# Pairwise Covariates-adjusted Block Model for Community Detection

arXiv: https://arxiv.org/abs/1807.03469 · HTML: https://mlsanity.com/paper/1807.03469/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jul 2018.

## Built on by (most cited first, 8 of 8)

- [Improved Algorithm and Bounds for Successive Projection](https://mlsanity.com/paper/2403.11013/) (2024)
- [Restricted Tweedie Stochastic Block Models](https://mlsanity.com/paper/2310.10952/) (2023)
- [Learning Joint and Individual Structure in Network Data with Covariates](https://mlsanity.com/paper/2406.08776/) (2024)
- [Poisson-Process Topic Model for Integrating Knowledge from Pre-trained Language Models](https://mlsanity.com/paper/2503.17809/) (2025)
- [Strongly Consistent Community Detection in Popularity Adjusted Block Models](https://mlsanity.com/paper/2506.07224/) (2025)
- [Perfect Spectral Clustering with Discrete Covariates](https://mlsanity.com/paper/2205.08047/) (2022)
- [Dynamic Network Sampling for Community Detection](https://mlsanity.com/paper/2208.13921/) (2022)
- [Covariate-assisted graph matching](https://mlsanity.com/paper/2512.11761/) (2025)

## Abstract (arXiv)

One of the most fundamental problems in network study is community detection. The stochastic block model (SBM) is a widely used model, for which various estimation methods have been developed with their community detection consistency results unveiled. However, the SBM is restricted by the strong assumption that all nodes in the same community are stochastically equivalent, which may not be suitable for practical applications. We introduce a pairwise covariates-adjusted stochastic block model (PCABM), a generalization of SBM that incorporates pairwise covariate information. We study the maximum likelihood estimates of the coefficients for the covariates as well as the community assignments. It is shown that both the coefficient estimates of the covariates and the community assignments are consistent under suitable sparsity conditions. Spectral clustering with adjustment (SCWA) is introduced to efficiently solve PCABM. Under certain conditions, we derive the error bound of community detection under SCWA and show that it is community detection consistent. In addition, we investigate model selection in terms of the number of communities and feature selection for the pairwise covariates, and propose two corresponding algorithms. PCABM compares favorably with the SBM or degree-corrected stochastic block model (DCBM) under a wide range of simulated and real networks when covariate information is accessible.