# Post-Processing Partitions to Identify Domains of Modularity Optimization
## 4 Discussion  
There are a number of features of CHAMP that make it a useful tool for community detection, as we have demonstrated by way of a variety of examples. By eliminating partitions that are non-admissible to the convex hull, CHAMP can greatly reduce the number of partitions remaining for consideration. By assessing the sizes of the domains of optimality of the partitions in the pruned admissible subset, and through direct pairwise comparisons of partitions in the admissible subset, CHAMP provides a framework for identifying stable parameter domains that signal robust community structures in the network. The set of input partitions can be obtained as a result of a community-detection method across a range of parameter choices (as we explored here) or from the comparison of different community-detection methods. Ideally the input set contains near-optimal partitions with relevance for the application at hand. Because each partition is allowed to compete across the whole space of resolution and coupling parameters, CHAMP can surmount some of the pathologies associated with modularity-based community detection heuristics. For example, CHAMP has uncovered several cases where there is a parameter range over which Louvain consistently identifies suboptimal partitions compared to partitions that Louvain itself identifies at other parameter values. In our study of the Human Protein Reactome network (see Section 3.2), we have seen that the stochasticity over multiple runs of the heuristic makes finding a plateau in the number of communities challenging: nevertheless, CHAMP is able to identify regions where a single partition is intrinsically stable, regardless of how frequently a particular detection algorithm uncovers such a partition. By identifying a manageable-sized and organized subset of admissible partitions with CHAMP, one can then apply a  
Figure 8: The domains of optimality for the time-varying U.S. Senate roll-call similarity network (as in Figure 6), colored by the layer-averaged AMI between the political-party affiliations of Senators and the community labels \(\{c_{i\varepsilon}\}\) for that layer. pairwise measure of similarity such as AMI to adjacent partitions to identify shifts in the landscape of optimal community structure. We in no way claim that CHAMP resolves all of the problems with modularity-based methods (see, e.g., the discussion in [3]). And CHAMP is certainly not the only way to try to process different results across various resolution parameters (see again the Introduction). However, by taking advantage of the underlying properties of modularity, including the fact that each partition defines a linear function for \(Q\) in terms of the resolution and interlayer coupling parameters, CHAMP provides a principled method built directly on the definition of modularity to make better sense of the parameter space when modularity methods are employed. In particular, many of the various other proposed approaches assess each partition at the particular parameter value input into the community detection heuristic that found the partition, that is treating each partition as a single point in \((\gamma,Q)\). In contrast, CHAMP returns to the underlying definition of modularity with a resolution parameter to recognize that each partition here is more completely represented as a line in \((\gamma,Q)\) [in the multilayer case, as a plane in \((\gamma,\omega,Q)\)]. The single point is on that line but does not completely explore the potential of that partition to compete against the other identified partitions. By using the full linear subspace associated with each partition, CHAMP prunes away the vast majority of partitions in practice. Importantly, CHAMP itself is not a method for partitioning a network, and as such its ability to highlight partitions is limited by the set of partitions given as input to the algorithm.