The Annals of Probability

Contact processes on random graphs with power law degree distributions have critical value 0

Shirshendu Chatterjee and Rick Durrett

Full-text: Open access

Abstract

If we consider the contact process with infection rate λ on a random graph on n vertices with power law degree distributions, mean field calculations suggest that the critical value λc of the infection rate is positive if the power α>3. Physicists seem to regard this as an established fact, since the result has recently been generalized to bipartite graphs by Gómez-Gardeñes et al. [Proc. Natl. Acad. Sci. USA 105 (2008) 1399–1404]. Here, we show that the critical value λc is zero for any value of α>3, and the contact process starting from all vertices infected, with a probability tending to 1 as n→∞, maintains a positive density of infected sites for time at least exp(n1−δ) for any δ>0. Using the last result, together with the contact process duality, we can establish the existence of a quasi-stationary distribution in which a randomly chosen vertex is occupied with probability ρ(λ). It is expected that ρ(λ)∼β as λ→0. Here we show that α−1≤β≤2α−3, and so β>2 for α>3. Thus even though the graph is locally tree-like, β does not take the mean field critical value β=1.

Article information

Source
Ann. Probab., Volume 37, Number 6 (2009), 2332-2356.

Dates
First available in Project Euclid: 16 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.aop/1258380791

Digital Object Identifier
doi:10.1214/09-AOP471

Mathematical Reviews number (MathSciNet)
MR2573560

Zentralblatt MATH identifier
1205.60168

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 05C80: Random graphs [See also 60B20]

Keywords
Contact process power-law random graph epidemic threshold

Citation

Chatterjee, Shirshendu; Durrett, Rick. Contact processes on random graphs with power law degree distributions have critical value 0. Ann. Probab. 37 (2009), no. 6, 2332--2356. doi:10.1214/09-AOP471. https://projecteuclid.org/euclid.aop/1258380791


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