Advances in Applied Probability

Infection spread in random geometric graphs

Ghurumuruhan Ganesan

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Abstract

In this paper we study the speed of infection spread and the survival of the contact process in the random geometric graph G = G(n, rn, f) of n nodes independently distributed in S = [-½, ½]2 according to a certain density f(·). In the first part of the paper we assume that infection spreads from one node to another at unit rate and that infected nodes stay in the same state forever. We provide an explicit lower bound on the speed of infection spread and prove that infection spreads in G with speed at least D1nrn2. In the second part of the paper we consider the contact process ξt on G where infection spreads at rate λ > 0 from one node to another and each node independently recovers at unit rate. We prove that, for every λ > 0, with high probability, the contact process on G survives for an exponentially long time; there exist positive constants c1 and c2 such that, with probability at least 1 - c1 / n4, the contact process ξ1t starting with all nodes infected survives up to time tn = exp(c2n/logn) for all n.

Article information

Source
Adv. in Appl. Probab., Volume 47, Number 1 (2015), 164-181.

Dates
First available in Project Euclid: 31 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.aap/1427814586

Digital Object Identifier
doi:10.1239/aap/1427814586

Mathematical Reviews number (MathSciNet)
MR3327320

Zentralblatt MATH identifier
1337.60242

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60C05: Combinatorial probability 62E10: Characterization and structure theory

Keywords
Random geometric graph speed of infection spread survival time of contact process

Citation

Ganesan, Ghurumuruhan. Infection spread in random geometric graphs. Adv. in Appl. Probab. 47 (2015), no. 1, 164--181. doi:10.1239/aap/1427814586. https://projecteuclid.org/euclid.aap/1427814586


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