Entrywise Approximate Laplacian Solving

TL;DR

This paper introduces improved algorithms for computing escape probabilities in random walks on graphs, using floating-point arithmetic to handle small probabilities efficiently and with better running times.

Contribution

It presents novel algorithms and analyses for weighted directed graphs that enhance the efficiency of computing escape probabilities, moving beyond fixed-point arithmetic limitations.

Findings

Achieved faster algorithms for escape probability computation.

Extended techniques to weighted directed graphs.

Potential broader impact on random walk computations.

Abstract

We study the escape probability problem in random walks over graphs. Given vertices, $s,t,$ and $p$, the problem asks for the probability that a random walk starting at $s$ will hit $t$ before hitting $p$. Such probabilities can be exponentially small even for unweighted undirected graphs with polynomial mixing time. Therefore current approaches, which are mostly based on fixed-point arithmetic, require $n$ bits of precision in the worst case. We present algorithms and analyses for weighted directed graphs under floating-point arithmetic and improve the previous best running times in terms of the number of bit operations. We believe our techniques and analysis could have a broader impact on the computation of random walks on graphs both in theory and in practice.