# How many asymmetric communities are there in multi-layer directed networks?

arXiv: https://arxiv.org/abs/2602.21569 · HTML: https://mlsanity.com/paper/2602.21569/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 24 of its 51 references that are in the map.
- First posted on arXiv 25 Feb 2026.

## Builds on (most cited first, 24 of 24)

- [Consistency of spectral clustering in stochastic block models](https://mlsanity.com/paper/1312.2050/) (2013)
- [Community detection in networks: A user guide](https://mlsanity.com/paper/1608.00163/) (2016)
- [Multilayer Networks](https://mlsanity.com/paper/1309.7233/) (2013)
- [Regularized Spectral Clustering under the Degree-Corrected Stochastic Blockmodel](https://mlsanity.com/paper/1309.4111/) (2013)
- [Clustering and Community Detection in Directed Networks: A Survey](https://mlsanity.com/paper/1308.0971/) (2013)
- [Layer aggregation and reducibility of multilayer interconnected networks](https://mlsanity.com/paper/1405.0425/) (2014)
- [Spectral and matrix factorization methods for consistent community detection in multi-layer networks](https://mlsanity.com/paper/1704.07353/) (2017)
- [Network cross-validation by edge sampling](https://mlsanity.com/paper/1612.04717/) (2016)
- [Impact of regularization on Spectral Clustering](https://mlsanity.com/paper/1312.1733/) (2013)
- [Coauthorship and Citation Networks for Statisticians](https://mlsanity.com/paper/1410.2840/) (2014)
- [Network Cross-Validation for Determining the Number of Communities in Network Data](https://mlsanity.com/paper/1411.1715/) (2014)
- [Mixed Membership Estimation for Social Networks](https://mlsanity.com/paper/1708.07852/) (2017)
- [Analysis of spectral clustering algorithms for community detection: the general bipartite setting](https://mlsanity.com/paper/1803.04547/) (2018)
- [Spectral Algorithms for Community Detection in Directed Networks](https://mlsanity.com/paper/2008.03820/) (2020)
- [Hypothesis Testing for Automated Community Detection in Networks](https://mlsanity.com/paper/1311.2694/) (2013)
- [Community detection in multi-relational data with restricted multi-layer stochastic blockmodel](https://mlsanity.com/paper/1506.02699/) (2015)
- [Co-citation and Co-authorship Networks of Statisticians](https://mlsanity.com/paper/2204.11194/) (2022)
- [Determining the Number of Communities in Degree-corrected Stochastic Block Models](https://mlsanity.com/paper/1809.01028/) (2018)
- [Estimating the number of communities in networks by spectral methods](https://mlsanity.com/paper/1507.00827/) (2015)
- [How Many Communities Are There?](https://mlsanity.com/paper/1412.1684/) (2014)
- [Using Maximum Entry-Wise Deviation to Test the Goodness-of-Fit for Stochastic Block Models](https://mlsanity.com/paper/1703.06558/) (2017)
- [Randomized spectral co-clustering for large-scale directed networks](https://mlsanity.com/paper/2004.12164/) (2020)
- [Community detection by spectral methods in multi-layer networks](https://mlsanity.com/paper/2403.12540/) (2024)
- [Discovering overlapping communities in multi-layer directed networks](https://mlsanity.com/paper/2407.16152/) (2024)

## Abstract (arXiv)

Estimating the asymmetric numbers of communities in multi-layer directed networks is a challenging problem due to the multi-layer structures and inherent directional asymmetry, leading to possibly different numbers of sender and receiver communities. This work addresses this issue under the multi-layer stochastic co-block model, a model for multi-layer directed networks with distinct community structures in sending and receiving sides, by proposing a novel goodness-of-fit test. The test statistic relies on the deviation of the largest singular value of an aggregated normalized residual matrix from the constant 2. The test statistic exhibits a sharp dichotomy: Under the null hypothesis of correct model specification, its upper bound converges to zero with high probability; under underfitting, the test statistic itself diverges to infinity. With this property, we develop a sequential testing procedure that searches through candidate pairs of sender and receiver community numbers in a lexicographic order. The process stops at the smallest such pair where the test statistic drops below a decaying threshold. For robustness, we also propose a ratio-based variant algorithm, which detects sharp changes in the sequence of test statistics by comparing consecutive candidates. Both methods are proven to consistently determine the true numbers of sender and receiver communities under the multi-layer stochastic co-block model.