2015/06/25 by Abdullah, Mohammed Amin, Bode, Michel, Fountoulakis, Nikolaos
#05C12 #05C80 #05C82 #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1506.07811
We study typical distances in a geometric random graph on the hyperbolic plane. Introduced by Krioukov et al.~\citear:Krioukov as a model for complex networks, N vertices are drawn randomly within a bounded subset of the hyperbolic plane and any two of them are joined if they are within a threshold hyperbolic distance. With appropriately chosen parameters, the random graph is sparse and exhibits power law degree distribution as well as local clustering. In this paper we show a further property: the distance between two uniformly chosen vertices that belong to the same component is doubly logarithmic in N, i.e., the graph is an ~ultra-small world. More precisely, we show that the distance rescaled by log log N converges in probability to a certain constant that depends on the exponent of the power law. The same constant emerges in an analogous setting with the well-known Chung-Lu model for which the degree distribution has a power law tail.