Random Subgraphs in Sparse Graphs

TL;DR

This paper determines the precise threshold probability for the connectivity of sparse Cartesian power graphs, showing it depends only on the degree sequence, thus extending known results for regular graphs.

Contribution

It provides a general threshold result for connectivity in sparse Cartesian power graphs based solely on degree sequences, broadening previous findings for regular graphs.

Findings

Threshold probability for connectivity is characterized by a simple formula.

The result applies to arbitrary connected graphs, not just regular ones.

Connectivity probability converges to an exponential function based on degree sequence.

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.