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

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

2009-12-09

arxiv.org/abs/0912.1699v1

Annals of Probability 2009, Vol. 37, No. 6, 2332-2356

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/09-AOP471 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/09-AOP471

If we consider the contact process with infection rate $\lambda$ on a random graph on $n$ vertices with power law degree distributions, mean field calculations suggest that the critical value $\lambda_c$ of the infection rate is positive if the power $\alpha>3$. Physicists seem to regard this as an established fact, since the result has recently been generalized to bipartite graphs by G\'{o}mez-Garde\~{n}es et al. [Proc. Natl. Acad. Sci. USA 105 (2008) 1399--1404]. Here, we show that the critical value $\lambda_c$ is zero for any value of $\alpha>3$, and the contact process starting from all vertices infected, with a probability tending to 1 as $n\to\infty$, maintains a positive density of infected sites for time at least $\exp(n^{1-\delta})$ for any $\delta>0$. 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 $\rho(\lambda)$. It is expected that $\rho(\lambda)\sim C\lambda^{\beta}$ as $\lambda \to0$. Here we show that $\alpha-1\le\beta\le2\alpha-3$, and so $\beta>2$ for $\alpha>3$. Thus even though the graph is locally tree-like, $\beta$ does not take the mean field critical value $\beta=1$.

Affiliated with

Also associated with

No associations

Rating

Contact processes on random graphs with power law degree distributions have critical value 0 does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Contact processes on random graphs with power law degree distributions have critical value 0, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Contact processes on random graphs with power law degree distributions have critical value 0 will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-275842