Permanental Graphs
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The two components for infinite exchangeability of a sequence of distributions (P_n) are (i) consistency, and (ii) finite exchangeability for each n. A consequence of the Aldous-Hoover theorem is that any node-exchangeable, subselection-consistent sequence of distributions that describes a randomly evolving network yields a sequence of random graphs whose expected number of edges grows quadratically in the number of nodes. In this note, another notion of consistency is considered, namely, delete-and-repair consistency; it is motivated by the sense in which infinitely exchangeable permutations defined by the Chinese restaurant process (CRP) are consistent. A goal is to exploit delete-and-repair consistency to obtain a nontrivial sequence of distributions on graphs (P_n) that is sparse, exchangeable, and consistent with respect to delete-and-repair, a well known example being the Ewens permutations <cit.>. A generalization of the CRP(α) as a distribution on a directed graph using the α-weighted permanent is presented along with the corresponding normalization constant and degree distribution; it is dubbed the Permanental Graph Model (PGM). A negative result is obtained: no setting of parameters in the PGM allows for a consistent sequence (P_n) in the sense of either subselection or delete-and-repair.
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