The diameter of KPKVB random graphs

Abstract

AbstractWe consider a random graph model that was recently proposed as a model for complex networks by Krioukovet al.(2010). 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 previously been 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 nodesN, which we think of as going to infinity, and$\alpha, \nu > 0$, which we think of as constant. Roughly speaking,$\alpha$controls the power-law exponent of the degree sequence and$\nu$the average degree. Earlier work of Kiwi and Mitsche (2015) has shown that, when$\alpha \lt 1$(which corresponds to the exponent of the power law degree sequence being$\lt 3$), the diameter of the largest component is asymptotically almost surely (a.a.s.) at most polylogarithmic inN. Friedrich and Krohmer (2015) showed it was a.a.s.$\Omega(\log N)$and 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.