Permanental Graphs
Daniel Xiang, Peter McCullagh

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 are (i) consistency, and (ii) finite exchangeability for each . 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 that is sparse, exchangeable, and consistent with respect to delete-and-repair, a well known example being the…
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Taxonomy
TopicsBayesian Methods and Mixture Models · Gene expression and cancer classification · Stochastic processes and statistical mechanics
