BOSTON UNIVERSITY

GRADUATE SCHOOL OF ARTS AND SCIENCES

Dissertation

Interacting Particle Systems on Graphs

Vishal Sood

B.Tech., Indian Institute of Technology Bombay, Mumbai India 2000

Submitted in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy

2007

Approved by

First ReaderSidney Redner, Ph.D.Professor of Physics

Second ReaderClaudio Chamon, Ph.D.Associate Professor of Physics

3

Acknowledgments

To spend ones youth as a student, sheltered from regular life, and meet the world

at the same time, is an opportunity that only a few can afford. For many years I

have managed to evade joining the work-force. Ten years after I started university, I

see the end of my life as a student. It is hard to imagine what comes after this, but

my beneficiaries will not shelter me any more. So they bestow this rank upon me

and I acknowledge their generosity.

Sid adopted me as his student, despite my accepting admission at another univer-

sity after rejecting his initial offer. For the last five years he has guided me at each

stage of a successful research project, from finding a problem to writing convincingly

and presenting the results. He read my manuscripts more carefully than I did and at

times worked harder than me at finishing the projects. I thank him for the guidance

and the home he provided to me to begin my research career.

Although I did not work with Paul on a research project, I learned a lot from

him. He was always available for a chat about the world and was always enthusiastic

about research. Interactions with him have contributed in many ways towards my

growth as a scientist. Many thanks to Tibor for listening patiently while I rambled

on, thinking loudly before him, and for carefully double checking the calculations.

Thanks to Pablo for his encouragement and advice. I wish him and his young family

the best.

When Sid dragged us to Los Alamos I expected one long year. But Feddy made it

easier to live in the desert. Feddy was a source of fun both at work and at leisure. And

he was always there to discuss physics and to hear my philosophical rants. I thank

him for listening and keeping the house clean. Best wishes for his future research in

the pleasant Mediterranean.

Wei, the nicest guy around, was always there as a pillar of hope. I thank him for

his help and the comradeship he provided, and for allowing us to make fun of him.

Thanks to Pradeep for sharing his experience and my unhealthy habits. His

paranoia was a source of much fun.

Thanks to Aaron for sharing his perspective on politics, economy and science.

Chats with him were like a Monte-Carlo walk, constantly shifting, searching for a

meaningful ground.

Thanks to Claudio and Anna for the many festivities and for filling a dull graduate

life with uplifting spirits.

Thanks to Saikat for providing the ingredients of a relaxing vacation and for

facilitating an altered view-point.

Thanks to Sameet for tolerating the mess and for sharing my cynicism.

Thanks to ERC for the caffeine and optimal noise necassary to make calculations.

I dedicate this thesis to my father, my mother and my brother, without whose

encouragement I would not be here. They would have preferred to see me become a

“real” doctor, nevertheless a doctor now I am.

iv

INTERACTING PARTICLE SYSTEMS ON GRAPHS

(Order No. )

VISHAL SOOD

Boston University Graduate School of Arts and Sciences, 2007

Major Professor: Sidney Redner, Professor of Physics

ABSTRACT

In this dissertation, the dynamics of socially or biologically interacting populations

are investigated. The individual members of the population are treated as particles

that interact via links on a social or biological network represented as a graph. The

effect of the structure of the graph on the properties of the interacting particle system

is studied using statistical physics techniques.

In the first chapter, the central concepts of graph theory and social and biologi-

cal networks are presented. Next, interacting particle systems that are drawn from

physics, mathematics and biology are discussed in the second chapter.

In the third chapter, the random walk on a graph is studied. The mean time

for a random walk to traverse between two arbitrary sites of a random graph is

evaluated. Using an effective medium approximation it is found that the mean first-

passage time between pairs of sites, as well as all moments of this first-passage time,

are insensitive to the density of links in the graph. The inverse of the mean-first

passage time varies non-monotonically with the density of links near the percolation

transition of the random graph. Much of the behavior can be understood by simple

heuristic arguments.

Evolutionary dynamics, by which mutants overspread an otherwise uniform pop-

ulation on heterogeneous graphs, are studied in the fourth chapter. Such a process

v

underlies epidemic propagation, emergence of fads, social cooperation or invasion of

an ecological niche by a new species. The first part of this chapter is devoted to

neutral dynamics, in which the mutant genotype does not have a selective advantage

over the resident genotype. The time to extinction of one of the two genotypes is de-

rived. In the second part of this chapter, selective advantage or fitness is introduced

such that the mutant genotype has a higher birth rate or a lower death rate. This

selective advantage leads to a dynamical competition in which selection dominates

for large populations, while for small populations the dynamics are similar to the

neutral case. The likelihood for the fitter mutants to drive the resident genotype to

extinction is calculated.

vi

Contents

1 Introduction 2

2 Graphs and Networks 7

2.1 Mathematical Representation . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Complex Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Erdos-Renyi Random Graphs . . . . . . . . . . . . . . . . . . . . . . 13

2.3.1 Structure of the Random Graph: A Galton Watson Branching Process 14

2.3.2 Rooted Geodesic Tree . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Scale Free Complex Networks . . . . . . . . . . . . . . . . . . . . . . 19

2.4.1 The Configuration or the Molloy-Reed(Molloy-Reed (MR)) model 20

2.5 Preferential Attachment models for growing networks . . . . . . . . . 29

2.5.1 Growing Network with Redirection(Growing Network with Redirection (GNR)) 32

3 Interacting Particle Systems 34

3.1 Random Walks on a Graph. . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Kinetic spin systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.1 Formulation of a Kinetic Spin System . . . . . . . . . . . . . . 44

3.2.2 The Out of Equilibrium Ising Model . . . . . . . . . . . . . . 50

3.2.3 The Voter Model . . . . . . . . . . . . . . . . . . . . . . . . . 53

vii

3.2.4 The Contact Process . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.5 The Ising Model at the Spinodal Critical Point . . . . . . . . . 57

3.2.6 Applications to Ecology: A Two Species Competition Model . 59

3.3 Kinetic Spin Systems on Degree-Heterogeneous Graphs. . . . . . . . . 60

4 Random Walks on Random Graphs 64

4.1 First passage characteristics of a graph . . . . . . . . . . . . . . . . . 66

4.2 Effective Medium Approach . . . . . . . . . . . . . . . . . . . . . . . 69

4.3 Simulation Results for the ER random graph . . . . . . . . . . . . . . 75

4.4 Spectrum of the Laplacian and the First-Passage Properties . . . . . 78

4.4.1 Bounds on transit times using the spectrum of the laplacian. . 84

4.5 Relation With Electrical Networks: Commute Times and Resistances. 88

4.5.1 Resistance and Commute times . . . . . . . . . . . . . . . . . 90

4.5.2 Escape probability . . . . . . . . . . . . . . . . . . . . . . . . 94

4.6 First-Passage Properties and the Structure of the Graph . . . . . . . 96

4.6.1 Commute Rate on the RGT . . . . . . . . . . . . . . . . . . . 99

4.6.2 Role of Loops on Commute Rate . . . . . . . . . . . . . . . . 101

4.7 First-Passage Time Fluctuations . . . . . . . . . . . . . . . . . . . . . 104

4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5 The Voter Model. 112

5.1 Voter Model Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.2 The Mean-Field Voter Model: Neutral Moran Process . . . . . . . . . 116

5.3 Voter Model on a Bipartite Graph . . . . . . . . . . . . . . . . . . . . 120

5.4 Voter Model on Heterogeneous-Degree Random Graphs . . . . . . . . 126

5.5 Invasion Process on Heterogeneous-Degree Random Graphs . . . . . . 132

viii

5.6 Evolutionary Dynamics With Selection . . . . . . . . . . . . . . . . . 135

5.6.1 Fixation in the biased Invasion Process . . . . . . . . . . . . . 144

6 Conclusions. 148

7 Bibliography 149

8 Curriculum Vitae 158

ix

List of Figures

2.1 A graph with two components. . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Examples of simple graphs. . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Diameter of the RGT . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1 The random walk kernel . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Forward propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Backward propagation . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4 Potential V (φ) for the φ4 theory. . . . . . . . . . . . . . . . . . . . . 63

4.1 Schematic decomposition of a random graph . . . . . . . . . . . . . . 71

4.2 Mean commute time (dashed) and mean commute rate (solid) . . . . 76

4.3 Mean commute rate R on a random graph . . . . . . . . . . . . . . . 77

4.4 Average two-point conductance . . . . . . . . . . . . . . . . . . . . . 94

4.5 Commute rates on the RGT () and on random graphs . . . . . . . . 99

4.6 Schematic representation of the random graph. . . . . . . . . . . . . . 102

4.7 Random graph structure between two sites . . . . . . . . . . . . . . . 103

4.8 Distribution functions for the first passage times in a random graph . 107

4.9 Distribution functions for the first passage times in a random graph . 108

4.10 Expected values of the first-passage time . . . . . . . . . . . . . . . . 109

4.11 Expected values of the first-passage time . . . . . . . . . . . . . . . . 110

x

4.12 Probability distribution function for return times and first-passage times111

5.1 The complete bipartite graph Ka,b. . . . . . . . . . . . . . . . . . . . 120

5.2 Subgraph densities ρb(t) versus ρa(t) . . . . . . . . . . . . . . . . . . 125

5.3 Consensus time TN versus N . . . . . . . . . . . . . . . . . . . . . . . 132

5.4 Update illustration for two specific sites. . . . . . . . . . . . . . . . . 137

5.5 Moments of the 1 density in the biased VM and biased IP . . . . . . 142

5.6 Scaling plot of fixation probabilities . . . . . . . . . . . . . . . . . . . 144

5.7 Fixation probability of a single mutant initially at a site of degree k . 145

xi

List of Abbreviations

IPS Interacting Particle System

ER Erdos-Renyi

GC Giant-Component

MR Molloy-Reed

SPT Shortest Path Tree

RGT Rooted Geodesic Tree

KR Krapivsky-Redner

GNR Growing Network with Redirection

VM Voter Model

CP Contact Process

DP Directed Percolation

RFT Reggeon Field Theory

IMS Ising Model at Spinodal critical point

MFPT Mean First Passage Time

xii

CDF Cumulative Distribution Function

PDF Probability Distribution Function

DF Death First

BF Birth First

LD Link Dynamics

1

Chapter 1

Introduction

Many natural systems can be understood in terms of their constituent parts. These

building blocks pull and push each other so as to lead to the observation of the

system as a whole. The behavior of iron fillings subject to the influence of a bar

magnet can be understood by modeling a filling as a box of spinning particles. The

spinning particles interact with each other and the external field of the bar magnet,

so that as a box the filling moves towards the magnet. As a guiding principle, this

reductionist viewpoint has helped us understand many natural phenomena. As a tool,

the reductionist viewpoint applied in reverse has helped conceptualize and build the

modern technological wonders.

After successfully modeling physical phenomena as Interacting Particle System

(IPS) , physicists and mathematicians have begun to shift their attention to problems

of socio-biological motivation. Recent innovations in laboratory technologies have

helped biologists collect abundant data for a quantitative treatment. For example,

much progress has been made in understanding how proteins, the building blocks of

life, interact with each other. Two proteins interact if they participate in the same

biochemical reaction or bind to the same genes on the DNA or RNA inside a cell.

Each protein is modeled as a site and a bond between two sites models the interaction

2

between the respective proteins. The resulting structure of sites and bonds is a graph

and is called the protein-protein interaction network. Similarly, graphs representing

reproductive or cooperative interaction networks can be used to model an ecology or

a society. These graphs, referred to as “real-world” networks, are structurally unalike

lattices, which have been employed traditionally to model physical phenomena. Study

of interacting particle systems on graphs, which approximate the real-world networks,

has begun to reveal that traditional statistical mechanics models, such as the Ising

model and percolation, can show unexpected novel behavior. The aim of this thesis is

to formulate and study interacting particle systems on general graphs with particular

emphasis on graphs which approximate real-world networks.

The particles in an IPS can be either mobile or stationary. A mobile IPS, such

as a random walk, consists of a graph and particles which reside and move randomly

on the sites of the graph. Two randomly moving particles in the event of a collision

interact. The rules governing the dynamics of the model are graph independent and

can be adapted to study a large variety of phenomena. Particles in a stationary IPS,

such as the Ising model, are fixed at sites of the graph. The particles are given a

finite number of states such as spin up or spin down. Though the particles do not

move, they can change their state via interactions with particles on neighboring sites.

The simplest mobile IPS is the random walk model. Here one studies the prop-

erties of a random walk, such as the time it takes to transit between two given sites

in the graph. Random walk is the basic ingredient of more complicated models.

Interaction between random walks has been studied using two basic mechanisms:

the coalescing random walks and the annihilating random walks. In the coalescing

random walks system two colliding particles coalesce into one, while in the annihi-

lating random walks system colliding particles annihilate each other. In the physics

3

literature the mobile IPS have been studied as reaction-diffusion problems. A typ-

ical example consists of reactants on a substrate. The reactants, which diffuse on

the substrate, interact via mechanisms similar to coalescence and annihilation. The

interactions used in reaction-diffusion models are motivated by applications in chem-

istry and biology. In solid state physics, stationary IPS arise naturally, with particles

identified with the ions sitting at the lattice sites. Ising model, a toy model used to

study magnetism, is a typical stationary IPS that has attracted a lot of attention in

both the physics and the mathematics communities.

In the social context, formation of opinions and rumor propagation can be mod-

eled as an IPS. Each member of the society corresponds to a site in the network.

Dispersal mechanisms, such as evening gossip, can be modeled by the movement of

the rumor as a branching random walk over the social network of acquaintances.

One can use this social network IPS to determine, for example, when and how a

rumor becomes public knowledge. A simple model to study formation of opinions in

a society is the voter model, a stationary IPS. In this model the individual voters

have to choose from one of the two opinions, for example they can choose either to

be liberal or to be conservative. The voters are assumed to lack any self-confidence,

and adopt the opinion of a randomly chosen acquaintance. Within this model one

can study, for example, the probability that all the voters will eventually reach the

same opinion.

Voter model like interacting particle systems have been employed in evolutionary

population biology. Interactions between the individuals determine the reproductive

mechanisms. Each reproducing individual passes the whole, or a part of its genome

to its offspring. Following the evolutionary process in forward time, one can study

the spread of a mutant in a population. We will formulate the voter model in Chap.

4

5 as an evolutionary model and study the probability that the mutant replaces the

resident type of a population. Following the evolutionary process in reverse time,

i.e. following the genealogy of the population, one can also look for the founder of a

particular mutation and estimate the time of the first appearance of the mutation.

An active research topic in computer science is the design of better algorithms for

search on a network of computers. An example is a peer to peer file network. Each

computer stores some information or data such as music files. An individual computer

might need a data file stored on one of the other computers in the network. The goal

is to design an optimal structure of the network and appropriate algorithms such that

searching for data files is fast. These problems can be modeled as interacting particle

systems in which random walkers move on the network searching for the required

information. Another application relevant to computer science is to the spread of

viruses on the Internet. Based on the infection rates one can determine whether a

significant fraction of the computers will become infected with the virus.

In Chap. 2 we introduce graphs and networks. We include the necessary graph

theoretical definitions, after which we discuss the complexity of real world networks.

We study two examples of complex networks: the Erdos-Renyi and Molloy-Reed

graphs, which are randomly constructed approximations of real world networks. We

also include a graph generation algorithm for the preferential attachment model of

complex networks.

In Chap. 3 we introduce interacting particle systems. We discuss various models

and the motivations behind them. We discuss the mean-field approach, which can be

used to understand generic features of a model fairly easily. We also show how the

mean-field theory can be augmented to a field-theory which serves as a scaling limit

of the discrete-spatial IPS. The mean-field theory also allows us to derive parallels

5

between disparate spatial models.

In Chap. 4 we discuss the random walk on a graph. We develop an effective-

medium approach to solve for the mean transit times between the sites of a random

graph. We test our predictions with numerical simulations. We then develop an

algebraic and a geometric technique to understand the behavior of the random walk

more accurately.

In Chap. 5 we discuss the voter model on a graph with large dispersity in its degree

distribution. We present the voter model as an evolutionary process to motivate the

latter half of that chapter, where we study a process by which a mutant species

overspreads a population living on a graph.

Chapter 2

Graphs and Networks

Many processes with pairwise interactions can be formulated as processes on graphs.

Consider for example the Ising model, a model used to study the ordering properties of

a ferromagnet. The Ising model consists of spins which reside on the sites of a lattice.

Each spin can have state ↓ or ↑ The dynamics depends on the interactions between

neighboring spins, with neighborhood modeled as a bond between two sites. For the

Ising model in 1D, the set of sites is identified with the set of integers Z, and a bond

exists between any two consecutive integers. Similarly, for a model of social opinion

formation the individuals of a society are identified as the sites and the bonds are the

social friendship interactions between the individuals. In this chapter we introduce

the concept of graphs, presenting the mathematical definitions in the Sec. 2.1 along

with examples of “simple-graphs”. Complex networks are introduced in Sec. . We

discuss the Erdos-Renyi (ER) random graph in Sec. . The ER random graph is

a complex network that is easy to define and study. Many structural properties of

the ER graph have been studied, and we discuss at length properties that will be

relevant to our study of the random walk in Chap. . In Sec. 2.4 we generalize the

ER random graph to accommodate dispersity in site-connectivity, and discuss the

resulting structural properties. We end the chapter with discussion of an algorithm

8

(Sec. 2.5) that we will use to generate graphs for numerical studies in later chapters.

2.1 Mathematical Representation

To build a mathematical representation of a graph, G, we begin with a set of sites

that we denote V [1]. Any two sites x,y ∈ V may be connected by a bond (or

a bond) e ≡ (x,y). The set of these bonds is denoted E. The graph is then the

pair of the sets of sites and bonds, G ≡ (V,E). The set of bonds is a subset of the

set of all the pairs of the sites, V2. Note that by definition we allow sites to have

self-bonds, i.e. a bond of the form (x,x). An bond may be directed when (x,y) is

not equivalent to (y,x). It is possible to include more than one copy of the same pair

(x,y) in the set of bonds. Such a bond is then called a multi-bond. In this thesis we

consider graphs that are undirected and do not have any self-bonds or multi-bonds.

We include some essential definitions in this section. There is a rich theory of graphs

and the reader is referred to [1] to enrich her knowlbond.

A graph can be algebraically defined as an adjacency matrix, A. A is a symmetric

matrix defined over the set of sites, V,

Axy = Ayx = 1 if(x,y) ∈ EAxy = Ayx = 0 if(x,y) /∈ E . (2.1)

The set of neighbors of a site x, is the set,Nx ≡ y ∈ V : Axy = 1. (2.2)

9

The degree of a site is the number of its neighbors,

kx ≡ |Nx| =∑

y

Axy. (2.3)

Two sites may be connected directly by a bond, or by a sequence of bonds with

common end points. In either case we will call the two sites connected. A set of

sites C is called a cluster if any pair of sites x,y ∈ C is connected. A graph may have

more than one cluster. The different clusters are also referred to as components of

the graph. A connected graph is a graph that contains only one component.

u

w

v

x

y

z

Figure 2.1: A graph with two components.

10

For example, the graph in Fig. 2.1 contains two disjoint clusters, with sites shown

as dark and light shaded circles . x and y are in the same component of the graph

and one sequence of bonds leading from x to y is shown by thick solid lines. Such a

sequence of bonds connecting two sites of a graph is called a path between the two

sites. A bond can appear only once in a path. There may be multiple paths between

the sites. A loop is a path that ends at the starting site. For example the sequence

of bonds shown in thick gray broken line that starts and ends at w is a loop.

Most physical problems are modeled as occurring on lattices, Zd. This choice is

natural, since physical processes do occur in a spatial geometry. A lattice is simple

in the sense that any site on Zd can be described in terms of d numbers which give

the position along the axis. This allows certain symmetries which can be used to

define concepts such as distance and momentum. The simplicity of the lattice and

the fact that it mimics spatial geometry makes possible the development of powerful

intuition.

Many statistical mechanics problems study the effect of interactions on the global

properties. Interactions are local i.e. between neighbors. In other words each site

evolves under the field generated by the collective state of its neighbors. The spatial

structure of a lattice can make these problems hard to solve. The first step is then

the mean-field assumption, i.e. to assume that the field at each site is the same. This

is equivalent to studying the problem for a complete graph. A complete graph , KN ,

on N sites is a graph whose bond set is complete, i.e. all sites are connected to each

other by bonds. Thus the whole graph forms the neighborhood of each site. The

field as a result is the same for each site making the mean-field exact for a complete

graph.

Sometimes instead of making a mean-field physicists study the dynamics on a

11

Bethe lattice, also referred to as the Cayley tree. Cayley tree is an example of a tree.

A tree is a connected graph without loops. Thus only one unique path exists between

any two sites of the graph. The absence of loops makes calculation of correlations

simple and adds to the results obtained for the complete graph. Certain processes can

show new phases on trees that are not possible on lattices. Trees also form a natural

representation for various data structures in computer science and search algorithms

on these data structures can be posed as statistical mechanics problems.

However, not all trees are simple. One example that qualifies as simple is that of

homogeneous trees in which each site is assigned a fixed number, d, of descendants.

Such a tree is referred to as a d-ary tree.

In computational studies of an Ising model one allows only one spin to flip at a

time. Consider a system of N binary spins, whose phase space consists of all the

possible configurations s of the N spins. Each spin x can be in either of two states,

↑ or ↓. Thus there are 2N possible configurations. Any two states (s, s′x), that

differ only at site x, can be reached from each other via one spin flip (that of x).

Representing each state s as a site in the phase space and connecting by a bond any

two sites that can be reached via one spin flip, we arrive at a graphical representation

of the configuration space and the single flip dynamics. The graph constructed in

this fashion is a hypercube. The same hyper-graph can be used to study the shuffling

of binary strings of length N .

2.2 Complex Networks

The graphs considered in the previous section are examples of simple graphs. To

construct any of these example graphs we start with a set of N sites and follow

simple deterministic rule. As a result the amount of information in the structure of

12

a

b

c

Figure 2.2: Examples of simple graphs.(a) A hypercube. (b) A complete graph with 8 sites. (c) A Cayley tree with coordi-nation number 3.

the graph is much smaller than N . These statements can be made precise in terms

of graph symmetries and automorphisms [2].

However, graphs derived from “real world” interaction networks can be far from

simple. Such graphs, termed “complex-networks”, have been the focus of much cur-

rent investigation [3]. Consider as an example the protein interaction network of an

organism. Each protein forms a site of the graph. Two proteins that participate in

the same biochemical reaction or bind to the same genes on the DNA or RNA are

said to interact and are connected by a bond. This graph is termed the protein inter-

13

action network. There are no known deterministic rules to construct such a graph.

Normally one needs information about all the bonds to represent the graph. Hence

the protein interaction network is a complex graph.

In order to deal with the complexity inherent in the real-world networks proba-

bilistic models have been developed. By choosing to look at certain structural prop-

erties of the network, one can study a class of probabilistically generated graphs.

The simplest of these is the ER random graph [1], using which one can study the

connectivity properties of graphs as a function of the average degree of the sites.

However ER graphs have a severe limitation as models of real world networks. The

degrees of the sites in an ER graph are Poisson distributed, while the site degrees

of many real world networks show a power-law distribution. These limitations can

be accommodated by a modification of the probabilistic model used to generate ER

graphs [4]. We now turn to a discussion of these probabilistic graph models.

2.3 Erdos-Renyi Random Graphs

The term Erdos-Renyi random graph is used for a graph in which the sites are

randomly wired up to form the bonds. The ER random graph is constructed by

taking N sites and introducing a bond between each pair of sites with probability

p. When p = 1, all possible bonds exist and this construction gives the complete

graph, where each site is connected to all the other N − 1 sites in the graph. As p

decreases, the random graph undergoes a percolation transition at p = pc = 1/N [1]

[5] that shares many common features with percolation on regular lattices. Another

important geometrical feature is a second connectivity transition at p1 = lnN/N .

For p > p1, all sites belong to a single component (in the limit N → ∞), while

for p < p1 disjoint clusters can exist. In this section we study the structure of the

14

random graph and derive the phase transition at p = pc.

2.3.1 Structure of the Random Graph: A Galton Watson Branch-

ing Process

Let us construct a component of the random graph using a breadth-first search pro-

cedure. Starting at a site o we reveal the component connected to o shell by shell. o

is designated as belonging to the zeroth shell S0. The first shell, S1, contains sites at

1 hop from o, i.e. those sites that are connected to o by a direct bond. After iden-

tifying the first shell, we gather the neighbors of the sites in the first shell. Filtering

out the sites that have already been identified to lie in the zeroth or the first shell,

we assign the remaining sites to the second shell. If we group the shells 0 through l

into the ball of radius l, we can define the l + 1-th shell to consist of sites that are

not inside in the l-ball and are connected to the sites in the l-th shell.

A site y not in the l-ball will not be added to the l + 1-th shell if none of the Sl

possible bonds with the sites in the l-th shell are present. Since each of these bonds

is absent with probability 1 − p, all of them are absent with probability (1 − p)Sl. A

site, neither in the l-ball nor added to the l+ 1-th shell, remains outside the l+ 1-th

ball as well. Since there are N − Vl sites outside the l ball, we have

N − Vl+1 = (N − Vl)(1 − p)Sl,

where Vl stands for the total number of sites in the l-ball and thus N − Vl is the

number of sites outside the l-ball. Since Sl+1 = Vl+1 − Vl,

N − Vl+1 = (N − Vl)(1 − p)Sl = (N − Vl)(1 − p)Vl−Vl−1 ,

15

Iterating the last result over l till we reach the starting site o, we get,

N − Vl+1 = (N − 1)(1 − p)Vl

When p = µ1/N , with µ1 = O(1) (finite compared to N), we can approximate,

(1 − p) ≈ e−µ1/N . Notice that µ1 = pN is the average degree of the sites. Since we

are interested in the N → ∞ limit, we divide the previous equation by N and keep

only the finite terms to get,

vl+1 = 1 − e−µ1vl,

where we used vl for the normalized volume Vl/N . To get the expected size of the

component connected to o notice that no more sites are added to the component

when we reach the last shell SL. Thus vL+1 = vL ≡ β(µ1) is given by

β = 1 − e−µ1β. (2.4)

When µ1 ≤ 1, the solution to Eq.(2.4) is β = 0, which means that no component

of the graph contains a finite fraction of the sites. However at µ1 = µc ≡ 1 a

finite component emerges. This finite component is called the Giant-Component

(GC). Precise distribution of the cluster sizes is also known but harder to derive

(see [1]). Thus there is a phase transition in the structure of the ER random graph

at p = pc ≡ 1/N . With increasing µ1 the GC captures more and more sites in the

graph. The value of µ1 for which the GC contains all the sites can be derived by

looking for the value of µ1 only an isolated site lies outside the GC. i.e. β = 1−1/N .

Using this in Eq.(2.4) we can see to the highest order, µ1 = lnN (or p1 = lnN/N).

The fact that µc = 1 is closely related to the fact that the structure of the random

16

graph around a site o can be interpreted as a random branching process [5]. This

kind of branching processes were initially used to study family trees and have been

termed the Galton-Watson (GW) processes [6]. o is the progenitor and every site

in the l-th generation (l-th shell) has an i.i.d. random number of daughters drawn

from a chosen distribution. For the ER random graph the number of daughters is

Poisson distributed with mean µ1. For a general GW process it can be proved that

the process is finite (the process goes extinct for a finite generation number) when

the mean number of daughters per site is µ1 < 1. For µ1 > 1 the process does not

go extinct with positive probability. The case of µ1 = 1 is critical and the most

interesting from a mathematical perspective [6].

When µ1 ≈ 1 the components of the random graph, including the GC, are trees

(there are no loops). However for larger values of µ1, when most of the sites already

lie in the GC, loops are formed. We can however extract a tree with a shell structure

similar to the shell structure derived above for the full random graph.

2.3.2 Rooted Geodesic Tree

Network flow problems arise in the study of commodity traffic and information trans-

fer. Graphs are a natural structure to study these problems. The efficiency of a flow

on a graph can be characterized by studying, instead of the whole graph, the spanning

tree of the graph. The spanning tree is a sub-graph which spans all the sites of the

graph and does not contain any loops. A spanning tree which minimizes the transfer

times across the graph is called the Shortest Path Tree (SPT) [7]. For the random

walk on the ER random graph in Chap. 4 we will use a version of the SPT, which we

term the Rooted Geodesic Tree (RGT) (introduced by the author and collaborators

in [8]). The RGT is a specific subset of a random graph cluster that contains the

17

root site o and (i) spans all the sites in the cluster, and (ii) the distance between o

and any other site y on the RGT is also the shortest distance between these two sites

in the random graph cluster. We can then build the rest of a random graph cluster

from the RGT. (See also Sec. 5.2 of [5] for a construction related to the RGT.)

In the previous sub-section we constructed the shells surrounding o one by one.

To generate the RGT we also add the bonds between the succesive shells. However,

since we want to generate a tree we cannot add all the bonds between the sites.

We construct the RGT shell by shell as before. While generating the l + 1-th shell

we include only one bond for each site x added to Sl+1. This bond is the one that

connects x to the l-th shell. The process continues until either no unassigned sites

remain or if all attempts to incorporate the available sites into the current shell fail.

There are two important subtleties associated with this construction algorithm

for an RGT. First, bonds that are not examined in the initial construction of the

RGT can only exist between sites in the same or in adjacent shells of the RGT. A

second important point is that in building the RGT, each examined bond was tested

one time only and is therefore included in the RGT with probability p.

Since the number of sites in successive shells of the RGT grows exponentially in

the number of steps away from the root, the radius of the GC is given by the criterion

µL1 ≈ βN . By the definition of the RGT, the fraction of sites in the largest cluster

in the GC and in its underlying RGT are identical. Thus the radius of the RGT is

given by

L ∼ lnN

lnµ1

+ln β

lnµ1

. (2.5)

Since β is a rapidly growing function of µ1 [1] [5], the radius of the RGT is an

increasing function of µ1 just above the percolation threshold µc=1. This increase in

radius occurs because the RGT acquires progressively more sites with increasing µ1.

18

On the other hand for sufficiently large µ1, the GC will contain almost all sites in the

graph and the radius of this component will decrease as µ1 is increased still further.

0

2

4

6

8

10

0 1 2 3 4µ

Figure 2.3: Diameter of the RGTDiameter of the RGT as a function of the mean degree µ1 is shown as the solid curve.Broken curve is the mean commute rate on the same tree, discussed in Chap. 3

Given a RGT, it is possible to augment the tree to generate a realization of a

random graph cluster. We merely attempt to add to the RGT each of the bonds

between sites on the cluster that were not previously considered in the construction

of the RGT itself. Each such bond addition attempt is carried out with probability

p. As a result of the fact that each of these newly-added bonds and each bond in

the RGT is present with probability p, all bonds in the full random graph cluster

are present with probability p. Furthermore , both the RGT and the corresponding

19

random graph have the same number of sites and radius for the same values of µ1

and N . As a result of this equivalence, the RGT undergoes the same percolation

transition as the random graph itself when µ1 passes through 1.

2.4 Scale Free Complex Networks

The degrees of the sites in ER random graphs are Poisson distributed [1] and in the

limit N → ∞ these graphs can be approximated by a regular degree graph, i.e. a

graph in which all sites have the same degree. This limits the use of ER graphs

as models for real world networks that arise in biological and social sciences. A

prominent feature of these real world networks is that the degrees of the sites are

broadly distributed as a power law [3]. Thus the probability that a randomly chosen

site has degree k is nk ∝ 1/kγ. The power law may be valid in a certain range of the

degree: kmin < k < kmax.

If γ ≤ 1, the degree-distribution is not normalizable. For 1 < γ < 2 the distri-

bution can be normalized, but the first moment of the distribution (mean degree)

diverges with system size. 2 < γ ≤ 3 is the most interesting regime for our prob-

lems, because for these values of the exponent, the graph has a finite mean degree

but the second moment of the distribution (indicative of the heterogeneity in the de-

gree distribution) diverges with the system size. Assuming a lower cut-off kmin, and

replacing the sum over k by an integral we get the normalized degree distribution

(when kmax → ∞)

nk =γ − 1

kmin

(kmin

k

)γ

. (2.6)

20

The mean degree µ1 ≡∑

k=kminknk is,

µ1 =γ − 1

γ − 2kmin (2.7)

So far we have assumed that the power law exponent γ > 2. For 2 < γ ≤ 3, the

second moment, µ2 ≡∑kmax

k=kmink2nk, of the degree distribution diverges with the

system size. For a large system size we can approximate,

µ2 ≈γ − 1

3 − γ

kγ−1min

k3−γmax

. (2.8)

A natural upper-bound for kmax can be derived by requiring that there be only

one site with degree larger than kmax, i.e.∑

k=kmaxnk = 1/N . This gives us

kmax ≤ O ( N1

γ−1

)(2.9)

However as we shall see, in certain cases it is necessary to bound the site degrees

with a value smaller than the right hand side of the above equation.

2.4.1 The Configuration or the Molloy-Reed(MR) model

It is possible to generalize the ER random graph model to generate random graphs

with any feasible values of the site degrees. We start with each site x assigned kx

stubs (or half bonds), where kx is the desired degree of x. Instead of choosing any

two sites for a possible bond, we choose two free stubs and match them to form

a bond. The chosen pair is rejected if it leads to multiple bonds between the sites

carrying the stubs or to self loops [4]. To generate a heterogeneous degree-distributed

random graph, the site degrees are drawn from the desired distribution. Graphs

21

generated in this fashion are maximally random graphs that satisfy a required degree

distribution. Thus the MR random graph can be used as a null model to study the

effect of degree-heterogeneity on the structural properties of complex networks, as

well as the effect on properties of interacting particle systems. In this sub-section

we study the structural properties of the MR random graph. We will reveal a shell

structure caused by the dispersity in the degree-distribution such that the high degree

sites form a densely connected core and are surrounded by shells containing sites of

sequentially decreasing degree. A similar shell-structure has been reported in the

structure of the Internet at the autonomous systems (AS) level [9].

Two sites x and y will not share a bond if none of the stubs connected to x is

chosen along with a stub attached to y. This happens with a probability,

P [Axy = 0] =

(1 − 1

µ1N

)kxky

∼ exp

(−kxky

µ1N

)(2.10)

The probability that x and y are connected by a bond is therefore

P [Axy = 1] = 1 − e−kxky/µ1N , (2.11)

where if we consider kxky ≪ µ1N,

P [Axy = 1] ≈ kxky

µ1N(2.12)

All x and y will satisfy the above equation if we put an artificial cut-off on kmax,

kmax ≤ O(N1/2). (2.13)

22

The mean degree of the neighbors of x is,

knn(x) ≡ 1

kx

∑

y

kyAxy. (2.14)

In the MR model all sites of degree k are equivalent. Thus we would expect knn(x)

to be a function of kx only. If it turns out that knn(x) is independent of x, the graph

is said to lack degree-degree correlations [10] [11]. This is the case if Eq.(2.12) is

accurate for all x and y, which yields,

knn =µ2

µ1. (2.15)

It might be the case that sites with a high degree have a higher probability to connect

to other sites of high degree. One can imagine this sort of “assortative mixing” in

social networks, such as scientific collaboration, in which more connected individuals

share more bonds with other highly connected individuals [10]. The other possibility

is when sites connect “dissassortatively”, i.e. sites of higher degree share more bonds

with sites of lower degree. This is the case for social networks such as the Internet and

the world-wide web and biological networks such as the protein interactions network

[10].

If we use kmax given in Eq.(2.9) for the MR model, the resulting graph turns out

to be dissassortatively mixed [11]. The MR random graph without a degree cut-off

has a very interesting shell structure with a highly connected core, a structure that

has also been observed in many real world networks [12] [13] [9] [14] [15] . However if

we want a random graph without degree-degree correlations then we should impose

the cut-off in Eq.(2.13). Since this cut-off causes the probability of bonds to take the

simple form in Eq.(2.12), it also facilitates analytical treatment of various processes.

23

Shell structure of the MR random graph without a degree cut-off

Similar to the shell structure of the ER random graph in Sec. 2.3, we now reveal the

shell structure of the MR random graph. Unlike the sites in the ER random graph,

the degrees of the sites in the power-law MR random graph will have be broadly

distributed. Thus we explore the structure of the MR random graph starting with

sites of highest degree. Consider the subset, Ck ⊂ V, consisting of sites with degree

larger than k. Using Eq.(2.10) we find that the probability that there is no bond

between two sites x,y ∈ Ck satisfies,

P [Axy = 0] < exp

(− k2

µ1N

), (2.16)

which gives us for the probability that the bond exists,

P [Axy = 1] > 1 − exp

(− k2

µ1N

), (2.17)

from which we can get the lower bound for the probability that the sub-graph con-

sisting of sites in Ck is complete,

P [Ck is complete] = P [Axy = 1 ∀x, y ∈ Ck]

=∏

x∈Ck

∏

y∈Ck

P [Axy = 1]

>

(1 − exp

(− k2

µ1N

))Ck(Ck−1)/2

∼ exp

(−1

2C2

ke−k2/µ1N

)

24

where Ck is the number of sites with degree larger than k,

Ck =∑

x:kx>k

1 = N∑

k′=k

nk′

∼ N

∫ kmax

k

dk

kγ

∝ N

kγ−1.

(2.18)

Thus the probability to find a complete clique in the network increases with k. If

k > O(k0), where

k0 ≡ ((3 − γ)µ1N lnN)1/2, (2.19)

the probability to find a complete clique becomes 1. The complete sub-graph con-

sisting of sites of degree larger than k0 is the core, O, of the graph. The core is

surrounded by sites of degree lower than k0. The total number of bonds coming out

of the sites in the core is,

L0 = N∑

k=k0

knk ≈ N

∫

k0

1

kγ−1∝ N

kγ−20

∝ N

(N lnN)(γ−2)/2(2.20)

Consider a site x of degree k such that k < O(k0). x does not share a bond with

any of the sites in the core if none of the k stubs attached to x are chosen along with

any of the L0 stubs attached to the sites in the core. Thus the probability that x is

not connected to any site in the core via a direct bond is,

P [x not connected to the core] =

(1 − 1

µ1N

)kL0

≈ e−kl0/µ1n,

25

which will go to zero in the limit N → ∞ if the exponent, kl0/µ1n, increases with

N . This is the case when k > O(k1) where

k1 ≡N

L0= (N lnN)(γ−2)/2. (2.21)

Thus the set of sites with their degree k in the range k1 < O(k) < k0 form a shell S1

around the core O, such that each site in this shell shares at least one direct bond

with the core.

We can further find a k2 such that a site with degree k > O(k2) is guaranteed to

have at least one bond to the core or the first shell. For this we need the number of

stubs attached to the sites in the core and the first shell,

L1 ∝N

kγ−21

∝ N

(N lnN)(γ−2)2/2. (2.22)

Proceeding by the same method used to find k1,

k2 ≡N

L1= (N lnN)(γ−2)2/2, (2.23)

This calculation can be iterated to define,

kl ≡ (N lnN)(γ−2)l/2, (2.24)

which defines the l-th shell and can be used to yield the diameter of the graph (largest

number of hops between any two sites). kl is a decreasing function of l. The sites

which will lie in the outer-most shell will be those which have a degree of O(1).

Since the decrease with l is super exponential, the sites furthest from the core will

26

be reached very quickly,

lmax = O(ln lnN), . (2.25)

Since, according to Lev Landau, “a chicken is not a bird and a log is not a function”,

the double log in the above equation means that the diameter of a MR random graph

with degree exponent in the range 2 < γ < 3 is small even for fairly large N .

The Rich Club Connectivity

We have seen that the power-law MR random graph with exponent in the range

2 < γ < 3 and without any degree cut-off has a completely connected core. This

prompts us to study the clique coefficient of a subset of sites in the graph. The

clique coefficient of a set S of sites measures how well an average site of S is connected

to the other sites of S.

q(S) = 2|E(S)|

|S|(|S| − 1)(2.26)

where E(S) is the bond set of the sub-graph consisting of the sites in S, i.e. it

contains the bonds between sites in S. Thus q(S) = 1, means that all the sites is S

are connected to each other. This is the case for the core in the MR graph. In [9] and

[13] the authors have discussed the clique coefficient of the Internet at the AS router

level as a function of the degrees of the vertices in the subset S. They have compared

the results to various models proposed for the Internet. Here we calculate the clique

coefficient for the MR random graph. We consider all the sites of degree larger than

or equal to k. There are Mk = N∑

k′=k nk′ = NA/(γ − 1)kγ−1 such sites. A is the

normalization constant for the distribution nk = A/kγ. Two sites of degrees k′ and

k′′ are connected by a bond with a probability 1− e−k′k′′/µ1N according to Eq.(2.11).

27

Thus the number of bonds from sites of degree at least k to other sites of degree at

least k is

Lk = (AN)2∑

k′,k′′=k

1 − e−k′k′′/µ1N

(k′k′′)γ,

while there are almost M2k possible bonds between these sites. Thus the clique coef-

ficient of the set consisting of sites of degree at least k is,

qk ≡ Lk

M2k

= (γ − 1)2k2(γ−1)∑

k′,k′′=k

1 − e−k′k′′/µ1N

(k′k′′)γ(2.27)

For small values of k, (k < Oõ1N), we can approximate the exponential to linear

order to get,

qk ≈(γ − 1

γ − 2

)2k2

µ1N. (2.28)

When k2 ≥ N lnN , e−k′k′′/µ1N < e−k2/µ1N < 1/N → 0, with N → ∞. This results in

qk = 1. We should have expected this result since the sites with k >√N lnN form

the completely connected core of the graph as observed in the precious subsection.

To evaluate the sum in Eq.(2.27) we replace the degrees k′ and k′′ by x = k′/k and

y = k′′/k and the sum by an integral,

qk ≈ (γ − 1)2

∫ ∞

1

dx dy1 − e−(k2/µ1N)xy

(xy)γ.

Since x, y > 1,

e−(k2/µ1N)xy < e−k2/µ1N ,

which enables us to lower-bound the integral for qk,

qk > 1 − e−k2/µ1N ,

28

which gives us the correct order of magnitude dependence on k2/µ1N for small k,

and that qk ≈ 1 when k ≥√µ1N lnN .

We can characterize the clique coefficient in terms of the connectivity-rank, r,

of the sites. The site with the highest degree has rank R = 1, while the lowest

connected site will have a rank equal to N , the total number of sites in the graph.

We can normalize the rank by the number of sites N to define the normalized rank,

r(k) ≡ R

N=∑

k′=k

nk =A

γ − 1

1

kγ−1. (2.29)

We can invert this expression to obtain k in terms of r and use the result in Eq.(2.28)

to get

q(r) ≈ µ1

r2/(γ−1)N, (2.30)

where we have used the normalization

A = ((γ − 2)γ−1/(γ − 1)γ−2)µγ−11 .

The result in Eq.(2.30) is valid when the normalized rank satisfies

1 > r ≫(

(γ − 2)√µ1

(γ − 1)√N

)γ−1

,

while for r < ((γ − 2)√µ1/(γ − 1)

√N lnN)γ−1, q(r) ≈ 1. Our results for the clique

coefficient for the MR random graph are comparable to those obtained for the Internet

at the AS-level and better than various models explored in [13] and [9].

We can conclude that certain connectivity patterns seen in real world networks,

such as the presence of a densely connected core surrounded by shells of sequentially

decreasing connectivity, can arise in randomly connected heterogeneous graphs as

29

well. This points to the need of careful analysis of data for real-world networks

while looking for structural regularities. In the next section we discuss some models

for growing networks using preferential attachment that lead naturally to power-law

behavior in the degree-distribution.

2.5 Preferential Attachment models for growing networks

Growth by preferential attachment has been proposed as a model for power-law dis-

tributed growing networks such as the web-graph and the Internet. Here we describe

the model and report major results and point the reader to the literature for more

information about their use and details of the calculations.

Consider a growing network in which sites are added one at a time. Each newly

added site attaches to m different previously existing sites. Each of these m sites

is selected independently with an attachment probability Ak that depends only on

the degree k of the “target” site. We restrict ourselves to the case m = 1 to state

the results for the growing network. Results for general m are analogous and can be

found in the literature.

Since at each time step we add one site to the graph, we can use the number of

sites N to parametrize time. The number of sites of degree k evolves according to

the following rate equation,

dNk

dN=Ak−1Nk−1 − AkNk

A+ δk,1. (2.31)

.

The newly introduced site might bond to a site of degree k − 1, which increases

the number of sites of degree k by 1. The probability for this event is Ak−1Nk−1/A,

30

where A =∑

j AjNj is the normalization. The second term in the above equation is

the loss term, which accounts for the event that the new site bonds to a site of degree

k. The last term is for the newly added sites which have degree 1. Notice that since

∑j Nj is the total number of sites in the graph,

∑

j

Nj = 1, (2.32)

as we add one site every time-step. Further∑

j jNj is twice the number of bonds in

the graph and since we add 1 bond ever time-step,

∑

j

jNj = 2. (2.33)

The last two expressions suggest that we should try Nj = njN as a solution. Here

nj is the fraction of sites of degree j. This allows us to replace the rate equation

Eq.(2.31) by,

nk =Ak−1nk−1 −Aknk

ξ+ δk,1 (2.34)

where ξ ≡∑

j Ajnj . Thus n1 = ξ/(ξ + A1) and for k > 1, the degree distribution

satisfies the recurrence relation,

nk =Ak−1

Ak

1

1 + ξ/Aknk−1, (2.35)

which can be solved to give,

nk =ξ

Ak

k∏

j=1

(1 +

ξ

Aj

)−1

. (2.36)

31

Specializing to the case of a linear kernel of the form Aj = j+w, we can see that

ξ =∑

j

(j + w)nj = 2 + w,

which allows us to write the recursion relation Eq.(2.35) as,

nk =k − 1 + w

k + 2 + 2wnk−1 =

1 + (w − 1)/k

1 + 2(w + 1)/knk−1. (2.37)

To calculate the degree distribution nk as k → ∞ we take the logarithm of both the

sides and rearrange the terms to find,

d lnnk

dk∼ lnnk − lnnk−1 = −3 + w

k,

which tells us that for large the degree distribution follows a power-law,

nk ∼ 1

kγ, γ = 3 + w. (2.38)

The particular case of w = 0 corresponds to the Barabasi-Albert model [16] with

scaling exponent γ = 3.

Generalization of the above results to the case in which the newly added site con-

nects to m > 1 pre-existing sites is straightforward [17]. The rate equation governing

the evolution of the degree-distribution is very similar to Eq.(2.31),

dNk

dN= m

Ak−1Nk−1 − AkNk

A+ δkm, (2.39)

and can be cast as a recurrence in similar way after noticing that for the shifted linear

32

kernel Ak = k + w, A =∑

j(j + w)Nj = (2m+ w)N ,

nk =1 + (w − 1)/k

1 + (w + 2 + w/m)/knk−1, (2.40)

for k > m. Thus for large k we have, akin to Eq.(2.38),

nk ∼ 1

kγ, γ = 3 +

w

m. (2.41)

The growing network has also been solved for sub-linear and super-linear kernels

[18] [17].

2.5.1 Growing Network with Redirection(GNR)

Since the probability to attach to an existing site depends on its degree, a straight-

forward algorithmic implementation of the rules described in the last section will

require us to track the global degree distribution as the network grows. Here we dis-

cuss an algorithm which reduces the computational requirements as well as motivates

its use as a model for various growth processes. We will call the graphs grown using

this algorithm Krapivsky-Redner (KR) graphs, after the authors who first suggested

this algorithm in [17].

While growing the graph each bond is given an orientation when it is formed. If

the newly added site n forms a bond with an existing site x, we can picture the bond

as an arrow from n to x and consider x as an ancestor of n. At each time step,

a new site n is added to the graph which forms bonds to m already existing sites.

For each of these m bonds a site x is first selected randomly from the set of existing

sites. With probability 1 − r n bonds to x, i.e. a bond is formed between n and x.

With the remaining probability r x redirects the bond to one of its ancestors. The

33

resulting rate equation for Nk is

dNk

dN= δkm +m(1 − r)

Nk−1 −Nk∑j Nj

+mr(k − 1 −m)Nk−1 − (k −m)Nk∑

j(j −m)Nj. (2.42)

The first term on the r.h.s. is due to the newly added sites which always form m

bonds to existing sites. The next two terms account for the undirected linking, while

the last two terms are due to the possibility of redirection. The factor m counts the

number of new bonds added. By recombining the various terms on the right we can

reduce Eq.(2.42) to the recurrence,

nk = δkm + r[(k − 1 +m

(1r− 2))nk−1 −

(k +m

(1r− 2))nk

],

from which we find that for k > m,

nk =1 + (m(1/r − 2) − 1)/k

1 + (1/r +m(1/r − 2))/knk−1

which has the same form as the recurrence for shifted linear attachment kernel in

Eq.(2.40). Using the method adopted to find the asymptotic solution to Eq.(2.37)

and Eq.(2.40) we find that the procedure of growing a network with redirection results

in a graph with a power-law degree distribution,

nk ∼ 1

kγ, γ = 1 +

1

r. (2.43)

Notice that the power-law exponent s independent of the number of bonds added for

every new site.

Chapter 3

Interacting Particle Systems

In this chapter we set out some formalism for interacting particle systems on graphs.

The first set of models that we consider are random walk models. In the second

section we develop a formalism for the study of kinetic spin systems and discuss

important concepts used in the study of spin systems on lattices. Following this

general exposition we review examples of kinetic spin systems drawn from diverse

topics in physics, mathematics and biology and point out important connections

between different models. In the third and the last section we discuss particularities

of spin systems on degree-heterogeneous graphs.

3.1 Random Walks on a Graph.

The dynamics are defined for a random walker, W , who occupies the vertexes or sites

of a graph. Here we limit our models to only single occupancy of the sites. Thus a

site x may be either occupied (η(x) = 1) by a random walker or vacant (η(x) = 0).

A random walker hops to one the neighboring sites of its current position. This

34

35

underlying rule can be written as the probability of jump from site y to site x,

pxy =1

ky

Axy, (3.1)

where the adjacency matrix element Axy takes into account the fact that for the

random walker to jump from y to x the two sites should be connected by a direct

site. The factor 1/ky is the probability of choosing x out of the ky neighbors of y.

The jump probability is the most basic quantity in this thesis. It not only forms the

definition of the random walk, but also the underlying mechanism of the kinetic spin

systems that we will discuss later.

x

__4

__ =1ky

pxy =

=p =1k

__ 1__yx

x 3

y

1

Figure 3.1: The random walk kernelThe random walk at y, which has degree 4, jumps to the neighbor x with probabilitypxy = 1/4 while the reverse transition probability from x to y is pyx = 1/3

We formulate the evolution of the random walk in continuous time. The random

36

walker spends an exponentially distributed time interval at its current position, after

which it jumps to one of the neighboring sites chosen according to Eq.(3.1). That is,

given that the random walker is at y at time t, the probability that it will exit y in

the time interval [t, t + δt) is δt. If we set δt = 1/N , an algorithmic interpretation

of these rules is: (i) at each elemental time step pick up a site y randomly; (ii) if

y is occupied move the random walker to a neighbor of y. Let c(x, t; s, 0) be the

transition probability of the random walk from site s to x in time t. Thus c(x, t; s, 0)

is the occupation probability of x by a random walk that starts at o. In a single time

step the random walk can arrive at x from one of its neighbors,

c(x, t+ δt; s, 0) =∑

y

c(x, t+ δt; y, t)c(y, t; s, 0). (3.2)

A cartoon explaining Eq.(3.2) is shown in Fig. 3.2. Using single update transition

rules we get,

c(x, t+ δt; s, 0) =1

N

∑

y

Axy

1

ky

c(y, t; s, 0) +

(1 − 1

N

)c(x, t; s, 0),

where 1/N is the probability of choosing a site. The first term sums over the events

in which the random walker moves to x from one of its neighbors. The second term

is for the case when the random walker, already at x, does not leave during the time

interval (t, t + δt). Subtracting c(x, t; s, 0) from both sides of the above equation,

37

=

δ

c(x,t+ t;s,0)δ

x y s

c(y,t;s,0)c(x,t+ ;y,t)Σy

Figure 3.2: Forward propagationThe random walk, starting at s will reach x in time t + δt by arriving at one ofthe neighbors (grouped as the shaded area) of x and then stepping to x in a singleupdate. The gray solid lines are bonds attached to x. The broken curves are fortransitions between the two sites at the ends of each curve, with the dashed curvedstanding for a single step of the random walk.

dividing by δt ≡ 1/N gives us,

d

dtc(x, t; s, 0) =

∑

y

Axy

1

ky

c(y, t; s, 0) − c(x, t; s, 0)

= −∑

y

(kxδxy −Axy)1

ky

c(y, t; s, 0)

≡ −∑

y

L⋆xyc(y, t; s, 0), (3.3)

38

where we have introduced the laplacian matrix L⋆, with elements,

L⋆xy = (kxδxy −Axy)

1

ky

. (3.4)

The manifest asymmetry in the laplacian, L⋆xy 6= L⋆

yx, arises because of the degree-

heterogeneity of the graph.

Eq.(3.3) is an instance of the Kolmogorov’s forward propagation equation or the

Fokker-Planck equation [19]. We could also investigate the effect of changing the

starting site s on the evolution of c(x, t; s, 0) (see Fig. 3.3),

c(x, t+ δt; s, 0) =∑

y6=s

c(x, t+ δt; y, δt)c(y, δt; s, 0) + c(x, t+ δt; s, δt)c(s, δt; s, 0)

=∑

y6=s

c(x, t; y, 0)c(y, δt; s, 0) + c(x, t; s, 0)c(s, δt; s, 0),

where the second line follows by using the Markov property of the random walk.

Since δt is an elemental time step, the random walk can reach y starting at s in only

a single hop,

c(y, t; s, 0) =1

NAys

1

ks

,

and the probability c(s, δt; s, 0) that a random walk at s does not leave that site in

an elemental time step is

c(s, δt; s, 0) = 1 − 1

N,

39

=

δ δ c(y, t;s,0)δ

c(x,t+ t;s,0)δ

x y s

Σy

c(x,t+ ;y, t)

Figure 3.3: Backward propagationThe random walk, starting at s will reach x in time t + δt by first stepping, in asingle update, to one of the neighbors (in the shaded area) of x and then arriving atx in the remaining time, t. The gray solid lines are bonds attached to b. The brokencurves are for transitions between the two sites at the ends of each curve, with thedashed curved standing for a single step of the random walk.

which enables us to write the evolution equation,

d

dtc(x, t; s, 0) =

∑

y

Ays

1

ks

c(x, t; y, 0)− c(x, t; s, 0)

= − 1

ks

∑

y

(ksδsy −Asy)c(x, t; y, 0)

= −∑

y

Lsyc(x, t; y, 0), (3.5)

40

where we identified another possible definition of the laplacian,

Lxy =1

kx

(kxδxy −Axy). (3.6)

Eq.(3.5) is the Kolmogorov’s backward propagation equation [19] and L the genera-

tor for backward propagation. The backward propagation equation tells us how the

starting state effects the transition probability to a given state. Backward propaga-

tion is extremely useful while calculating expected values such as the expected time

to arrive at (or the hitting time of ) a fixed site z starting at x, τx,z:

τx,z =1

kx

∑

y

Axy(τy,z + 1),

where we noticed that the random walk starting at x will jump to one of its neighbors

y with probability 1/kx. After spending a unit of time in the first step, the random

walk takes time τy,z from y to reach the target. Time to hit the target starting at z

itself is zero. The above equation for the hitting times can be cast in the form,

∑

y

Lxyτy,z = 1, (3.7)

where we used the alternate definition of the laplacian in Eq.(3.6). We will use the

backward propagation sketched here extensively in the later chapters to calculate

the hitting times, exit times as well as exit probabilities for the various processes we

study.

Unlike degree-regular graphs such as lattices, for which Eq.(3.4) and Eq.(3.6)

are equivalent, for degree-heterogeneous graphs we have to deal with two possible

choices for the laplacian. However in Chap. 3 we will avoid this need to work with

41

two definitions by defining a symmetrized form for the laplacian.

Consider the diffusion of a mass over a graph. The evolution of the mass distri-

bution is given by Eq.(3.3). The total mass is conserved, which can be checked by

summing Eq.(3.3) over x. For degree-regular graphs, since they have same number of

neighbors, all the sites are equally attractive for the random walker. However when

the degrees of the sites can vary, the stationary state occupation of the random walk

is proportional to the degree of the site,

πx =kx

µ1N. (3.8)

Thus more mass is concentrated on sites of higher degrees.

Possibility of the alternate form of the laplacian in Eq.(3.6) compels us to consider

an alternate mechanism for the diffusion of mass over the graph,

dc(x, t)

dt= −

∑

y

Lxycxy =1

kx

∑

y

Axyc(y, t) − c(x, t), (3.9)

The underlying process is: at each time instance, (i) a site x is chosen randomly and

(ii) the mass at x is set equal to the average of the masses at its neighbors. Unlike

the mass distribution evolving according to Eq.(3.3), the mass is not conserved,

d

dt

∑

x

c(x, t) 6= 0.

However the degree weighted mass is conserved,

d

dt

∑

x

kxc(x, t) = 0.

42

The generator on the right hand side of the Fokker-Planck Eq.(3.9) is, in fact, the

backward propagator in Eq.(3.5) of the the random walk. We will use the inherent

asymmetry in the random walk propagators for a degree-heterogeneous graph to write

two different mechanisms for the same process in Chap. 4.

3.2 Kinetic spin systems.

In this section we review some statistical mechanics models that we call kinetic spin

systems. The essential feature of these models is that the rules governing the ki-

netics of each spin are explicitly stated. Such models were first studied in physics

literature by Glauber to understand the time dependent behavior of the Ising model

[20]. The Ising model is a simple toy model using which various equilibrium as well

as off-equilibrium physical systems can be studied. Since the systems are physical,

the equilibrium distribution is the Maxwell-Boltzmann distribution, which motivates

the use of detailed balance to write down the Metropolis evolution rules to study the

off-equilibrium behavior. For non-equilibrium models motivated by biological phe-

nomena such as dynamics of an ecology, however, detailed balance cannot be used

due to the lack of an equilibrium state with a Maxwell-Boltzmann distribution. The

dynamics can however be written directly as an equation of motion for all the degrees

of freedom involved. These degrees of freedom are quantities such as the species of

the individual occupying the coordinates r, t. Many such models have been studied

in mathematical ecology as well as probability theory [21], [22].

We begin with a discussion of the Ising model in sub-section 1. We will formulate

the Ising model as a spin system with a prescribed dynamics. Despite the apparent

simplicity of the model, the non-linear dynamics make the Ising model notoriously

hard to solve. In sub-section 2 we will discuss the Voter Model (VM), a linear

43

spin system dynamics that can be solved exactly. The voter model is a pure non-

equilibrium model in the sense that an equilibrium phase cannot be defined. The

voter model studies the dynamics of two competing species, in which none of the two

species has any advantage over the other [21]. As a result the clusters in the voter

model lack surface tension and tend to be rarefied. The interface between the two

phases (areas occupied by the competing species) is also diffuse [23]. Phase ordering

thus turns out to share many features with the phase ordering in an Ising model at

its critical point.

The next model that we will discuss in sub-section 3 is the Contact Process (CP).

The CP is a simple model for infection spreading. The two phases are the infected

and the susceptible. CP is related to the Directed Percolation (DP) model studied in

the statistical mechanics literature and the Reggeon Field Theory (RFT) studied in

high energy physics literature. It also turns out that the field theory for the contact

process looks the same as the field theory for the Ising Model at Spinodal critical

point (IMS). The difference between the field theories for the IMS and the CP is in

the noise term. In sub-section 4 we will formulate the IMS field theory to make it

look similar to the field theory for the CP.

After the discussion of the basic models we will discuss an application in sub-

section 5. The focus will be on ecological models where phenomena similar to nucle-

ation have been observed ([24], [25]). A simple example from theoretical ecology is a

model to study the take over of a grassland by an alien (non-resident) species of grass.

Voter model or reaction-diffusion like dynamic rules can be readily written for the

evolution of the two species. Some studies [24] [25] have indicated classical-nucleation

like phenomenon in which colonization by the alien species is curvature driven. Thus

a droplet of the advantageous alien species, larger than a critical size, will eventually

44

take over the entire grassland, pushing the resident grass to extinction [26].

Dynamics of species abundances in tropical rain forests has been studied using

models similar to non-equilibrium statistical mechanics models. One has to generalize

from a two spin model to a multispin Potts like model. An outstanding problem has

been to understand the diversity of species which simple models fail to predict. A suc-

cessful theory is neutral, i.e. it requires that the different species be equally adapted

to the ecology [27], [28]. Any competitive advantage destroys the diversity. However,

the assumption about equivalence of the species is not empirically supported. Given

the extent of human interference in the mechanisms underlying ecology it is of in-

creasing importance to know how robust the mechanisms governing bio-diversity are.

In the chapter about the voter model we will briefly study the multi-state case, which

serves as the first step towards studying multi-species competition in an ecology.

3.2.1 Formulation of a Kinetic Spin System

Each site of a graph is endowed with a spin like variable with two or more possible

states. In the case of a binary spin variable these states will be ↑ and ↓. The kinetic

spin system is then specified by the rules governing the evolution of the spins. At

each time instance a site, x, is chosen at random. The spin at x, σx, is flipped with

a probability which is a function of the average of the spins in the neighborhood of

x. We call the average spin of the neighborhood the field at x,

φ(x) =1

kx

∑

y

Axyσy. (3.10)

45

Thus if σx = −1, the probability to flip is a function of the form,

p↓→↑ ≡ P [σx :↓→↑] = f (φ(x)) (3.11)

Notice that the argument of the function can also be interpreted as the fraction of

the neighbors of x in the spin state ↑. In order to have a global spin-flip symmetry

we require that the reverse spin flip probability is

p↑→↓ ≡ P [σx :↑→↓] = f (−φ(x)) (3.12)

For population biology models, the sites of the graphs will be considered as either

occupied (η(x) = 1) or unoccupied (η(x) = 0) by an individual of the population.

The dynamics consist of the occupants of the site dying and being replaced by the

off-springs of the neighbors. Let η be the state of the ecology. Then at any elemental

step η can change at only one site of the graph. We represent the state obtained by

flipping the state of site x by ηx:

ηx(y) =

η(y); y 6= x

1 − η(x); y = x

(3.13)

Representing by η the state of the ecology, the transitions can be specified by the

probability that the state of a site changes, an event which we will call a flip event,

P [η → ηx] = η(x)p1→0(ξ(x)) + (1 − η(x))p0→1(ξ(x)), (3.14)

where the first term is for the case when x is occupied while the second term for

the case when x is vacant. Thus p1→0(ξ(x)) is the probability that the occupant

46

of x dies, while p0→1(ξ(x)) is the probability that the vacant site x is occupied by

the offspring of one of its neighbors. Both these probabilities are functions of the

environment ξ(x) of site x:

ξ(x) ≡ 1

kx

∑

y

Axyη(y). (3.15)

The ecological formulation outlined above can also be interpreted as a competition

model between two species 0 and 1. In an ecology this would correspond to the

case when all the sites of the graph are occupied either by the resident type 0 or the

mutant type 1. The quantity of interest in this process is then the probability of

takeover by the mutant, a question we address in Chap. 4.

The simplest graph to study the interacting particle systems on is the complete

graph. The field in Eq.(3.10) at each site of the complete graph is the same and

is equal to the mean spin or mean occupation of the sites. The transition or jump

probabilities in Eqs.(3.11, 3.12 &3.14) can be written in the form,

F (φ) = P [φ→ φ+ δφ]

B(φ) = P [φ→ φ− δφ]. (3.16)

Let c(φ, t) be the probability that the mean spin is φ at time t. After one elemental

time step,

c(φ, t+δt) = F (φ−δφ)c(φ−δφ, t)+B(φ+δφ)c(φ+δφ, t)+(1−F (φ)−B(φ))c(φ, t),

(3.17)

where the first two terms are inputs to the state φ while the third term is the prob-

ability that the system leaves φ during the time interval (t, t + δt). We can expand

47

the transition probabilities to second order in φ,

F (φ− δφ) ≈ F (φ) − δφ∂

∂φF (φ) +

1

2(δφ)2 ∂

2

∂φ2F (φ),

B(φ− δφ) ≈ B(φ) + δφ∂

∂φB(φ) +

1

2(δφ)2 ∂

2

∂φ2B(φ), (3.18)

as well as the occupation probabilities,

c(φ− δφ, t) ≈ c(φ, t) − δφ ∂∂φc(φ, t) + 1

2(δφ)2 ∂2

∂φ2 c(φ, t)

c(φ+ δφ, t) ≈ c(φ, t) + δφ ∂∂φc(φ, t) + 1

2(δφ)2 ∂2

∂φ2 c(φ, t), (3.19)

Using in Eq.(3.17) the approximations, for the transition probabilities in Eq.(3.18)

and for the occupation probabilities in Eq.(3.19), we get the Fokker-Planck equation

for the spin system on a complete graph,

∂c(φ, t)

∂t= − ∂

∂φvd(φ)c(φ, t) +

∂2

∂φ2ζ(φ)c(φ, t), (3.20)

where

vd(φ) ≡ δφ

δt(F (φ) − B(φ)),

is the drift caused by the bias in the transition probabilities to move forward or

backward in φ, and the diffusion term,

ζ(φ) ≡ 1

2

(δφ)2

δt(F (φ) + B(φ)), (3.21)

48

is a measure of the noise caused by the stochasticity in the kinetics. We can also

write the rate at which φ changes as a Langevin equation,

∂φ

∂t= vd(φ) +

√ζ(φ)ν(t), (3.22)

where the stochasticity in the kinetics is encoded as the Gaussian white noise ν(t) of

unit intensity.

The Fokker-Planck equation (Eq.(3.20)) is the basic equation in our attempt to

understand the dynamics of kinetic spin systems on a graph. Much can be understood

about the model under study by comparing the magnitudes of the drift and the

diffusion terms in Eq.(3.20). For a complete graph with N sites, δφ = O(N). We

also set δt = δφ which accounts to flipping one spin in one elemental time step.

Then the factor outside the derivative in the second term in Eq.(3.20) (δφ)2/δt =

O(1/N) → 0 as N → ∞. Thus for large systems the noise (diffusion) term is small

and present as a finite size effect, while the drift dominates the stochastic evolution

as long as vd > O(1/N). From Eq.(3.21) it is clear that the diffusion term measures

the probability of a flip event. For systems which are driven by an external source

of noise, such as the Ising model in contact with a heat bath, the probability of a

flip event depends on an external parameter such as the temperature T of the heat

bath. This can be taken into account by setting the diffusion term artificially rather

than deriving it microscopically as we did in Eq.(3.20). For more general graphs the

Fokker-Planck equation will contain all the sites on its right hand side. instead of just

the mean-field in Eq.(3.20). However it is possible to write the dynamics in terms of

an effective field and reduce the Fokker-Planck equation to the form in Eq.(3.20), a

technique we will use in Chap. 4.

It is not always convenient to study the exact microscopic dynamics prescribed

49

by Eq.(3.11 & 3.12) or Eq.(3.14) on a lattice, due to the spatial constraints imposed

by the structure of the lattice, compared to the complete graph dynamics which are

much simpler and summarized by an equation of the form Eq(3.20). The effect of

the spatial geometry on the dynamics of the spin system can however be studied if

we replace the microscopic model by a field theory. We replace the underlying spin

variables by a coarse grained field φ(x), where x is now a continuous spatial variable.

The d-dimensional lattice is divided into boxes each of which contains Rd spins, with

R the radius of the coarse grained volume. For a given box we assume all spins to

interact with each other. This can be justified by assuming long distance interactions

instead of nearest neighbor interactions. Thus without interactions between the spins

in different boxes, the magnetization of each box, φ(x), will be governed by the

Langevin equation Eq.(3.20). Spatial heterogeneity is taken into account by allowing

interactions between nearest neighbor boxes via a laplacian term. We thus arrive at

the following Langevin equation for the field,

∂φ(x, t)

∂t−D∇2φ(x, t) = vd(φ) +

√ζ(φ(x, t))ν(x, t)) (3.23)

where D is related to the size of the coarse-grained volume, and ν(x, t) is a spatio-

temporal Gaussian white noise of unit intensity.

As formulated, Eq.(3.23) can be used to study various spatial stochastic systems,

ranging from Gibbsian equilibrium systems such as the Ising model to population

biology models.

50

3.2.2 The Out of Equilibrium Ising Model

The Ising model is used to study magnetic systems. Physical constraints require that

the chosen kinetics ensure the system will reach the Maxwell-Boltzmann distribu-

tion at equilibrium. The kinetics should also satisfy local time reversal symmetry at

equilibrium. The latter condition is guaranteed by imposing detailed balance condi-

tions. Within detailed balance plugging in the Maxwell-Boltzmann distribution at

equilibrium yields the following jump probabilities from state σ to state σ′ :

P [σ → σ′] =1

1 + e(H(σ′)−H(σ))/T ), (3.24)

where H(σ) is the energy of the state σ. We restrict ourselves to the case when all

spins in the system interact with each other, i.e. the spins are sitting on a complete

graph. Then the field around each spin is just the average magnetization, which we

call φ. Elemental changes in φ are δφ = 2/N , where N is the number of spins in the

system. The energy of the system when the magnetisation is φ is

H(φ) = −N(

1

2Jφ2 + hφ

),

where we scaled the spin-spin interaction by the number of spins N . Using Eq.(3.24)

we arrive at the following kinetics:

F (φ) = P [φ→ φ+ δφ] =

(1 − φ

2

)1

1 + e−2(Jφ+h)/T

B(φ) = P [φ→ φ− δφ] =

(1 + φ

2

)1

1 + e2(Jφ+h)/T(3.25)

The first line in the above equation is for the probability of the event that a down

spin flips to up. For this to occur a down spin needs to be chosen for flipping, the

51

probability for which is the first factor in the last expression on the right hand side.

The second factor is the success probability of a down to up flip. Similar arguments

lead to the second line, which gives the probability of an up to down flip. The

evolution of the system can now be written in terms of the Fokker-Planck equation,

which gives the time evolution of the occupation probabilities c(φ, t):

∂c(φ, t)

∂t= − ∂

∂φvd(φ, J, h, T )c(φ, t) +

∂2

∂φ2ζ(φ, J, h, T )c(φ, t), (3.26)

At equilibrium the drift vanishes and the dynamics is governed by the diffusion

term. The Ising model we study is coupled to a thermal bath at temperature T . Thus

the diffusion in Eq.(3.26) is modeled as a Gaussian white noise with an intensity that

depends on T and leads to the Langevin equation,

∂φ

∂t= vd(φ) +

√T ν(φ, t). (3.27)

We can also write the field equation for the Ising model using√T for the noise

term in Eq.(3.23),

∂φ(x, t)

∂t−D∇2φ(x, t) = vd(φ) +

√Tη(φ, t). (3.28)

Using the definition in Eq.(3.2.1) and the transition probabilities in Eq(3.25) we

can evaluate the drift velocity,

vd = tanh(Jφ+ h) − φ, (3.29)

where we set δφ/δt = 1.

52

We can define the potential,

V (φ) = −∫dφvd(φ). (3.30)

that drives the kinetics of the Ising model and facilitates intuition. The potential

corresponding to the drift velocity in Eq.(3.29) is the mexican hat potential for tem-

perature below a critical value Tc. This means that there are two values of φ at which

the system will be stable, while for T > Tc there is only one value of φ which is stable.

Since we are interested in the critical behavior of the Ising model we can replace the

Ising model drift in Eq.(3.29) by the φ4 theory potential,

vd = 2τφ− 4φ3 + h, (3.31)

which shows the same critical behavior as the Ising model drift. Here τ = (Tc−T )/Tc.

It can easily be verified that for h = 0, the φ4-drift vanishes for three values of φ,

φ = 0, φ =√τ/2 and φ = −

√τ/2, when τ is positive (T < Tc). Thus we know

that the system is driven to one of the two stable values φ = ±√τ/2. An off-

equilibrium situation can be created by preparing the system in one of the stable

states, say φ = +√τ/2, and applying a negative external field h < 0. This makes

φ ≈ −√τ/2(≡ φO) the stable state, while the starting state φ ≈ +

√τ/2( ≡ φM)

becomes metastable. The third value for which the drift vanishes is an unstable value

φ ≈ 0(≡ φU). φ has to overcome a potential barrier and pass through φU (at which

the potential is the highest) in order to reach the stable value φS. In an infinite mean-

field system this will never happen because the barrier will be very high. However in a

spatial system, governed by the Langevin equation Eq(3.23), the field is allowed to be

spatially heterogeneous. The system then relaxes to its stable state via a nucleation

53

process. Phase ordering is curvature driven, when a stable phase droplet of down

spins larger than a critical radius grows to overtake the entire system. Later we will

comment on a similar dynamics governing a biological competition model between

two species corresponds to the stable phase [24].

3.2.3 The Voter Model

The Ising model is too complicated to use to understand the generic features of

kinetic spin systems. We now address a model that can be solved exactly, namely

the voter model. The dynamics of the VM can be described in terms of voters trying

to decide one of two parties to vote for. However the voters lack self-confidence and

each voter looks up to a randomly chosen neighbor, whose opinion she adopts. For

the case when all voters interact with each other the forward and backward jump

probabilities can be cast in the form,

F (φ) =1 − φ

2

1 + φ

2

B(φ) =1 + φ

2

1 − φ

2

with the following interpretation for the first line: the first term on the right hand

side is the probability to chose a down spin and the second term is the probability

that this down spin chooses an up spin neighbor. Simplifying the jump probabilities

we get,

F (φ) = B(φ) =1 − φ2

4,

from which we can see that the drift velocity vd ∝ F −B = 0. The langevin equation

Eq.(3.22) will then contain only the noise term. Since the system is not a thermal

system, the noise term will not be temperature depended. Instead an appropriate

54

noise term can be derived from the underlying dynamics: ν ∝ F + B, which is the

probability that a spin flip event occurs during the elemental time step δt. The field

then evolves according to

∂φ(x, t)

∂t−D∇2φ(x, t) = Ω

√1 − φ2ν(φ, t). (3.32)

The voter model has been studied extensively in both statistical physics and mathe-

matical probability literature. There are no equilibrium like phases for this system.

However the behavior of the system can be characterized in terms of the probability

that the system will eventually reach consensus, a state when all spins are in the

same state. This probability is 1 for d = 1, 2 when the system is said to cluster [21].

For d > 2, the up and down phases can co-exist.

Absence of the drift term in Eq.(3.32) is reminiscent of the behavior of the Ising

model at its critical point (where the drift term vanishes). The effect of the absence

of drift is no surface tension in the voter model. A similar situation happens in the

critical IM. One crucial difference between the two models, however is the absence

of bulk noise in the VM, i.e. spins inside an up or a down cluster of spins are not

subject to any noise; while random thermal noise can cause bulk flips in the IM.

This is encoded in the langevin equation via the form of the noise term, which shows

that there are two absorbing states for the dynamics, φ = −1 and φ = +1. It is not

essential to write a langevin equation for the voter model, since the underlying lattice

model can be solved exactly. However given the exact solution one can evaluate the

accuracy of the langevin equation approach for non-equilibrium systems using the VM

as a toy model. Recent studies have shown that the voter model forms a separate

universality class of its own and can be generalized to a wider class of non-equilibrium

models with two absorbing states [29]. The simplicity of the VM is the reason we

55

choose this model as the focus of our interest to characterize interacting particle

systems on degree-heterogeneous graphs.

3.2.4 The Contact Process

The contact process (CP) is a model for the spread of an infection. Individuals of

a population can be in either of two states, infected (spin +1) and susceptible (spin

−1). At each elemental time step one individual is chosen. If the individual is infected

it will either get well and be susceptible for re-infection with probability 1/(1 + λ)

or infect a randomly chosen susceptible neighbor with probability λ/(1 + λ). From

a different perspective, one considers a lattice in which the sites may be occupied

by members of a population of plants (which are not be allowed any migration) or

animals (which can migrate between sites). Each site can be occupied by only one

individual. The individuals have a finite life time and die at rate 1/(1 + λ) and

give birth at rate λ/(1 + λ). However all births are not successful. The offspring

seek to populate a randomly chosen neighbor of the parent, and a successful birth

corresponds to the event when the offspring chooses a vacant neighboring site. On a

complete graph, the forward and backward jump probabilities of the global field can

be written,

F (φ) =1 − φ

2

1 + φ

2

λ

1 + λ

B(φ) =1 + φ

2

1

1 + λ. (3.33)

The drift velocity and the diffusion term of the Fokker Planck equation (Eq.(3.20))

now become,

vd =1 + φ

2

(ǫ− 1 + φ

2

)(3.34)

56

where we used the bias of birth over death events ǫ ≡ 1−1/λ, and have appropriately

rescaled time. For further analysis, instead of the coarse-grained spin variable φ we

use the density of active sites (or infected individuals), ρ = (1+φ)/2. From Eq.(3.34)

we can see that there is an active state for ǫ > ǫc ≥ 0, while for ǫ < ǫc the system is

attracted to the absorbing state ρ = 0(φ = −1). The critical behavior of the contact

process at the critical value ǫc has not been solved exactly for lattice models, however

via heuristic arguments and numerical simulations a lot is known about the critical

exponents (see [30] for a review of the contact process and other non-equilibrium

systems with absorbing states).

The intensity of the noise term can be derived in the same way as for the voter

model, ν ∝ F +B ∝ ρ(2− ǫ−ρ). However near criticality, when ρ is small, it suffices

to use ν = 2ρ. Thus we have the following langevin equation for the spatial contact

process,

∂

∂tρ(x, t) −D∇2ρ(x, t) = ρ(x, t)(ǫ− ρ(x, t)) +

√2αρ(x, t)ν(x, t) (3.35)

We can rescale the density by its value in the active phase when ǫ > 0, ψ ≡ ρ/ǫ. The

field equation in terms of the redefined field is,

∂

∂tψ(x, t) −D∇2ψ(x, t) = ǫψ(x, t)(1 − ψ(x, t)) +

√2θψ(x, t)η(x, t) (3.36)

In the next section we will show that the above langevin equation is very similar to

the langevin equation for the Ising model at its spinodal critical point.

57

3.2.5 The Ising Model at the Spinodal Critical Point

Let us consider the φ4 model with a negative applied magnetic field. We have already

discussed the presence of a metastable state of the Ising model in an applied field.

Here we consider the situation when the applied field is large. This model has been

studied in detail before (see [31] for details). Here we will show that the resulting

field equation is very similar to Eq.(3.35), except that the noise driving the system is

thermal. The stable phase φO < 0 corresponds to the absorbing state of the CP, while

the metastable phase φM > 0 is the active phase. The drift term can be written,

vd = −4(φ− φO)(φ− φU)(φ− φM),

where φU is the unstable value of φ for which the drift vanishes. These metastable

and the unstable values of φ are of the form,

φM = φ1 + λ1∆h1/2

φU = φ1 − λ1∆h1/2.

Here λ1 > 0 and ∆h is the departure of the applied magnetic field from its value,

hs, at the spinodal, ∆h = |h − hs|. In order to reach the stable state φ has to pass

through an energy barrier which is maximum at φ = φU . However at the spinodal,

∆h = 0, the metastable state merges with the unstable state. See Fig. 3.4 for a

graphical definition of the spinodal. In order to focus our attention for ∆h ≈ 0, we

replace φ by ψ = (φ − φU)/(φM − φU). Thus ψ tells us the fraction of the system

that has crossed over the barrier into the stable state. The drift in terms of ψ is of

58

the form,

vd = ǫψ(1 − ψ)(λ2 − ǫψ/2),

where ǫ ∝ ∆h, λ2 is a positive constant, and we have rescaled the variables appro-

priately. ǫ can be interpreted as the bias away from the unstable state, which tries

to keep the system in the metastable state preventing its escape over the unstable

state into the stable state. Since we are interested in the system when ∆h is small,

it suffices to keep terms up to O(ǫ). Thus we get for the drift,

vd = ǫψ(1 − ψ),

which yields the langevin equation,

∂ψ

∂t−D∇2ψ = ǫψ(1 − ψ) +

√γT ν(x, t). (3.37)

The above derived field equation is very similar to the langevin equation Eq.(3.36)

for the contact process. The difference is the form of the noise term. In terms of the

coarse grained field this means that regions of the lattice which have already escaped

into the stable (absorbing) state will not be subject to random noise in the case of

the CP, but are subject to random thermal noise in the spinodal Ising model. This

similarity between the two processes is reminiscient of the relation between the voter

model and the critical (h = 0) Ising model.

59

3.2.6 Applications to Ecology: A Two Species CompetitionModel

Let us consider a two species competition model that has been studied in [24] and [25].

The two species share the same niche in an ecology and thus compete for resources

to live and propagate. In isolation one of the two species might have reproductive

advantage over the other ( a higher birth rate). In [24] the authors discuss the case

when the two species have the same growth rate and in [25] they discuss the case when

one of the species has a higher growth rate. The idea is to understand the effect of the

interaction between the two species on their respective growth rates and the eventual

state of the ecology. As in the case of the Ising model, the individuals do better when

surrounded by other individuals of their own kind. The system is defined over R2.

Instead of explicitly considering sites and nearest neighbor interactions, the space is

divided into patches. All individuals within one patch interact with each other. Thus

the dynamics of a given patch is captured by mean-field like equations that we have

been using throughout this survey. For the ecological model the dynamics are for

the number of individuals of type 1 and 2 notated s1 and s2. The kinetics consist of

a birth terms and death terms. Probability that a 1 will give birth depends on the

contents of the patch that it lives in, i.e., the number of individuals of the type 1 and

2 that it interacts with. Both 1 and 2 die at rates which depend on the crowding of

the patch. The mean field equations are

ds1

dt= s1

[a

s1

s1 + s2+ b

s2

s1 + s2− κ(s1 + s2)

]

ds2

dt= s2

[c

s1

s1 + s2+ d

s2

s1 + s2− κ(s1 + s2)

](3.38)

60

a, b, c, d are elements of a game matrix which sets the rules of interactions between

two players, a notion borrowed from game theory [32]. Since each species does better

among its own kind, a > b and d > c. To simplify the dynamics the authors set

the cross terms equal b = c. Species 2 is made advantageous by setting d = a + ǫ.

The starting conditions are uniform distributions of the two species, ξ1 and ξ2. The

questions addressed are (1) the final state of the system as a function of the initial

relative abundances of the two species and (2) the time elapsed before one of the

two species vanishes. When the two species have the same growth rates (ǫ = 0), the

more abundant species finally takes over. However when the two species are equally

abundant critical slowdown is observed [24]. In [25] the authors discuss the case

when one of the two species has a higher growth rate. This corresponds to the case

of the Ising model with an applied magnetic field. The nucleation behavior observed

is also very similar to the curvature driven nucleation from the metastable to the

stable state observed for the Ising model [26]. The authors derive field theories that

are very similar in spirit that we have done in this survey, except that instead of one,

they have to two interacting fields (densities of the two species) to deal with. The

authors also discuss percolating structures of the two species relating them to the

theory of percolation.

3.3 Kinetic Spin Systems on Degree-Heterogeneous Graphs.

In the first section we discussed random walks and noticed that on degree-heterogeneous

graphs the Kolmogorov’s forward propagation equation (Eq.(3.3)) and the Kolmogorov’s

backward propagation equation (Eq.(3.5)) contain different forms of the laplacian

(Eq.(3.4) and Eq.(3.6)). These two forms of the laplacian can also be used to cal-

culate the field (Eq.(3.10)) around the spin x in two different ways. These two

61

possibilities arise naturally in ecological models. Consider a grassland ecology that

lives on a graph with N sites. The state of the system is represented η such that

η(x) = 1 means that site x is occupied by a plant, while η(x) = 0 means that the

site x is vacant. Each individual plant lives for an exponentially distributed time.

Birth events can occur in two different ways:

(i) Neighborhood density dependent birth rate: At each elemental time step a

randomly chosen site (i) if it is occupied, becomes vacant (the occupant dies) with

probability 1/(1+λ) or (ii) if the site is vacant, it becomes occupied with probability

((λ/(1 + λ))∑

yAxyη(y))/kx. Mathematically,

P [η → ηx] =1

N

[

(1 − ǫ)η(x) + (1 − η(x))1

kx

∑

y

Axyη(y)

]

(3.39)

=1

N

[

(1 − ǫ)η(x) + (1 − η(x))∑

y

(δxy − Lxy)η(y)

]

=1

N

[

(1 − ǫ)η(x) − (1 − η(x))∑

y

Lxyη(y)

]

, (3.40)

where in writing the second line we used the definition in Eq.(3.6) and we noticed

that (1 − η(x))η(x) = 0 in order to arrive at the third line. Thus we conclude that

the dynamics for this choice of birth events is intimately related to the choice of

laplacian in Eq.(3.6).

(ii) Spread the seed: At each elemental time step a randomly chosen site x (i)

if it is occupied, (a) becomes vacant with probability 1/(1 + λ) or (b) give birth to

an offspring which moves to a neighboring site of x, where it survives only if the site

62

is vacant, or (ii) if it is vacant there is no change. Mathematically we have,

P [η → ηx] =1

N

[(1 − ǫ)η(x) + (1 − η(x))

∑

y

Axy

1

ky

η(y)

](3.41)

=1

N

[(1 − ǫ)η(x) − (1 − η(x)

∑

y

L⋆xyη(y)

], (3.42)

where the alternate version of the laplacian in Eq.(3.4) appears

In Chap. 5 we discuss in detail the voter model where we also observe and solve

two different dispersal mechanisms analogous to those of Eq.(3.40 and Eq.(3.42).

63

0

0 φ

φM

φU

φS

0

0 φ

φM

φU

φS

ba

0

0 φ

φU = φM

φS

c

Figure 3.4: Potential V (φ) for the φ4 theory.(a) h = 0. There are two equivalent minima for V (φ) at which vd = 0. vd = 0 alsofor φ = φU , the unstable maxima between the two minima (see (Eq.(3.30)) for therelation between vd and V (φ)). (b) The symmetry is broken by applying a smallnegative field (h < 0). There is only one stable minimum, φS < 0. The minima atφM is a local minimum. Starting with field such that φU < φ < φM , the systemmoves to the metastable minimum at φM since vd > 0 (indicated by the thin arrows).Correspondingly for φS < φ < φU , the system moves to the stable minimum at φS.The thick arrow show the direction in which the extrema of the potential move when|h| is increased. (c) As the magnitude of the magnetic field is increased, the extremasof V (φ) at φU and φM approach each other and merge for h = hS, thus makingthe local minima an inflection point. This is the spinodal critical point for the Isingmodel [31].

Chapter 4

Random Walks on Random Graphs

In Chap. 2 we introduced the random walk on a graph. We presented Kolmogorov’s

forward and backward propagation equations. In this chapter we study in detail

the properties of random walks on graphs. Our goal will be to understand how the

basic properties of random walks are influenced by the disordered environment of a

complex network. Relaxation properties of the random walks on ER random graphs

have been studied previously in [33]. Properties of random walks on the small world

networks of Watts and Strogatz have been studied in [34] and [35]. Another property

that has been extensively studied, especially in mathematics literature, is the cover

time, defined as the expected value of the time required by the random walk to visit

every node in the graph. Bounds for cover times in sparse random graphs when p is

of order log(N)N

have been studied in [36].

Just as in percolation on lattice systems, the properties of random walks on

random graphs are of fundamental interest because they exhibit new types of scaling

properties. Due to the absence of a metric structure on a random graph, the most

natural properties of random walks to investigate are related to their first-passage

properties. In particular, we will focus on the behavior of the transit time between

two sites on the graph and study how this property depends on the concentration of

64

65

bonds p. From a statistical physics perspective, for a given graph, one natural way to

define this transit time is to average over all pairs of sites in the graph. However, this

average will necessarily diverge when the graph consists of more than one component

because sites in a different component (cluster) are not accessible by the random walk

starting at a site in a given component.

To obviate this difficulty, it is advantageous to consider the inverse of the mean

transit time – the mean transit rate – rather than the mean time itself. This is

analogous to considering the mean conductance of disordered conductor-insulator

mixture near percolation, rather than the mean resistance. Resistance between sites

in different clusters diverges causing divergence in the mean resistance. Conductance,

on the other, does not show any pathologies for any pair of sites in the graph and as

a result the mean conductance is well behaved, and in fact exhibits critical behavior

as p→ pc from above. In a similar fashion, we will focus on the behavior of the mean

transit rate as p→ pc from above on the random graph.

In the next section, we define the mean transit time and other associated quan-

tities that we study. We present the governing equation for mean transit times in

Sec. 1. In Sec. 2 we discuss an effective-medium approach to calculate first-passage

properties of the ER random graph and in Sec. 3 we present simulation results for the

first-passage times and rates. We show how the mean times on a graph are related to

the spectrum of the laplacian of the graph and what it tells us about the mean times

on the ER random graph in Sec. 4. In Sec. 5 we discuss the connection between

properties of random walks and electrical networks on a graph, using which we de-

velop a geometric approach to understand first-passage properties of the ER random

graph in Sec. 6. The results of the simulations are analyzed in view of the structure

of the random graph. Finally, in Sec. 7, we discuss the nature of the fluctuations in

66

the mean transit times. Our conclusions are given in Sec. 8.

Parts of this chapter were first published as [8].

4.1 First passage characteristics of a graph

The first passage time tab(X) of a random walk X starting at site a on a graph G

is the time elapsed till the first arrival of X at b. Using Xt to represent the position

of the random walk at time t, mathematically the first passage time is,

tab(X) ≡ mint

(t|X0 = a,Xt = b). (4.1)

Thus tab(X) is a random variable that depends on the path taken by the random

walk X to go from a to b. Average of the first passage time over all the possible

random walks gives the mean first passage time between a and b,

Tab = E[mint

(t|X0 = a,Xt = b)] (4.2)

Averaging over all the end points a, b of the random walk, we get the mean first

passage time T (G) for the graph G. If we are considering an ensemble of graphs such

as the Erdos-Renyi (ER) random graphs, a further average over the ensemble gives us

the Mean First Passage Time (MFPT) of the ensemble of random graphs. Beginning

next section we start characterizing MFPT for ER random graphs.

On a general graph the first passage time between two sites may not be symmetric.

To avoid the complications caused by this asymmetry we consider the symmetric

combination,

Kab ≡ Tab + Tba, (4.3)

67

which is the expected time the random walk takes to go from a to b and come back

to b, i.e. the commute time between a and b. The graph G may not be connected,

which happens for ER random graphs on N sites when the occupation probability

p < 1/N . For graphs with more than one component, when a and b are in different

components Tab is infinite, since there is no path for the random walk to reach b

starting at a. To avoid this complication we define the inverse of the commute time,

Rab ≡ 1

Kab

, (4.4)

which we call the commute rate. The commute rate tells us how often the random

walks return to a after visiting b at least once. If a and b are in different components

of the graph, rab = 0.

Another first-passage characteristic of a graph is the expected return time, rx, of

a random walk to its starting site x. Notice that rx 6= Txx, since by definition the

later quantity is zero. However we can relate rx to the first passage times Txy to x

from its neighbors y :

rx =1

kx

∑

y

Axy(Txy + 1) =1

kx

∑

y

AxyTxy + 1, (4.5)

since the random walk starting at x takes the first step to one of the neighbors of x,

from where the return to x is just a first-passage process. It turns out that return

times are determined only by the stationary distribution of the random walk,

rx =1

πx

, (4.6)

which is a special case of Kac’s formula for Markov processes [1]. In order to check

68

the validity of the formula above we briefly sketch the proof, which we have adapted

from [1]. If the random walk starting at x makes the l-th visit to its starting site at

time Yl, using the fact that the random walk between time l-th and the l+ 1-th visit

is independent of previous visits,

E[Yl+1 − Yl] = rx.

Thus after a long time t, the random walk will have visited x t/rx times. The number

of visits to x should tend to πxt as t → ∞, since the stationary distribution can be

interpreted as the fraction of time spent by the random walk at x. Thus

t

rx

= πxt,

which proves Eq.(4.6). In Eq.(3.8) we saw that the stationary distribution of the

random walk on a graph is,

πx =kx

2M,

where M is the total number of bonds in the graphs. Thus for the return times of a

random walk on a graph we have,

rx =2M

kx

. (4.7)

69

Using Eq.(4.5) and Eq.(4.7) we can calculate the average nearest-neighbor MFPT,

Tnn =1

2M

∑

x,y

AxyTxy

=1

2M

∑

x

∑

y

AxyTxy

=1

2M

∑

x

kx(rx − 1)

=1

2M

∑

x

(2M − kx)

=1

2M(2MN − 2M)

= N − 1, (4.8)

which is independent of the structure of the graph.

4.2 Effective Medium Approach

We now develop an effective medium approximation for the commute time of a

discrete-time random walk on the random graph. In a single time step, a walk

located at a site that is connected to k other sites can hop with probability 1/k to

any of these neighbors. To compute the mean time for such a random walk to go

between two arbitrary sites on any graph by a sequence of nearest-neighbor hops, we

use the underlying backward equation [37, 38]. This equation relates the transit time

from site a to site b to the transit times from the neighboring sites of a to site b as

follows:

Tab =∑

Π

P Π t Π =∑

y

pa→y(δt+ Tyb). (4.9)

70

The first sum is over all paths Π from a to b, PΠ is the probability for the random

walk to take the path Π, and tΠ is the transit time from a to b along this path. For

each path, we then decompose the full transit time into the time to go from a to an

intermediate site y after one step plus the time to go from y to b. Thus pa→y = 1/ka

is the probability of hopping from a to i in a single step, ka is the degree of a, and

δt is the time for each step of the random walk. Without loss of generality, we take

δt = 1.

Let us now construct an effective-medium approximation for the average transit

time on the random graph, under the assumption that the graph is connected. This

condition implicitly restricts the validity of our approach to the range p > p1, where

all nodes belong to a single component. A schematic representation of a random

graph, to illustrate our approach, is shown in Fig. 4.1. Between two sites a and b

on the graph, a direct bond to b may exist (thick line) with probability p. If there is

no such direct bond, then an indirect path must be followed. After a single step on

this indirect path (medium lines), there may be a direct bond to b with probability

p (dashed), or no direct bond with probability q = 1 − p.

Let us denote by τ the mean transit time to go from a to b under the assumption

that a direct bond exists, and τ ′ the transit time from a to b in the absence of a

direct bond. Then from Eq. (4.9) and following an effective-medium assumption, τ

obeys the recursion formula

τ =1

(N − 1)p+

[1 − 1

(N − 1)p

][ p(1 + τ) + q(1 + τ ′) ] . (4.10)

The first term accounts for the walk that goes directly from a to b. This contribution

corresponds, in Eq. (4.9), to the case where the intermediate site y coincides with b.

Since (N − 1)p bonds emanate from a on average, then according to the effective-

71

b

a

Y

X

Figure 4.1: Schematic decomposition of a random graphwith starting site a and sink site b. After one step, the walk may be in the group ofsites X that may have a direct connection to b (dashed), each with probability p oramong the sites Y without a direct connection.

medium approximation, the probability that a random walk steps along the direct

connection is just 1/(N − 1)p. The second set of terms accounts for those walks in

which the first step goes to an intermediate site y rather than hitting b directly. In

this case, we again apply an effective-medium approximation and posit that after

one step of the walk, a direct connection from y to b exists with probability p, or no

direct connection exists with probability 1 − p (Fig. 4.1).

To close this equation, we need an expression for τ ′, the first-passage time in the

absence of a direct connection to b. Applying the same effective-medium approxi-

mation as that used in Eq.(4.10), we assume that after the first step of the walk,

the terminal site b is directly reachable with probability p, while b is not directly

72

reachable with probability q. Thus τ ′ obeys

τ ′ = p(1 + τ) + q(1 + τ ′). (4.11)

Solving Eqs. (4.10) & (4.11) gives τ=N−1p

and τ ′=N . Finally, we average the

transit time over all pairs of terminal points and over all graph configurations. Again

in the spirit of an effective medium approximation, this average is simply

〈T 〉≡ 〈Tab 〉= pτ + (1 − p)τ ′ = N − 1. (4.12)

The backward equation for the mean transit time can be extended to any positive

integer moment of the transit time. Consider, for example, the mean-square transit

time. As in the case of the mean time, the governing equation can formally be written

as

T 2ab =

∑

Π

P Π t2Π, (4.13)

For each path we follow Eq.(4.9) and again write the transit time tΠ as 1+tΠ′ , namely,

the sum of the time for the first step and the time for the remainder of the path.

Thus

T 2ab =

∑

Π

P Π(1 + t Π′)2,

=∑

Π

P Π(1 + 2t Π′ + t2Π′),

=∑

y

pa→y(1 + 2τy + τ 2y). (4.14)

In going from the second to the last line of this equation, we use the fact that

P Π =∑

y pyP Π′, where py is the probability of hopping from the starting point to

73

one of its nearest neighbors y, and P Π′ is the probability for the remainder of the

path Π′ from y to b. In the last line, the quantities τy and τ 2y are the mean and

mean-square times to reach b when starting from y and the sum is over all neighbors

y of the starting point. Strictly speaking, we should write 〈τ〉 and 〈τ 2〉 for these

moments, so that it is obvious that 〈τ 2〉 6= 〈τ〉2. In the following, we drop these angle

brackets because the linear and quadratic powers of time always appear separately

and there is no ambiguity about where the angle brackets should appear.

The last line of Eq.(4.14) is now a backward equation for the second moment of

the first-passage time, in which the previously determined first moment is an input to

this equation. This construction for the mean-square transit time can be generalized

straightforwardly, albeit tediously, to any positive integer moment of the first-passage

time. For the random graph, the recursion formula for the mean-square transit time

is, in close analogy with Eqs. (4.10 and 4.11).

τ 2 =1

(N − 1)p+

[1 − 1

(N − 1)p

]×[p(1 + 2τ + τ 2) + q(1 + 2τ ′ + τ ′2)

]

τ ′2 = p(1 + 2τ + τ 2) + q(1 + 2τ ′ + τ ′2). (4.15)

Using our previously-derived results for the first moments, τ = N − 1p

and τ ′ =

N , these recursion formulas are easily solved. We then compute the configuration

averaged mean-square transit time, 〈T 2〉 ≡ pτ 2 + qτ ′2, and obtain 〈T 2〉 = (2N −

3)(N−1). Thus again, the second moment is independent of p and equals the second

moment of the transit time on the complete graph.

More generally, we show that the first-passage probability between any two sites

on a random graph, and thus all moments of the first-passage time, are independent

of p in the effective-medium approximation. As a preliminary, we first compute the

74

first-passage probability on the complete graph. Let f(t) be the probability that a

random walk hits the sink site for the first time at time t, and let f(z) =∑f(t)zt

be the corresponding generating function. For the complete graph, the generating

function obeys the recursion formula

f(z) =1

N − 1z +

N − 2

N − 1zf(z).

This equation encodes the fact that after a single step (the factor z) the walk hits

the sink site with probability 1/(N − 1), while with probability (N − 2)/(N − 1) the

walk hits another interior site of the graph, at which point the first-passage process

is renewed. The solution to this equation is

f(z) =z

N − 1

[1 −

(N − 2

N − 1

)]−1

from which

f(t) =1

N − 1

(N − 2

N − 1

)t

Now consider the random graph with bond occupation probability p. Let f(t) be

the first-passage probability from a to b when a bond is present between these two

sites, f ′(t) the first-passage probability when this bond is absent, and let f(z) and

f ′(z) be the respective generating functions. In the spirit of our effective medium

approximation given in Eqs. (4.10) and (4.11), we now have

f(z) =1

(N−1)pz+

[1− 1

(N−1)p

][pzf(z) + qzf ′(z)] ,

f ′(z) = pzf(z) + qzf ′(z).

75

From these two equations, the average first-passage probability 〈f(z)〉 = pf(z) +

qf ′(z) has the same form as the first-passage probability for the complete graph

(Eq. (4.2)). Hence all moments of the transit time are independent of p in the

effective medium approximation.

4.3 Simulation Results for the ER random graph

To test the effective-medium prediction for the mean first-passage time, we now turn

to numerical simulations. For very small systems (N ≤ 8) , we have obtained the

exact first-passage time by averaging over all configurations of random graphs, over all

pairs of endpoints, and over all random walks. For the graph configuration average,

each realization is weighted by the factor pkqE−k, where k is the number of occupied

bonds in the graph, E = N(N − 1)/2 is the total number of possible bonds, and

q = 1 − p. We then average over all pairs of endpoints directly. By this averaging,

the mean transit time is simply one-half of the mean commute time. Rather than

averaging over individual walks directly, we solve exactly the recursion formulas in

Eq. (4.9) for the transit times between all pairs of points.

For larger systems, the exact enumeration of all graph configurations is impracti-

cal. Instead we average over a finite number of graph realizations and endpoint pairs,

but still performed the exact average over all random walk trajectories by numerically

solving Eq. (4.9). For efficiency, we start our simulation with an empty graph, add

bonds one at a time and then update the commute times between all pairs of sites in

the graph after each bond addition. Each graph is then weighted by pkqE−k so that

we can obtain the commute time as a function of p. We repeat this sequential graph

construction over many realizations. The graphs that we obtain by this sequential

growth are the same as those obtained by a static construction in which each bond

76

is present with probability p = 2M/N(N − 1) when N is large (see [1] for different,

but equivalent ways of constructing random graphs).

10-1

100

10-3 10-2 10-1 100p

pc p1

Figure 4.2: Mean commute time (dashed) and mean commute rate (solid)versus bond occupation probability p for a random graph of N = 100 sites. Bothquantities are normalized to have the value 1 for the complete graph (p = 1). Av-erages over 103 graph realizations were performed for each p. Also shown are thelocations of pc = 0.01 and p1 ≈ 0.046.

For the average commute time, we only include connected graphs in the ensemble,

while for the average rate, the ensemble consists of all graph configurations. This

restriction plays a significant role only for p < p1, where the random graph normally

consists of multiple components. Typical results for a graph of 100 sites are shown

in Fig. 4.2. Above the connectivity threshold p1 = lnN/N , the average transit

time varies slowly with p, in agreement with our effective medium approach. The

77

apparent singularity of the average commute time at a value p < p1 stems from finite

size effects.

0

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2 2.5

R

µ

Figure 4.3: Mean commute rate R on a random graphfor N = 50 (⋄), N = 100 (), 200 (), 400 (), and 800 () sites as a functionof the average site degree µ = p(N − 1). These rates are normalized to one for thecomplete graph limit. Averages over 103 graphs were performed for each case.

The behavior of the mean commute rate is shown in Figs. ?? and 4.3. Unex-

pectedly, this rate is non-monotonic in p for p ≈ pc = 1/N , as shown in detail in

Fig. 4.3. For this plot, we use the average degree, µ = p(N − 1) as the dependent

variable, because it has the desirable feature that the percolation transition occurs at

the same value µc = 1 for all N . The fact that the non-monotonicity in the commute

rate occurs near µ = 1 suggests that this anomaly is connected with the percolation

78

transition of the random graph.

To understand this non-monotonicity, we study the laplacian formalism for the

random walk [39] [1]in the next section. We use the laplacian formalism to make a

connection between the commute rate and the electrical conductance on the same

network.

4.4 Spectrum of the Laplacian and the First-Passage Prop-erties

All the information about the first-passage properties can be obtained from the “lapla-

cian” of the graph in principle. In this section we define the laplacian and show how

the first-passage properties are derived from its spectrum. An analysis of the spec-

trum of a random graph laplacian will also help us understand why the results of

the effective-medium approach in Sec. 4.2 were not as crude as the effective-medium

approach itself. We arrive at our results by bounding the transit times in subsection

4.4.1, using the spectrum of the laplacian.

In Chap. 2 we derived Kolmogorov’s forward propagation equation for the evo-

lution of the transition probability from the starting site a to site x in time t,

d

dtc(x, t; a, 0) = −

∑

y

L⋆xyc(y, t; a, 0) (4.16)

where the matrix L⋆ is the laplacian,

L⋆xy = (kxδxy −Axy)

1

ky

. (4.17)

Eq.(4.16) relates the time derivative of the transition probability c(x, t; a, 0) to the

transition probabilities from a to the neighbors of x. We also derived Kolomogorov’s

79

backward propagation equation,

d

dtc(x, t; a, 0) = −

∑

y

Layc(x, t; y, 0), (4.18)

where the laplacian L is different from the one appearing in the forward equation

(Eq.(4.16)),

Lay =1

ka

(kaδay −Aay). (4.19)

Kolmogorov’s backward propagation equation relates the time evolution of c(x, t; a, 0)

to the transition probabilities from the neighbors of a to x.

In order to derive a symmetric form for the laplacian let us plug Eq.(4.17) into

Eq.(4.16) to write,

d

dtc(x, t; a, 0) =

∑

y

(Axy − kxδxy)1

ky

c(y, t; a, 0),

and dividing on both sides by√kx we get,

d

dt

c(x, t; a, 0)√kx

=1√kx

∑

y

(Axy − kxδxy)1√ky

c(y, t; a, 0)√ky

,

which suggests that we define a symmetrized laplacian L,

Lxy ≡ 1√kx

(Axy − kxδxy)1√ky

=Axy√kxky

− δxy, (4.20)

and weigh the occupation probability with the square root of the degree,

c(x, t; a, 0) ≡ c(x, t; a, 0)√kx

. (4.21)

80

With the symmetric laplacian and the new variable c Kolmogorov’s forward propa-

gation equation becomes,

d

dtc(x, t; a, 0) = −

∑

y

Lxyc(x, t; a, 0). (4.22)

If we use D for the diagonal matrix with degrees of the sites as its entries:

Dxy =√kxkyδxy, (4.23)

we can write the symmetrized laplacian in matrix form:

L = D− 1

2 L⋆D− 1

2 . (4.24)

We can now write the first passage properties of the random walk in terms of the

spectrum of the symmetrized laplacian.

Let el be the l-th eigenvector of the symmetrized laplacian with eigenvalue λl.

The eigenvalues of a graph laplacian are all positive except the smallest, which is

zero [2]. We order the eigenvalues such that, 0 = λ1 ≤ λ2 ≤ λ3... ≤ λN . The graph

may have more than one trivial eigenvalues with each zero eigenvalue corresponding

to a different component of the graph. A connected graph will have only one trivial

eigenvalue, which will be the case for the following arguments. The eigenvector with

eigenvalue zero on a connected graph has components,

ex1 =

√kx

2M, (4.25)

where M is the total number of edges in the graphs.

We can convert the differential equation Eq.(4.22) into an algebraic equation in

81

terms of the laplace transforms of the occupation probability,

ςxa(s) ≡∫ ∞

0

e−stc(x, t; a, 0)dt. (4.26)

Multiplying Eq.(4.22) by e−st on both sides and integrating over time we get,

sςxa(s) = −∑

y

Lxyςya(s) + c(x, 0; a, 0), (4.27)

after integrating on the left hand side by parts and using the definition in Eq.(4.26).

Since the random walk starts at a we can replace the second term on the right hand

side by a delta function,

sςxa(s) = −∑

y

Lxyςya(s) +δxa√kx

. (4.28)

We can now use the eigenvalue-spectrum of L ,

∑

y

Lxyey

l = λlexl , (4.29)

to solve the matrix equation Eq.(4.28). Multiplying Eq.(4.28) by exl and summing

over x we get

s∑

x

exl ςxa(s) = −

∑

y

(∑

x

exl Lxy

)

ςya(s) +∑

x

exl

δxa√kx

= −λl

∑

y

ey

l ςya(s) +1√ka

eal ,

where to get the second line we noticed that L is a symmetric matrix and used

82

Eq.(4.29). Thus we see that,

∑

y

eyl ςya(s) =

1

λl + s

1√ka

eal ,

which gives us ςxa(s) if multiply by exl , sum over l and use the ortho-normality of

the eigenvectors e on the left hand side,

ςxa(s) =1√ka

∑

l

1

λl + sea

l exl =

√kx

2Ms+

1√ka

∑

l>1

1

λl + sea

l exl ,

where in the second expression we separated the zero eigenvalue vector and used

Eq.(4.25). For the actual occupation probability we have,

cxa(s) =√kxςxa(s) =

kx

2Ms+

√kx

ka

∑

l>1

1

λl + sea

l exl , (4.30)

where we have abused notation to use c for the laplace transform of the occupation

probability, c(x, t; a, 0).

If the random walk starting at a reaches x (different from a) for the first instance

at time t′, the probability that it will be at x at a later time t is the same for the

random walk that is at x at time t − t′ starting at x itself. Thus we can relate the

occupation probability to first passage probability, f(x, t′; a, 0),

c(x, t; a, 0) =

∫ t

t′dt′f(x, t′; a, 0)c(x, t− t′; x, t′), (4.31)

from which after a multiplication with e−st and integrating over t we get,

83

cxa(s) =

∫ ∞

0

dte−st

∫ t

t′dt′f(x, t′; a, 0)c(x, t− t′; x, 0)

=

∫ ∞

0

dt′∫ ∞

t′dte−st′f(x, t′; a, 0)e−s(t−t′)c(x, t− t′; x, 0),

where the second line follows by interchanging the integrals over t and t′ and changing

the limit of the integrals appropriately. Calling t−t′ a new variable t′′ we can separate

the two integrals,

cxa(s) =

∫ ∞

0

dt′e−st′f(x, t′; a, 0)

∫ ∞

0

dt′′e−s(t−t′)c(x, t− t′; x, 0)

which allows us to relate the laplace transforms of the occupation and the first passage

probabilities,

cxa(s) = fxa(s)cxx(s), (4.32)

and using Eq.(4.30) we can write the laplace transform of the first passage probability

in terms of the eigenspectrum of the laplacian,

fxa(s) =cxa(s)

cxx(s)=

kx

2Ms+√

kx

ka

∑l>1

1λl+s

eal e

xl

kx

2Ms+∑

l>11

λl+s(ea

l )2. (4.33)

We can relate the first-passage time between sites a and x to the first passage

probability,

Tab =

∫ ∞

0

dt t f(x, t; a, 0)

= − d

dsfxa(s)

∣∣∣∣s=0

.

84

Using the expression for fxa(s) in Eq.(4.33) to calculation the above derivative we

find,

Txa = 2M∑

l>1

1

λl

exl√kx

(ex

l√kx

− eal√ka

), (4.34)

using which we can write for the commute times,

Kxa = Txa + Tax = 2M

[∑

l>1

1

λl

exl√kx

(ex

l√kx

− eal√ka

)+∑

l>1

1

λl

eal√ka

(ea

l√ka

− exl√kx

)]

which can be simplified to

Kxa = 2M∑

l>1

1

λl

(ex

l√kx

− eal√ka

)2

. (4.35)

4.4.1 Bounds on transit times using the spectrum of thelaplacian.

To average over the vertexes we need to decide how to sample the pairs of sites. The

random walk starts at a, which we shall call the source and stops at x, the sink. We

can average over the sources to find the dependence on the sink, or average over the

sinks to find the dependence on the source. We shall use a filled dot in the subscript

to indicate which of the two ends has been averaged over. So Tx• will mean that we

have averaged over all the sources with sink fixed to be x. There are four natural

choices for sampling the sites:

1. The source is sampled from the stationary distribution, kx/2M . After plugging

this probability into Eq.(4.34) and summing over all possible sources we get

Tx• ≡1

N

∑

a

Txa =1

kx

∑

l>1

[(ex

l )2

λl

∑

a

ka

]− 2M

∑

l>1

[ex

l√kx

(∑

a

√ka

2Mea

l

)]

85

Sum over the degrees of all the sites is just twice the number of edges, M . The

last sum in the above expression is e1 · el = 0 (see Eq.(4.25)), since l > 1 and the

eigenvectors are orthonormal. These observations allow us to simplify the expression,

Tx• =2M

kx

∑

l>1

exl

λl

. (4.36)

This result should be compared with the result for return times in Eq.(4.7). Both

these results lead us to conjecture that the most important quantity for the first-

passage properties of random walks on graphs is the degree of the sink. In Sec. 7

we shall present simulation results for fluctuations in the first-passage times on the

ER random graph and show the validity of our conjecture that the sink-degree is the

single important quantity for determining first passage properties of a graph.

2. The sink is sampled from the stationary distribution. Plugging in the station-

ary probabilities and summing over all possible sinks we get

T•a =∑

l>1

1

λl

[∑

x

(exl )2

]

−√

1

ka

∑

l>1

1

λlea

l

[∑

x

(√kxe

xl

)]

which can be simplified if we notice the norm of the l-th eigenvector in the first sum

and the dot product in the second sum,

T•a =∑

l>1

1

λl, (4.37)

which is independent of the source site a. Thus the starting site of the random walk

does not effect the first-passage times in the above average, an observation which

adds further weight to our conjecture following Eq.(4.36).

3. Both the sink and the source are sampled from the stationary distribution. In

86

this case the calculations are the same as above and the result

T•• =∑

l>1

1

λl, (4.38)

a result that does not contain any explicit information about the degree-distribution

of the graph.

4. The sink and the source are sampled from a uniform distribution.

T•• =1

2

2M

N ∗ (N − 1)

∑

a6=x

∑

l>1

1

λl

(ex

l√kx

− eal√ka

)2

(4.39)

(the factor of half enters because of double counting over the source sink pairs). For

the case of the random graph GN,p this equation will become

T•• =1

2p∑

a6=x

∑

l>1

1

λl

(ex

l√kx

− eal√ka

)2

. (4.40)

The eigenvalues of the symmetrized laplacian of a connected graph are non-

negative. The smallest eigenvalue is λ1 = 0, which gives us the stationary distri-

bution of the random walk on the graph. The smallest non-trivial eigenvalue, λ2, is

called the spectral gap, and we have for rest of the eigenvalues (l > 2) [2],

λ2 ≤ λl ≤ 2 (4.41)

which for the reciprocals of the eigenvalues becomes

1

2≤ 1

λl

≤ 1

λ2

87

which we insert in Eq.(4.35) and use orthonormality of the eigenvectors el to get

M

(1

kx

+1

ky

)≤ Kxy ≤ 2M

λ2

(1

kx

+1

ky

). (4.42)

Averaging over the end points x and y, and using M = µ1N/2, we have

µ1µ−1N ≤ K•• ≤2µ1µ−1

λ2N, (4.43)

where we used µ−1 = E[1/k]. Notice that since k > 0, µµ−1 = O(1) for all graphs.

The average commute time will go to infinity with the number of sites N . To study

how the commute time scales with N , we can divide the above inequality with N

and study the resulting quantity,

1 ≤ K••

N≤

√2

λ2(4.44)

The laplacian formulation becomes a very powerful method to determine the first-

passage properties of a random graph, as one can use the results developed in random

matrix theory. The most important result from random matrix theory concerns the

spectral gap λ2. Furedi and Komlos have shown that for the ER random graph

the eigenvalue spectrum of the adjacency matrix follows Wigner’s semi-circle law

[40] [41]. This result can be extended to show that the spectral gap, λ2, of the

laplacian does not become zero with systems size and hence the K ≤ O(N) even

when a large number of edges have been removed. Comparable results about the

spectral properties of random graphs with prescribed degree distribution have been

presented in [42], which show that K = O(N) for scale-free graphs. In light of these

results and the bounds for the average commute time in Eq.(4.44), we can assert that

88

K•• = O(N) for random graphs (including ER and MR random graphs discussed in

the first chapter). Thus no surprises should be expected for first-passage properties

of random graphs with power-law degree distributions.

4.5 Relation With Electrical Networks: Commute Timesand Resistances.

The laplacian formulation of the last section allows an algebraic characterization of

the first-passage properties. Here we show that the laplacian formalism also reveals

a connection between the random walk properties and conduction properties of an

electrical network on the same graph. The electrical network connection allows use

of geometric arguments about the structure of the graph to deduce properties of the

random walk [43]. We will illustrate our arguments by calculating the properties of

the random walk on a ER random graph.

To construct an electrical network from a graph, put a unit resistance across each

edge. Let Rab be the (effective) resistance between the sites a and b on this network.

We will relate Rab to the commute time Kab. When the sites a and b are hooked

up to a battery a current flow is set up across the network from a to b. The current

flowing from x to its neighbor y is related to the voltages at these two sites via Ohm’s

law,

vx − vy = jxy. (4.45)

The total current entering every site (other than the two sites hooked up to the

battery) is zero,

∑

y

Axyjxy = 0, (4.46)

89

A total current, J , flows into the network at a,

∑

y

Aayjay = J, (4.47)

and exits at b,∑

y

Abyjby = −J. (4.48)

Eqs.(4.46, 4.47 & 4.48) are the Kirchoff’s current law (KCL) for our network. The

KCL equations can be written in the form

∑

y

Axyjxy = J(δax − δbx). (4.49)

Using Ohm’s law (Eq.(4.45)) the KCL equations,

∑

y

Axy(vx − vy) = J(δax − δbx),

can be written,∑

y

(kxδxy −Axy)vy = J(δax − δbx), (4.50)

which when divided by kx yields

∑

y

1

kx

(kxδxy −Axy)vy = J1

kx

(δax − δbx),

where we can notice the laplacian of Eq.(4.19),

∑

y

Lxyvy = J1

kx

(δax − δbx), (4.51)

90

and impose the boundary condition that the voltage at b, the exit point for the

current, be zero,

vb = 0.

If we reverse the role of a and b, by symmetry we should have v′a = 0 and v′b = va,

where the primed quantities are the values of the voltages for the reversed current

flow. The voltages at other sites will also change

v′x = va − vx, (4.52)

which can be easily checked by inserting the last expression into Eq.(4.51) and fixing

the right hand side for the reverse current flow by using −J instead of J .

4.5.1 Resistance and Commute times

Let ux denote the expected number of visits to site x of the random walk that starts

at a and is absorbed at b. If x 6= a, to visit x the random walk has to pass through

one of its neighbors,

ux =∑

y

Axy

1

ky

uy

For the entry site a we have

ua = 1 +∑

y

Aay

1

ky

uy,

91

where the extra 1 is due to the fact that the random walk starts at a. The exit point

b will be visited only once,∑

y

Aby

1

ky

uy = 1,

which we cast as

ub = −1 +∑

y

Aby

1

ky

uy,

and impose the boundary condition,

ub = 0.

We can write the equations for the number of visits to each site derived above in a

condensed form,∑

y

(kxδxy − Axy)1

ky

uy = δxa − δxb, (4.53)

which in terms of the laplacian in Eq.(4.19) is

∑

y

L⋆xyuy = δxa − δxb. (4.54)

If we set J = 1 in Eq.(4.50) we can see that it is the same as Eq.(4.53) if we identify,

ux = kxvx. (4.55)

92

The expected time at which the random walk is absorbed at b is also the expected

first passage time from a to b, which allows us to write,

Tab =∑

x

ux (4.56)

=∑

x

kxvx (4.57)

Tab =∑

x

u′x (4.58)

=∑

x

kxv′x (4.59)

with the prime used for the reverse flow. Thus for the commute time we have

Kab = Tab + Tba (4.60)

=∑

x

kx (vx + v′x) (4.61)

= 2Mva (4.62)

where in the last line we used Eq.(4.52), and summed over the degrees to get twice

the number of bonds in the graph, M . Since the effective resistance Rab, between a

and b is the voltage generated at a when a unit current is passed between these two

sites Rab = va,

Kab = 2MRab. (4.63)

The conductance between a and b is Gab = 1/Rab and using the definition of the

commute rate Rab in Eq.(4.4),

Rab =Gab

2M. (4.64)

As we shall see, it is much easier to estimate the conductances rather than the

commute rates of a random graph by direct means. We will then rely on this con-

93

nection between Gab and Rab to determine the latter quantity.

For reasons of numerical convenience, we will often consider the following sum of

the rates

Ra ≡ 2∑

b6=a

Rab =1

M

∑

b

Gab. (4.65)

We include the factor of 2 in the definition because Ra then equals 1 for the complete

graph. We may also sum freely over all sites b in the system Eq. (4.65) because

Gab = 0 for any sites that are not in the same cluster as a. Finally, we obtain the

average commute rate for the graph by averaging over all initial sites a:

R ≡ 1

N

∑

a

Ra. (4.66)

In the limit of large µ1 all the sites in the ER random graph belong to the same

cluster and M = µ1N/2. Thus the average commute rate becomes

R =1

N

2

µ1N

∑

ab

Gab =2

µ1G, (4.67)

where G is the two-point conductance averaged over all pairs of graph endpoints.

For the conductance itself, it is worth noting that this function behaves anoma-

lously near the connectivity transition. Although the conductance must increase

monotonically with µ according the Ralyleigh’s monotonicity principle [43], the rate

of increase changes for µ1 in the critical range between 1 and lnN (Fig. 4.4). For

large µ, the conductance asymptotically approaches G = µ1/2 (dashed line), a result

that corresponds to the average commute rate approaching R = 1, in agreement with

the result of Fig. 4.2.

94

10-3

10-2

10-1

100

101

102

100 101 102

G

µ

Figure 4.4: Average two-point conductance(thick solid curve) on a random graph with N = 100 sites. The dotted line corre-sponds to G = µ1/2, the asymptotic large-µ1 form for the conductance.

4.5.2 Escape probability

Start a random walk at site a. What is the probability, pescab that the random walk

escapes to the site b before returning to a? This is the famous and much studied

question of recurrence/transience of a random walk. One may consider instead of a

single site b, a set of sites B. For example one may consider all the sites lying at

infinity to consist the set B, and ask for the probability to escape to infinity. In the

following we will restrict ourselves to escape probabilities between single sites and

relate this quantity to the electrical conductance and the commute rate.

Let pab(x) be the probability that a random walk starting at x will reach b before

95

visiting a. Since the random walk must pass through one of the neighbors of x,

pab(x) =1

kx

∑

y

Axypab(y) (4.68)

and

pab(a) = 0

pab(b) = 1. (4.69)

The structure of Eqs.(4.68 & 4.69) reminds of the electrical network equation Eq.(4.50),

when a unit voltage difference is induced across b and a (vb = 1, va = 0). For the

escape probability we have

pescab =

1

ka

∑

y

Aaypab(y). (4.70)

The sum on the r.h.s of the above equation corresponds to the current flowing into a

from its neighbors. Since the current flowing into the exit point a is the total current

J flowing in the network from b to a,

pescab =

1

ka

J =1

ka

vb − va

Rba

=1

kaRba

, (4.71)

which we can relate to the commute time using Eq.(4.63),]

pescab =

2M

Kbaka

(4.72)

96

and to the commute rate,

Rab =ka

2Mpesc

ab =kb

2Mpesc

ba (4.73)

where we used the symmetry rab = rba to get the second equality. Summing the last

equation over b we get the sum of commute rates between a and rest of the sites in

the graph, from which we can get the mean commute rate for the walks which start

at a.

Ra =ka

2M

∑

b

pescab , (4.74)

and

Ra =∑

b

kb

2Mpesc

ba . (4.75)

The right hand side of Eq.(4.74) can be interpreted as the mean escape probability

from the graph to a when the starting site b is chosen from the stationary distri-

bution, while the sum in Eq.(4.75) ] is the mean escape probability from a to a

uniformly chosen site in the graph. A further sum over a will give us the mean es-

cape probabilities when the random walk starts at a site chosen from the stationary

distribution and escapes to a site chosen from the uniform distribution.

R ≡ 1

N

∑

a,b

ka

2Mpesc

ab . (4.76)

4.6 First-Passage Properties and the Structure of the Graph

Conducting properties of an electrical network can be estimated by using geometric

arguments about the structure of the underlying graph. The most important result

in electrical network theory that we use is the Rayleigh’s monotonicity principle. To

97

state this principle, we need to distinguish between resistance across edges, that we

call edge-resistances, and the effective resistance between sites. Consider two sites

a and b that are connected by an edge of unit resistance. Thus we say that the

edge (ab) has resistance 1. If two vertexes c and d are not connected by an edge

directly, we can still assign an edge-resistance of magnitude infinity to the pair (cd).

The effective resistance is the voltage set up between a and b when a net current of

magnitude 1 is made to flow from a to b using a current source. Thus Rab is effected

by all the edge-resistances in the whole graph. Rayliegh’s monotonicity principle

states that the effective resistance Rab between two sites a and b in a graph is a

monotonically decreasing function of the edge-resistances between the sites in the

graph [43]. We will consider edge-resistances of magnitudes either 1 (the edge is

present) or ∞ (the edge is absent). Thus if we add edges anywhere in the graph, Rab

can only decrease.

Rayleigh’s monotonicity principle already allows us to derive an upper bound

for the commute times by considering the resistances on the spanning tree of the

graph. Removal of an edge from a graph is equivalent to replacing the corresponding

edge-resistance of 1 by ∞. Thus the effective resistances on the graph can only be

smaller than the effective resistances on the spanning tree (constructed by removal

of edges such that the resulting structure is a tree and contains all the sites in the

same component). If the maximum distance between two sites of the graph is L (the

diameter of the graph) we can immediately see that,

Rab ≤ L

since the effective resistance between two sites on a tree is just the distance between

the two sites (there is only one path between any two sites on a graph). Using

98

Eq.(4.63) we see that

Kab ≤ 2ML.

For the ER random graph, when the mean degree µ1 = O(1), we showed in Chap.

1 that L = O(lnN), which allows us to write

Kab 6 O(N lnN).

while on the other hand, once again using the Rayleigh principle, the effective-

resistance between two sites should be larger than that on a complete graph which

can be calculated easily to be 2/N ,

2

N≤ Rab ≤ L,

which allows us to write the following bounds for the commute times,

2µ1 < Kab < O(N lnN).

We now use the connection between commute rate and conductance to understand

the non-monotonicity in the commute rate for a random walk on a random graph,

that we have observed using a numerical simulation. To this end we employ the

results about the structure the ER random graph and its rooted geodesic tree (RGT)

derived in Chap. 1. As indicated in Fig. 4.5, there are three regimes for the commute

rate: (I) an initial increase with µ for small µ; (II) a decrease over an intermediate

range; and (III) an ultimate increase for large µ. For regimes I and II, the commute

rates on the RGT and the random graph are nearly identical and it is simpler to

consider the commute rate on the RGT. We then investigate how adding the bonds

99

to the RGT to create a random graph affects the commute rate.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.5 1 1.5 2 2.5 3 3.5 4

R

µ

I II III

Figure 4.5: Commute rates on the RGT () and on random graphs() for N = 100 sites based on averages over 105 realizations. The lower curveis our RGT prediction Ra = lnµ/ lnN for the large-µ limit. The upper curve inthe prediction for the random graph R = (µ − 1)/µ (Eq. (4.79)). The approximatelocations of regimes I, II, and III are indicated.

4.6.1 Commute Rate on the RGT

For a tree graph, the resistance between two sites is simply the path length between

these two sites. Thus the average commute rate in Eq.(4.64) has the form

Ra =1

V − 1

∑

b6=a

1

Dab

=1

V − 1

L∑

j=1

Sj

j. (4.77)

100

Here the number of bonds in a tree is one less than the total number of sites V , and

Dab is the distance between a and b. Thus Ra is the inverse moment of the distance

between the root a and all other sites in the tree. The second equality follows from

the shell structure of the RGT, where L is the radius of the tree. Thus we need only

the statistics of the shell sizes of the RGT to determine the commute rate.

Since each realization of the RGT is distinct, the number of sites V , the radius

L, and the shell sizes Sj fluctuate from realization to realization. To calculate the

configuration-averaged commute rate 〈Ra〉, we first use the algorithm of the previ-

ous section to generate RGTs. Then we solve the random walk problem on each

realization and average Eq. (4.77) over realizations to determine the commute rate

(Fig. 4.5).

To understand the non-monotonicity of the commute rate for the RGT, consider

first the small-µ1 limit. Because isolated sites contribute zero to the average rate,

the commute rate must initially increase with µ1, as small trees begin to form. Once

most sites are no longer isolated, the radii of typical RGTs then increase with µ1

due to the merging of small trees. This increase in radius causes a decrease in the

commute rate, as can be seen by writing the rate in Eq. (4.77) as

Ra(L) =

L∑

j=1

Sj

j

Ra(L) =PL

j=1Sj/j

PLj=1

Sj. (4.78)

In the analogous expression for Ra(L+1), the numerator increases by SL+1/(L+1)

while the denominator increases by SL+1. Thus Ra(L) is a decreasing function of L,

101

so that a tree with a larger radius will have a smaller commute rate.

As argued in Sec. V, a further increase in µ1 will cause the radius of the RGT

to eventually decrease with µ1. Correspondingly, the commute rate enters regime III

and increases with µ1. In this regime, we now use the fact that the number of sites

in successive shells of the RGT grows exponentially in the distance from the root.

Thus the shell at radius L contains almost all of the sites of the RGT. As a naive

approximation, we then replace the sum in Eq. (4.77) by the last term to give, in the

limit of large µ1,

(Ra)RGT ≈ 1

V

V

L∼ lnµ1

lnN.

This result agrees extremely well with numerical results for the commute rate on

RGTs, as shown in Fig. 4.5.

4.6.2 Role of Loops on Commute Rate

We now investigate how adding loops to the RGT to build a random graph affects

the behavior of the mean commute rate. Starting with a realization of an RGT

we generate a cluster of the random graph by adding missing bonds, following the

procedure discussed in Chap. 1. The addition of these bonds will create loops that

provide alternative paths between the root site and the endpoints of a random walk

(Fig. 4.6). The ostensible effect of these additional paths is to increase the commute

rate between the root and any endpoint.

To estimate the two-point conductance for a random graph for general µ > ln N ,

we start with the picture that the graph consists of two RGTs, one emanating from

a and the other from b (Fig. 4.7). For a graph of N sites, the radius of each tree is

102

b

a

Figure 4.6: Schematic representation of the random graph.The included RGT is also shown. Loops typically arise at a distance L = lnN/ lnµfrom the root. A site in this last shell will typically have µ independent paths to theroot.

of order L ∼ ln(N/2)/ lnµ. We argue that these two trees tend to join only at the

outermost shell because this is where most of the sites in the trees are located. We

further assume that, in the equivalent resistor network, all sites at the same distance

from the root are at the same potential. Thus the conductance of the two joining

RGTs is simply one-half of the conductance between the root and the last shell of a

single RGT.

For this last step, we approximate the RGT by an infinite Cayley tree with branch-

ing ratio µ. The resistance between the k-th and the (k + 1)st shell in this tree is

103

a b

Figure 4.7: Random graph structure between two sitesSchematic random graph structure to calculate the conductivity between two sitesa and b for mean degree µ > 1. An RGT is grown around both a and b. The twoRGTs meet at a distance O(lnN/ lnµ) from each of a and b. Broken lines are bondsbetween sites in the outermost shells of the two respective RGTs.

µ−(k+1), since the bonds between the two shells are in parallel. Because the shells

are in series, the resistance from the center to infinity is simply the geometric sum,

∑∞

k=0 µ−(k+1) = 1

µ−1. Thus the conductance between a and b is Gab = (µ − 1)/2.

Substituting this result in Eq. (4.67), then gives the commute rate

R =µ− 1

µ. (4.79)

This result converges to 1 as µ → ∞, in agreement with the effective medium ap-

104

proach in Sec II as well as our simulation results. Closer to the percolation threshold,

however, Eq. (4.79) and simulation results quantitatively disagree because our naive

picture for the structure of the random graph no longer applies.

Thus we observe that the eventual increase in the commute rate (regime III)

stems from the combined effect of the decrease in the radius of the underlying RGT

embedded within a random graph cluster and the emergence of loops that join two

RGTs in the random graph.

4.7 First-Passage Time Fluctuations

In this section we present simulation results for fluctuations in first-passage times

on a ER random graph. We start with the distributions of the mean first passage

times. In the complete graph all sites are equivalent, but after removing one edge

from the complete graph one has two classes of sites, x and x′ form one class and

the remaining sites, which still have all their bonds intact, form the other class. As a

result one observes 4 different first passage times in this graph . Further removal of

bonds breaks more of this symmetry, leading to the distributions shown in Fig. 4.8

and Fig 4.9.

When the number of bonds in the graph is large, the bumps in the cumulative

distribution are quite sharp (Fig. 4.8). As more edges are removed the distribution

gets smoother and broader (Fig. 4.9), indicating an increase in the heterogeneity

among the pairs of sites. (In these figures the x-axis has been normalized such that

1 corresponds to the value of the MT on a complete graph.)

The observation in Fig. 4.8 that the first passage times are clustered around

discrete values is explained by our conjecture following Eq.(4.36) that the single most

important quantity determining MFPT between two sites is the degree of the sink

105

or the destination site. We give further evidence for the validity of this conjecture in

Fig. 4.10, where we plot the dependence of the MFPT on the degree of the sink of

the random walk. The curve is 2M/k (k the degree of the terminal site), the value

of the return time to a site of degree k. This figure is a scatter plot and the extend

along the y-axis for a single value of k is indicative of the fluctuations one may expect

for a given k. However the general trend that the MFPT depends inversely on the

sink degree is visible, confirming the valdity of our conjecture. The fluctuation arise

due to secondary effects and decrease in size with increase in the sink degree as well

as the density of edges in the graph (Fig. 4.11).

4.8 Summary

We discussed random walks on general graphs and presented an algebraic approach

using the graph laplacian and a geometric approach using the connection with electri-

cal networks to study first-passage properties of the graph. We used these techniques

to study a basic first-passage characteristic of random walks on random graphs that

is related to the time for a walk to travel between two arbitrary points on a graph,

previously published in [8]. We first constructed an effective medium theory and a

small dilution approximation for this mean transit time. The former approach pre-

dicted that the mean transit time, and also all positive integer moments of the transit

time, are independent of the bond concentration p for p greater than the connectiv-

ity threshold p1 = lnN/N . The small dilution approximation also predicts a slow

dependence of the transit time on p near p = 1. Our numerical simulation results are

in qualitative accord with a transit time that is slowly varying in p for p > p1.

Below the connectivity threshold, the transit time is not well defined because the

mean time for a random walk to hop between sites on different components of a

106

disconnected graph is infinite. To avoid this pathology, we studied the inverse of the

commute time, namely, the commute rate. We developed a simple heuristic picture

for the behavior of the commute rate that relied on first identifying an embedded

rooted geodesic tree (RGT) within an arbitrary random graph cluster. For the RGT,

it is simple to compute the commute rate in terms of a geometric picture for the tree

and thus argue that this rate is a non-monotonic function of p in the critical regime.

We then presented a simple physical picture for the influence of loops on the

behavior of the commute rate. Qualitatively, the dependence of the radius of the

underlying RGT on the bond concentration explains the behavior of the commute

rate close to the percolation threshold. For larger µ loops become an important factor

and are ultimately responsible for the non-monotonic dependence of the commute

rate on p. While our arguments were heuristic and the approximations made are

uncontrolled, they provide an intuitive picture for the structure of random graphs

and also provide qualitative and satisfying agreement with simulation results for the

commute rate.

In the next chapter we turn to the voter model, another simple interacting particle

model that is well suited to study on a general graph.

107

0

0.2

0.4

0.6

0.8

1

0.6 0.8 1 1.2 1.4 1.6 1.8 2

PD

F

Ta b

Figure 4.8: Distribution functions for the first passage times in a random graphCumulative (CDF) and Probability (PDF) Distribution Functions for the first passagetimes on a random graph with 100 sites and 1000 bonds. Solid line is the CDF andthe dashed lines are histograms for the PDF( not to scale). On the x-axis is thevalue of the MT divided by the value for the complete graph and the y-axis is thefrequency (normalized) with which this time is observed in the graph. The graph isa series of non uniformly distribute discrete peaks, indicating that the first-passagetimes are clustered around discrete values.

108

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16 18

CD

F

Tab

Figure 4.9: Distribution functions for the first passage times in a random graphCumulative (CDF) and Probability (PDF) Distribution Functions for the first passagetimes between randomly chosen sites a and b on a random graph with 100 sites and100 bonds. Solid line is the CDF and the dashed lines are the histogram for the PDF(not to scale). On the x-axis is the value of the MT divided by the value for thecomplete graph and the y-axis is the frequency (normalized) with which this timeis observed in the graph. There are no bumps, and the the graph is very broad.One can see first passage times 10 times the value for the complete graph. The firstpassage times are more closely placed than the graph with a higher density of bonds(see Fig.4.8)

109

0

100

200

300

400

500

600

700

800

2 4 6 8 10 12 14

FPT

target degree

Figure 4.10: Expected values of the first-passage timeTba from starting site a to a sink site b when the starting site a is chosen uniformlyfrom the set of all the sites in the graph vs the degree kb of the sink, along with theerrorbars. The dotted line is the return time, 2M/kb to site b. The results shownhere are for the ER random graph with N = 100 and M = 200.

110

0

100

200

300

400

500

600

700

2 4 6 8 10 12 14 16 18 20

FPT

target degree

Figure 4.11: Expected values of the first-passage timeTba from starting site a to a sink site b when the starting site a is chosen uniformlyfrom the set of all the sites in the graph vs the degree kb of the sink, along with theerror bars. The dotted line is the return time, 2M/kb to site b. The results shownhere are for the ER random graph with N = 100 and M = 500.

111

0

0.2

0.4

0.6

0.8

1

0.6 0.8 1 1.2 1.4 1.6 1.8 2

PDF

Tab

Figure 4.12: Probability distribution function for return times and first-passagetimes, Tab between a pair of randomly selected sites a and b on a graph with 100 sites and1000 bonds. Tab has been scaled such that Tab is equal to 1 on a complete graph. Thedotted line is the FPT distribution and the solid lines the return time distribution.The return times are just a series of delta functions while the first passage times aredistributed around these delta functions.

Chapter 5

The Voter Model.

In this chapter we will use the formalism developed in Chap. 2 to study the voter

model. We show that the voter model has a a dramatically different behavior on

degree heterogeneous graphs than on regular lattices [21, 44]. Many recent stud-

ies of basic statistical mechanical models on heterogeneous graphs have begun to

understand how the dispersity in the site degree affects critical behavior. A repre-

sentative but incomplete set of examples include percolation [45], the Ising models

[46, 47, 48, 49], diffusion and random walks [35, 50, 34, 8], the contact process [51, 52]

as well as the voter model itself [53, 54, 55].

The voter model is perhaps the simplest example of cooperative behavior and its

simplicity is the reason for our interest in this system. In the model, each site of a

graph is endowed with two states – 0 and 1. The evolution consists of the following

two steps: (i) pick a random voter; (ii) the selected voter adopts the state of a

randomly-chosen neighbor. These steps are repeated until a finite system necessarily

reaches consensus. We first tackle the case of a neutral voter model, for which neither

of the states has an advantage, after which we turn to the biased voter model, for

which one of the two states has a larger probability to be adopted by a site. Primarily

a mathematical exercise within statistical physics, voter model like processes however

112

113

find applications in evolutionary population biology. Moran process, which is a simple

model for evolution [56, 57], is essentially a voter model on a complete graph. In the

next section we formulate the voter model as a Moran process. Our primary goal

will be to understand how the dispersity in the site degree affects critical behavior.

Parts of this chapter have appeared as [58, 59].

5.1 Voter Model Kinetics

We study a population of N individuals each of whom can be of one of the two

strains: 0, which we will call the resident type and 1, which we will call the mutant

type. The individuals live on the sites of a graph with N sites. Using the notation of

Chap. 1, we denote the state of the system by η. η(x) = 0 or 1 is the strain of the

occupant of site x. The two strains of the population may also be considered as two

types of spins, ↑↓. We call as flip the event of changing the strain of the occupant of

x; the resulting state is written ηx,

ηx(y) =

η(y); y 6= x

1 − η(x); y = x

, (5.1)

which we first introduced as Eq.(3.13). At each update event, two individuals are

chosen at random. One reproduces while the other dies and is replaced by the newly-

born offspring, so that N remains constant. In neutral dynamics, neither of the

strains has a selective advantage over the other and both the strains have the same

probability to replicate or die. In Sec. 4.6 we will consider the case where one of the

strains is selectively advantageous.

As a model of cooperative behavior, each site of the graph represents a voter who

114

can have either a positive opinion, 1, or a negative opinion, 0, about an issue. During

the evolution of the process the voters change their opinions based on the state of

their immediate neighbors on the graph.

The heterogeneity in the degree distribution allows us to consider two different

site-based updates,

Death First (Death First (DF)): A randomly-chosen individual dies and is

then replaced by the offspring of a randomly-chosen neighbor. Each individual in this

death/birth process can equivalently be viewed as a voter that adopts the opinion

of a randomly-selected neighbor. Mathematically the kinetics can be encoded as the

probability of flipping the occupant of x during an elemental time step,

P [η → ηx] =1

N

[

η(x)1

kx

∑

y

Axy(1 − η(y)) + (1 − η(x))1

kx

∑

y

Axyη(y)

]

. (5.2)

Birth First (Birth First (BF)): A randomly-chosen individual replicates and

its offspring invades and occupies a randomly-chosen neighboring site, replacing the

occupant there. Thus the probability that the occupant of x flips during an elemental

time step is

P [η → ηx] =1

N

[

η(x)∑

y

Axy

1

ky

(1 − η(y)) + (1 − η(x))∑

y

Axy

1

ky

η(y)

]

. (5.3)

We will also refer to the dynamics of the birth-first mechanism as the invasion

process dynamics, and reserve the term voter model dynamics for the death-first

mechanism.

The birth-first mechanism coincides with the death-first mechanism on degree-

regular graphs for which all sites have the same degree, but cause opposite bias on

115

degree-heterogeneous graphs such that the birth first favors the occupants of sites of

low degree and death first the occupants of sites of high degree. We can introduce

link based dynamics which eliminates this bias,

Link Dynamics (Link Dynamics (LD)): A link is selected at random. If the

individuals at the link ends are different, one of them is designated as the “donor” with

probability one-half. The replicate of the donor then replaces the other individual:

10 → 00 with probability 1/2 while 10 → 11 with probability 1/2. Probability of

flipping the occupant of x is,

P [η → ηx] =1

2M

[η(x)

∑

y

Axy(1 − η(y)) + (1 − η(x))∑

y

Axyη(y)

], (5.4)

where M is the total number of links in the system. The above flip probabilities for

the link dynamics coincide with the dynamics of the birth first and the death first

mechanisms on degree-regular graphs.

Two basic properties of the voter model are the exit probabilities and the mean

time until consensus. The former are the probabilities E1(ρ0) and E0(ρ0), for the

evolution to end with all sites occupied by only 1s or only 0s respectively. The exit

probabilities are functions of the initial density ρ0 of 1s. Because the mean density

of 1s (averaged over all realizations and all histories) on degree-regular graphs is con-

served [21, 38] , and because the only possible final states are consensus, E+(ρ0) = ρ0.

The consensus time TN depends fundamentally on the number of sites N ; the depen-

dence on ρ0 is weak unless ρ0 is close to 0 or 1 and we shall ignore this dependence

henceforth. With this caveat, it is well known that for a regular lattice in d dimen-

sions TN scales as N2 for d = 1, as N lnN for d = 2, and as N for d > 2 [21, 60].

Our main results are to characterize the route by which consensus is reached and to

116

determine TN for heterogeneous networks.

We start in the next section with a discussion of the voter model on a complete

graph, giving the reader a flavor of the primary techniques that we use for our analysis.

We follow in Sec. 5.3 with a discussion on a simple heterogeneous-degree graph, the

complete bipartite graph to begin to understand how dispersity in the site degree

affects voter model dynamics. In Sec. 5.4 we start discussing the voter model on

more general degree-heterogeneous graphs.

5.2 The Mean-Field Voter Model: Neutral Moran Process

Elementary evolutionary population biology models are defined for well-mixed pop-

ulations, which means that all individuals can interact with each other. In terms of

the voter model this means that the graph is complete, i.e. all sites are connected

by direct bonds. Thus an offspring can replace any individual on the graph. Since

the complete graph is regular, the three dynamics DF, BF and LD coincide. The

system can be defined by the density, ρ(t), of 1s in the population. The transition

probabilities take the form,

F (ρ) ≡ P [ρ→ ρ+ δρ] = (1 − ρ)ρ

B(ρ) ≡ P [ρ→ ρ− δρ] = ρ(1 − ρ), (5.5)

where the first line is the probability that an individual of strain 0 is replaced by

the offspring of an individual of strain 1, increasing the density ρ by δρ = 1/N for

a population of N individuals. The second line is for the replacement of a 1 by a 0.

117

We can see immediately that the expected value of the density is conserved,

d

dtE[ρ(t)] ≡ (F (ρ) − B(ρ))

δρ

δt= 0. (5.6)

Let E1(ρ) be the probability that as t → ∞ the population consists of only 1s

when the initial density of 1s in the population is ρ, and E0(ρ) = 1 − E1(ρ) be the

complementary probability that final state of the population is all 0s. From ρ the

system can go to the state ρ+δρ or ρ−δρ according to the probabilities in Eqs.(5.5),

E1(ρ) = F (ρ)E1(ρ+ δρ) + B(ρ)E1(ρ− δρ) + (1 − F (ρ) − B(ρ))E1(ρ), (5.7)

where the last term on the right hand side is for the event that ρ does not change

in an elementary time step. Rearranging the terms and plugging in the transition

probabilities of Eqs.(5.5) we get,

ρ(1 − ρ)(E1(ρ+ δρ) + E1(ρ− δρ) − 2E1(ρ)) = 0,

which we can rewrite as a differential equation after expanding to second order in δρ,

d2

dρ2E1(ρ) = 0,

which tells us that the exit probabilities are linear in ρ. If initially ρ = 1 or ρ = 0

there is no evolution and E1(1) = 1 and E0(1) = 0. Thus we conclude,

E1(ρ) = ρ. (5.8)

118

We did not need to use the above calculation to find the exit probabilities, which can

be inferred directly from Eq.(5.6). The expected density when t→ ∞ is

E[ρ∞] = 1 × E1(ρ0) + 0 × E0(ρ0) = E1(ρ0), (5.9)

when the systems starts with an initially density ρ0. Since the expected value of the

density is conserved by the stochastic dynamics (Eq.(5.6)), we can see that the exit

probability E1(ρ) is given by Eq.(5.8).

However derivation of the exit probability using Eq.(5.7) serves as a little example

of the method of Kolmogorov’s backward propagation. Use of a conserved quantity,

called a martingale in mathematical probability literature [19], to derive expected

values is another technique relevant to our work.

Let us now calculate the time to reach consensus, a state in which all the indi-

viduals are of the same strain. Denoting by T (ρ) the expected time to consensus

when the initial density is ρ, we can write a backward propagation equation similar

to Eq.(5.7),

T (ρ) = F (ρ)(T (ρ+ δρ) + δt) +B(ρ)(T (ρ− δρ) + δt) + (1−F (ρ)−B(ρ))(T (ρ) + δt),

where the first term is for the transition ρ→ ρ+ δρ, the second term for ρ→ ρ− δρ

and the third term for the event that ρ does not change during an elemental time

step. The δt in each term arises due to the time interval spent during a single update.

After rearranging the various terms and expanding to second order in δρ,

−1 =δρ

δt(F (ρ) − B(ρ))

d

dρT (ρ) +

1

2

δρ2

δt(F (ρ) + B(ρ))

d2

dρ2T (ρ),

119

which after using the transition probabilities of Eqs.(5.5) becomes,

ρ(1 − ρ)d2

dρ2T (ρ) = −N, (5.10)

where we set δt = δρ = 1/N . The solution to Eq.(5.10), under the boundary condi-

tions, T (0) = T (1) = 0, is

T (ρ) = N

((1 − ρ) ln

1

1 − ρ+ ρ ln

1

ρ

). (5.11)

Our choice of δt = δρ = 1/N is chosen such that during a unit time interval ev-

ery individual gets one chance at replication or, equivalently, one chance of being

replaced. Thus in one unit of time, on average, the whole current generation is re-

placed. The result of Eq.(5.11) then tells us that the evolutionary process defined

by Eq.(5.5) is very slow and takes N generations to eliminate one strain from the

populations. A population undergoing the dynamics of Eq.(5.5) is said to be un-

dergoing random genetic drift, instead of evolutionary selection. Evolutionary drift

should not be confused with the statistical mechanical concept of drift velocity of a

random walk introduced in Chap. 2. In fact evolutionary drift is associated with the

diffusion term in the Fokker-Plank equation,

∂c(ρ, t; ρ0, 0)

∂t= −δρ

δt

∂

∂ρ(F (ρ)−B(ρ))c(ρ, t; ρ0, 0)+

1

2

(δρ)2

δt

∂2

∂ρ2(F (ρ)+B(ρ))c(ρ, t; ρ0, 0),

for c(ρ, t; ρ0, 0), the transition probability from initial density ρ0 to density ρ at time

t. The first order derivative term on the right hand side is caused by a drift velocity,

vd(ρ) ≡δρ

δt(F (ρ) − B(ρ)) = 0, (5.12)

120

according to Eq.(5.5). The second order term is the diffusion responsible for evolu-

tionary drift. Using the transition probabilities defined in Eq.(5.5) the Fokker-Plank

equation takes the form,

∂

∂tc(ρ, t; ρ0, 0) =

1

N

∂2

∂ρ2ρ(1 − ρ)c(ρ, t; ρ0, 0). (5.13)

5.3 Voter Model on a Bipartite Graph

To understand how dispersity in the site degrees affects voter model dynamics, we

first consider the simple example of the complete bipartite graph Ka,b, with a + b

sites that are partitioned into two subgraphs of a and b sites (Fig. 5.1). Each site in

the a subgraph is connected to all sites in the b subgraph, and vice versa. Thus the

a sites all have degree b, while the b sites all have degree a.

degree ab sites

degree ba sites

Figure 5.1: The complete bipartite graph Ka,b.

Consider voter model evolution on this graph. Let Na.b be the respective number

of individuals of strain 1 on each subgraph and ρa = Na/a, ρb = Nb/b the subgraph

densities of 1s. In an update event, these numbers change according to transition

121

probabilities,

Fa ≡ P [ρa, ρb → ρa +1

a, ρb] =

a

a + b(1 − ρa)ρb,

Ba ≡ P [ρa, ρb → ρa −1

a, ρb] =

a

a + bρa(1 − ρb),

Fb ≡ P [ρa, ρb → ρa, ρb +1

b] =

b

a + b(1 − ρb)ρa,

Bb ≡ P [ρa, ρb → ρa, ρb −1

b] =

b

a + bρb(1 − ρa). (5.14)

The first equation is the probability to increase the number of 1s in subgraph a by

one, for which we need to choose a 0 in the subgraph a to be replaced with the

offspring of a 1 in subgraph b. The second equation is the probability of reducing the

number of 1s in subgraph a. The third and fourth equations similarly account for

the evolution of ρb. From these transition probabilities we can calculate the expected

change in the subgraph densities in one update event,

dρa =1

a(Fa − Ba) =

1

a+ b[(1 − ρa)ρb − ρa(1 − ρb)]

dρb =1

b(Fb − Bb) =

1

a + b[(1 − ρb)ρa − ρb(1 − ρb)] . (5.15)

Since the time increment for an event is proportional to 1/(a + b), the subgraph

densities obey, ρa,b = ρb,a − ρa,b with solution

ρa,b(t) =1

2(ρa,b(0) − ρb,a(0))e−2t +

1

2(ρa(0) + ρb(0)),

while the density of 1s in the entire graph satisfies,

ρ =1

a+ b(aρa + bρb) =

b− a

a+ b(ρa − ρb).

122

Although the mean density ρ is ostensibly not conserved, the bias in the rate

equations for ρa and ρb drive the subgraph densities to the common value ρa(∞) =

ρb(∞) → (ρa(0) + ρb(0))/2. The departure from density conservation vanishes as

this final state is approached. This density non-conservation in the voter model on

heterogeneous graphs was pointed out previously [55], and it was also shown that the

density is conserved if a link-based update rule is used.

However, it is possible to extract a combination of ρa and ρb which is conserved. A

quick glance at Eq.(5.15) reveals that dρa + dρb = 0, which means that the expected

change in

ω ≡ 1

2(ρa + ρb) (5.16)

during a single update event is zero. Thus ω is conserved by the stochastic dynamics

of Eqs.(5.14). These results for the subgraph densities immediately give the exit

probabilities of the voter mode. In the final state ω = 1 if all sites are occupied by

1s, and ω = 0 if all sites are occupied by 0s. If E1(ρa, ρb) be the probability that the

final state contains all 1s, we have

ω(∞) = E1(ρa, ρb),

and since ω is conserved,

E1(ρa, ρb) = 1 − E0(ρa, ρb) = ω(ρa, ρb) =1

2(ρa + ρb). (5.17)

Notice that when the initial populations on the two subgraphs are of different

strains, there is an equal probability of ending with all 1s or all 0s, independent of

the size of the subgraphs. In the extreme case of the star graph Ka,1, with a≫ 1 1s

123

at the periphery and a single 0 at the center, there is only a 50 % change that the

system will end with all 1. Thus a single individual with a macroscopic number of

neighbors plays a significant role in determining the final state.

We now study the mean time until consensus T (ρa, ρb) – either all 1s or all 0s

– as a function of ρa and ρb, the respective initial densities of 1s on the a and b

subgraphs. By enumerating all possible outcomes after a single spin-flip event, the

mean consensus time satisfies the recursion formula [37, 38]:

T (ρa, ρb) =a

a + b(1 − ρa)ρb[T (ρa +

1

a, ρb) + δt] +

a

a + bρa(1 − ρb)[T (ρa −

1

a, ρb) + δt]

+b

a + b(1 − ρb)ρa[T (ρa, ρb +

1

b) + δt] +

b

a+ bρb(1 − ρa)[T (ρa, ρb −

1

b) + δt]

+ (1 − ρa − ρb + 2ρaρb)[T (ρa, ρb) + δt], (5.18)

where δt = 1/(a + b) ≡ 1/N is the time step for a single replacement event. For

example, the first term accounts for picking a 0 from the a subgraph and then a 1

from the b subgraph, so that ρa → ρa + 1a. Similar explanations apply to the next

three terms. The last term accounts for the case where the two selected sites are

occupied by individuals of the same strain. This equation is subject to the boundary

conditions T (0, 0) = T (1, 1) = 0.

Expanding this recursion formula to second order, we get Kolmogorov’s backward

propagation equation for the time to consensus,

Nδt = (ρa−ρb)(∂a−∂b)T (ρa, ρb)+−1

2(ρa+ρb−2ρaρb)

(1

a∂2

a +1

b∂2

b

)T (ρa, ρb), (5.19)

where ∂a denotes a partial derivative with respect to ρa. The first term on the

right-hand side accounts for the convection that drives the system to equal subgraph

densities. This bias can be seen directly by writing the Fokker-Plank or the for-

124

ward propagation equation for the evolution of subgraph densities themselves. The

evolution of the occupation probabilities, c(ρa, ρb, t) can be written,

c(ρa, ρb, t+ δt) = Fa(ρa −1

a, ρb)c(ρa −

1

a, ρb, t) + Ba(ρa +

1

a, ρb)c(ρa +

1

a, ρb, t)

+Fb(ρa, ρb −1

b)c(ρa, ρb −

1

b, t) + Bb(ρa, ρb +

1

b)c(ρa, ρb +

1

b, t),

+(1 − Fa(ρa, ρb) − Ba(ρa, ρb) − Fb(ρa, ρb) − Bb(ρa, ρb))c(ρa, ρb, t)

where, analogous to Eq.(5.18), the first line accounts for change in ρa, the second

line for change in ρb and the third line for the events when state of the system

remains unchanged. While writing the above expressions we have suppressed the

dependence on the initial state, ρa(0), ρb(0). Once again we rearrange and expand

the probabilities above to second order to get the Fokker-Planck equation,

∂tc = − 1

aδt∂a(Fa − Ba)c−

1

bδt∂b(Fb − Bb)c

+1

2

1

a2δt∂2

a(Fa + Ba)c+1

2

1

b2δt∂2

b (Fb + Bb)c, (5.20)

where in the first line we can identify the drift velocities for the two subgraph densi-

ties,

va(ρa, ρb) ≡ a+ba

(Fa(ρa, ρb) − Ba(ρa, ρb)) = ρb − ρa,

vb(ρa, ρb) ≡ a+bb

(Fb(ρa, ρb) − Bb(ρa, ρb)) = ρa − ρb, (5.21)

using δt = 1/N = 1/(a + b). Unlike the complete graph in last section, the drift in

the voter model on the bipartite graph does not vanish. However, the drift velocities

in Eqs.(5.21) drive the system to equal subgraph densities ρa = ρb. The effects of

the two drift terms operate at time scales that are a and b times smaller than the

125

respective diffusion terms in Eq.(5.20). The drift term in Eqs.(5.19& 5.20) drives the

system to have equal subgraph densities in a time of the O(1). Subsequently, diffusive

fluctuations govern the ultimate approach to consensus, as illustrated in Fig. 5.2.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

ρ b

ρa

Figure 5.2: Subgraph densities ρb(t) versus ρa(t)for one realization of the voter model on a bipartite graph with a = b = 105 with theinitial conditions ρa(0) = 0 and ρb(0) = 1. The dotted curve is the motion causedby the convection term in Eq. (5.20) and the solid curve the diffusive motion whichfollows the initial transient.

While the transient caused by the mismatch of the initial subgraph densities

disappears, ω defined in Eq.(5.16) does not change. When the transient disappears,

ω = ρa = ρb. Thus we replace the ρs by ω in Eq.(5.19) to get,

ω(1 − ω)∂2ωT (ω) = − 4ab

a + b, (5.22)

126

which is of the same form as Eq.(5.10) with N replaced by an effective populations

size Neff ≡ 4ab/(a+ b) with a solution,

T (ω) =4ab

a+ b((1 − ω) ln

1

1 − ω) + ω lnω. (5.23)

The complete bipartite graphs is as a simple example of a structured population,

whose evolutionary dynamics can be reduced to that of a well-mixed population with

an effective population size [61]. Replacing the ρ in Eq.(5.20) by ω leads to the

following Fokker-Planck equation,

∂tc(ω, t) =4ab

a + b∂2

ωω(1 − ω)c(ω, t), (5.24)

which is the same as Eq.(5.13) for a well-mixed population with ω instead of ρ and

an effective population size Neff = 4ab/(a + b) instead of the actual population size

N .

5.4 Voter Model on Heterogeneous-Degree Random Graphs

Building on our approach for the bipartite graph, we now study graphs with an

arbitrary degree distribution. Defining the mean number of 1s in the neighborhood

of x,

ωη(x) ≡ 1

kx

∑

y

Axyη(y), (5.25)

we can cast the single update transition probabilities of Eq.(5.2) as,

Fx[η] ≡ P [x : 0 → 1] =1

N(1 − η(x))ωη(x),

Bx[η] ≡ P [x : 1 → 0] =1

Nη(x)(1 − ωη(x)). (5.26)

127

If we assume that the degree-heterogeneous graph is of the Molloy-Reed variety with-

out degree correlations, we can replace the adjacency matrix elements by the expected

values,

E[Axy] =kxky

µ1N. (5.27)

Using the expected values of the adjacency matrix elements in Eq.(5.25) we find,

ωη(x) = ω ≡ 1

µ1N

∑

y

kyη(y), (5.28)

for all sites x. Thus the assumption of Eq.(5.27) is a mean-field assumption such

that the average density of 1s around each site is the mean value ω in Eq.(5.28).

We define ρk as the density of up spins in the subset of sites with degree k,

ρk ≡ 1

Nk

∑

x:kx=k

η(x), (5.29)

where Nk is the number of sites of degree k. Single update transition probabilities in

Eq.(5.2) can be adapted to

Fk[ρk] ≡ P [ρk → ρk +1

Nk

] =1

N

∑

x:kx=k

(1 − η(x))1

kx

∑

y

Axyη(y)

Bk[ρk] ≡ P [ρk → ρk −1

Nk] =

1

N

∑

x:kx=k

η(x)1

kx

∑

y

Axy(1 − η(y)), (5.30)

which under the mean-field assumption of a MR graph become,

Fk =1

N

∑

x:kx=k

(1 − η(x))1

µ1N

∑

y

kyη(y) = nk(1 − ρk)ω,

Bk =1

N

∑

x:kx=k

η(x)1

µ1N

∑

y

ky(1 − η(y)) = nkρk(1 − ω), (5.31)

128

where we used nk ≡ Nk/N .

In a fashion similar to Eq.(5.18), the recursion formula for the mean consensus

time starting with initial densities ρk is

T (ρk) =∑

k

Fk[ρk](T (ρk + δk) + δt] +∑

k

Bk[ρk](T (ρk − δk) + δt)

+(1 −∑

k

Fk[ρk] −∑

k

Bk[ρk)(T (ρk) + δt), (5.32)

where the first term is the probability of increasing ρk, second the probability of

decreasing ρk and the last term of leaving ρk unchanged. We have abbreviated

the elemental change in ρk, 1/Nk = δk. Expanding to second order in δk we get

Kolmogorov’s backward propagation equation for the time to consensus,

∑

k

δkδtnk(ω − ρk)∂kT +

1

2

∑

k

δ2k

δtnk(ω + ρk − 2ωρk)∂

2kT = −1, (5.33)

while Kolmogorov’s forward propagation equation or the Fokker-Planck equation for

occupation probabilities c(ρk, t) takes the form,

∂tc(ρk, t) = −∑

k

δkδtnk∂k(ω−ρk)c(ρk, t)+

1

2

∑

k

δ2k

δtnk∂

2k(ω+ρk−2ωρk)c(ρk, t).

(5.34)

Since the phase space consists of the variables ρk, we have the corresponding drift

velocities,

vk =δkδtnk(ω − ρk) = ω − ρk, (5.35)

The drift velocities give us the expected change in the densities ρk during a single

129

update attempt, which allows us to write,

dρk

dt= ω − ρk, (5.36)

for the expected values, while ω is conserved in expected value,

dω

dt=

1

µ1N

∑

k

knkdρk

dt= 0. (5.37)

In fact ω can be proved to be conserved on any graph, not just the MR graphs to

which the current arguments apply. Conservation of ω allows us to solve Eq.(5.36),

ρk(t) = ω(0) − (ω(0) − ρk(0))e−t. (5.38)

The initial mismatch between the densities ρk, causes an initial transient governed

by the drift term in the Fokker-Plank equation (Eq.(5.34)). After the transient which

lasts for time O(1), the densities ρk become equal to the conserved quantity ω and

the propagation equations are reduced to

1

N

∑

k

(k2

µ21

nk

)ω(1 − ω)∂2

ωT = −1. (5.39)

and

∂tc(ω, t) =1

N

∑

k

(k2

µ21

nk

)∂2

ωω(1 − ω)c(ω, t), (5.40)

where we can identify the second moment of the degree distribution, µ2 =∑

k k2nk

and define the effective population size,

Neff =µ2

1

µ2N, (5.41)

130

to write the time to consensus,

TN(ω) = Neff [(1 − ω) ln1

1 − ω+ ω ln

1

ω. (5.42)

For a scale-free network [16] with degree distribution nk ∼ k−γ with γ > 2, the nth

moment is µn ∼∫ kmax knnk dk. Here kmax ∼ N1/(γ−1) is the maximal degree in a

finite network of N sites; this quantity may be obtained from the extremal condition∫

kmaxk−γ dk = 1

N[62]. Thus the second moment diverges at the upper limit for γ ≤ 3

while the first moment diverges for γ ≤ 2.

Assembling the results for the moments, the mean consensus time on a scale-free

graph has the N dependence

TN ∼

N γ > 3,

N lnN γ = 3,

N2(γ−2)/(3−γ) 2 < γ < 3,

(lnN)2 γ = 2,

O(1) γ < 2.

(5.43)

The prediction TN ∼ N/ lnN for ν = 3 may explain the apparent power-law

TN ∼ N0.88 in a previous simulation of the voter model on such a network [55].

To test Eq.(5.43), we simulated the voter model on a growing network with redi-

rection that is built by adding sites sequentially, where each new site attaches either

to a randomly-selected site with probability 1 − r or to the ancestor of this target

with probability r [17]. This growth rule gives a network with a power-law degree

distribution nk ∝ k−γ, with γ = 1 + 1r. Thus γ ranges between the values (2,∞)

as r is varied between 0 and 1. We chose the out degree of each site to be 4, and

131

redirection was applied to each outgoing bond of the new site

Fig. 5.3 shows the N dependence of the consensus time for representative values

of the degree exponent γ for both the MR network and the GNR. The results for the

two networks with the same γ are extremely close, suggesting that degree correlations

have a small effect on voter model dynamics. There is also curvature in the data that

originates from finite-N effects. Using the maximal degree kmax ∼ N1(ν−1) in the

definition of the moments ultimately lead to the exponent for TN being modified by

the corrections, for γ between 2 and 3,

d lnT

d lnN=

2(γ − 2)

γ − 1(1 − aN

2−γ

γ−1 + bNγ−3

γ−1 ), (5.44)

where a and b are of order 1.

For γ close to 2 or 3, the leading correction term decays slowly in N , causing

a discrepancy between our numerics and the theory. For example, for γ = 2.3 in

Fig. 5.3. the numerical best-fit slope to the data decreases from 0.53 to 0.48 as we

successively eliminate the first 18 data points. This accords well with the theoretical

prediction of 0.46 for the slope from Eq. (5.43). For γ = 2.5, the two correction

terms both decay at the same rate and have opposite sign. Here we may expect the

smallest corrections, as borne out by the data - the best-fit slope decreases from 0.680

to 0.671 as the first 18 data points are deleted, while the theoretical prediction for

the slope is 2/3. The case γ = 2.7 has the slowest decaying correction term and here

we observe the largest deviation between simulation and theory - the slope remains

in the range 0.77 - 0.79 as the first 18 points are deleted, while theory predicts a

slope of 0.82.

132

102

103

102 103 104

TN

N

Figure 5.3: Consensus time TN versus Non scale-free networks with degree distribution nk = k−ν for ν = 2.1(+), 2.3(×),2.5(∗), 2.7() and 2.9(•). Each data point is based on 100 realizations of the graphand 10 realizations of the voter model on each graph.

5.5 Invasion Process on Heterogeneous-Degree Random Graphs

We now turn to the invasion process or the birth-first mechanism for the voter model

dynamics, first studied by Castellano in [63] As discussed in Sec. 5.1, in this model

the occupant of a randomly-selected site replicates, and the replicant replaces the

occupant of a neighboring site. The single update transition probabilties are

P [η → ηx] =1

N

[

η(x)∑

y

1

ky

Axy(1 − η(y) + (1 − η(x))∑

y

1

ky

Axyη(y)

]

, (5.45)

133

with expected change in η(x),

E[∆η(x)] = (1 − 2η(x))P [η → η(x)]

=1

N

∑

y

[(1 − η(x))

1

ky

Axyη(y) − η(x)1

ky

Axy(1 − η(y))

]. (5.46)

The two terms inside the sum on the right hand side, corresponding to a 0 → 1 flip

and a 1 → 0 flip, are clearly asymmetric if we exchange x and y. This asymmetry

disappears if the graph is degree-regular. As a result, similar to the voter model

dynamics of Sec. 5.4, ρ is not conserved by the stochastic dynamics. However, we

can make the right hand side of Eq.( 5.46) symmetric if we divide both the terms by

kx. If we perform this division and sum over x we find,

E[∆η(x)

kx

] = 0,

from which we infer the normalized conserved quantity,

ω−1(η) ≡1

µ−1N

∑

x

η(x)

kx

, (5.47)

where µ−1 =∑

x k−1x /N , is the −1-th moment of the degree distribution. With the

knowledge of the conserved quantity we can immediately write the exit probability

as before,

E1(η) =1

µ−1N

∑

x

η(x)

kx

. (5.48)

134

Replacing the adjacency matrix elements by their expected values in Eq.(5.27) ,

Eq.(5.45) can be written as

P [η → ηx] =kx

µ1N[η(x)(1 − ρ) + (1 − η(x)ρ] ,

which can be used to write the single update transition rules in terms of the densities

ρk,

Fk ≡ P [ρk → ρk + δk] = nkk

µ1

(1 − ρk)ρ

Bk ≡ P [ρk → ρk − δk] = nkk

µ1ρk(1 − ρ). (5.49)

If we write the Fokker-Planck equation, akin to Eq.(5.34) for the death-first dynamics,

we will find that the drift velocities for each of the densities ρk:

vk =δkδt

(Fk − Bk) =k

µ1

(ρ− ρk),

which tells us that the densities approach the same value ρ over time scales of O(1).

However, ρ, unlike ω for the voter model dynamics, is not conserved by the stochastic

dynamics,

ρ =∑

k

nkvk = ρ− ω.

Once all ρk = ρ, ρ itself can be related to the conserved quantity ω−1 via Eq.( 5.47).

We find that ρ and as a result all the ρk approach ω−1.

Thus the IP dynamics has an initial transient due to the dispersion in the den-

sities ρk, after the disappearance of which we can replace the variables ρk by the

conserved quantity ω−1 to write Kolmogorov’s backward propagation equation for

135

time to consensus,

1

µ1µ−1Nω−1(1 − ω−1)∂

2ω−1

T = −1, (5.50)

where we can identify the effective system size,

Neff = µµ−1N . (5.51)

Unlike the VM dynamics the Neff does not show any dramatic dependence on the

structure of the graph and the consensus time,

TN ∼ O(N)

for all the cases enumerated in Eq.(5.43).

In summary, the voter model on a heterogeneous network approaches consensus by

a two-stage process of quick evolution to a homogeneous state followed by a diffusive

evolution to final consensus. By neglecting degree correlations, the consensus time

TN on scale-free graphs has the following dependence on the degree distribution

exponent γ: for γ < 2, TN ∼ O(1), while for γ > 3, TN ∼ N . In the intermediate

regime of 2 < γ < 3, TN ∼ N2(γ−2)/(γ−1). Generically, TN grows sub-linearly with

N ; that is, high degree sites greatly accelerate the approach to consensus. Finally,

the N -dependence of TN is virtually the same for networks without and with degree

correlations.

5.6 Evolutionary Dynamics With Selection

In this section we investigate the likelihood for fitter mutants to overspread an oth-

erwise uniform population on heterogeneous graphs by evolutionary dynamics. Such

136

a process underlies epidemic propagation [64, 65, 66], emergence of fads [67, 68, 69],

social cooperation [70], or invasion of an ecological niche by a new species [56, 71,

72, 73, 74]. The update dynamics at each elemental time step remains the same. We

also introduce a selective advantage, or fitness, such that each individual may be of

a unit-fitness strain 1 or of strain 0 with lower fitness 1 − s, with 0 < s < 1. These

fitnesses determine the replication or death rates of each individual. The selective

advantage leads to a dynamical competition in which selection dominates for large

populations, while random genetic drift [57, 75] occurs for small populations or weak

selection.

The three evolutionary models discussed in Sec. 5.1 in presence of evolutionary

selection can be formulated as:

Biased Link Dynamics (LD): A link is selected at random. If the individuals

at the link ends are different, one of them is designated as the “donor” with proba-

bility proportional to its fitness. The replicate of the donor then replaces the other

individual: 10 → 11 with probability 1/2 while 10 → 00 with probability (1-s)/2

(Fig. 5.4).

Death First with selection or Biased Voter Model (VM):

An individual dies with probability inversely proportional to its fitness, and is then

replaced by the offspring of a randomly-chosen neighbor. Equivalently, death occurs

randomly and replacement is proportional to the fitness of the donor. We imple-

ment the VM by updating a randomly-chosen genotype 0 with probability 1, while

the fitter genotype 1 is updated with a probability 1 − s. Each individual in this

death-first/birth-second process can equivalently be viewed as a voter that adopts

the opinion of a randomly-selected neighbor [21, 76, 58].

Birth First with selection or Biased Invasion Process (IP):

137

In this birth-first/death-second process, a randomly-chosen individual replicates with

probability proportional to its fitness, and its offspring then replaces an individual

at a randomly-chosen neighboring site [75, 56].

(c)

(IP)

(b)

(VM)

(a)

(LD)

1−s (1−s)/6 1/6 (1−s)/31/31

Figure 5.4: Update illustration for two specific sites.Strains 0 and 1 are denoted by and • respectively. Shown are the possible transitionsand their respective relative rates due to the interaction of two sites across a link forLD, VM and IP dynamics.

One strain ultimately replacing all other strains in the population is termed fix-

ation. An important, and easily checked fact is that these evolutionary models are

equivalent on degree-regular graphs; moreover, as we will show, the fixation proba-

bility in LD can be obtained exactly, independent of the underlying graph. However,

essential differences arise on degree-heterogeneous networks [58, 55, 63] that may lead

to an enhancement of the fixation probability, as discovered previously for the IP [74].

Here we cast LD, VM, and IP on degree-heterogeneous graphs within the same uni-

fying framework to understand the interplay between selection and random drift on

the fixation probability. By this approach, we show that on degree-heterogeneous

graphs the best strategy to reach fixation with VM dynamics is for the fitter strain

138

to be on high-degree sites. Conversely, for IP dynamics, it is best for the fitter stain

to be on low-degree sites.

We first study the evolution in VM dynamics. We symbolically represent the

state of the system by η. In an elemental time interval δt we choose a random site

x. If the genotype at this site at time t, denoted as ηt(x), equals 0, then site x is

updated by choosing a random neighbor y and setting η(t+δt)(x) = ηt(y) (Fig. 5.4).

However if ηt(x) = 1, the VM update is implemented with probability 1 − s. This

update rule can be written as

P[η→ηx] =∑

y

Axy

Nkx

[1−η(x)]η(y) + (1−s)η(x)[1−η(y)] (5.52)

The first term describes the update step for the case where (η(x), η(y)) = (0, 1)

and x,y are connected. Each of the nearest neighbors y of x may be selected with

probability Axy/kx. Here Axy is the adjacency matrix whose elements equal 1 if

xy are connected and zero otherwise. The second term in Eq. (5.52) is explained

analogously.

For degree-heterogeneous graphs, the density ρk of strain 1 at sites of degree k

increases by 1/Nk with probability Fk(η) and decreases by 1/Nk with probability

Bk(η) in an elemental update, where

Fk(η) =1

kN

′∑

xy

Axy[1 − η(x)]η(y)

Bk(η) =1 − s

kN

′∑

xy

Axyη(x)[1 − η(y)] (5.53)

are the forward (0 → 1) and backward (1 → 0) evolution rates. The primes on the

sums denote the restriction that the degree of sites x equals k. the sum over all k

139

then gives the total transition rate of Eq.(5.52). We seek the fixation probability Φ

to the state consisting entirely of strain 1 as a function of the initial densities of 1.

This probability obeys the backward Kolmogorov equation GΦ = 0 [37, 38], subject

to the boundary conditions Φ(0) = 0 and Φ(1) = 1. In the diffusion approximation,

the generator G of this equation may be expressed as a sum of the changes in ρk over

all k,

G =1

δt

∑

k

[δρk(Fk−Bk)∂k+

(δρk)2

2(Fk+Bk)∂

2k

], (5.54)

with δρk = 1/Nk = 1/(N nk) the change in ρk in a single update of a site of degree

k, and ∂k ≡ ∂∂ρk

.

For the special case of degree-regular graphs, where kx = k∀x, both sums in

Eq.(5.53) count the total number α of active links between different strains

α =1

Nµ1

∑

x,y

Axyη(x)[1 − η(y)], (5.55)

The generator thus reduces to

G = α

[s∂ρ +

1

N(1 − s

2)∂2

ρ

], (5.56)

where we use δρ = δt = 1/N . In this form, the drift and diffusion terms differ by

a factor O(sN). Thus selection dominates when the population N is larger than

O(1/s), while random genetic drift is important otherwise.

Notice that the probability of increasing the density of strain 1 at each update is

a factor 1/(1 − s) larger than the probability of decreasing the density. By its con-

struction, this same bias arises for LD on general networks. As a consequence of this

bias, the evolutionary process underlying fixation is the same as the absorption of a

140

uniformly biased random walk in a finite interval, from which the fixation probability

is [37, 38]

Φ(ρ) =1 − (1 − s)Nρ

1 − (1 − s)N→ 1 − e−sNρ/(1−s/2)

1 − e−sN/(1−s/2). (5.57)

The former is the exact discrete solution of GΦ = 0 on a finite network, while the

latter continuum limit represents the solution to GΦ = 0 in the diffusion approxima-

tion. These results apply for all three models on degree-regular graphs and for LD

on general graphs.

For degree-heterogeneous graphs, we found in Eq.(5.37) that the conserved quan-

tity for neutral dynamics (s = 0) was the average degree-weighted density ω, while

the overall density ρ of strain 1 is no longer conserved. The existence of this new

conservation law suggests that we study the time evolution of the expectation value

of ω. Since

ω(ηx) = ω(η) + kx(1 − 2η(x))/µ1N,

ω evolves in time according to,

∂tω =1

δt

∑

x

[ω(ηx) − ω(η)]P [η → ηx]

=s

µ1N

∑

x,y

Axyη(x)(1 − η(y)) = sα. (5.58)

Notice that ω is conserved in the absence of selection s = 0) a feature that ultimately

stems from the update rate being inversely proportional to site degree (Eq.(5.53)). To

evaluate the expression in Eq.(5.58) we make the mean-field assumption of replacing

the adjacency matrix elements by their expected values in Eq.(5.27). This assumption

141

simplifies Eq.(5.53) to

Fk(η) = nkω(η)(1 − ρk(η)),

Bk(η) = (1 − s)nk(1 − ω(η))ρk(η),

and Eq.(5.58) to,

∂tω = sω(1 − ω). (5.59)

Eq.(5.59) can be solved to give,

ω(t) =1

1 − [1 − ω(0)−1]e−st. (5.60)

The time evolution of the expectation value of ρk reduces to

∂tρk =δρk(Fk − Bk)

δt= ω − ρk + s(1 − ω)ρk. (5.61)

To solve this equation we combine it with Eq.(5.59) to yield

∂t(ω − ρk) = −(ω − ρk)(1 − s)(1 − ω)

with solution

ρk(t) = ω(t) − e−t[ω(0) − ρk(0)]ω(0) + [1 − ω(0)]e−st (5.62)

For small selective advantage (s≪ 1), this equation involves two distinct time scales.

On a time scale of order one, all the ρk become equal to ω, whereas the evolution of

ω occurs on a longer time scale of order s−1 ≫ 1 (Fig. 5.5).

142

0

0.1

0.2

0.3

0.4

0.5

0.6

0 1 2 3 4 5t

VM

IP

ω-1

ρ

ω

Figure 5.5: Moments of the 1 density in the biased VM and biased IPon a network of 104 sites with a power-law degree distribution nk ∼ k−ν (ν = 2.5),and no correlations between site degrees. Nodes with degree larger than the meandegree are initialized to 1 while all other sites are 0. For the VM, s = 8.5 × 10−4,while for the IP, s = 10−4.

We now determine the fixation probability simply by replacing the ρk by ω in the

forward and backward rates F and B in Eqs.( 5.59). In a similar vein, we replace

the derivative ∂k by (knk/µ1)∂ω as we did for the neutral case. Then the generator

in Eq.(5.54) becomes

G = s∑

k

(knk

µ1

)ω(1 − ω)∂ω +

1

N

(1 − s

2

)∑

k

(k2nk

µ21

)ω(1 − ω)∂2

ω

= ω(1 − ω)

[s∂ω +

µ2

µ21N

(1 − s

2

)∂2

ω

], (5.63)

143

which is the same as the generator for degree-regular graphs in Eq.(5.56) with the

actual population size N replaced by an effective population size,

Neff ≡ µ21

µ2

N ∼

Nγ > 3

N2(γ−2)/(γ−1)2 < γ < 3;

O(1)γ < 2,

(5.64)

with logarithmic corrections for γ = 2 and γ = 3. The effective population size

becomes much less than N when µ2 diverges; this occurs when γ > 3. A similar

change in the effective size of the population is observed for biological species evolving

in a spatially heterogeneous environment [56, 61].

The solution to GΦ = 0, with G given by Eq.(5.63) is

Φ(ω) =1 − e−sNeffω/(1−s/2)

1 − e−sNeff/(1−s/2). (5.65)

Our numerical data for the fixation probability shows both excellent scaling and

agreement with this functional form for Φ (Fig. 5.6). Eq.(5.65) also provides the

fixation probability when the system starts with a single mutant at a site of degree

k:

Φ1 =

k/Nµ1s≪ 1/Neff ;

skµ1/µ21/Neff ≪ s≪ 1.(5.66)

The crucial feature is that the fixation probability of a single fitter mutant is propor-

tional to the degree of the site that it initially occupies (Fig. 5.7). Notice also that

because the relative effect of selection versus random genetic drift is determined by

the variable combination sNeff , random genetic drift can be important for much larger

populations compared to the case of degree-regular graphs. In fact, for a power-law

144

graph with γ < 2, random genetic drift prevails for all population sizes

0.5

0.6

0.7

0.8

0.9

1

0.1 1 10

Φ

sNeff

Figure 5.6: Scaling plot of fixation probabilitiesfor VM (filled) and IP dynamics (open symbols).

5.6.1 Fixation in the biased Invasion Process

We now study fixation in the complementary biased invasion process (birth-first

dynamics). Here a randomly selected individual reproduces with probability propor-

tional to its fitness, hence the single update transition probability is,

P [η → ηx] =1

N

∑

y

Axy

ky

(1 − η(x))η(y) + (1 − s)η(x)(1 − η(y)). (5.67)

145

10-4

10-3

10-2

10-1

101 102

Φ1

k

Figure 5.7: Fixation probability of a single mutant initially at a site of degree kon an uncorrelated power-law degree distributed (nk ∼ k−ν , ν = 2.5) graph withN = 103 and µ1 = 8. The empty symbols correspond to IP dynamics with s = 0.004(), s = 0.008 () and s = 0.016 (); the filled symbols correspond to VM dynamicswith s = 0.01, (), s = 0.02 (•) and s = 0.08 (). The solid lines, with slopes +1 and−1, correspond to the second of Eqs.(5.66) and (5.71).

Notice an essential difference between VM and IP dynamics. In the VM the transition

rate is proportional to the inverse degree kx of the site of the disappearing strain

(Eq.(5.52)), while in the IP the transition rate is proportional to the inverse degree

ky of the site of the reproducing genotype (Eq.(5.67)).

For degree-uncorrelated graphs, the transition probabilities are

Fk(η) =k

µ1

nkρ(1 − ρk),

Bk(η) =k

µ1nk(1 − ρ)ρk. (5.68)

146

Consequently the time evolution of ρk is given by, in analogy with Eq.(5.61),

∂tρk =k

µ1(ρ− ρk + sρk(1 − ρ)),

from which low-order moments obey the equations of motion:

∂tω−1 =s

µ1µ−1ρ(1 − ρ),

∂tρ = ρ− ω + sω(1 − ρ),

∂tω =µ2

µ1

(ρ− ω2 + sω2(1 − ρ).

We saw in Sec. 5.5 that, in contrast to the VM dynamics, the conserved quantity in

the unbiased IP is ω−1, the inverse degree weighted frequency . For the biased IP,

ω−1 becomes the most slowly changing quantity (see Fig. 5.5). Hence we transform

all derivatives with respect to ρk in the generator to derivatives with respect to ω−1

to yield

G =ω−1(1 − ω−1)

µ1µ−1

[s

∂

∂ω−1+

1

N(1 − s

2)∂2

∂ω2−1

](5.69)

from which, in close analogy with our previous analysis of the VM, the fixation

probability is

Φ(ω−1) =1 − e−sNω−1/(1−s/2)

1 − e−sN/(1−s/2)(5.70)

From Eq.(5.69), the effective population size Neff equals N , contrary to VM dynamics

(Eq.(5.65)). More strikingly, the fixation probability of a single mutant acquires the

non-trivial dependence on the degree k of the occupied site (Fig. 5.7)

Φ1 =

1/(Nkµ−1)s≪ 1/N ;

s/(kµ−1)1/N ≪ s≪ 1(5.71)

To conclude, mutants are more likely to fixate in the voter model (VM) when they are

initially on high-degree sites [Eq.(5.66)], while in the invasion process (IP) fixation

is more probable when mutants start on low-degree sites [Eq.(5.71)]. This behavior

is understandable simply. In the VM, a well-connected individual is more likely to

be asked his opinion before he asks one of his neighbors. In the IP, a mutant on a

high-degree site is more likely to be invaded by a neighbor before the mutant itself

can invade. Thus network heterogeneity leads to effective evolutionary heterogeneity.

We can also understand the evolution when a mutant appears at a random site

on a graph. In the selection-dominated regime (sNeff ≫ 1) of the VM, we average

Eq.(5.66) over all sites and find that the fixation probability on degree-uncorrelated

graphs is smaller by a factor µ21/µ2 ≤ 1 than that on regular graphs. Thus a het-

erogeneous graph is an inhospitable environment for a mutant that evolves by VM

dynamics. Conversely, performing the same average of Eq.(5.71) over all sites, the

fixation probability for the IP is the same on all degree-uncorrelated graphs. Finally,

in the small-selection limit (sNe ≪ 1), the site average fixation probability is the

same for both the VM and IP on degree-uncorrelated graphs.

Chapter 6

Conclusions.

In this thesis we studied dynamical processes defined on graphs. Both the graphs

and the processes are motivated by phenomena of a socio-biological motivation. We

find that both the random walk as well as the voter model on real world networks

show novel behavior which is not observed on lattices. Our results complement work

done for other interacting particle systems, such as percolation, the Ising model and

the contact process.

For the random walk on the ER random graph we found that transit times show

the same system size scaling as the transit times for the random walk on the complete

graph. Similar results have been obtained elsewhere for the random walk on degree-

heterogeneous graphs.

We then developed a formalism to study the voter model on degree-heterogeneous

graphs. Unlike the random walk, the voter model shows a drastic dependence on the

structure of the graph. To begin with, one has to distinguish between the birth-

first ( death-later ) and the death-first ( birth-later ) versions of the model. We

identified fast varying quantities, which converge to an average value in time of O(1).

The remaining evolution of the system resembles the evolution on a complete graph,

enabling us to derive asymptotically accurate results. We applied this methodology to

148

149

the voter model and the Moran model. For the voter model we found that the opinion

of a highly connected site is more likely to be adopted by the other sites in the graph,

and that the time to consensus increases with the increase in the dispersity of the

degree distribution. For the Moran model we discovered that the mutant’s fixation

probability depends on the specific birth/death mechanism used. For the death-first

(voter model dynamics) mutants are more likely to fixate when they are initially on

high-degree sites, while in the birth-first (invasion process dynamics) version fixation

is more probable when the mutants start on low degree sites.

We are currently trying to extend the methodology developed here to other in-

teracting particle systems, such as the contact process and the exclusion process.

The effect of heterogeneity other than dispersity in the degree distribution is also

worthy of attention. Despite the difficulties involved in encoding structural hetero-

geneity, efforts to understand the behavior of interacting particle systems on general

heterogeneous graphs are underway.

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Curriculum Vitae

Vishal Sood

E-mail: vsood@buphy.bu.eduWeb: http://physics.bu.edu/∼vsoodWork: +1 617 353 3845Home: +1 857 540 1479

Research Interests:Statistical mechanics, stochastic processes, random walks, population biology,

dynamics of social systems, computer networks, graph theory, evolutionary dynamicsand game theory.Education:

01/07: PhD. Physics (Requirements completed 07/06).Boston University, Boston, MA, USA.

07/00-05/01: Graduate study in physics.University of Pennsylvania, Philadelphia, PA, USA.

09/96-06/00: Bachelor of Technology in Engineering Physics, August 2000.Indian Institute of Technology Bombay, Mumbai, India.

Fellowships:

2002–2003 Teaching Fellow, Department of Physics, Boston University

2001–2006 Research Assistant, Department of Physics, Boston University

2004-2005 Research Assistant, CNLS, LANL, Los Alamos New Mexico.

Contributed and Invited Talks:

1. V. Sood. “First-Passage Properties of Erdos-Renyi Random Graph”, PosterSession at Stat-Phys-23 Bangalore, India, July 2004.

2. V. Sood. “Voter model on Heterogeneous Graphs”, contributed talk, APSMarch Meeting, Los Angeles, CA, USA, March 2004.

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3. V. Sood. “Voter model on Heterogeneous Graphs”, contributed talk, FirstCornell Summer School in Probability, Ithaca, NY, USA, July 2005.

4. V. Sood. “Evolutionary Dynamics on Degree-Heterogeneous Graphs”, con-tributed talk, Second Cornell Summer School in Probability, Ithaca, NY, USA,June 2006.

Research papers (refereed journals):

PhD Thesis Research:

1. V. Sood, S. Redner, and D. ben-Avraham. “First-Passage Properties of theErdos-Renyi Random Graph”, J. Phys. A 38, 109-123 (2005).

2. V. Sood and S. Redner. “Voter Model on Heterogeneous Graphs”, Phys. Rev.Lett. 94, 178701 (2005).

3. T. Antal, S. Redner, and V. Sood. “Evolutionary Dynamics on Degree-Heterogeneous Graphs”, Phys. Rev. Lett. 96, 188104 (2006).