Localization of dominant eigenpairs and planted communities by means of Frobenius inner products

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• 作者：Dario Fasino ; Francesco Tudisco
• 关键词：dominant eigenpair ; cone of matrices ; spectral method ; community detection
• 刊名：Czechoslovak Mathematical Journal
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：66
• 期：3
• 页码：881-893
• 全文大小：165 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Analysis
  Convex and Discrete Geometry
  Ordinary Differential Equations
  Mathematical Modeling and IndustrialMathematics
  
• 出版者：Springer Netherlands
• ISSN：1572-9141
• 卷排序：66

文摘

We propose a new localization result for the leading eigenvalue and eigenvector of a symmetric matrix A. The result exploits the Frobenius inner product between A and a given rank-one landmark matrix X. Different choices for X may be used, depending on the problem under investigation. In particular, we show that the choice where X is the all-ones matrix allows to estimate the signature of the leading eigenvector of A, generalizing previous results on Perron-Frobenius properties of matrices with some negative entries. As another application we consider the problem of community detection in graphs and networks. The problem is solved by means of modularity-based spectral techniques, following the ideas pioneered by Miroslav Fiedler in mid-’70s.