Cliques in hyperbolic random graphs

Most complex real-world networks display scale-free features. This motivated the study of numerous random graph models with a power-law degree distribution. There is, however, no established and simple model which also has a high clustering of vertices as typically observed in real data. Hyperbolic random graphs bridge this gap. This natural model has recently been introduced by Papadopoulos, Krioukov, Boguñá, Vahdat (INFOCOM, pp. 2973-2981, 2010) and has shown theoretically and empirically to fulfill all typical properties of real-world networks, including power-law degree distribution and high clustering. We study cliques in hyperbolic random graphs G and present new results on the expected number of k-cliques E[K_k] and the size of the largest clique ω(G). We observe that there is a phase transition at power-law exponent γ = 3. More precisely, for γ ε (2,3) we prove E[K_k] = n^k(3-γ)/2 Θ(k)^-k and ω(G) = Θ(n^(3-γ)/2) while for γ ≥ 3 we prove E[K_k] = nΘ(k)^-k and ω(G) = Θ(log(n)/log log n). We empirically compare the ω(G) value of several scale-free random graph models with real-world networks. Our experiments show that the ω(G)-predictions by hyperbolic random graphs are much closer to the data than other scale-free random graph models.