# Post-Processing Partitions to Identify Domains of Modularity Optimization
## 3 Results
### Human Protein Reactome Network  
We employed CHAMP to map the domains of modularity optimization for a larger example: the undirected (single-layer) network representation of the Human Protein Reactome [54; 55], with 6327 nodes representing human proteins and 147,547 edges signifying common biological reactions. We ran the Louvain heuristic 20,000 times on this network with \(\gamma\in[0,4]\) uniformly spaced, generating 19,980 unique partitions. For this example, each run of Louvain required an average of 2.6 s, generating the input set of partitions in approximately 140 CPU hours. CHAMP pruned this input set of partitions down to 39 admissible partitions in the convex hull over the original parameter search space (\(\gamma\in[0,4]\)). Similar to the figures of the previous example, Figure 3A shows the spread in the modularities and the numbers of communities identified across all instances of the heuristic, along with the domains of optimization and the number of communities for the admissible subset (see the red step function). Contrasting Figures 1A and 3A, we observe in the latter that the red step function decreases with increasing \(\gamma\) at some points. Importantly, these decreases are not because of our choice to plot the number of communities that contain at least 5 nodes. The numbers of communities is provably monotonically non-decreasing with increasing resolution parameter in the special case where the null model \(P_{ij}=\gamma\) is a constant independent of \(i\) and \(j\)[29], but we are unaware of any similarly rigorous condition for the Newman-Girvan null model used in Equation (1). Nevertheless, one typically observes the number of communities to be non-decreasing with increasing \(\gamma\), so the results here may indicate values of the resolution parameter near which additional runs of the heuristic might be more likely to identify higher quality partitions. Figure 2: (**A**) ForceAtlas2 [52] layout, created with [53], of the unweighted NCAA Division I-A (2000) college football network. Nodes are colored according to the dominant 12-community partition with the widest \(\gamma\)-domain \(\gamma\in[1.45,3.89]\), with node shapes and border indicating their conference labels; (**B**) Pairwise adjusted mutual information (N = AMI) between all partitions in the admissible subset identified by CHAMP, arranged by their corresponding \(\gamma\)-domains of optimality. Dashed lines indicate the transition values of \(\gamma\) identified by CHAMP. The number of communities in the initial set of partitions is highly variable, even for small adjustments in \(\gamma\), as shown by the yellow triangles in Figure 3A. It would be difficult to extract any range of stability from such a plot. However, when we consider the admissible subset of partitions, we see a few wide domains of optimality in the figure, the two most prominent being \(\gamma\in[0.77,1.17]\) and \(\gamma\in[2.62,3.11]\). Layouts of the network colored according to the partitions of these two broadest domains are shown in Figure 4. The pairwise AMI of the admissible partitions are shown in Figure 3B. Unlike the college football network, where pairwise AMI appears to indicate two well separated groups of highly similar partitions, the communities here appear to be diffusely similar throughout. Partitions of adjacent domains are fairly similar but there is no clear divide into groups of partitions. Figure 4: ForceAtlas2 layout [52], created with [53], of the Human Reactome Network (edges downsampled to 50,000), colored according to the partitions with the two widest \(\gamma\)-domains of optimization identified by CHAMP from \(20,000\) runs of Louvain. Figure 3: (**A**) Modularity \(Q(\gamma)\) given by Equation (1) v. resolution parameter \(\gamma\) for \(20,000\) runs (25% of results shown) of Louvain [42; 47] on the Human Protein Reactome network [54]. Small, grey triangles indicate the number of communities that include \(\geq 5\) nodes in each run, while the dark green step function shows the number in the optimal partition in each domain.