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

TL;DR

This paper proves an exact equivalence between modularity maximization with a resolution parameter and maximum likelihood estimation in the planted partition model, clarifying their relationship and providing a principled derivation of modularity.

Contribution

It establishes a rigorous mathematical link between two popular community detection methods, enhancing understanding and guiding their application.

Findings

Proves the equivalence between modularity maximization and maximum likelihood in the planted partition model.

Provides an explicit formula for the optimal resolution parameter.

Clarifies the assumptions and conditions under which modularity is valid.

Abstract

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.