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Simultaneous estimation of connectivity and dimensionality in samples of networks

arXiv 2508.12483 ↗First posted 17 Aug 2025stat.ME math.ST stat.MLcited by 0

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Abstract (arXiv)

An overarching objective in contemporary statistical network analysis is extracting salient information from datasets consisting of multiple networks. To date, considerable attention has been devoted to node and network clustering, while comparatively less attention has been devoted to downstream connectivity estimation and parsimonious embedding dimension selection. Given a sample of potentially heterogeneous networks, this paper proposes a method to simultaneously estimate a latent matrix of connectivity probabilities and its embedding dimensionality or rank after first pre-estimating the number of communities and the node community memberships. The method is formulated as a convex optimization problem and solved using an alternating direction method of multipliers algorithm. We establish estimation error bounds under the Frobenius norm and nuclear norm for settings in which observable networks have blockmodel structure, even when node memberships are imperfectly recovered. When perfect membership recovery is possible and dimensionality is much smaller than the number of communities, the proposed method outperforms conventional averaging-based methods for estimating connectivity and dimensionality. Numerical studies empirically demonstrate the accuracy of our method across various scenarios. Additionally, analysis of a primate brain dataset demonstrates that posited connectivity is not necessarily full rank in practice, illustrating the need for flexible methodology.

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References · 21

  1. 2017Community Detection and Stochastic Block Modelscited by 319
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  3. 2013Stochastic blockmodel approximation of a graphon: Theory and consistent estimationcited by 39
  4. 2020Community Detection on Mixture Multi-layer Networks via Regularized Tensor Decompositioncited by 30
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  6. 2017Spectral and matrix factorization methods for consistent community detection in multi-layer networkscited by 27
  7. 2016Network cross-validation by edge samplingcited by 26
  8. 2020Exact Clustering in Tensor Block Model: Statistical Optimality and Computational Limitcited by 23
  9. 2014Network Cross-Validation for Determining the Number of Communities in Network Datacited by 21
  10. 2014A goodness-of-fit test for stochastic block modelscited by 20
  11. 2015Community detection in multi-relational data with restricted multi-layer stochastic blockmodelcited by 14
  12. 2015Estimating the number of communities in networks by spectral methodscited by 9
9 more
  1. 2017Asymptotically efficient estimators for stochastic blockmodels: the naive MLE, the rank-constrained MLE, and the spectralcited by 9
  2. 2019On Low-rank Trace Regression under General Sampling Distributioncited by 9
  3. 2020Randomized Spectral Clustering in Large-Scale Stochastic Block Modelscited by 9
  4. 2021ALMA: Alternating Minimization Algorithm for Clustering Mixture Multilayer Networkcited by 8
  5. 2021Clustering of Diverse Multiplex Networkscited by 8
  6. 2019Simultaneous Dimensionality and Complexity Model Selection for Spectral Graph Clusteringcited by 4
  7. 2021Network Estimation by Mixing: Adaptivity and Morecited by 4
  8. 2022Sparse Subspace Clustering in Diverse Multiplex Network Modelcited by 4
  9. 2022Consensus Knowledge Graph Learning via Multi-view Sparse Low Rank Block Modelcited by 3

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