Figure 1

Detection of modules using multiscale ensemble clustering. (a) For a given adjacency matrix $A$, the diffusion character matrix $D$, the dissimilarity matrix $L$, and the co-membership frequency matrix $F$ are obtained sequentially: $A\to D\to L\to F$. For details, refer to the text of Secs. 2 and 3a. $A$ is the adjacency matrix of a hierarchically modular (HM) network with $N=600$ nodes and a three-level hierarchy: two modules (top level), six modules (middle level), and 19 modules (bottom level). (b) Dendrogram showing the modular structure of the network. The color code denotes the index of the module to which one node belongs. (c) Identified significant modules corresponding to the modules in each level of the HM network. The four colors represent, respectively, the one-, two-, six-, and 19-module partitions indicated by the arrows of the same colors in (d). (d) Top panel: The number of modules as a function of $\eta$. Only those modules which remain undivided at $\eta$ are counted. Bottom panel: Modularity $Q\left(\eta \right)$ defined by Eq. (5). $\alpha =0.99$, $T=5000$, $S_{\min }=10$ are used for the analysis, where $\alpha$, $T$, and $S_{\min }$ are the decay parameter of the diffusion, the number of times node grouping is performed in the procedure $L\to F$, and the minimum module size, respectively. See the text of Sec. 2 for a definition of the parameters.Reuse & Permissions

Figure 2

Analysis of a network with four cliques (complete graphs). (a) Adjacency matrix $A$ [30]. (b) Top panel: Typical partition with $Q=0.5426$ identified by the Louvain method [19] based on modularity optimization. This partition is obtained 1000 times out of 1000 trials. Bottom panel: Typical partition identified by $K$-means clustering [21]. Our dissimilarity matrix $L$ and $K=4$ are used. Partitions with the two five-node cliques ($K_5$'s) merged and one of the two 20-node cliques ($K_{20}$'s) divided into two modules of comparable sizes are obtained 658 times out of 1000; partitions with the two $K_5$'s merged and one of the two $K_{20}$'s divided into two modules of vastly different sizes (e.g., 19 and 1) are obtained 318 times. (c) Dendrogram by our method. The color code is as in Fig. 1b. $Q$ values for the partitions indicated by arrows are shown along the arrows. $\alpha =0.99$, $T=3000$, $S_{\min }=2$. (d) The hierarchy of the network identified by our method. The same colors of (c) and (d) represent the same modules.Reuse & Permissions

Figure 3

Analysis of a network with two modules of significantly different sizes. (a) Adjacency matrix $A$. $N=500$. For details, refer to the text of Sec. 3c. (b) Top panel: Typical partition found by the Louvain method [19]. $Q=0.1189$, $n\left({\mathcal{M}}_1\right)=175$, $n\left({\mathcal{M}}_2\right)=108$, $n\left({\mathcal{M}}_3\right)=100$, $n\left({\mathcal{M}}_4\right)=117$. This partition has the maximum $Q$ among the obtained partitions. Similar four-module partitions with the small module identified and the big module divided into three modules with comparable sizes are obtained 819 times out of 1000 trials. Bottom panel: A typical partition found by $K$-means clustering. Our dissimilarity matrix $L$ and $K=2$ are used. $n\left({\mathcal{M}}_1\right)=234$, $n\left({\mathcal{M}}_2\right)=266$. The same partition is obtained 679 times out of 1000 trials and similar partitions with modules with nearly equal sizes are obtained 145 times. (c) Dendrogram by our method. $\alpha =0.99$, $T=3000$, $S_{\min }=10$. (d) Top panel: The number of modules as a function of $\eta$. Bottom panel: Modularity $Q\left(\eta \right)$. $Q_{\max }=0.106$ for the two-module partition.Reuse & Permissions

Figure 4

Performance tests. (a) Girvan-Newman benchmark [13]. Normalized mutual information $I_{\mathrm{norm}}$ between the given partition and the found partition as a function of mixing parameter $\mu$ defined in the text. (b) LFR benchmark [11, 37]. $N$ is the number of nodes in the benchmark network; S and B stand for two different ranges of module sizes. For details, refer to the text. $\alpha =0.99$, $S_{\min }=4$, and $T=5000$ for (a) and $T=3000$ for (b).Reuse & Permissions

Figure 5

Major League Baseball network for the 2010 regular season. 30 teams, 162 games for each team. (a) Adjacency matrix $A$. $A_{ij}$ is the number of games between the two teams $i$ and $j$. (b) Adjacency matrix reordered according to the partition by our method. (c) Dendrogram showing the hierarchy of the MLB. (d) Top panel: The number of modules as a function of $\eta$. Bottom panel: Modularity $Q\left(\eta \right)$. $\alpha =0.99$, $T=5000$, $S_{\min }=2$.Reuse & Permissions

Figure 6

American college football network for the 2000 NCAA Division I-A football season [13, 40, 41]. We use the corrected data [40, 41]. 115 teams, 613 games, 11 conferences, and eight IA independents. (a) Adjacency matrix $A$. $A_{ij}$ is the number of games between the two teams $i$ and $j$. $A_{ij}=0$ or 1. (b) Adjacency matrix reordered according to the partition by our method. (c) Dendrogram by our method. The numbers in the dendrogram are the module indices for those modules found in the ranges indicated by the arrows in (d). $\alpha =0.99$, $T=3000$, $S_{\min }=4$. (d) Top panel: The number of modules as a function of $\eta$. Bottom panel: Modularity $Q\left(\eta \right)$. (e) Correspondence between modules found in (b) and the conferences. The superscripts N, S, W, and E preceding some symbols represent the divisions of the conferences denoted by the symbols.Reuse & Permissions