On total weight exiting finite, strongly connected sets in shift-invariant weighted directed graphs on $\mathbb{Z}$

TL;DR

This paper derives explicit formulas for the minimal weight exiting finite, strongly connected sets in shift-invariant weighted directed graphs on integers, with applications to understanding trapping phenomena in reinforced random walks.

Contribution

It provides a formula for the minimal exit weight as a finite minimum of integer combinations of edge weights for various graphs, advancing understanding of trap formation in random walks.

Findings

The minimal exit weight is always attained by an actual set.

Explicit formulas are derived for several graph types.

The results relate to the strength of traps in reinforced random walks.

Abstract

For a shift-invariant weighted directed graph with vertex set $\mathbb{Z}$, we examine the minimal weight $\kappa_0$ exiting a finite, strongly connected set of vertices. Although $\kappa_0$ is defined as an infimum, it has been shown that the infimum is always attained by an actual set of vertices. We show that for each underlying directed graph (prior to assignment of the weights), there is a formula for $\kappa_0$ as a minimum of finitely many integer combinations of the edge weights. We find this formula for several different directed graphs. Motivation for this problem comes from random walks in Dirichlet environments (equivalently, directed edge reinforced random walks), where the size of $\kappa_0$ has been shown to determine the strength of finite traps where the walk can get stuck for a long time.