Random Subgraphs in Sparse Graphs

Abstract: We investigate the threshold probability for connectivity of sparse graphs under weak assumptions. As a corollary this completely solve the problem for Cartesian powers of arbitrary graphs. In detail, let $G$ be a connected graph on $k$ vertices, $G^n$ the $n$-th Cartesian power of $G$, $\alpha_i$ be the number of vertices of degree $i$ of $G$, $\lambda$ be a positive real number, and $G^n_p$ be the graph obtained from $G^n$ by deleting every edge independently with probability $1-p$. If $\sum_{i}\alpha_i(1-p)^i=\lambda^{\frac{1}{n}}$, then $\lim_{n\rightarrow \infty}\mathbb{P}[G^n_p {\rm\ is\ connected}]=\exp(-\lambda)$. This result extends known results for regular graphs. The main result implies that the threshold probability does not depend on the graph structure of $G$ itself, but only on the degree sequence of the graph.