On power-law relationships of the Internet topology
Proceedings of the conference on Applications, technologies, architectures, and protocols for computer communication
The degree sequence of a scale-free random graph process
Random Structures & Algorithms
Random Structures & Algorithms
Random Evolution in Massive Graphs
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Graphs over time: densification laws, shrinking diameters and possible explanations
Proceedings of the eleventh ACM SIGKDD international conference on Knowledge discovery in data mining
First to market is not everything: an analysis of preferential attachment with fitness
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Proceedings of the forty-first annual ACM symposium on Theory of computing
Concentration of Measure for the Analysis of Randomized Algorithms
Concentration of Measure for the Analysis of Randomized Algorithms
Models for the Compressible Web
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Greedy forwarding in dynamic scale-free networks embedded in hyperbolic metric spaces
INFOCOM'10 Proceedings of the 29th conference on Information communications
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Recently, Papadopoulos, Krioukov, Boguñá and Vahdat [Infocom'10] introduced a random geometric graph model that is based on hyperbolic geometry. The authors argued empirically and by some preliminary mathematical analysis that the resulting graphs have many of the desired properties for models of large real-world graphs, such as high clustering and heavy tailed degree distributions. By computing explicitly a maximum likelihood fit of the Internet graph, they demonstrated impressively that this model is adequate for reproducing the structure of such with high accuracy. In this work we initiate the rigorous study of random hyperbolic graphs. We compute exact asymptotic expressions for the expected number of vertices of degree k for all k up to the maximum degree and provide small probabilities for large deviations. We also prove a constant lower bound for the clustering coefficient. In particular, our findings confirm rigorously that the degree sequence follows a power-law distribution with controllable exponent and that the clustering is nonvanishing.