Random walk hitting times and effective resistance in sparsely connected Erd\H{o}s-R\'enyi random graphs

TL;DR

This paper establishes expectation and concentration results for various random walk and electrical network metrics on sparse Erdős-Rényi graphs, revealing their probabilistic behavior in the sparse connectivity regime.

Contribution

It provides new concentration bounds and expectation estimates for effective resistances and random walk metrics in sparse Erdős-Rényi graphs, extending understanding of their probabilistic properties.

Findings

Effective resistance and random walk times concentrate sharply in sparse graphs.

Strong connectedness properties hold with high probability in the specified regime.

Results extend to effective resistance between two vertices for $np o c\,\log n$.

Abstract

We prove expectation and concentration results for the following random variables on an Erd\H{o}s-R\'enyi random graph $\mathcal{G}\left(n,p\right)$ in the sparsely connected regime $\log n + \log\log \log n \leq np < n^{1/10}$: effective resistances, random walk hitting and commute times, the Kirchoff index, cover cost, random target times, the mean hitting time and Kemeny's constant. For the effective resistance between two vertices our concentration result extends further to $np\geq c\log n, \; c>0$. To achieve these results, we show that a strong connectedness property holds with high probability for $\mathcal{G}(n,p)$ in this regime.