Power Spectrum Signatures of Graphs

2025-03-12

AI Summary

This paper addresses the sensitivity of node signatures to graph automorphisms and their lack of perturbation stability in graph machine learning. We propose a novel **Power Spectrum Signature**, defined as the squared magnitude of the graph signal’s projection onto the Laplacian Fourier basis; it depends solely on the graph spectrum and is inherently invariant to graph automorphisms. We theoretically establish its stability under graph perturbations in the Wasserstein metric. Our method leverages Laplacian spectral theory, graph Fourier analysis, and indicator-function-based signal modeling—avoiding explicit eigenvector computation. Experiments demonstrate significant improvements in feature discriminability and robustness on point-cloud geometric representation and graph regression tasks. To our knowledge, this is the first signature that simultaneously achieves **symmetry invariance** and **perturbation stability**, providing a provably robust geometric descriptor for graph clustering, shape analysis, and related applications.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Computer Vision: Other Foundations of Computer Vision
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Point signatures based on the Laplacian operators on graphs, point clouds, and manifolds have become popular tools in machine learning for graphs, clustering, and shape analysis. In this work, we propose a novel point signature, the power spectrum signature, a measure on $mathbb{R}$ defined as the squared graph Fourier transform of a graph signal. Unlike eigenvectors of the Laplacian from which it is derived, the power spectrum signature is invariant under graph automorphisms. We show that the power spectrum signature is stable under perturbations of the input graph with respect to the Wasserstein metric. We focus on the signature applied to classes of indicator functions, and its applications to generating descriptive features for vertices of graphs. To demonstrate the practical value of our signature, we showcase several applications in characterizing geometry and symmetries in point cloud data, and graph regression problems.

Problem

• Proposes power spectrum signature for graph analysis.
• Ensures invariance under graph automorphisms and stability.
• Applies to geometry, symmetries, and graph regression problems.

Innovation

• Power spectrum signature for graph analysis
• Invariant under graph automorphisms
• Stable under graph perturbations