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).

When I was a graduate student in the mid-1960s, the mathematical theory underlying analysis of variance and regression became clear to me after I read a draft of William Kruskal's monograph on the so-called coordinate-free, or geometric, approach to these subjects. Alas, with Kruskal's demise, this excellent treatise will never be published.

From time to time during the 1970s, 80s, and early 90s, I had the good fortune to teach the coordinate-free approach to linear models, more precisely, to Model I analysis of variance and linear regression with nonrandom predictors. While doing so, I evolved my own set of lecture notes, presented here. With regard to inspiration and content, my debt to Kruskal is clear. However, my notes differ from Kruskal's in many ways. To mention just a few, my notes are intended for a one- rather than three-quarter course. The notes are aimed at statistics graduate students who are already familiar with the basic concepts of linear algebra, such as linear subspaces and linear transformations, and who have already had some exposure to the matricial formulation of the GLM, perhaps through a methodology course, and who are interested in the underlying theory. I have also included Tjur experimental designs and some of the highlights of the optimality theory for estimation and testing in linear models under the assumption of normality, feeling that the elegant setting provided by the coordinate-free approach is a natural one in which to place these jewels of mathematical statistics.

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.