# Post-Processing Partitions to Identify Domains of Modularity Optimization
## 3 Results
### U.S. Senate Roll Call Voting Network
Near \(\omega=1\), the widths in \(\gamma\) of the domains appear larger than those at smaller \(\omega\), suggesting perhaps that the stability of identified communities  
Figure 6: (**A**) Domains of optimization for the pruned set of partitions, colored by the number of communities within each partition. The set of partitions was generated from \(240,000\) runs of GenLouvain [41] on a \(600\times 400\) uniform grid over \([0.3,2]\times[0,2]\) in \((\gamma,\omega)\). The largest partitions are labeled “\(X.Y\)” with \(X\) the number of communities with \(\geq 5\) nodes and \(Y\) the rank of the domain area (that is, in terms of size) for that given number of communities (e.g., “5.2” is the second-largest domain corresponding to 5-community partitions). The partitions of each labeled domain are visualized in Appendix A; (**B**) Weighted-average AMI of each partition with its neighboring domains’ partitions, weighted by the length of the borders between neighboring domains. is being enhanced by coupling between the layers. As \(\gamma\) increases only slightly past 1, the number of communities in each partition rapidly increases, with the majority of partitions past \(\gamma=1.2\) having over 100 communities. At the lower right corner we see the domains are small and highly fragmented in both the \(\gamma\) and \(\omega\) directions. We also aim to identify parameter regions corresponding to similar partitions. For single-layer networks, we directly visualized the whole set of pairwise AMI's ordered by \(\gamma\). Given two parameters here, we calculate the weighted average AMI of each partition with its neighbors, with weight proportional to the length of the border with the neighboring domain along which the two partitions have the same value of multilayer modularity. The resulting neighbor-averaged AMI of each partition is shown by color in Figure 6B. We again observe at least three distinct regions of high pairwise similarity, separated by much lower neighbor-averaged AMI, aligned with the different regions in Figure 6A discussed above: (1) the region below \(\gamma\approx 0.8\); (2) the region just below \(\gamma=1\), with particularly high neighbor-averaged AMI for \(\omega\in[0.6,0.9]\); and (3) the many-community partitions for \(\gamma>1.2\). Indeed, we see a shift in the types of partitions with increasing \(\gamma\) across this \(\gamma\approx 0.8\) transition boundary. The qualitative difference in community structure between these regions is demonstrated in Figure 7, highlighting in Figure 6A the two partitions labeled 5.1 (Figure 7A) and 8.1 (Figure 7C). Recall, that these are the partitions with the largest domains of optimality with 5 and with 8 communities, respectively. Most of the Congress layers in the 8-community partition include only a single community label per Congress (see Figure 7D). In contrast, the 5-community partition divides the Senators both across time and within each Congress, typically into 2 communities in each Congress. These intralayer divisions that extend across time are additionally highlighted by the individual Senator layout in Figure 7A showing distinct branches, because the Senators have been sorted here first by community label and then, within each community by time. Layouts for the other domains labeled in white in Figure 6A further demonstrate qualitatively similar patterns, as shown in Appendix A.  
Figure 7: Time-varying community structure for the U.S. Senate from 1789 to 2008 according to the (**A**,**B**) 5-community and (**C**,**D**) 8-community partitions with widest domains of optimality (see labels 5.1 and 8.1 in Figure 6A); (**A**,**C**) The vertical axis indicates individual Senators, sorted by community label and time. The AMI reported here is the average over layers (Congresses) of the AMIs in each layer between the identified communities in that layer and political party labels.