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Spectral clustering is a fast and popular algorithm for finding clusters innetworks. Recently, Chaudhuri et al. (2012) and Amini et al.(2012) proposedinspired variations on the algorithm that artificially inflate the node degreesfor improved statistical performance. The current paper extends the previousstatistical estimation results to the more canonical spectral clusteringalgorithm in a way that removes any assumption on the minimum degree andprovides guidance on the choice of the tuning parameter. Moreover, our resultsshow how the "star shape" in the eigenvectors--a common feature of empiricalnetworks--can be explained by the Degree-Corrected Stochastic Blockmodel andthe Extended Planted Partition model, two statistical models that allow forhighly heterogeneous degrees. Throughout, the paper characterizes and justifiesseveral of the variations of the spectral clustering algorithm in terms ofthese models.

Regularized Spectral Clustering under the Degree-Corrected Stochastic Blockmodel