Localization of Electrical Flows

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.