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-neighbour energy-level spacing distribution |$P(s)$| and the entropic eigenfunction localization length || 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~ 1|⁠, that is, two-dimensional geometries, but also approach the limit of quasi-one-dimensional wires when |a1|⁠. In general, we look for the scaling properties of |$P(s)$| and || as a function of |a|⁠, |r| and |N|⁠. We find that the ratio |r/N^γ|⁠, with |γ(a)≈ -1/2|⁠, fixes the properties of both RGGs and RRGs. Moreover, when |a≥ 10| we show that spectral and eigenfunction properties of weighted RRGs are universal for the fixed ratio |r/CN^γ|⁠, with |C(a)≈ a|⁠.