Distance-Preserving Embeddings in Inhomogeneous Random Graphs

2026-07-10

AI Summary

This study addresses the challenge of learning low-dimensional node embeddings in complex networks that simultaneously preserve both local and global shortest-path distances, overcoming the overly pessimistic distortion inherent in traditional worst-case analyses. Focusing on inhomogeneous random graphs, the work proposes a landmark-based embedding approach that captures network heterogeneity through multi-type branching processes and constructs a virtual graph spanner to optimize the trade-off between embedding dimensionality and distortion. The authors introduce a unified metric sandwiching framework, enabling the first general average distortion bounds for L² kernel models—including those generating heavy-tailed and power-law networks—under both finite-type and continuous latent spaces. To avoid costly exact shortest-path computations, they design a GNN-augmented neural surrogate model. Experiments demonstrate that the method achieves or surpasses the distance fidelity of classical landmark embeddings on large-scale real-world networks, with small-scale trained models capable of generalizing to extract universal distance-preserving features.

Abstract

Graph machine learning provides powerful tools for understanding complex networks and learning meaningful node representations. A central challenge, however, is designing embeddings with minimal distortion of both local and global functionals, such as shortest path lengths. Prior distortion guarantees for distance-preserving embeddings are worst-case in nature, producing overly pessimistic bounds that fail to capture the structure of typical large-scale networks. To address this, we analyze shortest-path approximation via landmark-based embeddings on inhomogeneous random graphs, a general model with type-dependent edge probabilities. By retaining shortest paths to a small set of reference nodes called landmarks, landmark-based methods effectively function as virtual graph spanners, where structural heterogeneity and controlled neighborhood expansion modeled via multi-type branching processes enable significantly tighter dimension-distortion trade-offs than classical worst-case bounds. We extend these guarantees to global, component-wide averages and unify the analysis across finite-type and continuous latent spaces through a novel metric sandwiching framework, establishing universal distortion bounds for general $L^2$ kernel models, including heavy-tailed and power-law networks. Finally, we introduce a GNN-augmented variant that replaces rigid, computationally expensive exact shortest-path queries with flexible, structure-aware neural surrogates. By leveraging the inherent alignment between graph neural message-passing and the dynamic programming principles of shortest-path algorithms, our approach demonstrates that models trained on small-scale random graphs learn to extract universal distance-preserving features, achieving robust generalization to large-scale, real-world networks that match or exceed the fidelity of classical, exact landmark-based embeddings.

Problem

• distance-preserving embeddings
• inhomogeneous random graphs
• shortest path distortion
• graph representation learning
• landmark-based embeddings

Innovation

• landmark-based embeddings
• inhomogeneous random graphs
• metric sandwiching
• graph neural networks
• distance-preserving representations