Cliques in Hyperbolic Random Graphs

Abstract

Most complex real world networks display scale-free features. This characteristic 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 Krioukov et al. (in Phys Rev E 82(3):036106, 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 $\mathbb {E}\left[ K_k\right]$ and the size of the largest clique $\omega (G)$. We observe that there is a phase transition at power-law exponent $\beta = 3$. More precisely, for $\beta \in (2,3)$ we prove $\mathbb {E}\left[ K_k\right] =n^{k (3-\beta )/2} \varTheta (k)^{-k}$ and $\omega (G)=\varTheta (n^{(3-\beta )/2})$, while for $\beta \geqslant 3$ we prove $\mathbb {E}\left[ K_k\right] =n \, \varTheta (k)^{-k}$ and $\omega (G)=\varTheta (\log (n)/ \log \log n)$. Furthermore, we show that for $\beta \geqslant 3$, cliques in hyperbolic random graphs can be computed in time $\mathcal {O}(n)$. If the underlying geometry is known, cliques can be found with worst-case runtime $\mathcal {O}(m \cdot n^{2.5})$ for all values of $\beta$.