Weighted random--geometric and random--rectangular graphs: Spectral and eigenfunction properties of the adjacency matrix

Within a random-matrix-theory approach, we use the nearest-neighbor energy level spacing distribution $P(s)$ and the entropic eigenfunction localization length $\mathrm{\ell }$ 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 $1\mathrm{/}a$, 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 $a\sim 1$, i.e.~two-dimensional geometries, but also approach the limit of quasi-one-dimensional wires when $a\gg 1$. In general we look for the scaling properties of $P(s)$ and $\mathrm{\ell }$ as a function of $a$, $r$ and $N$. We find that the ratio $r\mathrm{/}{N}^{\gamma }$, with $\gamma (a)\approx -1\mathrm{/}2$, 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 $r\mathrm{/}\mathcal{C}{N}^{\gamma }$, with $\mathcal{C}\approx a$.