Investigating the performance of different methods is a fundamental problem in graph partitioning. In this paper, we estimate the so-called detectability threshold for the spectral method with both unnormalized and normalized Laplacians in sparse graphs. The detectability threshold is the critical point at which the result of the spectral method is completely uncorrelated to the planted partition. We also analyze whether the localization of eigenvectors affects the partitioning performance in the detectable region. We use the replica method, which is often used in the field of spin-glass theory, and focus on the case of bisection. We show that the gap between the estimated threshold for the spectral method and the threshold obtained from Bayesian inference is considerable in sparse graphs, even without eigenvector localization. This gap closes in a dense limit.
Graph energy as a measure of community detectability in networks
Frankfurt School of Finance and Management（法兰克福金融管理学院）；University of Florida（佛罗里达大学）；University of California, Los Angeles（加州大学洛杉矶分校）；Santa Fe Institute（圣塔菲研究所）；Indiana University（印第安纳大学）
California State University, Fullerton（加州州立大学富勒顿分校）；USC Information Sciences Institute（南加州大学信息科学研究所）；Claremont Graduate University（克莱蒙特研究生大学）；HP Labs（惠普实验室）
On the Robustness of Spectral Algorithms for Semirandom Stochastic Block Models
University of Utah（犹他大学）；University of Chicago（芝加哥大学）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Toyota Technological Institute at Chicago（芝加哥丰田理工学院）
