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 l 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 similar to 1, that is, two-dimensional geometries, but also approach the limit of quasi-one-dimensional wires when a >> 1. In general, we look for the scaling properties of P(s) and l as a function of a, r and N. We find that the ratio r/N-gamma, with gamma (a) approximate to -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 gamma, with C(a) approximate to a