Random rectangular graphs

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, such as connectivity, average degree, average path length and clustering, 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 ! 1 the very elongated rectangle generated resembles a one-dimensional RGG. We provided computational and analytical evidence that the topological properties of the RRG differ significantly from those of the RGG. The connectivity of the RRG depends not only on the number of nodes as in the case of the RGG, but also on the side length of the rectangle. As the rectangle is more elongated the critical radius for connectivity increases following first a power-law and then a linear trend. Also, as the rectangle becomes more elongated the average distance between the nodes of the graphs increases, but the local cliquishness of the graphs also increases thus producing graphs which are relatively long and highly locally connected. Finally, we found the analytic expression for the average degree in the RRG as a function of the rectangle side lengths and the radius. For different values of the side length, the expected and the observed values of the average degree display excellent correlation, with correlation coefficients larger than 0.9999