Consider a network where the nodes split into K different communities. The community labels for the nodes are unknown and it is of major interest to estimate them (i.e., community detection). Degree Corrected Block Model (DCBM) is a popular network model. How to detect communities with the DCBM is an interesting problem, where the main challenge lies in the degree heterogeneity.
We propose a new approach to community detection which we call the Spectral Clustering On Ratios-of-Eigenvectors (SCORE). Compared to classical spectral methods, the main innovation is to use the entry-wise ratios between the first leading eigenvector and each of the other leading eigenvectors for clustering. Let X be the adjacency matrix of the network. We first obtain the K leading eigenvectors, say, ˆη1, . . . , ηˆK, and let Rˆ be the n×(K−1) matrix such that Rˆ(i, k) = ηˆk+1(i)/ηˆ1(i), 1 ≤ i ≤ n, 1 ≤ k ≤ K −1. We then use Rˆ for clustering by applying the k-means method.
The central surprise is, the effect of degree heterogeneity is largely ancillary, and can be effectively removed by taking entry-wise ratios between ˆηk+1 and ˆη1, 1 ≤ k ≤ K − 1.
The method is successfully applied to the web blogs data and the karate club data, with error rates of 58/1222 and 1/34, respectively. These results are much more satisfactory than those by the classical spectral methods. Also, compared to modularity methods, SCORE is computationally much faster and has smaller error rates.
We develop a theoretic framework where we show that under mild conditions, the SCORE stably yields successful community detection. In the core of the analysis is the recent development on Random Matrix Theory (RMT), where the matrix-form Bernstein inequality is especially helpful.