# Post-Processing Partitions to Identify Domains of Modularity Optimization
## 4 Discussion
Given the many available heuristics, the computational complexity of maximizing modularity [17], and the potentially large number of near-optimal partitions [18], it is possible that interesting and important community features may be missing from the provided input set. CHAMP as developed here is restricted to processing hard partitions of nodes into community labels, whereas overlapping communities and background nodes (those not belonging to any community) can be important for some applications. One may also reasonably worry about the potential value of partitions in the input set that are near-optimal over a wide domain of the parameters but yet never achieve admission to the convex hull itself and are thus discarded by the algorithm. With the introduction of CHAMP presented here, we have left open many other possible uses of this general approach that may be worth exploring. Although we apply Louvain to discover partitions, CHAMP is agnostic to the detection method used to generate the set of partitions. The partitions input into CHAMP do not even need to be generated by modularity-maximizing heuristics; for example, one may also include new partitions as generated by ensemble learning [35] or consensus clustering [34; 36]. By comparing the results between sets of partitions generated by different methods, CHAMP might be useful as an additional method for making comparisons between these methods. Of course, even with a resolution parameter, modularity may not be a good measure for what constitutes a good \"community\" in some networks, and one could investigate whether other quality functions with parameters might be explored with an analogous approach. Even  
Additionally, it would be interesting to study the consistency of optimality domains output from the application of CHAMP to different input sets of partitions in order to possibly provide insight about how quickly the convex intersection of half-spaces shrinks to the underlying true but unknown upper envelope as the set of input partitions grows. For the networks tested here, the numbers of admissible partitions remaining in the pruned subset were only a very small fraction of the numbers in the input sets. In our experience, the numbers of partitions in the final pruned admissible subset appeared to increase slowly as the size of the input set was increased, but the position of the larger domains appeared to remain relatively consistent in practice. The number of initial partitions needed to get a good mapping of the parameter space undoubtedly depends on the structure of the network and the computational heuristics used. It may also be possible to use a variant of CHAMP to iteratively steer the parameters at which additional partitions might be sought. For instance, input parameters that consistently give rise to dominant partitions with broad domains could be targeted for more runs in an iterative fashion. In summary, we have presented the CHAMP algorithm as a post-processing tool for pruning a set of network partitions down to the admissible subset in the convex hull that optimizes modularity at different parameters. We have demonstrated the utility of CHAMP on various single-layer networks and on a multilayer network, identifying partitions and their associated domains of optimality in the parameter space. Further research may focus on how the sizes of these domains and the comparisons between domains can be best used to ascertain confidence in identified community structures, to explore subgraphs of a network, and to further process the admissible subset for consensus clustering, as well as other uses of the pruned subset identified by CHAMP. **Acknowledgments:** This project was supported by the James S. McDonnell Foundation 21 st Century Science Initiative -Complex Systems Scholar Award grant #220020315, by the National Institutes of Health through Award Numbers R01 HD075712, R56 DK111930 and T32 GM067553, and by the CDC Prevention Epicenter Program.