Spectral Embeddings Leak Graph Topology: Theory, Benchmark, and Adaptive Reconstruction

2026-04-22

AI Summary

This work addresses the challenge of effectively deploying graph neural networks in decentralized, localized, noisy, and privacy-sensitive graph data settings by proposing a unified framework to tackle graph fragmentation. The authors introduce the LoGraB benchmark to simulate local graph environments and develop the AFR method for adaptive, high-fidelity graph reconstruction based on spectral embeddings. Key contributions include the first formal identification of topological information leakage through spectral embeddings—termed “spectral leakage”—along with theoretical guarantees for deterministic reconstruction. The approach avoids forced global reconstruction by incorporating island-wise coherence assessment and an adaptive stitching strategy. It further integrates heat kernel edge recovery, Davis–Kahan stability analysis, RANSAC-Procrustes alignment, Bundle Adjustment optimization, and differential privacy mechanisms. Experiments demonstrate state-of-the-art F1 scores on 7 out of 9 benchmarks and retain 75% of non-private performance under (ε=2, δ)-Gaussian differential privacy, confirming its robustness and practicality.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Data transparency and provenance
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Graph Neural Networks (GNNs) excel on relational data, but standard benchmarks unrealistically assume the graph is centrally available. In practice, settings such as Federated Graph Learning, distributed systems, and privacy-sensitive applications involve graph data that are localized, fragmented, noisy, and privacy-leaking. We present a unified framework for this setting. We introduce LoGraB (Local Graph Benchmark), which decomposes standard datasets into fragmented benchmarks using three strategies and four controls: neighborhood radius $d$, spectral quality $k$, noise level $σ$, and coverage ratio $p$. LoGraB supports graph reconstruction, localized node classification, and inter-fragment link prediction, with Island Cohesion. We propose AFR (Adaptive Fidelity-driven Reconstruction), a method for noisy spectral fragments. AFR scores patch quality via a fidelity measure combining a gap-to-truncation stability ratio and structural entropy, then assembles fragments using RANSAC-Procrustes alignment, adaptive stitching, and Bundle Adjustment. Rather than forcing a single global graph, AFR recovers large faithful islands. We prove heat-kernel edge recovery under a separation condition, Davis--Kahan perturbation stability, and bounded alignment error. We establish a Spectral Leakage Proposition: under a spectral-gap assumption, polynomial-time Bayesian recovery is feasible once enough eigenvectors are shared, complementing AFR's deterministic guarantees. Experiments on nine benchmarks show that LoGraB reveals model strengths and weaknesses under fragmentation, AFR achieves the best F1 on 7/9 datasets, and under per-embedding $(ε,δ)$-Gaussian differential privacy, AFR retains 75% of its undefended F1 at $ε=2$. Our anonymous code is available at https://anonymous.4open.science/r/JMLR_submission

Problem

• Graph Fragmentation
• Privacy-Preserving Graph Learning
• Spectral Embeddings
• Distributed Graph Data
• Graph Reconstruction

Innovation

• Spectral Embeddings
• Graph Reconstruction
• Federated Graph Learning
• Differential Privacy
• Local Graph Benchmark