  Spectral clustering methods are widely used for detecting clusters in
networks for community detection, while a small change on the graph Laplacian
matrix could bring a dramatic improvement. In this paper, we propose a dual
regularized graph Laplacian matrix and then employ it to three classical
spectral clustering approaches under the degree-corrected stochastic block
model. If the number of communities is known as $K$, we consider more than $K$
leading eigenvectors and weight them by their corresponding eigenvalues in the
spectral clustering procedure to improve the performance. Three improved
spectral clustering methods are dual regularized spectral clustering (DRSC)
method, dual regularized spectral clustering on Ratios-of-eigenvectors
(DRSCORE) method, and dual regularized symmetrized Laplacian inverse matrix
(DRSLIM) method. Theoretical analysis of DRSC and DRSLIM show that under mild
conditions DRSC and DRSLIM yield stable consistent community detection,
moreover, DRSCORE returns perfect clustering under the ideal case. We compare
the performances of DRSC, DRSCORE and DRSLIM with several spectral methods by
substantial simulated networks and eight real-world networks.
