晶格图结构非均匀渗流的谱统计
Spectral Statistics of Lattice Graph Structured, Non-uniform Percolations
AI总结：
本文研究了晶格图结构非均匀渗流的谱统计特性，利用Girko发展的随机主方程方法，推导出随机图的谱分布的确定性等价分布，并通过模拟验证结果。
摘要：
为图形信号处理设计滤波器受益于对编码图的矩阵的谱分解的知识，例如用于定义位移操作符的邻接矩阵和拉普拉斯矩阵。对于具有实特征值的位移矩阵（出现在对称图中），经验谱分布捕捉了特征值的位置。在现实情况下，随机影响通常会影响网络结构，从而影响位移矩阵的经验谱分布。然而，确定性函数往往可以被发现近似随机矩阵经验谱分布的渐进行为。本文利用Girko发展的随机主方程方法，推导出由结构非均匀渗流形成的D维超图随机图的经验谱分布的确定性等价分布。包含的模拟展示了样本参数的结果。
英文摘要：
Design of filters for graph signal processing benefits from knowledge of the spectral decomposition of matrices that encode graphs, such as the adjacency matrix and the Laplacian matrix, used to define the shift operator. For shift matrices with real eigenvalues, which arise for symmetric graphs, the empirical spectral distribution captures the eigenvalue locations. Under realistic circumstances, stochastic influences often affect the network structure and, consequently, the shift matrix empirical spectral distribution. Nevertheless, deterministic functions may often be found to approximate the asymptotic behavior of empirical spectral distributions of random matrices. This paper uses stochastic canonical equation methods developed by Girko to derive such deterministic equivalent distributions for the empirical spectral distributions of random graphs formed by structured, non-uniform percolation of a D-dimensional lattice supergraph. Included simulations demonstrate the results for sample parameters.
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