# Community detection in networks: Modularity optimization and maximum likelihood are equivalent

arXiv: https://arxiv.org/abs/1606.02319 · HTML: https://mlsanity.com/paper/1606.02319/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jun 2016.

## Built on by (most cited first, 10 of 10)

- [Generative hypergraph clustering: from blockmodels to modularity](https://mlsanity.com/paper/2101.09611/) (2021)
- [Community Detection in the Stochastic Block Model by Mixed Integer Programming](https://mlsanity.com/paper/2101.12336/) (2021)
- [Analyzing Modularity Maximization in Approximation, Heuristic, and Graph Neural Network Algorithms for Community Detection](https://mlsanity.com/paper/2310.10898/) (2023)
- [Informative core identification in complex networks](https://mlsanity.com/paper/2101.06388/) (2021)
- [Semi-Supervised Clustering with Inaccurate Pairwise Annotations](https://mlsanity.com/paper/2104.02146/) (2021)
- [Heuristic Modularity Maximization Algorithms for Community Detection Rarely Return an Optimal Partition or Anything Similar](https://mlsanity.com/paper/2302.14698/) (2023)
- [Term-community-based topic detection with variable resolution](https://mlsanity.com/paper/2103.13550/) (2021)
- [Adversarial contamination of networks in the setting of vertex nomination: a new trimming method](https://mlsanity.com/paper/2208.09710/) (2022)
- [NISQ-ready community detection based on separation-node identification](https://mlsanity.com/paper/2212.14717/) (2022)
- [ModTGCN: Modularity-aware Graph Neural Networks for Text Classification](https://mlsanity.com/paper/2606.23694/) (2026)

## Abstract (arXiv)

We demonstrate an exact equivalence between two widely used methods of community detection in networks, the method of modularity maximization in its generalized form which incorporates a resolution parameter controlling the size of the communities discovered, and the method of maximum likelihood applied to the special case of the stochastic block model known as the planted partition model, in which all communities in a network are assumed to have statistically similar properties. Among other things, this equivalence provides a mathematically principled derivation of the modularity function, clarifies the conditions and assumptions of its use, and gives an explicit formula for the optimal value of the resolution parameter.