# Community detection by spectral methods in multi-layer networks

arXiv: https://arxiv.org/abs/2403.12540 · HTML: https://mlsanity.com/paper/2403.12540/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 66 references that are in the map.
- First posted on arXiv 19 Mar 2024.

## Builds on (most cited first, 21 of 21)

- [Community Detection and Stochastic Block Models](https://mlsanity.com/paper/1703.10146/) (2017)
- [Consistency of spectral clustering in stochastic block models](https://mlsanity.com/paper/1312.2050/) (2013)
- [Multilayer Networks](https://mlsanity.com/paper/1309.7233/) (2013)
- [The structure and dynamics of multilayer networks](https://mlsanity.com/paper/1407.0742/) (2014)
- [Regularized Spectral Clustering under the Degree-Corrected Stochastic Blockmodel](https://mlsanity.com/paper/1309.4111/) (2013)
- [Inference for multiple heterogeneous networks with a common invariant subspace](https://mlsanity.com/paper/1906.10026/) (2019)
- [Layer aggregation and reducibility of multilayer interconnected networks](https://mlsanity.com/paper/1405.0425/) (2014)
- [Community Detection on Mixture Multi-layer Networks via Regularized Tensor Decomposition](https://mlsanity.com/paper/2002.04457/) (2020)
- [Spectral and matrix factorization methods for consistent community detection in multi-layer networks](https://mlsanity.com/paper/1704.07353/) (2017)
- [Impact of regularization on Spectral Clustering](https://mlsanity.com/paper/1312.1733/) (2013)
- [Estimating Mixed Memberships with Sharp Eigenvector Deviations](https://mlsanity.com/paper/1709.00407/) (2017)
- [Spectral Algorithms for Community Detection in Directed Networks](https://mlsanity.com/paper/2008.03820/) (2020)
- [Convexified Modularity Maximization for Degree-corrected Stochastic Block Models](https://mlsanity.com/paper/1512.08425/) (2015)
- [Community detection in multi-relational data with restricted multi-layer stochastic blockmodel](https://mlsanity.com/paper/1506.02699/) (2015)
- [Spectral clustering in the dynamic stochastic block model](https://mlsanity.com/paper/1705.01204/) (2017)
- [Determining the Number of Communities in Degree-corrected Stochastic Block Models](https://mlsanity.com/paper/1809.01028/) (2018)
- [Combinatorial Analysis of Multiple Networks](https://mlsanity.com/paper/1303.4986/) (2013)
- [Community detection for weighted bipartite networks](https://mlsanity.com/paper/2109.10319/) (2021)
- [Subsampling Spectral Clustering for Large-Scale Social Networks](https://mlsanity.com/paper/2110.13613/) (2021)
- [Bipartite mixed membership distribution-free model. A novel model for community detection in overlapping bipartite weighted networks](https://mlsanity.com/paper/2211.00912/) (2022)
- [Finding mixed memberships in categorical data](https://mlsanity.com/paper/2312.01565/) (2023)

## Built on by (most cited first, 3 of 3)

- [How many asymmetric communities are there in multi-layer directed networks?](https://mlsanity.com/paper/2602.21569/) (2026)
- [Individual-heterogeneous sub-Gaussian Mixture Models](https://mlsanity.com/paper/2604.05337/) (2026)
- [A Subsampled Davis-Kahan Bound for Large-Scale Eigenspace Estimation](https://mlsanity.com/paper/2609.09211/) (2026)

## Abstract (arXiv)

Community detection in multi-layer networks is a crucial problem in network analysis. In this paper, we analyze the performance of two spectral clustering algorithms for community detection within the framework of the multi-layer degree-corrected stochastic block model (MLDCSBM) framework. One algorithm is based on the sum of adjacency matrices, while the other utilizes the debiased sum of squared adjacency matrices. We also provide their accelerated versions through subsampling to handle large-scale multi-layer networks. We establish consistency results for community detection of the two proposed methods under MLDCSBM as the size of the network and/or the number of layers increases. Our theorems demonstrate the advantages of utilizing multiple layers for community detection. Our analysis also indicates that spectral clustering with the debiased sum of squared adjacency matrices is generally superior to spectral clustering with the sum of adjacency matrices. Furthermore, we provide a strategy to estimate the number of communities in multi-layer networks by maximizing the averaged modularity. Substantial numerical simulations demonstrate the superiority of our algorithm employing the debiased sum of squared adjacency matrices over existing methods for community detection in multi-layer networks, the high computational efficiency of our accelerated algorithms for large-scale multi-layer networks, and the high accuracy of our strategy in estimating the number of communities. Finally, the analysis of several real-world multi-layer networks yields meaningful insights.