Abstract

We consider the contact process on the model of hyperbolic random graph, in the regime when the degree distribution obeys a power law with exponent χ∈(1,2) (so that the degree distribution has finite mean and infinite second moment). We show that the probability of nonextinction as the rate of infection goes to zero decays as a power law with an exponent that only depends on χ and which is the same as in the configuration model, suggesting some universality of this critical exponent. We also consider finite versions of the hyperbolic graph and prove metastability results, as the size of the graph goes to infinity.

Highlights

  • It has been empirically observed that complex networks such as social networks, scientific collaborator networks, citation networks, computer networks and others typically are scale-free and exhibit a non-vanishing clustering coefficient

  • A model of complex networks that naturally exhibits these properties is the random hyperbolic model introduced by [29]: one convincing demonstration of this fact was given by Boguna, Papadopoulos, and Krioukov in [8] where a compelling maximum likelihood fit of autonomous systems of the internet graph in hyperbolic space was computed

  • 3 4 corresponds to a change of regime in the local clustering coefficient averaged over all vertices of degree exactly k

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Summary

Introduction

It has been empirically observed that complex networks such as social networks, scientific collaborator networks, citation networks, computer networks and others (see [2]) typically are scale-free and exhibit a non-vanishing clustering coefficient. Theorem 1.4 makes use of the idea that if the process on the infinite graph starting from only the root infected, survives for a long time and only it will escape from a large neighborhood of the root The proof of this idea is based on self-duality of the contact process, and by applying the first and second moment methods to the number of vertices escaping from a large neighborhood (for corresponding upper and lower bounds, respectively); The hyperbolic shapes of the neighborhoods, and in particular, the existence of very high-degree vertices make this basic idea a bit delicate at times

Discussion of results
Hyperbolic graph model
Contact process
Survival probability: lower bounds
Infection paths and ordered traces
Convergence of density
Discussion and outlook gave a complete picture of metastability
Full Text
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