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The contact process on the complete graph with random vertex-dependent infection rates

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  • Peterson, Jonathon

Abstract

We study the contact process on the complete graph on n vertices where the rate at which the infection travels along the edge connecting vertices i and j is equal to [lambda]wiwj/n for some [lambda]>0, where wi are i.i.d. vertex weights. We show that when there is a phase transition at [lambda]c>0 such that for [lambda] [lambda]c the contact process lives for an exponential amount of time. Moreover, we give a formula for [lambda]c and when [lambda]>[lambda]c we are able to give precise approximations for the probability that a given vertex is infected in the quasi-stationary distribution. Our results are consistent with a non-rigorous mean field analysis of the model. This is in contrast to some recent results for the contact process on power law random graphs where the mean field calculations suggested that [lambda]c>0 when in fact [lambda]c=0.

Suggested Citation

  • Peterson, Jonathon, 2011. "The contact process on the complete graph with random vertex-dependent infection rates," Stochastic Processes and their Applications, Elsevier, vol. 121(3), pages 609-629, March.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:3:p:609-629
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    Cited by:

    1. Xue, Xiaofeng, 2017. "Law of large numbers for the SIR model with random vertex weights on Erdős–Rényi graph," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 486(C), pages 434-445.
    2. Xue, Xiaofeng, 2016. "Critical value for the contact process with random recovery rates and edge weights on regular tree," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 462(C), pages 793-806.
    3. Xue, Xiaofeng, 2016. "Critical value for contact processes on clusters of oriented bond percolation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 448(C), pages 205-215.
    4. Xue, Xiaofeng, 2013. "Contact processes with random connection weights on regular graphs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(20), pages 4749-4759.

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