Estimating graph parameters with random walks

An algorithm observes the trajectories of random walks over an unknown graph $G$, starting from the same vertex $x$, as well as the degrees along the trajectories. For all finite connected graphs, one can estimate the number of edges $m$ up to a bounded factor in $O\left(t{\mathrm{rel}}^{{3/4}\sqrt{m/d}\right)$} steps, where $t{\mathrm{rel}}$ is the relaxation time of the lazy random walk on $G$ and $d$ is the minimum degree in $G$. Alternatively, $m$ can be estimated in $O\left(t{\mathrm{unif}} +t{\mathrm{rel}}^{{5/6}\sqrt{n}\right)$,} where $n$ is the number of vertices and $t{\mathrm{unif}}$ is the uniform mixing time on $G$. The number of vertices $n$ can then be estimated up to a bounded factor in an additional $O\left(t{\mathrm{unif}}\frac{m}{n}\right)$ steps. Our algorithms are based on counting the number of intersections of random walk paths $X,Y$, i.e. the number of pairs $(t,s)$ such that $Xt=Ys$. This improves on previous estimates which only consider collisions (i.e., times $t$ with $Xt=Yt$). We also show that the complexity of our algorithms is optimal, even when restricting to graphs with a prescribed relaxation time. Finally, we show that, given either $m$ or the mixing time of $G$, we can compute the "other parameter" with a self-stopping algorithm.