Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userExplicit bounds for critical infection rates and expected extinction times of the contact process on finite random graphs
162
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We introduce a method to prove metastability of the contact process on ErdH{o}s-Renyi graphs and on configuration model graphs. The method relies on uniformly bounding the total infection rate from below, over all sets with a fixed number of nodes. Once this bound is established, a simple comparison with a well chosen birth-and-death process will show the exponential growth of the extinction time. Our paper complements recent results on the metastability of the contact process: under a certain minimal edge density condition, we give explicit lower bounds on the infection rate needed to get metastability, and we have explicit exponentially growing lower bounds on the expected extinction time.
rate research
Read More
We consider the contact process on the model of hyperbolic random graph, in the regime when the degree distribution obeys a power law with exponent $chi in(1,2)$ (so that the degree distribution has finite mean and infinite second moment). We show that the probability of non-extinction as the rate of infection goes to zero decays as a power law with an exponent that only depends on $chi$ and which is the same as in the configuration model, suggesting some universality of this critical exponent. We also consider fini
We show that the contact process on the rank-one inhomogeneous random graphs and Erdos-R{e}nyi graphs with mean degree large enough survives a time exponential in the size of these graphs for any positive infection rate. In addition, a metastable result for the extinction time is also proved.
The key to our investigation is an improved (and in a sense sharp) understanding of the survival time of the contact process on star graphs. Using these results, we show that for the contact process on Galton-Watson trees, when the offspring distribution (i) is subexponential the critical value for local survival $lambda_2=0$ and (ii) when it is geometric($p$) we have $lambda_2 le C_p$, where the $C_p$ are much smaller than previous estimates. We also study the critical value $lambda_c(n)$ for prolonged persistence on graphs with $n$ vertices generated by the configuration model. In the case of power law and stretched exponential distributions where it is known $lambda_c(n) to 0$ we give estimates on the rate of convergence. Physicists tell us that $lambda_c(n) sim 1/Lambda(n)$ where $Lambda(n)$ is the maximum eigenvalue of the adjacency matrix. Our results show that this is not correct.
We consider the extinction time of the contact process on increasing sequences of finite graphs obtained from a variety of random graph models. Under the assumption that the infection rate is above the critical value for the process on the integer line, in each case we prove that the logarithm of the extinction time divided by the size of the graph converges in probability to a (model-dependent) positive constant. The graphs we treat include various percolation models on increasing boxes of Z d or R d in their supercritical or percolative regimes (Bernoulli bond and site percolation, the occupied and vacant sets of random interlacements, excursion sets of the Gaussian free field, random geometric graphs) as well as supercritical Galton-Watson trees grown up to finite generations.
In this note we give a new method for getting a series of approximations for the extinction probability of the one-dimensional contact process by using the Grobner basis.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
