Fig. 2

The formation process of a protein complex. The four type of nodes are marked by different colors. a the deep red protein represents the seed protein; b these red proteins inside the red dotted circle constitute a complex core; c these green proteins inside the green dotted circle represent peripheral proteins; d the yellow proteins inside the yellow dotted circle represent overlapping proteins; e the chocolate yellow proteins represent interspersed node; f complex core, peripheral proteins, and overlapping proteins inside the blue circle constitute a protein complex; An example illustrates the clustering process. This simple network has 22 nodes, and each edge has weight 0.2 except (0,1),(0,2),..., and (3,4). The node 0 is taken as a seed protein and the initial cluster {0} is constructed. In the greedy search process, the neighbors of the node 0 include {1,2,3,4,5,8,9}. The node 1 has the highest support function \(\frac {0.98}{0.98+0.87+0.87+0.2*3}=0.295\) according to support function (Eq. (7)). We add node 1 to the cluster, and if the value of local modularity score increases, then this cluster is {0,1}. Similarly, the nodes 2, 3, and 4 are added to the cluster in sequence and now the neighbors of node 0 include 5, 8, 9 are left, the node 5 has the highest support function, but when the node 5 is added to the cluster {0,1,2,3,4}, its local modularity score decrease. Thus the node 5 is removed from the cluster and this greedy is terminated. Now the cluster {0,1,2,3,4} constitutes the complex core. We do the next greedy search to extend the complex core to form the whole complex. Furthermore, for the complex core {0,1,2,3,4}, its neighboring nodes have the nodes 5, 6, 7, 8, and 9, we repeat iteration this process for the cluster until the cluster isn’t change and save it as the first cluster. Similar, the next search will start from the next seed node to expand the next cluster