## Internet Mathematics

- Internet Math.
- Volume 4, Number 4 (2007), 329-356.

### Percolation on Sparse Random Graphs with Given Degree Sequence

#### Abstract

We study the two most common types of percolation processes on a sparse random graph with a given degree sequence. Namely, we examine first a bond percolation process where the edges of the graph are retained with probability $p$, and afterwards we focus on site percolation where the vertices are retained with probability $p$. We establish critical values for $p$ above which a giant component emerges in both cases. Moreover, we show that, in fact, these coincide. As a special case, our results apply to power-law random graphs. We obtain rigorous proofs for formulas derived by several physicists for such graphs.

#### Article information

**Source**

Internet Math., Volume 4, Number 4 (2007), 329-356.

**Dates**

First available in Project Euclid: 27 May 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.im/1243430810

**Mathematical Reviews number (MathSciNet)**

MR2522948

**Zentralblatt MATH identifier**

1206.68234

#### Citation

Fountoulakis, N. Percolation on Sparse Random Graphs with Given Degree Sequence. Internet Math. 4 (2007), no. 4, 329--356. https://projecteuclid.org/euclid.im/1243430810