Figure 1

(Color online) By partitioning the links of a network into communities, one may uncover overlapping communities for the nodes by noting that a node belongs to the communities of its links. In this toy example, a meaningful partition consists in dividing the links into two groups (straight blue lines and the dashed red lines). In that case, the central node belongs to the two communities because it is at the interface between these link communities.Reuse & Permissions

Figure 2

(Color online) Illustration of the two types of random walk considered in this paper. In both cases, the walkers are situated on the links of a graph, here starting from the central red dashed link. In (a) the link-link random walk is shown where the walker jumps (the green dashed arrows) to any of the adjacent links with equal probability. In (b) a link-node-link random walk is illustrated. In this case the walker moves first to a neighboring node with equal probability and then moves on to a new link chosen with equal probability from those new links incident at the node.Reuse & Permissions

Figure 3

(Color online) The information of the bow tie graph in (a), as encoded by the adjacency matrix $\mathsf{A}$ of Eq. (7), has other equivalent graph representations. In (b) the incidence matrix [$\mathsf{B}$ of Eq. (7)] of the bow tie is shown as a bipartite network, the incidence graph $I\left(G\right)$. The line graph of the bow tie, $L\left(G\right)$, is the unweighted version of the graph labeled (c),(d) with adjacency matrix $\mathsf{C}$ of Eq. (8). The weighted version in diagram (c),(d) has an adjacency matrix $\mathsf{D}$ of Eq. (11). The weighted line graph with self-loops labeled (e) has an adjacency matrix $\mathsf{E}$ of Eq. (14). Circles represent entities that correspond to nodes of the original graph, while triangles come from links in the original graph.Reuse & Permissions

Figure 4

(Color online) Optimal node partitions for the unweighted Karate club data of Zachary, notation as in [2]. On the left is the partition into two communities made by Zachary [2] using the Ford-Fulkerson binary community algorithm [25]. It is also produced by optimizing $R\left(\mathsf{A},11\right)$ of Eq. (5). The right-hand figure shows the node partition with optimal $Q\left(\mathsf{A}\right)=0.420$ [26], which contains four communities.Reuse & Permissions

Figure 5

(Color online) The optimal link partitions of (c) $Q\left(\mathsf{C}\right)$, (d) $Q\left(\mathsf{D}\right)$, and (e) $Q\left(E_1\right)$ for the Karate club. They contain four, seven, and three communities, respectively. The two smallest communities in the center of (d) consist of the links: (a) {(3,10), (10,34)}, (b) {(0,20), (1,20), (2,20)}.Reuse & Permissions

Figure 6

(Color online) Optimal link partition into two communities of the stability $R\left(\mathsf{D},10\right)$ of the Karate club.Reuse & Permissions

Figure 7

(Color online) The simple graph created from the South Florida Free Association Norms data [28], in the manner of [14]. The link partition shown is produced by optimizing a modified version of the modularity $Q\left(\mathsf{D}\right)$ where the null model factor was $10.0\times \left(k_{\alpha }{k}_{\beta }\right)/W^2$. This controls the number of communities found [9]. The subgraph shown contains the word “bright” along with nodes that have at least 90% of their links in one of the communities connected to “bright.”Reuse & Permissions