Hitting and commute times in large graphs are often misleading

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 Huv is the expected time it takes a random walk to travel from u to v. The commute time is its symmetrized version Cuv = Huv + Hvu. 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 → ∞, under mild assumptions, 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 Huv converges to 1/dv and the commute time to 1/du +1/dv where du and dv 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 (kNN-graphs, ε-graphs, Gaussian similarity graphs) and random graphs with given expected degrees (in particular, Erdős-Rényi graphs with and without planted partitions).