  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.
