通过弗罗贝尼乌斯内积定位主导特征对和种植社区
Localization of dominant eigenpairs and planted communities by means of Frobenius inner products
AI总结：
本文提出利用弗罗贝尼乌斯内积定位对称矩阵主导特征值和特征向量，通过不同地标矩阵的选择改进社区检测方法，提供新的质量保证。
摘要：
我们提出了一种新的定位结果，用于对称矩阵A的主导特征值和特征向量。该结果利用A与给定的秩一地标矩阵X之间的弗罗贝尼乌斯内积。不同的X选择取决于所研究的问题。特别是，我们展示当X是全一矩阵时，可以估计A主导特征向量的符号，推广了之前关于具有部分负元素矩阵的佩罗-弗罗贝尼乌斯性质的结果。作为另一个应用，我们考虑图和网络中的社区检测问题，通过基于模块度的谱技术解决，借鉴了Miroslav Fiedler在70年代中期开创的思路。我们证明了适当选择X可以为遵循随机块模型的网络提供新的质量保证。
英文摘要：
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 upon 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. We show that a suitable choice of $X$ can be used to provide new quality guarantees of those techniques, when the network follows a stochastic block model.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
