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.