Hitting and commute times in large graphs are often misleading

TL;DR

This paper demonstrates that in large graphs, hitting and commute times become unreliable as they converge to simple degree-based expressions, ignoring the global structure of the graph.

Contribution

It provides theoretical analysis showing the asymptotic behavior of hitting and commute times in large graphs, revealing their limitations as structural indicators.

Findings

Hitting times converge to inverse degree of the target vertex.

Commute times converge to sum of inverse degrees of both vertices.

These metrics become misleading for large graphs, ignoring global structure.

Abstract

Next to the shortest path distance, the second most popular distance function between vertices in a graph is the commute distance (resistance distance). For two vertices u and v, the hitting time H_{uv} is the expected time it takes a random walk to travel from u to v. The commute time is its symmetrized version C_{uv} = H_{uv} + H_{vu}. In our paper we study the behavior of hitting times and commute distances when the number n of vertices in the graph is very large. We prove that as n converges to infinty, hitting times and commute distances converge to expressions that do not take into account the global structure of the graph at all. Namely, the hitting time H_{uv} converges to 1/d_v and the commute time to 1/d_u + 1/d_v where d_u and d_v denote the degrees of vertices u and v. In these cases, the hitting and commute times are misleading in the sense that they do not provide information about the structure of the graph. We focus on two major classes of random graphs: random geometric graphs (k-nearest neighbor graphs, epsilon-graphs, Gaussian similarity graphs) and random graphs with given expected degrees (in particular, Erdos-Renyi graphs with and without planted partitions)