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arXiv · 2205.07414

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

Daniel J. Slonim
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Abstract

For a shift-invariant weighted directed graph with vertex set $\mathbb{Z}$, we examine the minimal weight $κ_0$ exiting a finite, strongly connected set of vertices. Although $κ_0$ 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 $κ_0$ 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 $κ_0$ has been shown to determine the strength of finite traps where the walk can get stuck for a long time.

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BibTeXRIS

Daniel J. Slonim. 2022-05-16. On total weight exiting finite, strongly connected sets in shift-invariant weighted directed graphs on $\mathbb{Z}$. https://arxiv.org/abs/2205.07414

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