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 $\exp(\Omega(|V|^{1/3}))$ and $\exp(O(|V|^{1/3+\epsilon}))$ with high probability for any $\epsilon > 0$. For a hyperbolic random graph, we give the bounds of $\exp(\Omega(|V|^{(3-\gamma)/6}))$ and $\exp(O(|V|^{(3-\gamma+\epsilon)/6)}))$ where $\gamma$ is the power-law degree exponent between 2 and 3.

翻译：最大团枚举出现在各类现实网络（如社交网络和蛋白质相互作用网络）中，具有多种应用场景。对于一般图输入，最大团数量可达 $3^{|V|/3}$。然而，先前许多研究表明，现实网络中的最大团数量远小于该值，且多项式延迟算法使我们能在实际时间跨度内完成枚举。为弥合最坏情况与实际应用之间的差距，我们研究了两种主流现实网络模型中最大团的数量：欧几里得随机几何图和双曲随机图。我们证明，欧几里得随机几何图中最大团数量以高概率满足下界 $\exp(\Omega(|V|^{1/3}))$ 和上界 $\exp(O(|V|^{1/3+\epsilon}))$ 对任意 $\epsilon > 0$。对于双曲随机图，我们给出下界 $\exp(\Omega(|V|^{(3-\gamma)/6}))$ 和上界 $\exp(O(|V|^{(3-\gamma+\epsilon)/6}))$，其中 $\gamma$ 为介于2和3之间的幂律度指数。