Fast community detection by SCORE

TL;DR

The paper introduces SCORE, a spectral clustering method that uses entry-wise ratios of eigenvectors to effectively detect communities in networks with degree heterogeneity, outperforming classical spectral and modularity methods.

Contribution

SCORE is a novel spectral clustering approach that removes degree heterogeneity effects via eigenvector ratios, improving community detection accuracy and computational efficiency.

Findings

SCORE achieves lower error rates than classical spectral methods.

SCORE is easier to implement and faster than modularity methods.

Theoretical analysis confirms SCORE's consistency under mild conditions.

Abstract

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 $A$ be the adjacency matrix of the network. We first obtain the $K$ leading eigenvectors of $A$, say, $\hat{\eta}_1,\ldots,\hat{\eta}_K$, and let $\hat{R}$ be the $n\times (K-1)$ matrix such that $\hat{R}(i,k)=\hat{\eta}_{k+1}(i)/\hat{\eta}_1(i)$, $1\leq i\leq n$, $1\leq k\leq K-1$. We then use $\hat{R}$ 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 $\hat{\eta}_{k+1}$ and $\hat{\eta}_1$, $1\leq k\leq K-1$. 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 more satisfactory than those by the classical spectral methods. Additionally, compared to modularity methods, SCORE is easier to implement, computationally faster, and also has smaller error rates. We develop a theoretic framework where we show that under mild conditions, the SCORE stably yields consistent 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.