Modulus on graphs as a generalization of standard graph theoretic quantities

This paper presents new results for the modulus of families of walks on a graph—a discrete analog of the modulus of curve families due to Beurling and Ahlfors. Particular attention is paid to the dependence of the modulus on its parameters. Modulus is shown to generalize (and interpolate among) three important quantities in graph theory: shortest path, effective resistance, and max-flow or min-cut.