  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.
