Exact and Approximate Mean First Passage Times on Trees and other Necklace Structures: a Local Equilibrium Approach

TL;DR

This paper introduces a local-equilibrium based method to accurately compute mean first-passage times on trees and similar graphs, providing exact results for certain structures and useful approximations for others.

Contribution

It presents a novel local-equilibrium approach for calculating MFPTs that is exact for specific graph classes and offers approximations for more complex structures, extending existing theoretical tools.

Findings

Exact MFPTs for graphs coarse-grainable into 1D lattices.

Generalized essential edge lemma for irreversible walks.

Effective approximations for non-ideal graph structures.

Abstract

In this work we propose a novel method to calculate mean first-passage times (MFPTs) for random walks on graphs, based on a dimensionality reduction technique for Markov State Models, known as local-equilibrium (LE). We show that for a broad class of graphs, which includes trees, LE coarse-graining preserves the MFPTs between certain nodes, upon making a suitable choice of the coarse-grained states (or clusters). We prove that this relation is exact for graphs that can be coarse-grained into a one-dimensional lattice where each cluster connects to the lattice only through a single node of the original graph. A side result of the proof generalises the well-known essential edge lemma (EEL), which is valid for reversible random walks, to irreversible walkers. Such a generalised EEL leads to explicit formulae for the MFPTs between certain nodes in this class of graphs. For graphs that do not fall in this class, the generalised EEL provides useful approximations if the graph allows a one-dimensional coarse-grained representation and the clusters are sparsely interconnected. We first demonstrate our method for the simple random walk on the $c$-ary tree, then we consider other graph structures and more general random walks, including irreversible random walks.