Localization of electrical flows

Abstract

We show that in any graph, the average length of a ow path in an electrical ow between the endpoints of a random edge is O(log2 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(log2 n). This result implies a simple oblivious routing scheme based on electrical ows in the case of transitive graphs.