Minimum stationary values of sparse random directed graphs

TL;DR

This paper analyzes the stationary distribution of simple random walks on sparse directed graphs, revealing how minimum stationary values scale with graph size and degree distribution, and linking these to hitting and cover times.

Contribution

It provides a precise asymptotic characterization of the minimum stationary values in directed configuration models with bounded degrees, incorporating effects of atypical neighborhoods and trajectories.

Findings

Minimum positive stationary value scales as n^{-(1+C+o(1))}

Hitting and cover times are both n^{1+C+o(1)} with high probability

Results extend understanding of stationary distributions beyond logarithmic fluctuations

Abstract

We consider the stationary distribution of the simple random walk on the directed configuration model with bounded degrees. Provided that the minimum out-degree is at least $2$, with high probability (whp) there is a unique stationary distribution. We show that the minimum positive stationary value is whp $n^{-(1+C+o(1))}$ for some constant $C \ge 0$ determined by the degree distribution. In particular, $C$ is the competing combination of two factors: (1) the contribution of atypically "thin" in-neighbourhoods, controlled by subcritical branching processes; and (2) the contribution of atypically "light" trajectories, controlled by large deviation rate functions. Additionally, our proof implies that whp the hitting and the cover time are both $n^{1+C+o(1)}$. Our results complement those of Caputo and Quattropani who showed that if the minimum in-degree is at least 2, stationary values have logarithmic fluctuations around $n^{-1}$.