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 at a fixed vertex $x$. For regular graphs $G$, one can estimate the number of vertices $n_G$ and the $\ell^2$ mixing time of $G$ from $x$ in $\widetilde{O}(\sqrt{n_G}\,(t_{\rm unif}^G)^{3/4})$ steps, where $t_{\rm unif}^G$ is the uniform mixing time on $G$. The algorithm is based on the number of intersections of random walk paths $X,Y$, i.e. the number of times $(t,s)$ such that $X_t=Y_s$. Our method improves on previous methods by various authors which only consider collisions (i.e. times $t$ with $X_t=Y_t$). 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 $m_G$ and the $\ell^2$ mixing time from the starting vertex $x$ in $\widetilde{O}(\sqrt{m_G}\,(t_{\rm unif}^G)^{3/4})$ 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 $m_G$ or the mixing time of $G$, we can compute the "other parameter" with a self-stopping algorithm.