Weighted random-geometric and random-rectangular graphs: spectral and eigenfunction properties of the adjacency matrix. (30th October 2017) - Alonso, L; Méndez-Bermúdez, J A; González-Meléndrez, A; Moreno, Yamir
Abstract: 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 $\mathcal{C}(a)\approx a$ .