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Spectral bounds in random graphs applied to spreading phenomena and percolation

Part of: Markov processes Special processes Graph theory

Published online by Cambridge University Press:  26 July 2018

Rémi Lemonnier ,
Kevin Scaman  and
Nicolas Vayatis
Rémi Lemonnier*
Affiliation:
École Normale Supérieure Paris-Saclay
Kevin Scaman*
Affiliation:
École Normale Supérieure Paris-Saclay
Nicolas Vayatis*
Affiliation:
École Normale Supérieure Paris-Saclay
*
* Postal address: 55 rue de Rome, 75008 Paris, France. Email address: [email protected]
** Postal address: 18 Quai du Point du Jour, 92100 Boulogne-Billancourt, France. Email address: [email protected]
*** Postal address: Centre de Mathématiques et de Leurs Applications (CMLA), École Normale Supérieure Paris-Saclay, 61 Avenue President Wilson, F-94230 Cachan, France. Email address: [email protected]
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Abstract

In this paper we derive nonasymptotic upper bounds for the size of reachable sets in random graphs. These bounds are subject to a phase transition phenomenon triggered by the spectral radius of the hazard matrix, a reweighted version of the adjacency matrix. Such bounds are valid for a large class of random graphs, called local positive correlation (LPC) random graphs, displaying local positive correlation. In particular, in our main result we state that the size of reachable sets in the subcritical regime for LPC random graphs is at most of order O(√n), where n is the size of the network, and of order O(n2/3) in the critical regime, where the epidemic thresholds are driven by the size of the spectral radius of the hazard matrix with respect to 1. As a corollary, we also show that such bounds hold for the size of the giant component in inhomogeneous percolation, the SIR model in epidemiology, as well as for the long-term influence of a node in the independent cascade model.

Keywords

Random graphlocal positive correlationhazard matrixpercolationepidemiologyinformation cascade

MSC classification

Primary: 05C80: Random graphs
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 92D30: Epidemiology 60J60: Diffusion processes
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 2 , June 2018 , pp. 480 - 503
Copyright
Copyright © Applied Probability Trust 2018 

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