Contact processes on random graphs with power law degree distributions have critical value 0
By Shirshendu Chatterjee, R. Durrett. 2009
188 citations recorded by the source corpus.
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 λ c 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 Gomez-Gardenes et al. [Proc. Natl. Acad. Sci. USA 105 (2008) 1399-1404]. Here, we show that the critical value λ c 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(n 1-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) ~ Cλ β as → 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.
Subfields: Condensed Matter Physics, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/c1f22d6dff5ee00989558ef2710096d4e1c3c704
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In Condensed Matter Physics
In Mathematical Physics
In Statistical and Nonlinear Physics
Citations in the corpus
- Random Graphs and Complex Networks (2016, background): “15 can be found in (Chatterjee and Durrett, 2009, Lemma 1.2). We could not find the precise version stated in Theorem 4.15. Theorem 4.18 is proved by Federico and van der Hofstad (2017). This paper also allows for a number of vertices n1 of degree 1 satisfying n1 = ρ √ n.”