Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany.
Departamento de Matemática Aplicada e Estatística, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo - Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos, São Paulo, Brazil and Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla 72570, México.
Phys Rev E. 2019 Dec;100(6-1):062309. doi: 10.1103/PhysRevE.100.062309.
We perform an extensive numerical analysis of β-skeleton graphs, a particular type of proximity graphs. In a β-skeleton graph (BSG) two vertices are connected if a proximity rule, that depends of the parameter β∈(0,∞), is satisfied. Moreover, for β>1 there exist two different proximity rules, leading to lune-based and circle-based BSGs. First, by computing the average degree of large ensembles of BSGs we detect differences, which increase with the increase of β, between lune-based and circle-based BSGs. Then, within a random matrix theory (RMT) approach, we explore spectral and eigenvector properties of random BSGs by the use of the nearest-neighbor energy-level spacing distribution and the entropic eigenvector localization length, respectively. The RMT analysis allows us to conclude that a localization transition occurs at β=1.
我们对β-骨架图（一种特殊类型的邻近图）进行了广泛的数值分析。在β-骨架图（BSG）中，如果满足依赖于参数β∈（0，∞）的邻近规则，则两个顶点就会相连。此外，对于β＞1，存在两种不同的邻近规则，导致基于月形和基于圆形的 BSG。首先，通过计算大量 BSG 的平均度数，我们检测到基于月形和基于圆形的 BSG 之间存在差异，这些差异随着β的增加而增加。然后，在随机矩阵理论（RMT）方法中，我们通过使用最近邻能级间距分布和熵特征向量局域化长度分别研究了随机 BSG 的谱和特征向量特性。RMT 分析使我们能够得出结论，在β=1 处发生局域化转变。
