# Directed degree corrected mixed membership model and estimating community memberships in directed networks

arXiv: https://arxiv.org/abs/2109.07826 · HTML: https://mlsanity.com/paper/2109.07826/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 47 references that are in the map.
- First posted on arXiv 16 Sep 2021.

## Builds on (most cited first, 23 of 23)

- [Spectral Methods for Data Science: A Statistical Perspective](https://mlsanity.com/paper/2012.08496/) (2020)
- [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)
- [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)
- [Detecting Overlapping Communities in Networks Using Spectral Methods](https://mlsanity.com/paper/1412.3432/) (2014)
- [Estimating Mixed Memberships with Sharp Eigenvector Deviations](https://mlsanity.com/paper/1709.00407/) (2017)
- [Network Cross-Validation for Determining the Number of Communities in Network Data](https://mlsanity.com/paper/1411.1715/) (2014)
- [A goodness-of-fit test for stochastic block models](https://mlsanity.com/paper/1412.4857/) (2014)
- [Robust and computationally feasible community detection in the presence of arbitrary outlier nodes](https://mlsanity.com/paper/1404.6000/) (2014)
- [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)
- [Hypothesis Testing for Automated Community Detection in Networks](https://mlsanity.com/paper/1311.2694/) (2013)
- [On Mixed Memberships and Symmetric Nonnegative Matrix Factorizations](https://mlsanity.com/paper/1607.00084/) (2016)
- [Semidefinite Programming Based Preconditioning for More Robust Near-Separable Nonnegative Matrix Factorization](https://mlsanity.com/paper/1310.2273/) (2013)
- [Convex Relaxation Methods for Community Detection](https://mlsanity.com/paper/1810.00315/) (2018)
- [Overlapping Clustering Models, and One (class) SVM to Bind Them All](https://mlsanity.com/paper/1806.06945/) (2018)
- [A Tensor Approach to Learning Mixed Membership Community Models](https://mlsanity.com/paper/1302.2684/) (2013)
- [Estimating the number of communities in a network](https://mlsanity.com/paper/1605.02753/) (2016)
- [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)
- [Bayesian estimation of the latent dimension and communities in stochastic blockmodels](https://mlsanity.com/paper/1904.05333/) (2019)
- [A Spectral Algorithm with Additive Clustering for the Recovery of Overlapping Communities in Networks](https://mlsanity.com/paper/1506.04158/) (2015)

## Built on by (most cited first, 2 of 2)

- [A useful criterion on studying consistent estimation in community detection](https://mlsanity.com/paper/2109.14950/) (2021)
- [Overlapping and nonoverlapping models](https://mlsanity.com/paper/2111.01392/) (2021)

## Abstract (arXiv)

This paper considers the problem of modeling and estimating community memberships of nodes in a directed network where every row (column) node is associated with a vector determining its membership in each row (column) community. To model such directed network, we propose directed degree corrected mixed membership (DiDCMM) model by considering degree heterogeneity. DiDCMM is identifiable under popular conditions for mixed membership network when considering degree heterogeneity. Based on the cone structure inherent in the normalized version of the left singular vectors and the simplex structure inherent in the right singular vectors of the population adjacency matrix, we build an efficient algorithm called DiMSC to infer the community membership vectors for both row nodes and column nodes. By taking the advantage of DiMSC's equivalence algorithm which returns same estimations as DiMSC and the recent development on row-wise singular vector deviation, we show that the proposed algorithm is asymptotically consistent under mild conditions by providing error bounds for the inferred membership vectors of each row node and each column node under DiDCMM. The theory is supplemented by a simulation study.