  The concept of $p$-modulus gives a way to measure the richness of a family of
objects on a graph. In this paper, we investigate the families of connecting
walks between two fixed nodes and show how to use $p$-modulus to form a
parametrized family of graph metrics that generalize several well-known and
widely-used metrics. We also investigate a characteristic of metrics called the
"antisnowflaking exponent" and present some numerical findings supporting a
conjecture about the new metrics. We end with explicit computations of the new
metrics on some selected graphs.
