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The Hierarchy of Block Models


There exist various types of network block models such as the Stochastic Block Model (SBM), the Degree Corrected Block Model (DCBM), and the Popularity Adjusted Block Model (PABM). While this leads to a variety of choices, the block models do not have a nested structure. In addition, there is a substantial jump in the number of parameters from the DCBM to the PABM. The objective of this paper is formulation of a hierarchy of block model which does not rely on arbitrary identifiability conditions. We propose a Nested Block Model (NBM) that treats the SBM, the DCBM and the PABM as its particular cases with specific parameter values, and, in addition, allows a multitude of versions that are more complicated than DCBM but have fewer unknown parameters than the PABM. The latter allows one to carry out clustering and estimation without preliminary testing, to see which block model is really true.

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Both authors of the paper were partially supported by National Science Foundation (NSF) grants DMS-1712977 and DMS-2014928

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Correspondence to Majid Noroozi.

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Noroozi, M., Pensky, M. The Hierarchy of Block Models. Sankhya A 84, 64–107 (2022).

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Keywords and phrases

  • Stochastic block model
  • Degree corrected block model
  • Popularity adjusted block model
  • Sparse subspace clustering
  • Spectral clustering with k-median

AMS (2000) subject classification

  • 05C80
  • 62F12
  • 62H30