SCOREH+: A High-Order Node Proximity Spectral Clustering on Ratios-of-Eigenvectors Algorithm for Community Detection

The research on complex networks has achieved significant progress in revealing the mesoscopic features of networks. Community detection is an important aspect of understanding real-world complex systems. We present in this paper a High-order node proximity Spectral Clustering on Ratios-of-Eigenvectors (SCOREH+) algorithm for locating communities in complex networks. The algorithm improves SCORE and SCORE+ and preserves high-order transitivity information of the network affinity matrix. We optimize the high-order proximity matrix from the initial affinity matrix using the Radial Basis Functions (RBFs) and Katz index. In addition to the optimization of the Laplacian matrix, we implement a procedure that joins an additional eigenvector (the $(k+1)^{th}$ leading eigenvector) to the spectrum domain for clustering if the network is considered to be a "weak signal" graph. The algorithm has been successfully applied to both real-world and synthetic data sets. The proposed algorithm is compared with state-of-art algorithms, such as ASE, Louvain, Fast-Greedy, Spectral Clustering (SC), SCORE, and SCORE+. To demonstrate the high efficacy of the proposed method, we conducted comparison experiments on eleven real-world networks and a number of synthetic networks with noise. The experimental results in most of these networks demonstrate that SCOREH+ outperforms the baseline methods. Moreover, by tuning the RBFs and their shaping parameters, we may generate state-of-the-art community structures on all real-world networks and even on noisy synthetic networks.