Fully Dynamic Random-Walk Quantities on Undirected Expanders and $κ$-Uniform Directed Graphs

Abstract: We study fully dynamic data structures for random-walk quantities on undirected expanders and fast-mixing directed graphs. For an unweighted graph whose normalized Laplacian has spectral gap at least $γ$, we give a $(1\pm\varepsilon)$ effective-resistance data structure with expected update time $$O(γ^{{-4}\varepsilon^{-2}\log} n\log<sup>4(n/\varepsilon))$$ and expected $O(1)$ query time. For directed graphs, somewhat surprisingly, the OMv conjecture precludes a fast data structure based on rapid mixing conditions alone. To address this, we introduce the notion of $κ$-uniform directed graphs, which requires the stationary distribution to $κ$-approximate the uniform distribution, and show that in this setting, random walk based data structures with $\text{poly}(κ, \log{n}, \varepsilon<sup>{-1})$ update/query times are possible.