# Modularity-like objective function in annotated networks

## TL;DR

This paper introduces a modularity-like objective function for annotated networks that balances modularity and metadata influence, enabling adjustable community detection and expanding modularity methods.

## Contribution

It proposes a novel objective function that integrates metadata influence into modularity optimization, allowing adjustable community detection in annotated networks.

## Key findings

- The objective function is a linear combination of modularity and conditional entropy.
- Adjustable influence of metadata enables recovery of metadata-driven communities.
- Transition exists between metadata-dominant and modularity-dominant partitions.

## Abstract

We ascertain the modularity-like objective function whose optimization is equivalent to the maximum likelihood in annotated networks. We demonstrate that the modularity-like objective function is a linear combination of modularity and conditional entropy. In contrast with statistical inference methods, in our method, the influence of the metadata is adjustable; when its influence is strong enough, the metadata can be recovered. Conversely, when it is weak, the detection may correspond to another partition. Between the two, there is a transition. This paper provides a concept for expanding the scope of modularity methods.