Estimating graph parameters via random walks with restarts.

In this paper we discuss the problem of estimating graph parameters from a random walk with restarts. In this setting, an algorithm observes the trajectory of a random walk over an unknown graph G, starting from a vertex x. The algorithm also sees the degrees along the trajectory. The only other power that the algorithm has is to request that the random walk be reset to its initial state at any given time, based on what it has seen so far. Our main results are as follows. For regular graphs G, one can estimate the number of vertices nG and the ℓ2 mixing time of G from x in [EQUATION] steps, where tGunif is the uniform mixing time of the random walk on G. The algorithm is based on the number of intersections of random walk paths X, Y, ie. the number of times (t, s) such that Xt = Ys. Our method improves on previous methods by various authors which only consider collisions (ie. times t with Xt = Yt). We also show that the time complexity of our algorithm is optimal (up to log factors) for 3-regular graphs with prescribed mixing times. For general graphs, we adapt the intersections algorithm to compute the number of edges mG and the ℓ2 mixing time from the starting vertex x in [EQUATION] steps. Under mild additional assumptions (which hold e.g. for sparse graphs) the number of vertices can also be estimated by this time. Finally, we show that these algorithms, which may take sublinear time, have a fundamental limitation: it is not possible to devise a sublinear stopping time at which one can be reasonably sure that our parameters are well estimated. On the other hand, we show that, given either mG or the mixing time of G, we can compute the "other parameter" with a self-stopping algorithm.