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Contagions in random networks with overlapping communities

Part of: Mathematical sociology Graph theory Combinatorial probability

Published online by Cambridge University Press:  21 March 2016

Emilie Coupechoux  and
Marc Lelarge
Emilie Coupechoux*
Affiliation:
Université Nice Sophia Antipolis and INRIA-ENS
Marc Lelarge*
Affiliation:
INRIA-ENS
*
∗ Postal address: Laboratoire I3S, Université Nice Sophia Antipolis, CS 40121, 06903 Sophia Antipolis Cedex, France. Email address: emilie.coupechoux@gmail.com
∗∗ Postal address: INRIA-ENS, 23, avenue d'Italie, CS 81321, 75214 Paris Cedex 13, France.
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Abstract

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We consider a threshold epidemic model on a clustered random graph model obtained from local transformations in an alternating branching process that approximates a bipartite graph. In other words, our epidemic model is such that an individual becomes infected as soon as the proportion of his/her infected neighbors exceeds the threshold q of the epidemic. In our random graph model, each individual can belong to several communities. The distributions for the community sizes and the number of communities an individual belongs to are arbitrary. We consider the case where the epidemic starts from a single individual, and we prove a phase transition (when the parameter q of the model varies) for the appearance of a cascade, i.e. when the epidemic can be propagated to an infinite part of the population. More precisely, we show that our epidemic is entirely described by a multi-type (and alternating) branching process, and then we apply Sevastyanov's theorem about the phase transition of multi-type Galton-Watson branching processes. In addition, we compute the entries of the mean progeny matrix corresponding to the epidemic. The phase transition for the contagion is given in terms of the largest eigenvalue of this matrix.

Keywords

Random graphsthreshold epidemic modelbranching processesclustering

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 05C80: Random graphs 91D30: Social networks
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 4 , December 2015 , pp. 973 - 988
Copyright
Copyright © Applied Probability Trust 2015 

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