Permanental Graphs

Abstract: 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 \cite{tavare}. A generalization of the CRP$(\alpha)$ as a distribution on a directed graph using the $\alpha$-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.