Elisabetta Candellero

Clustering and the Hyperbolic Geometry of Complex Networks

Abstract Clustering is a fundamental property of complex networks and it is the mathematical expression of a ubiquitous phenomenon that arises in various types of self-organized networks such as biological networks, computer networks, or social networks. In this article, we consider what is called the global clustering coefficient of random graphs on the hyperbolic plane. This model of random graphs was proposed recently by Krioukov and colleagues as a mathematical model of complex networks, under the fundamental assumption that hyperbolic geometry underlies the structure of these networks. We give a rigorous analysis of clustering and characterize the global clustering coefficient in terms of the parameters of the model. We show how the global clustering coefficient can be tuned by these parameters and we give an explicit formula for this function.

Branching random walks on free products of groups

We study certain phase transitions of branching random walks (BRW) on Cayley graphs of free products. The aim of this paper is to compare the size and structural properties of the trace, that is, the subgraph that consists of all edges and vertices that were visited by some particle, with those of the original Cayley graph. We investigate the phase when the growth parameter λ is small enough such that the process survives, but the trace is not the original graph. A first result is that the box-counting dimension of the boundary of the trace exists, is almost surely constant and equals the Hausdorff dimension which we denote by Φ(λ). The main result states that the function Φ(λ) has only one point of discontinuity which is at λc=R where R is the radius of convergence of the Green function of the underlying random walk. Furthermore, Φ(R) is bounded by one half the Hausdorff dimension of the boundary of the original Cayley graph and the behaviour of Φ(R)−Φ(λ) as λ ↑ R is classified. In the case of free products of infinite groups the end-boundary can be decomposed into words of finite and words of infinite length. We prove the existence of a phase transition such that if λ≤λc, the end boundary of the trace consists only of infinite words and if λ>λc, it also contains finite words. In the last case, the Hausdorff dimension of the set of ends (of the trace and the original graph) induced by finite words is strictly smaller than the one of the ends induced by infinite words.

Phase transitions for random walk asymptotics on free products of groups

Suppose we are given finitely generated groups Γ1,…,Γm equipped with irreducible random walks. Thereby we assume that the expansions of the corresponding Green functions at their radii of convergence contain only logarithmic or algebraic terms as singular terms up to sufficiently large order (except for some degenerate cases). We consider transient random walks on the free product Γ1* … *Γm and give a complete classification of the possible asymptotic behaviour of the corresponding n-step return probabilities. They either inherit a law of the form ϱnδn**math-image** log **math-image**n from one of the free factors Γi or obey a ϱnδn−3/2-law, where ϱ \documentclass{article} \usepackage{amsmath, amsthm, amssymb, amsfonts}\pagestyle{empty}\begin{document}

Bootstrap percolation and the geometry of complex networks

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Clustering in random geometric graphs on the hyperbolic plane

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