# ESTIMATING GRAPH PARAMETERS WITH RANDOM WALKS

Abstract : 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 $O(t_rel^{3/4}\sqrt{m/d})$ steps, where $t_rel$ 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 $O(t_unif +t_rel^{5/6}\sqrt{n})$, where $n$ is the number of vertices and $t_unif$ is the uniform mixing time on $G$. The number of vertices $n$ can then be estimated up to a bounded factor in an additional $O(t_unif \frac{m}{n})$ 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 $X_t=Y_s$. This improves on previous estimates which only consider collisions (i.e. times $t$ with $X_t=Y_t$). 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.
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https://hal.archives-ouvertes.fr/hal-01598914
Contributor : Anna Ben-Hamou <>
Submitted on : Wednesday, September 19, 2018 - 6:45:07 PM
Last modification on : Friday, March 27, 2020 - 3:05:54 AM

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• HAL Id : hal-01598914, version 2

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Anna Ben-Hamou, Roberto Oliveira, Yuval Peres. ESTIMATING GRAPH PARAMETERS WITH RANDOM WALKS. 2018. ⟨hal-01598914v2⟩

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