Nearest-neighbor directed random hyperbolic graphs

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 neighbors. 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 to 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. We demonstrate that reciprocity of the network can be regulated by adjusting an additional temperature-like parameter without changing other global properties of the network.