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Abstract:
We study the short term behavior of random walks on graphs,
in particular, the rate at which a random walk
discovers new vertices and edges.
We prove a conjecture by
Linial that the expected time to find $\cal N$ distinct vertices is $O({\cal N} ^ 3)$.
We also prove an upper bound of
$O({\cal M} ^ 2)$ on the expected time to traverse $\cal M$ edges, and
$O(\cal M\cal N)$ on the expected time to either visit $\cal N$ vertices or
traverse $\cal M$ edges (whichever comes first).
