# Post-Processing Partitions to Identify Domains of Modularity Optimization
## 3 Results
### NCAA Division I-A College Football Network  
Figure 1A visualizes a computational scan of the \(\gamma\) resolution domain for the Division I-A college football network of 115 nodes representing teams and 613 (unweighted) edges representing that at least one game was played between two teams. Additionally, each team has a label identifying its athletic conference, a subgroup of teams that generally share a geographic region and compete for a conference championship. One would expect that a good partition of the network reflects the conference structure. For input to CHAMP, we ran the Louvain heuristic [42; 47] 50,000 times on the network. The modularity and number of communities found for each run is plotted at the \(\gamma\) resolution parameter used, which were uniformly spaced on \(\gamma\in[0,6]\). We observe in particular the wide range of \(\gamma\) over which one finds 12-community partitions, but note that the range also includes results with other numbers of communities, with ambiguity about which partition is the better choice. By considering each partition as a line over the full domain of \(\gamma\) as shown in Figure 1B, we find the set of lines that form the convex hull of all the modularity functions and the intervals in which each partition is optimal, indicated by the red step function in Figure 1A, with the steps at the transition values of \(\gamma\) indicated by blue triangles in Figure 1B. These 50,000 runs of the heuristic generated 384 unique partitions, with the average run time for each cycle of the Louvain heuristic was 0.02 s. After application of CHAMP, there were only 19 partitions in the pruned admissible subset associated with the original parameter search space (\(\gamma\in[0,6]\)). Moreover, CHAMP identifies a wide \(\gamma\)-domain of optimality of the 12-community partition, running from \(\gamma\doteq 1.45\) to just below 4. Throughout the paper, we use an information theoretic metric to assess how much the partitions are varying across the dominant domains. Mutual information (see also normalized mutual information [33]) quantifies the decrease in entropy for one random variable that comes from knowing the value of a second random variable. Here the two random variables are discrete, community labelson the nodes. If two partitions are highly similar, knowledge of the community label of node \(i\) in the first partition drastically reduces the uncertainty of the label of node \(i\) in the second partition. In this paper, we use a more stringent, normalized version of the metric introduced by [PERSON] et al. (2017) called Adjusted Mutual Information (AMI),  
\[AMI(X,Y)=\frac{MI(X,Y)-E(MI(X,Y))}{\max\left(H(X),H(Y)\right)-E(MI(X,Y))}\,, \tag{8}\]  
where \(MI(X,Y)\) is the mutual information between random variables \(X\) and \(Y\) and \(H(X)\) is the entropy of the random variable \(X\). The expected value, \(E(MI(X,Y))\), is calculated over random partitions sampled from a hypergeometric null distribution (see ([PERSON] and [PERSON], 2017) for details). The AMI between two partitions equals 1 to indicate perfect concordance, with the value 0 representing alignment no better than random. There are various other clustering and label prediction metrics that could also be used, including pairwise counting scores such as Adjusted Rand Index (ARI) ([PERSON] and [PERSON], 2017) (see also the discussion in ([PERSON] and [PERSON], 2017)), Variation of Information (VI) ([PERSON] and [PERSON], 2017; [PERSON] and [PERSON], 2017), and F\({}_{1}\)-score ([PERSON] and [PERSON], 2017). This 12-community partition, visualized in Figure 2B, aligns very closely with the conference labels of the teams as measured by Adjusted Mutual Information (AMI \(\doteq\) 0.92). Further increasing \(\gamma\), we see this 12-community partition domain is followed immediately by a smaller (but still sizeable) domain of optimality for a 13-community partition.