# Perfect Clustering in Very Sparse Diverse Multiplex Networks

arXiv: https://arxiv.org/abs/2507.19423 · HTML: https://mlsanity.com/paper/2507.19423/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 28 references that are in the map.
- First posted on arXiv 25 Jul 2025.

## Builds on (most cited first, 18 of 18)

- [Statistical inference on random dot product graphs: a survey](https://mlsanity.com/paper/1709.05454/) (2017)
- [A statistical interpretation of spectral embedding: the generalised random dot product graph](https://mlsanity.com/paper/1709.05506/) (2017)
- [Tensor SVD: Statistical and Computational Limits](https://mlsanity.com/paper/1703.02724/) (2017)
- [Rate-Optimal Perturbation Bounds for Singular Subspaces with Applications to High-Dimensional Statistics](https://mlsanity.com/paper/1605.00353/) (2016)
- [Inference for multiple heterogeneous networks with a common invariant subspace](https://mlsanity.com/paper/1906.10026/) (2019)
- [Community Detection on Mixture Multi-layer Networks via Regularized Tensor Decomposition](https://mlsanity.com/paper/2002.04457/) (2020)
- [Community Detection for Hypergraph Networks via Regularized Tensor Power Iteration](https://mlsanity.com/paper/1909.06503/) (2019)
- [The multilayer random dot product graph](https://mlsanity.com/paper/2007.10455/) (2020)
- [Latent Space Model for Higher-order Networks and Generalized Tensor Decomposition](https://mlsanity.com/paper/2106.16042/) (2021)
- [Inference for Low-rank Tensors -- No Need to Debias](https://mlsanity.com/paper/2012.14844/) (2020)
- [ALMA: Alternating Minimization Algorithm for Clustering Mixture Multilayer Network](https://mlsanity.com/paper/2102.10226/) (2021)
- [Clustering of Diverse Multiplex Networks](https://mlsanity.com/paper/2110.05308/) (2021)
- [Estimating Higher-Order Mixed Memberships via the ℓ2,∞ Tensor Perturbation Bound](https://mlsanity.com/paper/2212.08642/) (2022)
- [A Sharp Blockwise Tensor Perturbation Bound for Orthogonal Iteration](https://mlsanity.com/paper/2008.02437/) (2020)
- [Davis-Kahan Theorem in the two-to-infinity norm and its application to perfect clustering](https://mlsanity.com/paper/2411.11728/) (2024)
- [Optimal Clustering of Discrete Mixtures: Binomial, Poisson, Block Models, and Multi-layer Networks](https://mlsanity.com/paper/2311.15598/) (2023)
- [Statistical Inference for Low-Rank Tensors: Heteroskedasticity, Subgaussianity, and Applications](https://mlsanity.com/paper/2410.06381/) (2024)
- [Signed Diverse Multiplex Networks: Clustering and Inference](https://mlsanity.com/paper/2402.10242/) (2024)

## Abstract (arXiv)

The paper studies the DIverse MultiPLEx Signed Generalized Random Dot Product Graph (DIMPLE-SGRDPG) network model (Pensky (2024)), where all layers of the network have the same collection of nodes. In addition, all layers can be partitioned into groups such that the layers in the same group are embedded in the same ambient subspace but otherwise matrices of connection probabilities can be all different. This setting includes majority of multilayer network models as its particular cases. The key task in this model is to recover the groups of layers with unique subspace structures, since the case where all layers of the network are embedded in the same subspace has been fairly well studied. Until now, clustering of layers in such networks was based on the layer-per-layer analysis, which required the multilayer network to be sufficiently dense. Nevertheless, in this paper we succeeded in pooling information in all layers together and providing a tensor-based methodology that ensures perfect clustering for a much sparser network. Our theoretical results, established under intuitive non-restrictive assumptions, assert that the new technique achieves perfect clustering under sparsity conditions that, up to logarithmic factors, coincide with the computational lower bound derived for a much simpler model.