On the Number of Maximal Cliques in Two-Dimensional Random Geometric Graphs: Euclidean and Hyperbolic
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Maximal clique enumeration appears in various real-world networks, such as social networks and protein-protein interaction networks for different applications. For general graph inputs, the number of maximal cliques can be up to 3^|V|/3. However, many previous works suggest that the number is much smaller than that on real-world networks, and polynomial-delay algorithms enable us to enumerate them in a realistic-time span. To bridge the gap between the worst case and practice, we consider the number of maximal cliques in two popular models of real-world networks: Euclidean random geometric graphs and hyperbolic random graphs. We show that the number of maximal cliques on Euclidean random geometric graphs is lower and upper bounded by exp(Ω(|V|^1/3)) and exp(O(|V|^1/3+ϵ)) with high probability for any ϵ > 0. For a hyperbolic random graph, we give the bounds of exp(Ω(|V|^(3-γ)/6)) and exp(O(|V|^(3-γ+ϵ)/6))) where γ is the power-law degree exponent between 2 and 3.
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