Meeting times on graphs in near-cubic time

TL;DR

This paper presents a fast algorithm for computing meeting times of two random walkers on large graphs, reducing complexity from $O(N^6)$ to $O(N^4)$ and further to $O(N^3 ext{log}^2 N)$ with structural exploitation.

Contribution

The authors develop a novel algorithm exploiting the structure of the linear system to compute meeting times efficiently, improving computational complexity significantly.

Findings

Exact meeting times can be computed in $O(N^4)$ operations.

The method can be improved to $O(N^3 ext{log}^2 N)$ using Cauchy structure.

Applications include efficient calculation of fixation probabilities in evolutionary dynamics.

Abstract

The expected meeting time of two random walkers on an undirected graph of size $N$, where at each time step one walker moves and the process stops when they collide, satisfies a system of $\binom{N}{2}$ linear equations. Na\"{i}vely, solving this system takes $O\left(N^{6}\right)$ operations. However, this system of linear equations has nice structure in that it is almost a Sylvester equation, with the obstruction being a diagonal absorption constraint. We give a simple algorithm for solving this system that exploits this structure, leading to $O\left(N^{4}\right)$ operations and $\Theta\left(N^{2}\right)$ space for exact computation of all $\binom{N}{2}$ meeting times. While this practical method uses only standard dense linear algebra, it can be improved (in theory) to $O\left(N^{3}\log^{2}N\right)$ operations by exploiting the Cauchy structure of the diagonal correction. We generalize this result slightly to cover the Poisson equation for the absorbing "lazy" pair walk with an arbitrary source, which can be solved at the same cost, with $O\left(N^{3}\right)$ per additional source on the same graph. We conclude with applications to evolutionary dynamics, giving improved algorithms for calculating fixation probabilities and mean trait frequencies.