Percolation and Random Graphs

Abstract

In this section, we define percolation and random graph models, and survey the features of these models. 1.1 Introduction and notation In this section, we discuss random networks. In Section 1.2, we study perco-lation, which is obtained by independently removing vertices or edges from a graph. Percolation is a model of a porous medium, and is a paradigm model of statistical physics. Think of the bonds in an infinite graph that are not removed as indicating whether water can flow through this part of the medium. Then, the interesting question is whether water can percolate, or, alternatively, whether there is an infinite connected component of bonds that are kept? As it turns out, the answer to this question depends sensitively on the fraction of bonds that are kept. When we keep most bonds, then the kept or occupied bonds form most of the original graph. In particular, an infinite connected component may exist, and if this happens, we say that the system percolates. On the other hand, when most bonds are removed or vacant, then the connected components tend to be small and insignificant. Thus, percolation admits a phase transition. Despite the simplicity of the model, the results obtained up to date are far from complete, and many aspects of percolation, particularly of its critical behavior, are ill understood. In Section 1.2 we shall discuss the basics of percolation, and highlight some important open questions. The key challenge in percolation is to uncover the relation between the percolation critical behavior and the properties of the underlying graph from which we obtain percolation by removing edges. In Section 1.3, we discuss random graphs. While in percolation, the random network considered naturally lives on an infinite graph, in random graph theory one considers random finite graphs. Thus, all random graphs are obtained by removing edges from the complete graph, or by adding edges to an empty graph. An important example of a random graph is obtained by independently removing bonds from a finite graph, which makes it clear that there is a strong link to percolation. However, also other mechanisms are possible to generate a random graph. We shall discuss some of the basics of random graph theory, focussing 2 Percolation and random graphs on the phase transition of the largest connected component and the distances in random graphs. The random graph models studied here are inspired by applications , and we shall highlight real-world networks that these random graphs aim to model to some extent. The fields that this contribution covers, percolation and random graph theory , have attracted tremendous attention in the past decades, and enormous progress has been made. It is impossible to cover all material appearing in the literature, and we believe that one should not aim to do so. Thus, we have strived to cover the main results which have been proved, as well as recent results in which we expect that more progress shall be made in the (near?) future, and we list open problems which we find of interest. We ...

Marked metric measure spaces

“…Recently, random configurations and random dynamics on metric spaces in the form of random graphs have been studied as well (see [8]). Two examples are percolation [20] and epidemic models on random graphs [6].…”

Recent advances in percolation theory and its applications

“…Denoting the Cheeger constant of a graph G by h(G) = inf V ⊂V (G):|V |<∞ |∂ E V ||V | , that is the minimal ratio of boundary to bulk of its nontrivial subgraphs, a graph is called amenable when h(G) = 0, and non-amenable otherwise. The simplest example of a non-amenable graph is the Bethe lattice with z ≥ 3 for which the Cheeger constant is h = z − 2[140,144]. It is also shown[140] that pc(G) ≤ 1/(h(G) + 1), so that for every non-amenable graph pc(G) < 1.…”

Generating Infinite Random Graphs

“…We refer the reader to the survey paper of Cameron [7] for further interesting properties of the random graph along this direction. Connections between percolation theory and the random graphs can be found in the survey paper of van der Hofstad [18]. That the Rado graph can be topologically 2-generated with a great deal of flexibility was shown by the second author and Mitchell [8].…”