## The Annals of Probability

- Ann. Probab.
- Volume 38, Number 1 (2010), 150-183.

### Percolation on dense graph sequences

Béla Bollobás, Christian Borgs, Jennifer Chayes, and Oliver Riordan

#### Abstract

In this paper we determine the percolation threshold for an arbitrary sequence of dense graphs (*G*_{n}). Let *λ*_{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/*λ*_{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, Lovász, Sós and Vesztergombi.

In addition to using basic properties of convergence, we make heavy use of the methods of Bollobás, 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.

#### Article information

**Source**

Ann. Probab., Volume 38, Number 1 (2010), 150-183.

**Dates**

First available in Project Euclid: 25 January 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1264433995

**Digital Object Identifier**

doi:10.1214/09-AOP478

**Mathematical Reviews number (MathSciNet)**

MR2599196

**Zentralblatt MATH identifier**

1190.60090

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 05C80: Random graphs [See also 60B20]

**Keywords**

Percolation cut metric random graphs

#### Citation

Bollobás, Béla; Borgs, Christian; Chayes, Jennifer; Riordan, Oliver. Percolation on dense graph sequences. Ann. Probab. 38 (2010), no. 1, 150--183. doi:10.1214/09-AOP478. https://projecteuclid.org/euclid.aop/1264433995