We investigate the problem of identifying planted cliques in random geometric graphs, focusing on two distinct algorithmic approaches: the first based on vertex degrees (VD) and the other on common neighbors (CN). We analyze the performance of these methods under varying regimes of key parameters, namely the average degree of the graph and the size of the planted clique. We demonstrate that exact recovery is achieved with high probability as the graph size increases, in a specific set of parameters. Notably, our results reveal that the CN-algorithm significantly outperforms the VD-algorithm. In particular, in the connectivity regime, tiny planted cliques (even edges) are correctly identified by the CN-algorithm, yielding a significant impact on anomaly detection. Finally, our results are confirmed by a series of numerical experiments, showing that the devised algorithms are effective in practice.

翻译：本文研究了在随机几何图中识别植入团块的问题，重点分析了两种不同的算法方法：一种基于顶点度数（VD），另一种基于共同邻居（CN）。我们分析了这些方法在关键参数（即图的平均度数和植入团块的大小）不同机制下的性能。我们证明，随着图尺寸的增加，在特定参数集合下，能够以高概率实现精确恢复。值得注意的是，我们的结果表明CN算法显著优于VD算法。特别是在连通性机制中，CN算法能够正确识别微小的植入团块（甚至边），这对异常检测具有重要影响。最后，通过一系列数值实验验证了我们的结果，表明所设计的算法在实践中是有效的。