Minimal Subfamilies and the Probabilistic Interpretation for Modulus on Graphs
نویسنده
چکیده
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.
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