Figure 1. The explanation for an overlapping community structure.

(A) A schematic network with overlapping communities. Communities are represented by different colors. Overlapped nodes shared by more than one community are emphasized in red. Connections between communities are shown by gray lines. (B) The definition of a clique. A clique (k-clique) is a complete subgraph of size k. (C) The flowchart for the process of determining an overlapping community structure. In the cover (ii) of the original example network (i), each superordinate maximal clique (k = 3, 4) is a cluster, and each subordinate maximal clique (k<3) forms a cluster consisting of only one vertex (here, k is set to 3 in the calculation). The maximal clique with k = 4 (red arrow) is not a subset of any other clique. The maximal clique network (iv) is obtained according to the belonging coefficient α (iii) of each vertex (1–11) to its corresponding clusters (a–g). The optimal partition of the maximal clique network (v) is computed by an efficient modularity optimization algorithm, and it can be mapped to the optimal cover of the original network (vi). In the cover (vi), it holds the information about the overlapping community structure of the original network; overlapping communities are represented by different colors, and an overlapped node is emphasized in red. Some parts of this figure are reproduced, with permission, from Shen et al., 2009b ©2009, IOP Publishing and SISSA.