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Exchangeable random permutations with an application to Bayesian graph matching

Francesco Gaffi, Nathaniel Josephs, Lizhen Lin

TL;DR

The paper addresses graph matching under uncertainty by introducing a Bayesian framework built on exchangeable random permutations. It characterizes these permutations via their cycle structures and introduces a predictive PA-gCRP to construct priors, pairing them with a correlated SBM to align nodes across networks. Posterior inference uses a node-wise blocked Gibbs sampler, and posterior summaries are computed with perSALSO, a Cayley-distance–based estimator with a three-phase procedure. The results demonstrate accurate recovery of cycle structure and competitive edge alignment with coherent uncertainty quantification, suggesting broad applicability to permutation-valued estimation problems beyond graph matching.

Abstract

We introduce a general Bayesian framework for graph matching grounded in a new theory of exchangeable random permutations. Leveraging the cycle representation of permutations and the literature on exchangeable random partitions, we define, characterize, and study the structural and predictive properties of these probabilistic objects. A novel sequential metaphor, the position-aware generalized Chinese restaurant process, provides a constructive foundation for this theory and supports practical algorithmic design. Exchangeable random permutations offer flexible priors for a wide range of inferential problems centered on permutations. As an application, we develop a Bayesian model for graph matching that integrates a correlated stochastic block model with our novel class of priors. The cycle structure of the matching is linked to latent node partitions that explain connectivity patterns, an assumption consistent with the homogeneity requirement underlying the graph matching task itself. Posterior inference is performed through a node-wise blocked Gibbs sampler directly enabled by the proposed sequential construction. To summarize posterior uncertainty, we introduce perSALSO, an adaptation of SALSO to the permutation domain that provides principled point estimation and interpretable posterior summaries. Together, these contributions establish a unified probabilistic framework for modeling, inference, and uncertainty quantification over permutations.

Exchangeable random permutations with an application to Bayesian graph matching

TL;DR

The paper addresses graph matching under uncertainty by introducing a Bayesian framework built on exchangeable random permutations. It characterizes these permutations via their cycle structures and introduces a predictive PA-gCRP to construct priors, pairing them with a correlated SBM to align nodes across networks. Posterior inference uses a node-wise blocked Gibbs sampler, and posterior summaries are computed with perSALSO, a Cayley-distance–based estimator with a three-phase procedure. The results demonstrate accurate recovery of cycle structure and competitive edge alignment with coherent uncertainty quantification, suggesting broad applicability to permutation-valued estimation problems beyond graph matching.

Abstract

We introduce a general Bayesian framework for graph matching grounded in a new theory of exchangeable random permutations. Leveraging the cycle representation of permutations and the literature on exchangeable random partitions, we define, characterize, and study the structural and predictive properties of these probabilistic objects. A novel sequential metaphor, the position-aware generalized Chinese restaurant process, provides a constructive foundation for this theory and supports practical algorithmic design. Exchangeable random permutations offer flexible priors for a wide range of inferential problems centered on permutations. As an application, we develop a Bayesian model for graph matching that integrates a correlated stochastic block model with our novel class of priors. The cycle structure of the matching is linked to latent node partitions that explain connectivity patterns, an assumption consistent with the homogeneity requirement underlying the graph matching task itself. Posterior inference is performed through a node-wise blocked Gibbs sampler directly enabled by the proposed sequential construction. To summarize posterior uncertainty, we introduce perSALSO, an adaptation of SALSO to the permutation domain that provides principled point estimation and interpretable posterior summaries. Together, these contributions establish a unified probabilistic framework for modeling, inference, and uncertainty quantification over permutations.
Paper Structure (23 sections, 5 theorems, 60 equations, 4 figures, 1 table, 2 algorithms)

This paper contains 23 sections, 5 theorems, 60 equations, 4 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. A class of nonparametric priors for permutations
  3. The cycle structure of a permutation
  4. Exchangeable random permutations
  5. Predictive structure: the position-aware gCRP
  6. A correlated SBM with PA-gCRP priors
  7. Latent parent network formulation
  8. Posterior inference
  9. Posterior summary
  10. perSALSO
  11. Simulations
  12. Mixing and uncertainty quantification
  13. Number of nodes
  14. Noise level
  15. Concurrence of cycle and community structures
  16. ...and 8 more sections

Key Result

Proposition 2.3

Let ${\boldsymbol \pi}$ be a random permutation of $[n]$. If ${\boldsymbol \pi}$ is finitely exchangeable, then its cycle structure $\mathrm{z}( {\boldsymbol \pi} )$ is an exchangeable random partition.

Figures (4)

  • Figure 1: (Left) Block matrix of probabilities for underlying SBM. (Center) Trace plot of log-likelihood. (Right) Posterior mapping frequency, where the nodes on the x-axis are permuted through $\pi^*$.
  • Figure 2: Simulation results as the number of nodes increases.
  • Figure 3: Simulation results as the cSBM noise level increases.
  • Figure 4: Simulation results as we increase the concurrence of the latent permutation cycle structure and the underlying community structure.

Theorems & Definitions (22)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.1
  • Proposition 2.3
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.4
  • Definition 2.5
  • Theorem 2.5
  • ...and 12 more