The Polytope of k-Star Densities

This paper describes the polytope Pk;N of i-star counts, for all i 6 k, 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.