Planted clique recovery in random geometric graphs

2025-10-14

AI Summary

This paper addresses the exact recovery of planted cliques in random geometric graphs. We propose two algorithms—vertex-degree-based (VD) and common-neighbor-based (CN)—and derive their theoretical recovery thresholds under varying connectivity regimes. Our analysis shows that, as the graph size tends to infinity, the CN algorithm achieves exact recovery of arbitrarily small planted cliques—including single edges—with high probability under appropriate parameter conditions, whereas the VD algorithm exhibits significantly weaker performance. Numerical experiments corroborate the robustness and efficiency of the CN algorithm across both sparse and dense regimes. To our knowledge, this is the first work to rigorously establish the strong discriminative power of common-neighbor statistics for detecting microscopic structural anomalies in random geometric graphs, thereby providing novel methodological insights and theoretical guarantees for anomalous subgraph detection in high-dimensional Euclidean spaces.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Anomaly/Outlier Detection
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

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.

Problem

• Identifying planted cliques in random geometric graphs
• Comparing vertex degree and common neighbor algorithms
• Analyzing exact recovery performance under varying parameters

Innovation

• Uses vertex degrees for clique identification
• Employs common neighbors for clique detection
• Achieves exact recovery with high probability