In this paper, we study the maximum clique problem on hyperbolic random graphs. A hyperbolic random graph is a mathematical model for analyzing scale-free networks since it effectively explains the power-law degree distribution of scale-free networks. We propose a simple algorithm for finding a maximum clique in hyperbolic random graph. We first analyze the running time of our algorithm theoretically. We can compute a maximum clique on a hyperbolic random graph $G$ in $O(m + n^{4.5(1-\alpha)})$ expected time if a geometric representation is given or in $O(m + n^{6(1-\alpha)})$ expected time if a geometric representation is not given, where $n$ and $m$ denote the numbers of vertices and edges of $G$, respectively, and $\alpha$ denotes a parameter controlling the power-law exponent of the degree distribution of $G$. Also, we implemented and evaluated our algorithm empirically. Our algorithm outperforms the previous algorithm [BFK18] practically and theoretically. Beyond the hyperbolic random graphs, we have experiment on real-world networks. For most of instances, we get large cliques close to the optimum solutions efficiently.

翻译：本文研究双曲随机图上的最大团问题。双曲随机图是分析无标度网络的数学模型，因其能有效解释无标度网络的幂律度分布。我们提出一种在双曲随机图中寻找最大团的简单算法。首先从理论上分析算法运行时间：在给定几何表示的情况下，可在期望时间$O(m + n^{4.5(1-\alpha)})$内计算出双曲随机图$G$的最大团；若未给定几何表示，则期望时间为$O(m + n^{6(1-\alpha)})$，其中$n$和$m$分别表示$G$的顶点数和边数，$\alpha$为控制$G$度分布幂律指数的参数。此外，我们对算法进行实现与实证评估。与既有算法[BFK18]相比，本算法在理论和实践层面均表现更优。除双曲随机图外，我们还在真实网络数据上开展实验。在多数实例中，算法能够高效获取接近最优解的大规模团。