Geometrical and spectral study of beta-skeleton graphs

Alonso, Lázaro; Mendez-Bermudez, J. A.; Estrada, Ernesto

doi:10.1103/PhysRevE.100.062309

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Geometrical and spectral study of beta-skeleton graphs

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Alonso, Lázaro
Max Planck Institute for the Physics of Complex Systems, Max Planck Society;

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Zitation

Alonso, L., Mendez-Bermudez, J. A., & Estrada, E. (2019). Geometrical and spectral study of beta-skeleton graphs. Physical Review E, 100(6): 062309. doi:10.1103/PhysRevE.100.062309.

Zitierlink: https://hdl.handle.net/21.11116/0000-0005-DFCF-8

Zusammenfassung

We perform an extensive numerical analysis of beta-skeleton graphs, a particular type of proximity graphs. In beta-skeleton graph (BSG) two vertices are connected if a proximity rule, that depends of the parameter beta is an element of (0, infinity), is satisfied. Moreover, for beta > 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 beta, 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 beta = 1.