The Skeleton of Hyperbolic Graphs for Greedy Navigation

Random geometric (hyperbolic) graphs are important modeling tools in analyzing real-world complex networks. Greedy navigation (routing) is one of the most promising information forwarding mechanisms in complex networks. This paper is dealing with greedy navigability of complex graphs generated by using a metric (hyperbolic) space. Greedy navigability means that every source-destination pairs in the graph can communicate in such a way that every node passes the information towards that neighboring node which is "closest" to the destination in terms of node coordinates in the metric space. A set of compulsory links in greedy navigable graphs called Greedy Skeleton is identified. Because the two-dimensional hyperbolic plane (H^2, also known as the two dimensional Bolyai-Lobachevsky Space bolyai) turned out to be extremely useful in modelling and generating real-like networks, we deal with the statistical properties of the Greedy Skeleton when the metric space is H^2. Some examples of numerical studies and simulation results supporting the analytical formulae are also performed. The significance of the results lies in that every (either artificial or natural) network formation process aiming at greedy navigability must contain this Greedy Skeleton. Furthermore, this could be an important step towards the formal argumentation of the very high greedy navigability of some models observed only experimentally for the time being, and also to analyze equilibrium of greedy network navigation games on H^2.