Means of Hitting Times for Random Walks on Graphs: Connections, Computation, and Optimization

TL;DR

This paper explores the relationships between mean hitting times and the Kemeny constant in random walks on graphs, introduces efficient algorithms for their estimation, and addresses the NP-hard problem of selecting optimal vertex groups.

Contribution

It establishes a connection between mean hitting times and the Kemeny constant, develops nearly linear time algorithms for their estimation, and proposes greedy methods for optimal vertex set selection.

Findings

Efficient algorithms estimate mean hitting times and the Kemeny constant in nearly linear time.

New bounds and relationships between hitting times, the Kemeny constant, and vertex centrality.

Greedy algorithms achieve near-optimal solutions with proven approximation guarantees.

Abstract

For random walks on graph $\mathcal{G}$ with $n$ vertices and $m$ edges, the mean hitting time $H_j$ from a vertex chosen from the stationary distribution to vertex $j$ measures the importance for $j$, while the Kemeny constant $\mathcal{K}$ is the mean hitting time from one vertex to another selected randomly according to the stationary distribution. In this paper, we first establish a connection between the two quantities, representing $\mathcal{K}$ in terms of $H_j$ for all vertices. We then develop an efficient algorithm estimating $H_j$ for all vertices and \(\mathcal{K}\) in nearly linear time of $m$. Moreover, we extend the centrality $H_j$ of a single vertex to $H(S)$ of a vertex set $S$, and establish a link between $H(S)$ and some other quantities. We further study the NP-hard problem of selecting a group $S$ of $k\ll n$ vertices with minimum $H(S)$, whose objective function is monotonic and supermodular. We finally propose two greedy algorithms approximately solving the problem. The former has an approximation factor $(1-\frac{k}{k-1}\frac{1}{e})$ and $O(kn^3)$ running time, while the latter returns a $(1-\frac{k}{k-1}\frac{1}{e}-\epsilon)$-approximation solution in nearly-linear time of $m$, for any parameter $0<\epsilon <1$. Extensive experiment results validate the performance of our algorithms.