随机子图的刚度与刚度矩阵的特征值

T. Jordán, Shin-ichi Tanigawa
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引用次数: 8

摘要

在随机子图模型中,我们考虑图G的随机子图G(t),得到如下:对于G中的每条边,我们独立决定保留概率为t的边,丢弃概率为1 - t的边,对于某些0≤t≤1。该模型的一个特例是Erdős-Rényi随机图模型,其中主图为完全图Kn。本文分析了随机子图的刚性性质,给出了t≥t0时Gt为a.a.s.刚性或全局刚性的阈值t0的新上界。通过将我们的结果专门化到完整的主图,我们获得,除其他外,Erdős-Rényi随机图在Rd中是a.a.s.全局刚性的,如果t≥Cd logn,对于某些常数Cd。我们还考虑(条形和关节)框架的随机子框架,这是我们图的几何实现。我们的随机子图的刚度阈值的界限是根据框架的刚度矩阵的最小非零特征值,这是它的归一化刚度矩阵的格拉曼。在这种联系的激励下,我们引入了图的d维代数连通性的概念,并给出了几种基本图类的这个值的上界或下界。d = 1的情况对应于众所周知的代数连通性,即图的第二小拉普拉斯特征值。我们还考虑了随机分子图中的刚性阈值,也称为键弯曲网络,用于研究分子的刚性特性。在这个模型中,我们考虑了某图g的平方图的刚性,并根据主图的代数连通性给出了随机子图的平方的刚性阈值的上界。这使我们能够推导出稀疏主图的刚性阈值的上界。Eötvös大学运筹系和MTA-ELTE Egerváry组合优化研究小组,Pázmány p2013.sétány 1/C,匈牙利布达佩斯1117。e-mail: jordan@cs.elte.hu日本东京文京区本乡东京大学数学信息系。电子邮件:tanigawa@mist.i.u-tokyo.ac.jp
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Rigidity of Random Subgraphs and Eigenvalues of Stiffness Matrices
In the random subgraph model we consider random subgraphs G(t) of a graph G obtained as follows: for each edge in G we independently decide to retain the edge with probability t and discard the edge with probability 1 − t, for some 0 ≤ t ≤ 1. A special case of this model is the Erdős-Rényi random graph model, where the host graph is the complete graph Kn. In this paper we analyze the rigidity properties of random subgraphs and give new upper bounds on the threshold t0 for which Gt is a.a.s. rigid or globally rigid when t ≥ t0. By specializing our results to complete host graphs we obtain, among others, that an Erdős-Rényi random graph is a.a.s. globally rigid in Rd if t ≥ Cd logn n for some constant Cd. We also consider random subframeworks of (bar-and-joint) frameworks, which are geometric realizations of our graphs. Our bounds for the rigidity threshold of random subgraphs are in terms of the smallest non-zero eigenvalue of the stiffness matrix of the framework, which is the Gramian of its normalized rigidity matrix. Motivated by this connection, we introduce the concept of ddimensional algebraic connectivity of graphs and provide upper or lower bounds for this value of several fundamental graph classes. The case d = 1 corresponds to the well-known algebraic connectivity, that is, the second smallest Laplacian eigenvalue of the graph. We also consider the rigidity threshold in random molecular graphs, also called bond-bending networks, which are used in the study of rigidity properties of molecules. In this model we are concerned with the rigidity of the square graph of some graph G. We give an upper bound for the rigidity threshold of the square of random subgraphs in terms of the algebraic connectivity of the host graph. This enables us to derive an upper bound for the rigidity threshold for sparse host graphs. Department of Operations Research, Eötvös University, and the MTA-ELTE Egerváry Research Group on Combinatorial Optimization, Pázmány Péter sétány 1/C, 1117 Budapest, Hungary. e-mail: jordan@cs.elte.hu Department of Mathematical Informatics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo, Japan. email: tanigawa@mist.i.u-tokyo.ac.jp
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