Several consequences of adding or removing an edge from an electric network

TL;DR

This paper explores how adding or removing edges in electric networks affects random walk properties, providing new proofs and methods for calculating hitting and return times in modified networks.

Contribution

It introduces novel insights into the impact of edge modifications on electric networks and random walks, including a new proof for expected return times and methods for calculating hitting times.

Findings

New proof for expected return time formula

Methods to calculate hitting times in nearly symmetric networks

Insights into how edge changes influence electric resistance and random walks

Abstract

In certain instances an electric network transforms in natural ways by the addition or removal of an edge. This can have interesting consequences for random walks, in light of the known relationships between electric resistance and random walks. We exhibit several instances in which this can be used to prove facts or simplify calculations. In particular, a new proof is given for the formula for the expected return time of a random walk on a graph. We also show how hitting times can be calculated in certain instances when a network differs from a highly symmetric one by one edge.