Meeting, coalescence and consensus time on random directed graphs

TL;DR

This paper analyzes the meeting, coalescence, and consensus times of random walks and voter models on large directed configuration model graphs, showing these times are approximately exponential and depend linearly on graph size.

Contribution

It provides the first first-order approximation of these times on typical large directed graphs, linking them to degree sequence statistics.

Findings

Meeting time distribution is approximately exponential.

Expected meeting time scales linearly with graph size.

Degree sequence influences coalescence and consensus times.

Abstract

We consider Markovian dynamics on a typical realization of the so-called Directed Configuration Model (DCM), that is, a random directed graph with prescribed in- and out-degrees. In this random geometry, we study the meeting time of two random walks on a typical realization of the graph starting at stationarity, the coalescence time for a system of coalescent random walks, and the consensus time of the voter model. Indeed, it is known that the latter three quantities are related to each other when the underlying sequence of graphs satisfies certain mean field conditions. Such conditions can be summarized by requiring a fast mixing time of the random walk and some anti-concentration of its stationary distribution: properties that a typical random directed graph is known to have under natural assumptions on the degree sequence. In this paper we show that, for a typical large graph from the DCM ensemble, the distribution of the meeting time is well-approximated by an exponential random variable and we provide the first-order approximation of its expectation, showing that the latter is linear in the size of the graph, and the preconstant depends on some easy statistics of the degree sequence. As a byproduct, we are able to analyze the effect of the degree sequence in changing the meeting, coalescence and consensus time. Our approach follows the classical idea of converting meeting into hitting times of a proper collapsed chain, which we control by the so-called First Visit Time Lemma.