The convex support of the k-star model
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Submission date: 11. Sep. 2012
MSC-Numbers: 52B11, 05C80, 05C35, 05C07
Keywords and phrases: polytope, $k$-star model, exponential random graph model, vertex degrees, convex support
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This paper describes the polytope Pk;N of i-star counts, for all i ≤ k, for graphs on N vertices. 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.