Permanental Partition Models and Markovian Gibbs Structures

Crane, Harry

doi:10.1007/s10955-013-0855-0

Cited by 6 publications

(3 citation statements)

References 31 publications

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“…Figure 1: Pictured above (top) is an example of a directed graph on vertex set [4] projected down to a directed graph on [3] according to the delete-and-repair operation (4), and (bottom) the same graph projected down according to subselection.…”

Section: Two Notions Of Consistencymentioning

confidence: 99%

Permanental Graphs

Daniel¹,

McCullagh²

2020

Preprint

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The two components for infinite exchangeability of a sequence of distributions (Pn) 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 (Pn) that is sparse, exchangeable, and consistent with respect to delete-and-repair, a well known example being the Ewens permutations [10]. 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 (Pn) in the sense of either subselection or delete-and-repair.

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Section: Two Notions Of Consistencymentioning

confidence: 99%

Permanental Graphs

Daniel¹,

McCullagh²

2020

Preprint

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“…Ewens [12] first derived (1) while studying the sampling theory of neutral alleles, but the Ewens sampling formula also occurs in purely mathematical contexts, e.g., as the asymptotic distribution of large prime factors [9] and as a special case of the α-permanent of a matrix [6,7].…”

Section: Preliminaries: Random Partitionsmentioning

confidence: 99%

Reversible Markov structures on divisible set partitions

Crane

McCullagh

2015

J. Appl. Probab.

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“…where #n := n j=1 λ j is the number of parts of n and θ ↑n := θ(θ + 1) • • • (θ + n − 1). Ewens [12] first derived (1) while studying the sampling theory of neutral alleles, but the Ewens sampling formula also occurs in purely mathematical contexts, e.g., as the asymptotic distribution of large prime factors [9] and as a special case of the αpermanent of a matrix [6,7].…”

Section: Preliminaries: Random Partitionsmentioning

confidence: 99%

Reversible Markov structures on divisible set partitions

Crane

McCullagh

2015

Journal of Applied Probability

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We study k-divisible partition structures, which are families of random set partitions whose block sizes are divisible by an integer k = 1, 2, . . .. In this setting, exchangeability corresponds to the usual invariance under relabeling by arbitrary permutations; however, for k > 1, the ordinary deletion maps on partitions no longer preserve divisibility, and so a random deletion procedure is needed to obtain a partition structure. We describe explicit Chinese restauranttype seating rules for generating families of exchangeable k-divisible partitions that are consistent under random deletion. We further introduce the notion of Markovian partition structures, which are ensembles of exchangeable Markov chains on k-divisible partitions that are consistent under a random process of Markovian deletion. The Markov chains we study are reversible and refine the class of Markov chains introduced in J. Appl. Probab. 48(3):778-791.

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Permanental Partition Models and Markovian Gibbs Structures

Cited by 6 publications

References 31 publications

Permanental Graphs

Permanental Graphs

Reversible Markov structures on divisible set partitions

Reversible Markov structures on divisible set partitions

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