  The notion of $p$-modulus of a family of objects on a graph is a measure of
the richness of such families. We develop the notion of minimal subfamilies
using the method of Lagrangian duality for $p$-modulus. We show that minimal
subfamilies have at most $|E|$ elements and that these elements carry a weight
related to their "importance" in relation to the corresponding $p$-modulus
problem. When $p=2$, this measure of importance is in fact a probability
measure and modulus can be thought as trying to minimize the expected overlap
in the family.
