Watts and Strogatz Model

As explained in more detail in Section 1.3, our next model was inspired by the popular concept of “six degrees of separation,” which is based on the notion that every one in the world is connected to everyone else through a chain of at most six mutual acquaintances. Now an Erdös–Rényi random graph for n = 6 billion people in which each individual has an average of μ = 42.62 friends would have average pairwise distance (log n)/(log μ) = 6, but would have very few triangles, while in social networks if A and B are friends and A and C are friends, then it is fairly likely that B and C are also friends.

To construct a network with small diameter and a positive density of triangles, Watts and Strogatz (1998) started from a ring lattice with n vertices and k edges per vertex, and then rewired each edge with probability p, connecting one end to a vertex chosen at random. This construction interpolates between regularity (p = 0) and disorder (p = 1). The disordered graph is not quite an Erdös–Rényi graph, since the degree of a node is the sum of a Binomial(k, 1/2) and an independent Poisson(k/2).

Chapter 1 will explain what this book is about. Here I will explain why I chose to write the book, how it is written, when and where the work was done, and who helped.

Why. It would make a good story if I was inspired to write this book by an image of Paul Erdös magically appearing on a cheese quesadilla, which I later sold for thousands of dollars on eBay. However, that is not true. The three main events that led to this book were (i) the use of random graphs in the solution of a problem that was part of Nathanael Berestycki's thesis; (ii) a talk that I heard Steve Strogatz give on the CHKNS model, which inspired me to prove some rigorous results about their model; and (iii) a book review I wrote on the books by Watts and Barabási for the Notices of the American Math Society.

The subject of this book was attractive for me, since many of the papers were outside the mathematics literature, so the rigorous proofs of the results were, in some cases, interesting mathematical problems. In addition, since I had worked for a number of years on the properties of stochastic spatial models on regular lattices, there was the natural question of how the behavior of these systems changed when one introduced long-range connections between individuals or considered power law degree distributions.

In this chapter we will introduce and study the random graph model introduced by Erdös and Rényi in the late 1950s. This example has been extensively studied and a very nice account of many of the results can be found in the classic book of Bollobás (2001), so here we will give a brief account of the main results on the emergence of a giant component, in order to prepare for the analysis of more complicated examples. In contrast to other treatments, we mainly rely on methods from probability and stochastic processes rather than combinatorics.

To define the model, we begin with the set of vertices V = {1, 2, … n}. For 1 ≤ x < y ≤ n let ηx,y be independent = 1 with probability p and 0 otherwise. Let ηy,x = ηx,y. If ηx,y = 1 there is an edge from x to y. Here, we will be primarily concerned with situation p = λ/n and in particular with showing that when λ < 1 all of the components are small, with the largest O(log n), while for λ > 1 there is a giant component with ~ g(λ)n vertices. The intuition behind this result is that a site has a Binomial(n – 1, λ/n) number of neighbors, which has mean ≈ λ.

Definitions and Heuristics

In an Erdös–Rényi random graph, vertices have degrees that have asymptotically a Poisson distribution. However, as discussed in Section 1.4, in social and communication networks, the distribution of degrees is much different from the Poisson and in many cases has a power law form, that is, the fraction of vertices of degree k, pk ~ Ck-β as k → ∞. Molloy and Reed (1995) were the first to construct graphs with specified degree distributions. We will use the approach of Newman, Strogatz, and Watts (2001, 2002) to define the model.

Let d1,…dn be independent and have P(di = k) = pk. Since we want di to be the degree of vertex i, we condition on En = {d1 + … + dn is even}. If the probability P(E1) ∊ (0, 1) then P(En) → ½ as n → ∞ so the conditioning will have little effect on the finite-dimensional distributions. If d1 is always even then P(En) = 1 for all n, while if d1 is always odd, P(E2n) = 1 and P(E2n+1) = 0 for all n.

To build the graph we think of di half-edges attached to i and then pair the half-edges at random. The picture gives an example with eight vertices.

Introduction to the Introduction

The theory of random graphs began in the late 1950s in several papers by Erdös and Rényi. However, the introduction at the end of the twentieth century of the small world model of Watts and Strogatz (1998) and the preferential attachment model of Barabási and Albert (1999) have led to an explosion of research. Querying the Science Citation Index in early July 2005 produced 1154 citations for Watts and Strogatz (1998) and 964 for Barabási and Albert (1999). Survey articles of Albert and Barabási (2002), Dorogovstev and Mendes (2002), and Newman (2003) each have hundreds of references. A book edited by Newman, Barabási, and Watts (2006) contains some of the most important papers. Books by Watts (2003) and Barabási (2002) give popular accounts of the new science of networks, which explains “how everything is connected to everything else and what it means for science, business, and everyday life.”

While this literature is extensive, many of the papers are outside the mathematical literature, which makes writing this book a challenge and an opportunity. A number of articles have appeared in Nature and Science. These journals with their impressive impact factors are, at least in the case of random graphs, the home of 10 second sound bite science.

Barabási–Albert Model

As we have noted, many real-world graphs have power law degree distributions. Barabási and Albert (1999) introduced a simple model that produces such graphs. They start with a graph with

Bollobás, Riordan, Spencer, and Tusnády (2001) complain: “The description of the random graph process quoted above is rather imprecise. First as the degrees are initially zero, it is not clear how the process is supposed to get started. More seriously, the expected number of edges linking a new vertex to earlier vertices is ΣiΠ(ki) = 1, rather than m. Also when choosing in one go a set S of m earlier vertices as neighbors of v, the distribution is not specified by giving the marginal probability that each vertex lies in S.”

As we will see below there are several ways to make the process precise and all of them lead to the same asymptotic behavior.

Percolation theory was founded by Broadbent and Hammersley [1957], in order to model the flow of fluid in a porous medium with randomly blocked channels. Interpreted narrowly, it is the study of the component structure of random subgraphs of graphs. Usually, the underlying graph is a lattice or a lattice-like graph, which may or may not be oriented, and to obtain our random subgraph we select vertices or edges independently with the same probability p. In the quintessential examples, the underlying graph is ℤd.

The aim of this chapter is to introduce the basic concepts of percolation theory, and some easy fundamental results concerning them.

We shall use the definitions and notation of graph theory in a standard way, as in Bollobás [1998], for example. In particular, if ∧ is a graph, then V(∧) and E(∧) denote the sets of vertices and edges of ∧, respectively. We write x ∈ ∧ for x ∈ V(∧). We also use standard notation for the limiting behaviour of functions: for f = f(n) and g = g(n), we write f = o(g) if f/g → 0 as n → ∞, f = O(g) if f/g is bounded, f = Ω(g) for g = O(f), and f = Θ(g) if f = O(g) and g = O(f).

The standard terminology of percolation theory differs from that of graph theory: vertices and edges are called sites and bonds, and components are called clusters.

Our first aim in this chapter is to present a result of Aizenman, Kesten and Newman [1987] that, under mild conditions, above the critical probability there is a unique infinite open cluster; Burton and Keane [1989] have given a very simple and elegant proof of this result. Together with Menshikov's Theorem, this uniqueness result gives an alternative proof of the Harris–Kesten Theorem; this proof is easily adapted to determine the critical probabilities of certain other lattices. The key consequence of uniqueness is that, under a symmetry assumption, the critical probabilities for bond percolation on a planar lattice and on its dual must sum to 1. We shall prove this, along with a corresponding result for site percolation, assuming only order two symmetry. Finally, we discuss the star-delta transformation, which may be used to find the critical probabilities for certain lattices that are not self-dual.

Uniqueness of the infinite open cluster – the Aizenman–Kesten–Newman Theorem

Throughout this chapter the underlying graph ∧ will be unoriented, and of finite-type. In other words, there are finitely many equivalence classes of sites under the relation in which two sites x and y are equivalent if there is an automorphism of ∧ mapping x to y, in which case we write x ∼ y. As usual, ∧ will be infinite, locally finite, and connected.

We start this chapter with the result of Aizenman, Kesten and Newman [1987] that, in this setting, with probability 1, there is at most one infinite open cluster.

The celebrated ‘conformal invariance’ conjecture of Aizenman and Langlands, Pouliot and Saint-Aubin [1994] states, roughly, that if ∧ is a planar lattice with suitable symmetry, and we consider percolation on ∧ with probability p = pc(∧), then as the lattice spacing tends to zero certain limiting probabilities are invariant under conformal maps of the plane ℝ2 ≅ ℂ. This conjecture has been proved for only one standard percolation model, namely independent site percolation on the triangular lattice. The aim of this chapter is to present this remarkable result of Smirnov [2001a; 2001b], and to discuss briefly some of its consequences.

In the next section we describe the conformal invariance conjecture, in terms of the limiting behaviour of crossing probabilities, and present Cardy's explicit prediction for these conformally invariant limits. In Section 2, we present Smirnov's Theorem and its proof; as we give full details of the proof, this section is rather lengthy. Finally, we shall very briefly describe some consequences of Smirnov's Theorem concerning the existence of certain ‘critical exponents.’

Crossing probabilities and conformal invariance

Throughout this chapter we identify the plane ℝ2 with the set ℂ of complex numbers in the usual way. A domain D ⊂ ℂ is a non-empty connected open subset of ℂ. If D and D′ are domains, then a conformal map from D to D′ is a bijection f : D → D′ which is analytic on D, i.e., analytic at every point of D. Note that f-1 is then analytic on D′.