Module identification in bipartite and directed networks

Modularity is one of the most prominent properties of real-world complex networks. Here, we address the issue of module identification in two important classes of networks: bipartite networks and directed unipartite networks. Nodes in bipartite networks are divided into two nonoverlapping sets, and the links must have one end node from each set. Directed unipartite networks only have one type of node, but links have an origin and an end. We show that directed unipartite networks can be conveniently represented as bipartite networks for module identification purposes. We report on an approach especially suited for module detection in bipartite networks, and we define a set of random networks that enable us to validate the approach.

Figure 1

(Color online) Model random bipartite networks with modular structure. (a) Nodes are divided into two sets, actors (circles) and teams (rectangles). Each color represents a different module in the actors’ set, and teams of a given color are more likely to contain actors of their color (see text). (b) Two sample networks with $N_M=4$ modules, with 16 actors (circles) each, and $N_T=64$ teams (diamonds), with $m=7$ actors each. The network on the left-hand side has a strong modular structure, $p=0.9$, while the modular structure is less well defined on the right-hand side, $p=0.5$ (see text for the definition of $p$).

Figure 2

(Color online) Algorithm performance as a function of (a) team homogeneity $p$ (simulation parameters, $N_M=4$, $S_s=32$ for all modules); (b) number of teams $N_T$ (simulation parameters, $N_M=4$, $S_s=32$ for all modules); (c) module size homogeneity $h$ (simulation parameters $N_M=6$, $132$ nodes); and (d) mean team size $\mu$ (simulation parameters, $N_M=4$, $S_s=32$ for all modules). Error bars indicate the standard error.

Figure 3

(Color online) Modular structure of the Southern women dataset [32, 47]. Circles represent women and diamonds represent social events. A woman and an event are connected if the woman attended the event. (a) Modular structure as obtained from the unweighted projection (UWP) approach. (b) Modular structure as obtained from the weighted projection (WP) approach and the bipartite $\left(B\right)$ approach. The UWP approach fails to capture the real modular structure of the network.

Figure 4

(Color online) Application of the bipartite approach to the identification of modules in directed networks. (a), (b) A directed model network. A link from node $i$ to node $j$ is established according to the probabilities in the matrix in (a). For example, there is a probability $p_i$ that there is a link from node 1 to node 13. In particular, we use $p_i=0.45>p_o=0.05$ to generate the directed network in (b). (c) Bipartite representation of the network in (b). Each node $i$ in (b) is represented by two nodes here, a circle $A_i$ and a square $B_i$. All links in the bipartite network run between circles and diamonds, and a link between $A_i$ and $B_j$ corresponds to a link from $i$ to $j$ in the directed network. (d) Modules identified in the bipartite network. (e) Modules identified from the directed network disregarding link direction. Here, we use the same color for $A_i$ and $B_i$, since this approach does not make distinctions between incoming and outgoing links.