Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Minimal subfamilies and the probabilistic interpretation for modulus on
  graphs   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.

14
The categories of this paper are: math-ph
4
The categories of this paper are: cond-mat
12
The categories of this paper are: nucl-th
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
34
The categories of this paper are: chem-ph
21
The categories of this paper are: adap-org
26
The categories of this paper are: bayes-an
10
The categories of this paper are: q-bio
25
The categories of this paper are: patt-sol
0
The categories of this paper are: hep-ph
17
The categories of this paper are: q-fin
27
The categories of this paper are: comp-gas
23
The categories of this paper are: solv-int
18
The categories of this paper are: econ
7
The categories of this paper are: hep-th