Potential Distribution on Random Electrical Networks

TL;DR

This paper investigates the potential distribution in random electrical networks modeled by random graphs, showing that interior vertices tend to have nearly uniform potential with high probability under certain conditions.

Contribution

It proves that in such networks, the potentials of interior vertices are almost constant with high probability when the graph is sufficiently connected.

Findings

Interior potentials are nearly uniform in large random networks.

High probability results hold for graphs with edge probability p=c*ln(n)/n, c>1.

Potential distribution concentrates around a constant value.

Abstract

Let $G=(V,E)$ be a random electronic network with the boundary vertices which is obtained by assigning a resistance of each edge in a random graph in $\mathbb{G}(n,p)$ and the voltages on the boundary vertices. In this paper, we prove that the potential distribution of all vertices of $G$ except for the boundary vertices are very close to a constant with high probability for $p=\frac{c\ln n}{n}$ and $c>1$.