Polytopes from Subgraph Statistics

Abstract

We study polytopes that are convex hulls of vectors of subgraph densities. Many graph theoretical questions can be expressed in terms of these polytopes, and statisticians use them to understand exponential random graph models. Relations among their Ehrhart polynomials are described, their duals are applied to certify that polynomials are non-negative, and we find some of their faces. For the general picture we inscribe cyclic polytopes in them and calculate volumes. From the volume calculations we conjecture that a variation of the Selberg integral indexed by Schur polynomials has a combinatorial formula. We inscribe polynomially parametrized sets, called curvy zonotopes, in the polytopes and show that they approximate the polytopes arbitrarily close.

Keywords

• polytopes
• subgraph statistics
• exponential random graph models
• curvy zonotopes
• graph limits
• [math.math-co] mathematics [math]/combinatorics [math.co]
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]