  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 to
m. We study local bond length, in-degree and reciprocity in these networks as a
function of spatial 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.
