## Internet Mathematics

- Internet Math.
- Volume 6, Number 3 (2009), 331-347.

### Percolation in General Graphs

Fan Chung, Paul Horn, and Linyuan Lu

#### Abstract

We consider a random subgraph $G_p$ of a host graph $G$ formed by retaining each edge of $G$ with probability $p$. We address the question of determining the critical value $p$ (as a function of $G$) for which a giant component emerges. Suppose $G$ satisfies some (mild) conditions depending on its spectral gap and higher moments of its degree sequence. We define the second-order average degree $\overline{d}$ to be $\overline{d} = \sum_v d^2_v/(\sum_v d_v)$, where $d_v$ denotes the degree of $v$. We prove that for any $\epsilon > 0$, if $p > (1 + \epsilon)/\overline{d}$, then asymptotically almost surely, the percolated subgraph $G_p$ has a giant component. In the other direction, if $p < (1 − \epsilon)/\overline{d}$, then almost surely, the percolated subgraph $G_p$ contains no giant component. An extended abstract of this paper appeared in the WAW 2009 proceedings. The main theorems are strengthened with much weaker assumptions.

#### Article information

**Source**

Internet Math., Volume 6, Number 3 (2009), 331-347.

**Dates**

First available in Project Euclid: 10 October 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.im/1318269501

**Mathematical Reviews number (MathSciNet)**

MR2798108

**Zentralblatt MATH identifier**

1238.05244

#### Citation

Chung, Fan; Horn, Paul; Lu, Linyuan. Percolation in General Graphs. Internet Math. 6 (2009), no. 3, 331--347. https://projecteuclid.org/euclid.im/1318269501