Planted clique recovery in random geometric graphs

TL;DR

This work studies exact recovery of a planted clique in hard random geometric graphs $G(n,r_n)$, where a clique of size $k_n$ is embedded into an otherwise random geometric graph. It analyzes two polynomial-time algorithms: a vertex-degrees based VD-algorithm and a common-neighbors CN-algorithm, deriving recovery thresholds in terms of the average degree $\mu_n = n \phi_d r_n^d$ and the dimension $d$; notably, CN achieves recovery over a much wider regime, including tiny cliques and planted edges, while VD is limited to larger cliques in some regimes. Theoretical results show CN can recover planted cliques even when $k_n$ is sublinear in $n$ in connectivity and sparse regimes, and experimental results corroborate the superiority of CN over VD in finite samples. The findings leverage the geometric structure to enable reliable detection of artificial dense subgraphs, with potential applications in anomaly detection and data integrity, and open avenues for extending to soft graphs and high-dimensional limits. Overall, the paper demonstrates that geometry greatly enhances planted-clique detection and provides actionable algorithms with provable guarantees and practical validations.

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.

Figures (4)

• Figure 1: Representation in dimension $d=2$ of the regions $R_1$, $R_2$. Vertices with coordinates in $R_1$ cannot connect to vertices with coordinates in $R_2$, unless they are part of the planted clique.
• Figure 2: Qualitative representation of recovery results, in the $\mu$--$k$ plane, according to Theorems \ref{['thm:successVDalgo']}, \ref{['thm:negative_VD_algo']} and \ref{['thm:successCNalgo']}. The pictures are shown for $n=10^9$, on a log-log scale. Green areas: exact recovery of the planted clique. Red areas: VD algorithm fails. Beige area: no theoretical guarantees are currently available. Gray vertical bands: boundaries between the sparse, connectivity, and dense regimes of geometric random graphs. The gray vertical bands separate different regimes of the average degree $\mu$: in Figure \ref{['thm:successVDalgo']}, the gray band separates the two regions where condition \ref{['eq:cond2successCN']} is satisfied or not.
• Figure 3: Percentage of runs (out of 1000) in which the VD- and CN-algorithms returned $K$ correctly for $G_k(n,r)$ with $n = 10^4$.
• Figure 4: Representation for $d=2$ of the area that can contain common neighbors of $i$ and $j$

Theorems & Definitions (17)

• Theorem 1: kuvcera1995expected
• Theorem 2: VD-algorithm -- positive result
• Theorem 3: VD-algorithm -- negative result
• Theorem 4: CN algorithm -- positive result
• proof : Proof of Theorem \ref{['thm:successVDalgo']}
• Lemma 1
• proof
• Lemma 2
• proof
• proof : Proof of Theorem \ref{['thm:negative_VD_algo']}