Permanental Graphs

TL;DR

This paper introduces the Permanental Graph Model (PGM), a new approach to modeling sparse, exchangeable graphs using the alpha-weighted permanent, but finds limitations in achieving full consistency.

Contribution

It proposes the PGM based on the alpha-weighted permanent and explores its properties, highlighting the challenges in achieving certain types of consistency.

Findings

The PGM generalizes the Chinese restaurant process to directed graphs.

No parameter setting in PGM yields a fully consistent sequence of graph distributions.

The model offers a new perspective on exchangeability and sparsity in graph sequences.

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.