Working with Hyperbolic Random Graphs part1

Abstract: Undirected hyperbolic graph models have been extensively used as models of scale-free small-world networks with high clustering coefficient. Here we presented a simple directed hyperbolic model, where nodes randomly distributed on a hyperbolic disk are connected to a fixed number m of their nearest spatial neighbours. We introduce also a canonical version of this network (which we call network with varied connection radius''), where maximal length of outgoing bond is space-dependent and is determined by fixing the average out-degree at m. We study local bond length, in-degree and reciprocity in these networks as a function of spacial coordinates of the nodes, and show that the network has a distinct core-periphery structure. We show that for small densities of nodes the overall in-degree has a truncated power law distribution.