2017/07/29 by Müller, Tobias, Staps, Merlijn · 1 citation
#05C80 #05C82 #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1707.09555
We consider a model for complex networks that was recently proposed as a model for complex networks by Krioukov et al. In this model, nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes are connected if they are at most a certain hyperbolic distance from each other. It has been previously shown that this model has various properties associated with complex networks, including a power-law degree distribution and a strictly positive clustering coefficient. The model is specified using three parameters : the number of nodes N, which we think of as going to infinity, and α, ν> 0 which we think of as constant. Roughly speaking α controls the power law exponent of the degree sequence and ν the average degree. Earlier work of Kiwi and Mitsche has shown that when α< 1 (which corresponds to the exponent of the power law degree sequence being < 3) then the diameter of the largest component is a.a.s.~polylogarithmic in N. Friedrich and Krohmer have shown it is a.a.s.~Ω(log N) and they improved the exponent of the polynomial in log N in the upper bound. Here we show the maximum diameter over all components is a.a.s.~O(log N) thus giving a bound that is tight up to a multiplicative constant.