# Post-processing partitions to identify domains of modularity optimization

## TL;DR

The paper presents CHAMP, an algorithm that refines network community structures by identifying the most robust partitions across parameter spaces, improving the selection of meaningful communities.

## Contribution

CHAMP introduces a novel convex hull-based method to prune and prioritize community partitions across multi-dimensional parameters, enhancing robustness analysis.

## Key findings

- CHAMP reduces the set of candidate partitions by up to 1785 times.
- The algorithm effectively identifies admissible partitions with maximal modularity.
- Demonstrated utility on various network examples.

## Abstract

We introduce the Convex Hull of Admissible Modularity Partitions (CHAMP) algorithm to prune and prioritize different network community structures identified across multiple runs of possibly various computational heuristics. Given a set of partitions, CHAMP identifies the domain of modularity optimization for each partition ---i.e., the parameter-space domain where it has the largest modularity relative to the input set---discarding partitions with empty domains to obtain the subset of partitions that are "admissible" candidate community structures that remain potentially optimal over indicated parameter domains. Importantly, CHAMP can be used for multi-dimensional parameter spaces, such as those for multilayer networks where one includes a resolution parameter and interlayer coupling. Using the results from CHAMP, a user can more appropriately select robust community structures by observing the sizes of domains of optimization and the pairwise comparisons between partitions in the admissible subset. We demonstrate the utility of CHAMP with several example networks. In these examples, CHAMP focuses attention onto pruned subsets of admissible partitions that are 20-to-1785 times smaller than the sets of unique partitions obtained by community detection heuristics that were input into CHAMP.