us to claim that we reached our proposed goal to discover what game transformations

occur for some different topologies and challenges. For 5/6 of the graphics, we were able to relate

the observed FFC with the changes in the AGoS. We can also state that the AGoS is a tool able to

provide useful insight regarding behavior in multilayers, as expected, and can be applied to several

other problems.

6.2 Future work

In the end, we look back and can see that we only scratched the surface, if we consider the amount

of experiments we can realize and the amount of variables that we could consider. Here we present a

profusion of interesting extensions to this work:

• More network topologies In this work we only studied HR and SFBA in multiplex networks, yet

there are numerous other types with interesting properties that can be used.

• Mixing topologies One could also use multilayers that were comprised of different topologies

between layers, e.g the interaction between nodes playing in a SFBA and a HR.

• N-person games We applied the AGoS to 2-person games. It can also be used to study the

gradients in games occurring in larger groups.

• More values of B We have already argued that the way B affects cooperation is far from trivial,

therefore a more precise and exhaustive study would perhaps provide useful insight.

57

• More layers We already showed that the multilayers with L = 2 have an influence over the out-

come. It would be interesting to study the AGoS for more layers.

• Finer degree tuning We only studied SFBA networks which were either copies or random permu-

tations of one another. It would perhaps be interesting to study them with finer degree correlations.

• Edge overlap We applied the changes in degree by switching node i with node j in the network.

It would also be interesting to see how behavior would change if we maintained the degree and

randomized a certain amount of the edges of the nodes.

6.3 Final remarks

Generally speaking, the simpler a model, while accurately preserving the key properties of the com-

plex system one wants to address, the “better” it is. This allows for the development of a treatable or

transparent model, enough to be comprehended, and later on complexified, if needed. To a great extent,

this is what we tried to do here. Cooperation among self-regarding entities is a complex system com-

posed of many interacting components, with an emerging outcome difficult to predict, even in its simplest

form, as we have seen. This thesis represents an attempt to resort to such high-level abstractions to

characterize the complex interplay between cooperative actions and the network structure that underlies

our social interactions. There are, however, aspects that are impossible to address with the present

approach. In other words, less is not always more.

Thus, it is worth pointing out that our model does not capture the metaphysical motifs that are lin-

gering on top of the behavior observed. That is, we can analyze the gradient (or any other hypothetical

metric) and see what changes occur and try to justify behavior with them. But we do not think that we

are able to get to the driving forces behind decision-making at an individual level, the ”what” that moves

individuals to act the way they do. We think that there are structures in the human brain that guide our

interactions, decisions and the way we experience the world and perceive others, in a way that is far

from trivial and that, possibly, is unverifiable, through the study of behavior alone. We believe that this is

a realm where biology, psychology, and philosophy, have more to offer than we can.

Despite these doubts, we believe that they are a mere reflection of the limitations of the act of mod-

elling itself, and not the limitations of our work so to speak. These emerge from the fact that we chose

a simple update rule that (as any other much more complex rule) is unable to capture to perfection the

ever so complex behavior and nature of humans. Yet, if we regard the goals we had, the model proved

to be sufficient to allow us to study the behavior on a macro scale, also due to the fact that, even though

humans are individuals and can have tremendous variability in many different behavioral aspects, on

average we can take a pretty good picture using simple models, which is exactly what we did.

58

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62

AExample runs

Here we show 4 example runs that provide evidence for where would the quasy-stationary state be

(in the case of HR networks) or where would the unstable root be (in the case of SFBA networks), if we

were able to consider 100 generations, and not merely 30.

In figure A.1 we show the evolution of the fraction of cooperators in a HR network as a function of

time (in rounds), using B = 1.005 and starting from two different initial fractions fi = 0.8 and fi = 0.2. As

you can see we are able to reach a fraction of cooperators of fc ' 0.52, which is larger than our obtained

stable root xobtained ' 0.42, and closer to xexpected.

In figure A.2 we show a similar evolution but for a SFBA network with B = 1.25, starting from fi = 0.4.

To justify our difference between obtained and expected roots we use an inverted logic from previously.

Assuming we started with an initial fraction where the unstable root was (according to our obtained

AGoS), in xobtained ' 0.45 , we would expect that random runs starting in this same fraction would

evolve towards 0 or 1. This is not what we observed. They evolved solely towards 1, leading us to

the conclusion that the actual root should be in fact smaller than xobtained = 0.45. In fact, after some

attempts of different initial fractions we found out that the largest point that guaranteed reaching 0 or 1

was, as in the figure, fi = 0.4. This information, in association with the results of [6] that showed that the

63

Figure A.1: Fraction of cooperators per round, in a HR network, for B=1.005.

unstable root xexpected was larger for the first generations, allows us to conclude that, again, if we were

averaging over a larger number of generations we would obtain a root much closer to xexpected than the

one we observed.

Figure A.2: Fraction of cooperators per round, in a SFBA network ,for B=1.25.

64

BThe expected results, according to

previous papers

Here we present all the results we were aiming to recreate in our thesis, as a way to validate and

make sure that our whole implementation was correct.

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(a) Complete (b) Single-Scale

(c) Scale-free random (d) Scale-free BA

(e) HR (S varies from −2 to 1, unlikethe other figures)

Figure B.1: Expected FFC, according to [7].

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(a) HR, with β = 1

(b) SFBA, with β = 0.1

Figure B.2: Expected AGoS over 100 generations, for different values of B from [6].

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Figure B.3: Expected FFC as a function of the number of layers and the degree correlations, for a) Harmony gameand b) Prisoner’s dilemma, according to [8].

Figure B.4: Expected FFC as a function of the initial fraction of cooperators and the degree correlations, for a)Harmony game and b) Prisoner’s dilemma, according to [8].

68