Penalized versions of the Newman-Girvan modularity and their relation to normalized cuts and $k$-means clustering

Two penalized—balanced and normalized—versions of the Newman-Girvan modularity are introduced and estimated by the non-negative eigenvalues of the modularity and normalized modularity matrix, respectively. In this way, the partition of the vertices that maximizes the modularity can be obtained by applying the $k$-means algorithm for the representatives of the vertices based on the eigenvectors belonging to the largest positive eigenvalues of the modularity or normalized modularity matrix. The proper dimension depends on the number of the structural eigenvalues of positive sign, while dominating negative eigenvalues indicate an anticommunity structure; the balance between the negative and the positive eigenvalues determines whether the underlying graph has a community, anticommunity, or randomlike structure.

Figure 1

The network of social connections in the karate club network of Zachary[29]. Circles and squares represent nodes of the two clusters with sizes proportional to their degrees. The shaded nodes are the administrator (1) and the instructor (34, covering 33). The separation of the two clusters found by our spectral algorithm maximizing the balanced modularity coincides with the real-life separation of the club members found in the original paper and denoted by the dashed line.

Figure 2

The network of social connections between 40 bottlenose dolphins retained for association analysis by Lusseau et al.[30]. Squares and circles represent individuals of the two main clusters obtained by our spectral clustering algorithm maximizing the balanced modularity with $k=2$. In the $k=3$ case, the squares remained in the same cluster, while circles separated into the shaded and open ones. The dense parts of these clusters coincide with the three communities described in[30]. The two main communities observed in the original paper are separated by dashed lines, while the intermediate low degree nodes are not classified uniquely by the original paper.