The Polytope of k-Star Densities

  • Johannes Rauh
Keywords: Polytopes, k-star Model, Exponential Random Graph Model, Vertex Degrees, Convex Support

Abstract

This paper describes the polytope Pk;N of i-star counts, for all ik, for graphs on N nodes.  The vertices correspond to graphs that are regular or as regular as possible.  For even N the polytope is a cyclic polytope, and for odd N the polytope is well-approximated by a cyclic polytope.  As N goes to infinity, Pk;N approaches the convex hull of the moment curve. The affine symmetry group of Pk;N contains just a single non-trivial element, which corresponds to forming the complement of a graph.

The results generalize to the polytope PI;N of i-star counts, for i in some set I of non-consecutive integers.  In this case, PI;N can still be approximated by a cyclic polytope, but it is usually not a cyclic polytope itself.

Polytopes of subgraph statistics characterize corresponding exponential random graph models.  The elongated shape of the k-star polytope gives a qualitative explanation of some of the degeneracies found in such random graph models.
Published
2017-01-20
Article Number
P1.4