The study of complex networks has emerged over the past several years as a theme spanning many disciplines, ranging from mathematics and computer science to the social and biological sciences. A significant amount of recent work in this area has focused on the development of random graph models that capture some of the qualitative properties observed in large-scale network data; such models have the potential to help us reason, at a general level, about the ways in which real-world networks are organized. We survey one particular line of network research, concerned with small-world phenomena and decentralized search algorithms, that illustrates this style of analysis. We begin by describing a well-known experiment that provided the first empirical basis for the "six degrees of separation" phenomenon in social networks; we then discuss some probabilistic network models motivated by this work, illustrating how these models lead to novel algorithmic and graph-theoretic questions, and how they are supported by recent empirical studies of large social networks.

Abstract We study how correlations in the random fitness assignment may affect the structure of fitness landscapes. We consider three classes of fitness models. The first is a continuous phenotype space in which individuals are characterized by a large number of continuously varying traits such as size, weight, color, or concentrations of gene products which directly affect fitness. The second is a

ω-periodic graphs are introduced and studied. These are graphs which arise as the limits of periodic extensions of the nearest neighbor graph on the integers. We observe that all bounded degree ω-periodic graphs are ameanable. We also provide examples of ω-periodic graphs which have exponential volume growth, non-linear polynomial volume growth and intermediate volume growth. 1

We consider a family of long-range percolation models (Gp)p>0 on Z d allowing dependence between edges and having these connectivity properties for p ∈ (1/d, ∞): (i) the degree distribution of vertices in Gp has a power law distribution, (ii) the graph distance between points x and y is bounded by a multiple of logpd logpd |x − y | with probability 1 − o(1), and (iii) an adversary can delete a relatively small number of nodes from Gp(Zd ∩ [0, n] d) resulting in two disconnected large subgraphs. 1

Abstract. A discussion regarding aspects of several quite different random planar metrics and related topics is presented. 1.

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We formulate and study a model for inhomogeneous long-range percolation on Zd. Each vertex x ∈ Zd is assigned a non-negative weight Wx, where (Wx) x∈Zd are i.i.d. random variables. Conditionally on the weights, and given two parameters α, λ> 0, the edges are independent and the probability that there is an edge between x and y is given by pxy = 1 − exp{−λWxWy/|x − y | α}. The parameter λ is the percolation parameter, while α describes the long-range nature of the model. We focus on the degree distribution in the resulting graph, on whether there exists an infinite component and on graph distance between remote pairs of vertices. First, we show that the tail behavior of the degree distribution is related to the tail behavior of the weight distribution. When the tail of the distribution of Wx is regularly varying with exponent τ − 1, then the tail of the degree distribution is regularly varying with exponent γ = α(τ − 1)/d. The parameter γ turns out to be crucial for the behavior of the model. Conditions on the weight distribution and γ are formulated for the existence of a critical value λc ∈ (0, ∞) such that the graph contains an infinite component when λ> λc and no infinite component when λ < λc. Furthermore, a phase transition is established for the graph distances between vertices in the infinite component at the point γ = 2, that is, at the point where the degrees switch from having finite to infinite second moment. The model can be viewed as an interpolation between long-range percolation and models for inhomogeneous random graphs, and we show that the behavior shares the interesting features of both these models.

Quenched invariance principle for simple random walk on percolation clusters Abstract. We consider the simple random walk on the (unique) infinite cluster of supercritical bond percolation in Z d with d ≥ 2. We prove that, for almost every percolation configuration, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. Our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square-integrable martingale. The size of the deformation, expressed by the so called corrector, is estimated by means of ergodicity arguments. 1.

Abstract. We study asymptotic percolation as N → ∞ in an infinite random graph GN embedded in the hierarchical group of order N, with connection probabilities depending on an ultrametric distance between vertices. GN is structured as a cascade of finite random subgraphs of (approximate) Erd ′ ′ os-Rényi type. However, the results are different from those of classical random graphs, e.g., the average length of paths in the giant component of an ultrametric ball is much longer than in the classical case. We give a criterion for percolation, and show that percolation takes place along giant components of giant components at the previous level in the cascade of subgraphs for all consecutive hierarchical distances. The proof involves a hierarchy of “doubly stochastic ” random graphs with vertices having an internal structure and random connection probabilities. 1.

Quenched invariance principle for simple random walk on percolation clusters Abstract. We consider the simple random walk on the (unique) infinite cluster of supercritical bond percolation in Z d with d ≥ 2. We prove that, for almost every percolation configuration, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. Our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square-integrable martingale. The size of the deformation, expressed by the so called corrector, is estimated by means of ergodicity arguments. 1.