Weighted random-geometric and random-rectangular graphs: spectral and eigenfunction properties of the adjacency matrix
L Alonso, Jose Antonio Mendez-Bermudez, A. Gonzalez-Melendrez, Yamir Moreno · Journal of Complex Networks · 2017
Within a random-matrix theory approach, we use the nearest-neighbour energy-level spacing distribution |$P(s)$| and the entropic eigenfunction localization length |$\ell$| to study spectral and eigenfunction properties (of adjacency matrices) of weighted random-geometric and random-rectangular graphs. A random-geometric graph (RGG) considers a set of vertices uniformly and independently distributed on the unit square, while for a random-rectangular graph (RRG) the embedding geometry is a rectangle. The RRG model depends on three parameters: The rectangle side lengths |$a$| and |$1/a$|, the connection radius |$r$| and the number of vertices |$N$|. We then study in detail the case |$a=1$|, which corresponds to weighted RGGs and explore weighted RRGs characterized by |$a\sim 1$|, that is, two-dimensional geometries, but also approach the limit of quasi-one-dimensional wires when |$a\gg1$|. In general, we look for the scaling properties of |$P(s)$| and |$\ell$| as a function of |$a$|, |$r$| and |$N$|. We find that the ratio |$r/N^\gamma$|, with |$\gamma(a)\approx -1/2$|, fixes the properties of both RGGs and RRGs. Moreover, when |$a\ge 10$| we show that spectral and eigenfunction properties of weighted RRGs are universal for the fixed ratio |$r/\mathcal{C}N^\gamma$|, with |$\mathcal{C}(a)\approx a$|.