Random Rectangular Graphs

TL;DR

This paper introduces random rectangular graphs as a generalization of random geometric graphs, analyzing how their properties vary with rectangle shape and connection radius, supported by theoretical formulas and simulations.

Contribution

It provides analytical expressions for key topological properties of RRGs and explores how these properties change with rectangle elongation and connection radius.

Findings

Clustering coefficient peaks at slight elongation of the rectangle.

Most properties depend monotonically on connection radius and side length.

Simulations confirm the accuracy of the theoretical models.

Abstract

A generalization of the random geometric graph (RGG) model is proposed by considering a set of points uniformly and independently distributed on a rectangle of unit area instead of on a unit square [0,1]^2. The topological properties of the random rectangular graphs (RRGs) generated by this model are then studied as a function of the rectangle sides lengths a and b=1/a, and the radius r used to connect the nodes. When a=1 we recover the RGG, and when a-->infinity the very elongated rectangle generated resembles a one-dimensional RGG. We obtain here analytical expressions for the average degree, degree distribution, connectivity, average path length and clustering coefficient for RRG. These results provide evidence that show that most of these properties depend on the connection radius and the side length of the rectangle, usually in a monotonic way. The clustering coefficient, however, increases when the square is transformed into a slightly elongated rectangle, and after this maximum it decays with the increase of the elongation of the rectangle. We support all our findings by computational simulations that show the goodness of the theoretical models proposed for RRGs.