Abstract
We consider the random hyperbolic graph model introduced by [KPK+10] and then formalized by [GPP12]. We show that, in the subcritical case α>1, the size of the largest component is asymptotically almost surely n1∕(2α)+o(1), thus strengthening a result of [BFM15] which gave only an upper bound of n1∕α+o(1).
Highlights
Introduction and statement of resultIn the last decade, the model of random hyperbolic graphs introduced by Krioukov et al in [KPK+10] was studied quite a bit due to its key properties observed in large real-world networks
The model of random hyperbolic graphs introduced by Krioukov et al in [KPK+10] was studied quite a bit due to its key properties observed in large real-world networks
In [BnPK10] the authors showed empirically that the network of autonomous systems of the Internet can be very well embedded in the model of random hyperbolic graphs for a suitable choice of parameters
Summary
The model of random hyperbolic graphs introduced by Krioukov et al in [KPK+10] was studied quite a bit due to its key properties observed in large real-world networks. Construct the following graph G = (V, E): the set of vertices V is the point set of the Poisson process and for u, u ∈ V , u = u , there is an edge with endpoints u and u provided the hyperbolic distance d(u, u ) between u and u is such that d(u, u ) ≤ R, where d(u, u ) is obtained by solving (1.1). In the original model of Krioukov et al [KPK+10], n points, corresponding to vertices, are chosen uniformly and independently in the disk Bh(O, R) of the hyperbolic space of curvature −α2, but since from a probabilistic point of view it is arguably more natural to consider the Poissonized version of this model, we consider the ECP 26 (2021), paper 14.
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