Localization of Electrical Flows

TL;DR

This paper establishes that in any graph, the average electrical flow path length is polylogarithmic, leading to efficient routing schemes based on spectral properties of the transfer impedance matrix.

Contribution

It introduces a bound on the spectral norm of the transfer impedance matrix, enabling simple oblivious routing schemes in transitive graphs.

Findings

Average flow path length is O(log^2 n)

Spectral norm of transfer impedance matrix is O(log^2 n)

Oblivious routing scheme based on electrical flows is effective in transitive graphs

Abstract

We show that in any graph, the average length of a flow path in an electrical flow between the endpoints of a random edge is $O(\log^2 n)$. This is a consequence of a more general result which shows that the spectral norm of the entrywise absolute value of the transfer impedance matrix of a graph is $O(\log^2 n)$. This result implies a simple oblivious routing scheme based on electrical flows in the case of transitive graphs.