Regularized Spectral Clustering under the Degree-Corrected Stochastic Blockmodel

TL;DR

This paper extends spectral clustering theory to degree-heterogeneous networks, explaining empirical eigenvector patterns and guiding parameter choices without minimum degree assumptions.

Contribution

It generalizes previous results to the canonical spectral clustering algorithm under the Degree-Corrected Stochastic Blockmodel, removing degree constraints and explaining eigenvector features.

Findings

Theoretical guarantees for spectral clustering without minimum degree assumptions.

Explanation of the 'star shape' eigenvector pattern in empirical networks.

Guidance on tuning parameter selection for improved clustering performance.

Abstract

Spectral clustering is a fast and popular algorithm for finding clusters in networks. Recently, Chaudhuri et al. (2012) and Amini et al.(2012) proposed inspired variations on the algorithm that artificially inflate the node degrees for improved statistical performance. The current paper extends the previous statistical estimation results to the more canonical spectral clustering algorithm in a way that removes any assumption on the minimum degree and provides guidance on the choice of the tuning parameter. Moreover, our results show how the "star shape" in the eigenvectors--a common feature of empirical networks--can be explained by the Degree-Corrected Stochastic Blockmodel and the Extended Planted Partition model, two statistical models that allow for highly heterogeneous degrees. Throughout, the paper characterizes and justifies several of the variations of the spectral clustering algorithm in terms of these models.