Percolation on dense graph sequences

Details Percolation on dense graph sequences Percolation on dense graph sequences

2007-01-12

arxiv.org/abs/math/0701346v3

Annals of Probability 2010, Vol. 38, No. 1, 150-183

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/09-AOP478 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/09-AOP478

In this paper we determine the percolation threshold for an arbitrary sequence of dense graphs $(G_n)$. Let $\lambda_n$ be the largest eigenvalue of the adjacency matrix of $G_n$, and let $G_n(p_n)$ be the random subgraph of $G_n$ obtained by keeping each edge independently with probability $p_n$. We show that the appearance of a giant component in $G_n(p_n)$ has a sharp threshold at $p_n=1/\lambda_n$. In fact, we prove much more: if $(G_n)$ converges to an irreducible limit, then the density of the largest component of $G_n(c/n)$ tends to the survival probability of a multi-type branching process defined in terms of this limit. Here the notions of convergence and limit are those of Borgs, Chayes, Lov\'asz, S\'os and Vesztergombi. In addition to using basic properties of convergence, we make heavy use of the methods of Bollob\'as, Janson and Riordan, who used multi-type branching processes to study the emergence of a giant component in a very broad family of sparse inhomogeneous random graphs.

Affiliated with

Also associated with

No associations

Rating

Percolation on dense graph sequences 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 Percolation on dense graph sequences, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Percolation on dense graph sequences will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-218468