We investigate the number of maximal cliques, i.e., cliques that are not contained in any larger clique, in three network models: Erd\H{o}s-R\'enyi random graphs, inhomogeneous random graphs (also called Chung-Lu graphs), and geometric inhomogeneous random graphs. For sparse and not-too-dense Erd\H{o}s-R\'enyi graphs, we give linear and polynomial upper bounds on the number of maximal cliques. For the dense regime, we give super-polynomial and even exponential lower bounds. Although (geometric) inhomogeneous random graphs are sparse, we give super-polynomial lower bounds for these models. This comes form the fact that these graphs have a power-law degree distribution, which leads to a dense subgraph in which we find many maximal cliques. These lower bounds seem to contradict previous empirical evidence that (geometric) inhomogeneous random graphs have only few maximal cliques. We resolve this contradiction by providing experiments indicating that, even for large networks, the linear lower-order terms dominate, before the super-polynomial asymptotic behavior kicks in only for networks of extreme size.

翻译：我们研究了三种网络模型中最大团（即不被任何更大团包含的团）的数量：Erdős–Rényi随机图、非齐次随机图（亦称Chung-Lu图）以及几何非齐次随机图。对于稀疏且非过于稠密的Erdős–Rényi图，我们给出了最大团数量的线性与多项式上界。在稠密区域，我们得到了超多项式甚至指数级下界。尽管（几何）非齐次随机图是稀疏的，但我们仍为这些模型给出了超多项式下界。这一结果源于这些图具有幂律度分布，导致存在一个稠密子图，其中包含大量最大团。这些下界似乎与以往实证证据相矛盾——后者表明（几何）非齐次随机图仅包含少量最大团。我们通过实验解决了这一矛盾：实验表明，即使对于大规模网络，线性低阶项仍占主导地位，而超多项式渐近行为仅在极端规模网络中才会显现。