# Post-Processing Partitions to Identify Domains of Modularity Optimization
## 3 Results
### U.S. Senate Roll Call Voting Network  
We demonstrate the use of CHAMP to explore the parameter space for a multilayer network using the roll-call-voting similarity network for the U.S. Senate from 1789 to 2008 (Congresses 1 to 110) as defined in [56] and studied with multilayer modularity in [16; 57]. This data represents the similarities of voting patterns within each two-year Congress between the 1884 distinct U.S.  
Figure 5: (**A**) Modularity \(Q(\gamma)\) v. \(\gamma\) for \(100,000\) runs (5% of results shown) of Louvain [42; 47] on the Caltech Facebook network [31]. Orange triangles indicate the number of communities that include \(\geq 5\) nodes in each run, while the red step function shows the number in the optimal partition in each domain. The dashed green curve is the piecewise-linear modularity function for the optimal partitions, with the transition values marked by blue triangles. The condensed layout of communities (created with [53]) here visualizes the optimal partition found for \(\gamma\in[0.908,1.09]\), with each pie-chart corresponding to a community, fractionally colored according to the House membership of the nodes in the community. The AMI between this partition and House labels (including the missing label) is 0.513; (**B**) Pairwise AMI between all partitions in the admissible subset identified by CHAMP, arranged by their corresponding \(\gamma\)-domains of optimality. Senators who served across the first 110 Congresses. As in [16; 57], each two-year Congress starting in the early January following the biennial Congressional elections is represented as a layer, with interlayer connections only between the multiple appearances of each Senator when they appear in nearest-neighbor layers; as such, multilayer modularity directly handles additions and removals of Senators over time. Self-loops within each layer are zeroed out, since these only represent perfect agreement of a Senator with herself during the same two-year period. This representation of the voting data is useful for describing legislative voting activity because the community structures typically group together Senators who vote similarly, providing relatively accessible and intuitive examples of communities that are related to the underlying political alignments as expressed by the Senators through voting, independent of their nominally declared party affiliations. The temporal extents of the communities found by multilayer modularity can then indicate different periods of stability in these political alignments. We ran the GenLouvain [41] heuristic 240,000 times, on a 600-by-400 uniform grid over \([0.3,2]\times[0,2]\) in \((\gamma,\omega)\), generating 197,879 unique partitions of the network. Each run of GenLouvain required approximately 5 s for a total of 340 hours of CPU time. CHAMP pruned this set to 1447 partitions admissible in the convex hull of modularity. We note that there were 267 additional partitions with corresponding domains of optimality that were completely outside the selected parameter range \([0.3,2]\times[0,2]\). In Figure 6 we visualize the \((\gamma,\omega)\)-domains of optimality within this region of parameter space. In Figure 6A, a domain's color indicates the numbers of communities for its corresponding optimal partition, whereas in Figure 6B domain color indicates the average AMI between the corresponding partition and the neighboring optimal partitions (weighted by the lengths of borders between domains). The trivial 1-community partition dominates the left of the panels in Figure 6 at small \(\gamma\). Increasing \(\gamma\) outside of this domain, most of the (non-trivial) domains here appear to be relatively long in the \(\omega\) direction and much narrower in \(\gamma\). Interestingly, we observe a range of \(\gamma\) from roughly 0.8 to just above 1 where the domains visually widen in the \(\gamma\) direction while also corresponding to a smaller number of communities than partitions below \(\gamma\approx 0.8\).