Remco van der Hofstad

Random Graphs and Complex Networks

In this chapter, we draw motivation from real-world networks, and formulate random graph models for them. We focus on some of the models that have received the most attention in the literature, namely, the Erdős-Rényi random graph, Inhomogeneous random graphs, the configuration model and preferential attachment models. We also discuss some of their extensions that have the potential to yield more realistic models for real-world networks. We follow van der Hofstad (2017), which we refer to as [Volume 1], both for the motivation as well as for the introduction of the random graph models involved. We then continue to discuss an important technique in this book, called local-weak convergence. Looking back, and ahead In Volume 1 of this pair of books, we have discussed various models having flexible degree sequences. The generalized random graph and the configuration model give us static flexible models for random graphs with various degree sequences. Preferential attachment models give us a convincing explanation of the abundance of power-law degree sequences in various applications. In [Volume 1, Chapters 6– 8], we have focussed on the properties of the degrees of such graphs. However, we have noted in [Volume 1, Chapter 1] that many real-world networks not only have degree sequences that are rather different from the ones of the Erdős-Rényi random graph, also many examples are small worlds and have a giant connected component. In Chapters 3–8, we shall return to the models discussed in [Volume 1, Chapters 6–8], and focus on their connected components as well as on the distances in these random graph models. Interestingly, a large chunk of the non-rigorous physics literature suggests that the behavior in various different random graph models can be described by only a few essential parameters. The key parameter of each of these models in the power-law degree exponent, and the physics literature predicts that the behavior in random graph models with similar degree sequences is similar. This is an example of the notion of universality, a notion which is central in statistical physics. Despite its importance, there are only few example of universality that can be rigorously proved. In Chapters 3–8, we investigate the level of universality present in random graph models. We will often refer to Volume 1. When we do, we write [Volume 1, Theorem 2.17] to mean that we refer to Theorem 2.17 in van der Hofstad (2017). Organisation of this chapter This chapter is organised as follows. In Section 1.1, we discuss real-world networks the inspiration that they provide. In Section 1.2, we then discuss how graph

On the efficiency of multicast

The average number of joint hops in a shortest-path multicast tree from a root to m arbitrary chosen group member nodes is studied. A general theory for all graphs, hence including the graph representation of the Internet, is presented which quantifies the multicast reduction in network links compared to m times unicast. For two special types of graphs, the random graph Gp(N) and the k-ary tree, exact and asymptotic results are derived. Comparing these explicit results with previously published Internet measurements [13] indicates that the number of routers in the Internet that can be reached from a root grows exponentially in the number of hops with an effective degree of approximately 3.2.

Random subgraphs of finite graphs. II. The lace expansion and the triangle condition

In a previous paper we defined a version of the percolation triangle condition that is suitable for the analysis of bond percolation on a finite connected transitive graph, and showed that this triangle condition implies that the percolation phase transition has many features in common with the phase transition on the complete graph. In this paper we use a new and simplified approach to the lace expansion to prove quite generally that, for finite graphs that are tori, the triangle condition for percolation is implied by a certain triangle condition for simple random walks on the graph. The latter is readily verified for several graphs with vertex set {0, 1, ..., r - 1} n , including the Hamming cube on an alphabet of r letters (the n-cube, for r = 2), the n-dimensional torus with nearest-neighbor bonds and n sufficiently large, and the n-dimensional torus with n > 6 and sufficiently spread-out (long range) bonds. The conclusions of our previous paper thus apply to the percolation phase transition for each of the above examples.

The Universality Classes in the Parabolic Anderson Model

We discuss the long time behaviour of the parabolic Anderson model, the Cauchy problem for the heat equation with random potential on

Convergence of critical oriented percolation to super-Brownian motion above 4+1 dimensions

\mathbb{Z}^{d}

Diameters in Preferential Attachment Models

. We consider general i.i.d. potentials and show that exactly four qualitatively different types of intermittent behaviour can occur. These four universality classes depend on the upper tail of the potential distribution: (1) tails at ∞ that are thicker than the double-exponential tails, (2) double-exponential tails at ∞ studied by Gärtner and Molchanov, (3) a new class called almost bounded potentials, and (4) potentials bounded from above studied by Biskup and König. The new class (3), which contains both unbounded and bounded potentials, is studied in both the annealed and the quenched setting. We show that intermittency occurs on unboundedly increasing islands whose diameter is slowly varying in time. The characteristic variational formulas describing the optimal profiles of the potential and of the solution are solved explicitly by parabolas, respectively, Gaussian densities. Our analysis of class (3) relies on two large deviation results for the local times of continuous-time simple random walk. One of these results is proved by Brydges and the first two authors in [BHK04], and is also used here to correct a proof in [BK01].

Uncovering disassortativity in large scale-free networks

We study first-passage percolation on the random graph Gp(N) with exponentially distributed weights on the links. For the special case of the complete graph, this problem can be described in terms of a continuous-time Markov chain and recursive trees. The Markov chain X(t) describes the number of nodes that can be reached from the initial node in time t. The recursive trees, which are uniform trees of N nodes, describe the structure of the cluster once it contains all the nodes of the complete graph. From these results, the distribution of the number of hops (links) of the shortest path between two arbitrary nodes is derived.We generalize this result to an asymptotic result, as N → ∞, for the case of the random graph where each link is present independently with a probability pN as long as NpN/(log N)3 → ∞. The interesting point of this generalization is that (1) the limiting distribution is insensitive to p and (2) the distribution of the number of hops of the shortest path between two arbitrary nodes has a remarkable fit with shortest path data measured in the Internet.

Random Subgraphs Of Finite Graphs: III. The Phase Transition For The n -Cube

Abstract We consider oriented bond percolation on Z d × N , at the critical occupation density pc, for d>4. The model is a “spread-out” model having long range parameterised by L. We consider configurations in which the cluster of the origin survives to time n, and scale space by n1/2. We prove that for L sufficiently large all the moment measures converge, as n→∞, to those of the canonical measure of super-Brownian motion. This extends a previous result of Nguyen and Yang, who proved Gaussian behaviour for the critical two-point function, to all r-point functions (r⩾2). We use lace expansion methods for the two-point function, and prove convergence of the expansion using a general inductive method that we developed in a previous paper. For the r-point functions with r⩾3, we use a new expansion method.