Advances in Applied Probability

Contagions in random networks with overlapping communities

Emilie Coupechoux and Marc Lelarge

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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.

Article information

Adv. in Appl. Probab., Volume 47, Number 4 (2015), 973-988.

First available in Project Euclid: 11 December 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability
Secondary: 05C80: Random graphs [See also 60B20] 91D30: Social networks

Random graphs threshold epidemic model branching processes clustering


Coupechoux, Emilie; Lelarge, Marc. Contagions in random networks with overlapping communities. Adv. in Appl. Probab. 47 (2015), no. 4, 973--988. doi:10.1239/aap/1449859796.

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