  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)
