# Spectral and matrix factorization methods for consistent community detection in multi-layer networks

arXiv: https://arxiv.org/abs/1704.07353 · HTML: https://mlsanity.com/paper/1704.07353/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Apr 2017.

## Built on by (most cited first, 27 of 27)

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- [Consensus Knowledge Graph Learning via Multi-view Sparse Low Rank Block Model](https://mlsanity.com/paper/2209.13762/) (2022)
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- [A Latent Space Model for Multilayer Network Data](https://mlsanity.com/paper/2102.09560/) (2021)
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- [Community detection in multiplex networks based on orthogonal nonnegative matrix tri-factorization](https://mlsanity.com/paper/2205.00626/) (2022)
- [Privacy-Preserving Community Detection for Locally Distributed Multiple Networks](https://mlsanity.com/paper/2306.15709/) (2023)
- [Learning Joint and Individual Structure in Network Data with Covariates](https://mlsanity.com/paper/2406.08776/) (2024)
- [Fundamental Limits of Community Detection in Contextual Multi-Layer Stochastic Block Models](https://mlsanity.com/paper/2602.08173/) (2026)
- [rMultiNet: An R Package For Multilayer Networks Analysis](https://mlsanity.com/paper/2302.04437/) (2023)
- [Mixture of multilayer stochastic block models for multiview clustering](https://mlsanity.com/paper/2401.04682/) (2024)
- [VEC-SBM: Optimal Community Detection with Vectorial Edges Covariates](https://mlsanity.com/paper/2402.18805/) (2024)
- [Latent class analysis for multi-layer categorical data](https://mlsanity.com/paper/2408.05535/) (2024)
- [Spectral clustering for dependent community Hawkes process models of temporal networks](https://mlsanity.com/paper/2505.21845/) (2025)
- [Joint estimation of smooth graph signals from partial linear measurements](https://mlsanity.com/paper/2505.23240/) (2025)
- [Simultaneous estimation of connectivity and dimensionality in samples of networks](https://mlsanity.com/paper/2508.12483/) (2025)
- [How many asymmetric communities are there in multi-layer directed networks?](https://mlsanity.com/paper/2602.21569/) (2026)
- [T-GINEE: A Tensor-Based Multilayer Graph Representation Learning](https://mlsanity.com/paper/2605.28300/) (2026)
- [A Subsampled Davis-Kahan Bound for Large-Scale Eigenspace Estimation](https://mlsanity.com/paper/2609.09211/) (2026)
- [Degree-Corrected Joint Matrix Factorization for Multilayer Community Detection](https://mlsanity.com/paper/2610.01361/) (2026)

## Abstract (arXiv)

We consider the problem of estimating a consensus community structure by combining information from multiple layers of a multi-layer network using methods based on the spectral clustering or a low-rank matrix factorization. As a general theme, these "intermediate fusion" methods involve obtaining a low column rank matrix by optimizing an objective function and then using the columns of the matrix for clustering. However, the theoretical properties of these methods remain largely unexplored. In the absence of statistical guarantees on the objective functions, it is difficult to determine if the algorithms optimizing the objectives will return good community structures. We investigate the consistency properties of the global optimizer of some of these objective functions under the multi-layer stochastic blockmodel. For this purpose, we derive several new asymptotic results showing consistency of the intermediate fusion techniques along with the spectral clustering of mean adjacency matrix under a high dimensional setup, where the number of nodes, the number of layers and the number of communities of the multi-layer graph grow. Our numerical study shows that the intermediate fusion techniques outperform late fusion methods, namely spectral clustering on aggregate spectral kernel and module allegiance matrix in sparse networks, while they outperform the spectral clustering of mean adjacency matrix in multi-layer networks that contain layers with both homophilic and heterophilic communities.