Generalized Eigenvectors of Isospectral Transformations,Spectral Equivalence and Reconstruction of Original Networks

Eigenvectors of isospectral graph transformations

L.A. Bunimovich and B.Z. Webb developed a theory for isospectral graph reduction. We make a simple observation regarding the relation between eigenvectors of the original graph and its reduction, that sheds new light on this theory. As an application we propose an updating algorithm for the maximal eigenvector of the Markov matrix associated to a large sparse dynamical network.

非负矩阵的等谱约化

Isospectral Reductions of Non-negative Matrices

本文提出基于等谱约化逼近随机矩阵平稳测度的算法方案，并通过数值实验与谱间隙分析说明其在多个近1特征值情形下优于迭代方法。

Isospectral Graph Transformations, Spectral Equivalence, and Global Stability of Dynamical Networks

In this paper we present a general procedure that allows for the reduction or expansion of any network (considered as a weighted graph). This procedure maintains the spectrum of the network's adjacency matrix up to a set of eigenvalues known beforehand from its graph structure. This procedure can be used to establish new equivalence relations on the class of all weighted graphs (networks) where two graphs are equivalent if they can be reduced to the same graph. Additionally, dynamical networks (or any finite dimensional, discrete time dynamical system) can be analyzed using isospectral transfo…

Isospectral Compression and Other Useful Isospectral Transformations of Dynamical Networks

It is common knowledge that a key dynamical characteristic of a network is its spectrum (the collection of all eigenvalues of the network's weighted adjacency matrix). In \cite{BW10} we demonstrated that it is possible to reduce a network, considered as a graph, to a smaller network with fewer vertices and edges while preserving the spectrum (or spectral information) of the original network. This procedure allows for the introduction of new equivalence relations between networks, where two networks are spectrally equivalent if they can be reduced to the same network. Additionally, using this t…

Isospectral Reductions of Dynamical Networks

We present a general and flexible procedure which allows for the reduction (or expansion) of any dynamical network while preserving the spectrum of the network's adjacency matrix. Computationally, this process is simple and easily implemented for the analysis of any network. Moreover, it is possible to isospectrally reduce a network with respect to any network characteristic including centrality, betweenness, etc. This procedure also establishes new equivalence relations which partition all dynamical networks into spectrally equivalent classes. Here, we present general facts regarding isospect…

Birkhoff台球桌的标记长度等谱集的紧性

Compactness of Marked Length Isospectral Sets of Birkhoff Billiard Tables

证明在C^∞拓扑下，标记长度等谱的Birkhoff台球桌的等价类构成紧集，通过导出Marvizi-Melrose积分不变量或Mather β函数泰勒系数的层次结构，并作为副产品给出严格凸平面台球桌Laplace等谱集紧性的独立证明。