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

Journal Article

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)~ Cλβ 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. © Institute of Mathematical Statistics, 2009.

### Full Text

### Duke Authors

### Cited Authors

- Chatterjee, S; Durrett, R

### Published Date

- 2009

### Published In

### Volume / Issue

- 37 / 6

### Start / End Page

- 2332 - 2356

### International Standard Serial Number (ISSN)

- 0091-1798

### Digital Object Identifier (DOI)

- 10.1214/09-AOP471