This book is concerned with certain aspects of discrete probability on infinite graphs that are currently in vigorous development. Of course, finite graphs are analyzed as well, but usually with the aim of understanding infinite graphs and networks. These areas of discrete probability are full of interesting, beautiful, and surprising results, many of which connect to other areas of mathematics and theoretical computer science. Numerous fascinating questions are still open.

Our major topics include random walks and their intimate connection to electrical networks; uniform spanning trees, their limiting forests, and their marvelous relationships with random walks and electrical networks; branching processes; percolation and the powerful, elegant mass-transport technique; isoperimetric inequalities and how they relate to both random walks and percolation; minimal spanning trees and forests and their connections to percolation; Hausdorff dimension, capacity, and how to understand them via trees; and random walks on Galton-Watson trees. Connections among our topics are pervasive and rich, making for surprising and enjoyable proofs.

There are three main classes of graphs on which discrete probability is most interesting, namely, trees, Cayley graphs of groups (or, more generally, transitive, or even quasi-transitive, graphs), and planar graphs. More classical discrete probability has tended to focus on the special and important case of the Euclidean lattices, Zd, which are prototypical Cayley graphs. This book develops the general theory of various probabilistic processes on graphs and then specializes to the three broad classes listed, always seeing what we can say in the case of Zd.

Besides their intrinsic interest, there are several reasons for a special study of trees. Since in most cases, analysis is easier on trees, analysis can be carried further. Then one can often either apply the results from trees to other situations or transfer to other situations the techniques developed by working on trees. Trees also occur naturally in many situations, either combinatorially or as descriptions of compact sets in Euclidean space, R d.

In choosing our topics, we have been swayed by those results we find most striking as well as by those that do not require extensive background. Thus, the only prerequisite is basic knowledge of Markov chains and conditional expectation with respect to a σ-algebra.

In this first chapter, we give an introduction to random graphs and complex networks. We discuss examples of real-world networks and their empirical properties, and give a brief introduction to the kinds of models that we investigate in the book. Further, we introduce the key elements of the notation used throughout the book.

Motivation

The advent of the computer age has incited an increasing interest in the fundamental properties of real-world networks. Due to the increased computational power, large data sets can now easily be stored and investigated, and this has had a profound impact on the empirical studies of large networks. As we explain in detail in this chapter, many real-world networks are small worlds and have large fluctuations in their degrees. These realizations have had fundamental implications for scientific research on networks. Network research is aimed to both to understand why many networks share these fascinating features, and also to investigate what the properties of these networks are in terms of the spread of diseases, routing information and ranking of the vertices present.

The study of complex networks plays an increasingly important role in science. Examples of such networks are electrical power grids and telecommunication networks, social relations, the World-Wide Web and Internet, collaboration and citation networks of scientists, etc. The structure of such networks affects their performance. For instance, the topology of social networks affects the spread of information or disease (see, e.g., Strogatz (2001)). The rapid evolution and success of the Internet have spurred fundamental research on the topology of networks. See Barabási (2002) and Watts (2003) for expository accounts of the discovery of network properties by Barabási, Watts and co-authors. In Newman et al. (2006), you can find some of the original papers detailing the empirical findings of real-world networks and the network models invented for them. The introductory book by Newman (2010) lists many of the empirical properties of, and scientific methods for, networks.

One common feature of complex networks is that they are large. As a result, a global description is utterly impossible, and researchers, both in the applications and in mathematics, have turned to their local description: how many vertices they have, by which local rules vertices connect to one another, etc. These local rules are probabilistic, reflecting the fact that there is a large amount of variability in how connections can be formed

In this chapter, we investigate graphs with fixed degrees. Ideally, we would like to investigate uniform random graphs having a prescribed degree sequence, i.e, a degree sequence which is given to us beforehand. An example of such a situation could arise from a real-world network of which we know the degree sequence, and we would be interested in generating a random, or even uniform, graph with precisely the same degrees.

Such a random graph can be created by uniformly matching half-edges, at the expense of possibly creating self-loops and multiple edges, and the result is called the configuration model. The main results in this chapter describe the degree structure in such random graphs, as well as the probability for the random graph to be simple. We also draw connections between the configuration model and the generalized random graphs defined in Chapter 6.

Motivation for the Configuration Model

The configuration model is a model in which the degrees of vertices are fixed beforehand. Such a model is more flexible than the generalized random graph. For example, the generalized random graph always has a positive proportion of vertices of degree 0, 1, 2, etc. (see Exercises 6.15–6.16). In some real-world networks, however, it is natural to investigate graphs where every vertex has at least one or two neighbors. We start by discussing a few examples where random graphs with prescribed degrees appear naturally:

Example 7.1 (Facebook wall posts) In Facebook wall posts, the vertices of the network are Facebook users, and each directed edge represents one post, linking the user writing the post to the user on whose wall the post is written. This gives rise to a (directed) network of a small subset of posts to the walls of other users on Facebook. Since users may write multiple posts on a wall, the network allows multiple edges between pairs of vertices. Since users may write on their own wall, the network contains loops. The degree distribution in log-log scale is displayed in Figure 7.1, and we see that the degree distribution resembles a power law. We see that the degrees are, by definition, all at least 1. Thus, this data set is poorly modeled by the generalized random graph, which always has many vertices of degree 0.