Stationary distribution and cover time of sparse directed configuration models

TL;DR

This paper analyzes the cover time and stationary distribution of sparse directed graphs generated by the configuration model, revealing that cover time is nearly linear and depends on degree sequences through a logarithmic exponent.

Contribution

It provides new bounds on cover time and stationary distribution behavior for sparse directed configuration models, linking degree sequences to graph properties.

Findings

Cover time is linear up to poly-logarithmic factors.

Stationary distribution is nearly uniform, with extremal values characterized by degree sequences.

Diameter matches the typical distance between vertices.

Abstract

We consider sparse digraphs generated by the configuration model with given in-degree and out-degree sequences. We establish that with high probability the cover time is linear up to a poly-logarithmic correction. For a large class of degree sequences we determine the exponent $\gamma \ge 1$ of the logarithm and show that the cover time grows as $n\log^{\gamma}(n)$, where $n$ is the number of vertices. The results are obtained by analysing the extremal values of the stationary distribution of the digraph. In particular, we show that the stationary distribution $\pi$ is uniform up to a poly-logarithmic factor, and that for a large class of degree sequences the minimal values of $\pi$ have the form $\frac1n\log ^{1-\gamma}(n)$, while the maximal values of $\pi$ behave as $\frac1n\log ^{1-\kappa}(n)$ for some other exponent $\kappa\in[0,1]$. In passing, we prove tight bounds on the diameter of the digraphs and show that the latter coincides with the typical distance between two vertices.