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Mixture of multilayer stochastic block models for multiview clustering

Kylliann De Santiago, Marie Szafranski, Christophe Ambroise

TL;DR

This work introduces mimi-SBM, a Bayesian mixture of multilayer SBMs designed for multiview clustering where observations share a global partition while each view contributes through a latent component. The model jointly treats nodes and views in a tensor of adjacency matrices, with $K$ consensus clusters and $Q$ view components, and employs a variational Bayes EM algorithm to estimate parameters and perform model selection via ELBO-based criteria. Identifiability is established under mild conditions, and the framework accommodates conjugate priors for tractable posterior updates. Extensive simulations show competitive clustering performance and robust model selection, while application to Worldwide Food Trading Networks reveals meaningful country clusters and product-group structures consistent with trade dynamics. The approach provides a principled, scalable pathway to integrative, model-based consensus clustering across heterogeneous information sources.

Abstract

In this work, we propose an original method for aggregating multiple clustering coming from different sources of information. Each partition is encoded by a co-membership matrix between observations. Our approach uses a mixture of multilayer Stochastic Block Models (SBM) to group co-membership matrices with similar information into components and to partition observations into different clusters, taking into account their specificities within the components. The identifiability of the model parameters is established and a variational Bayesian EM algorithm is proposed for the estimation of these parameters. The Bayesian framework allows for selecting an optimal number of clusters and components. The proposed approach is compared using synthetic data with consensus clustering and tensor-based algorithms for community detection in large-scale complex networks. Finally, the method is utilized to analyze global food trading networks, leading to structures of interest.

Mixture of multilayer stochastic block models for multiview clustering

TL;DR

This work introduces mimi-SBM, a Bayesian mixture of multilayer SBMs designed for multiview clustering where observations share a global partition while each view contributes through a latent component. The model jointly treats nodes and views in a tensor of adjacency matrices, with consensus clusters and view components, and employs a variational Bayes EM algorithm to estimate parameters and perform model selection via ELBO-based criteria. Identifiability is established under mild conditions, and the framework accommodates conjugate priors for tractable posterior updates. Extensive simulations show competitive clustering performance and robust model selection, while application to Worldwide Food Trading Networks reveals meaningful country clusters and product-group structures consistent with trade dynamics. The approach provides a principled, scalable pathway to integrative, model-based consensus clustering across heterogeneous information sources.

Abstract

In this work, we propose an original method for aggregating multiple clustering coming from different sources of information. Each partition is encoded by a co-membership matrix between observations. Our approach uses a mixture of multilayer Stochastic Block Models (SBM) to group co-membership matrices with similar information into components and to partition observations into different clusters, taking into account their specificities within the components. The identifiability of the model parameters is established and a variational Bayesian EM algorithm is proposed for the estimation of these parameters. The Bayesian framework allows for selecting an optimal number of clusters and components. The proposed approach is compared using synthetic data with consensus clustering and tensor-based algorithms for community detection in large-scale complex networks. Finally, the method is utilized to analyze global food trading networks, leading to structures of interest.
Paper Structure (51 sections, 11 theorems, 76 equations, 14 figures, 1 table, 1 algorithm)

This paper contains 51 sections, 11 theorems, 76 equations, 14 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contribution.
  3. Organization of the paper.
  4. Background
  5. Consensus clustering
  6. Block models for multiview clustering
  7. Multilayer SBM
  8. VEM inference.
  9. Spectral clustering.
  10. Multiway block models
  11. Mixture of multilayer SBM.
  12. Tensor block models.
  13. Mixture of Multilayer SBM
  14. Observations.
  15. Latent structures.
  16. ...and 36 more sections

Key Result

Theorem 1

Let $N \geq \max(2K,4Q)$ and $V \geq 2K$. Assume that for any $k,l \in \{1,\dots,K \}$ and every $s \in \{1,\dots,Q \}$, the coordinates of $\boldsymbol{\pi}^T \boldsymbol{\alpha}_{k..} \boldsymbol{\rho}$ are all different, $(\boldsymbol{\pi}^T \boldsymbol{\alpha}_{..s} \boldsymbol{\pi})_{s=1:Q}$

Figures (14)

  • Figure 1: Illustration of mimi-SBM. Left: Four adjacency matrices $\mathbf{A}^{(1)}, \cdots, \mathbf{A}^{(4)}$ coming from four different views organized into two components. Right: identification of the two components from the views (local and complementary information) and clustering of the observations described by the classification matrix $\mathbf{Z}$ (global and consensus information).
  • Figure 2: Illustration of mimi-SBM with bayesian notations
  • Figure 3: Diagram of links between different model selection criteria.
  • Figure 4: Diagram of the simulation process. Example of adjacency matrices resulting from mixing and traversing clusters across views, with potentially label-switching, for $K = 5$. For each view component, a number of clusters $K^q$ is randomly drawn (discrete uniform distribution). Each cluster in the $q^{\textrm{th}}$ component is then linked to certain clusters in the final partition. For this component, $K^q=3$, and final clusters $1$ (respectively $3$) and $4$ (resp. $5$) are merged into cluster $2$ (resp. $3$) of the component, and the first cluster of the component corresponds perfectly to the final consensus cluster $2$. Afterwards, these links are represented by a very strong connectivity within the $\boldsymbol{\alpha}_{..q}$ matrix ($p = 0.99$) and a very weak one ($p=0.01$) for the others.
  • Figure 5: Bar plots of model selection criteria on $50$ simulations with $10$ true clusters for observations and $5$ true components for views. Figure (a) (resp. (b)) indicates the number of times the $K$ (resp. $Q$) value selected while the other parameters is set to the true value. Figures (c) and (d) show the same information when hyperparameters are optimized at the same time.
  • ...and 9 more figures

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 2
  • Proposition 3: Invertibility of $\mathbf{R}$
  • Proposition 4: Relations between $\mathbf{R}$, $\mathbf{M}$ and $\boldsymbol{\pi}$
  • Proposition 5
  • Proposition 6
  • Proposition 7: Invertibility of $\mathbf{T}$
  • Proposition 8: Relations between $\mathbf{T}$, $\tilde{\mathbf{M}}$ and ${\boldsymbol{\rho}}$
  • Proposition 9
  • Remark 10
  • ...and 2 more