Cliques in hyperbolic random graphs

Tobias Friedrich, Anton Krohmer
2015 2015 IEEE Conference on Computer Communications (INFOCOM)  
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(INFOCOM, pp. -2981(INFOCOM, pp. , 2010 and has shown
more » ... ically 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 powerlaw exponent γ = 3. More precisely, for γ ∈ (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 ] = n Θ(k) −k and ω(G) = Θ(log(n)/ log log n). We empirically compare the ω(G) value of several scalefree 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.
doi:10.1109/infocom.2015.7218533 dblp:conf/infocom/FriedrichK15 fatcat:wnp3xv4xbvbufi3uxzeabahwc4