Powers of Magnetic Graph Matrix: Fourier Spectrum, Walk Compression, and Applications

2025-06-09

AI Summary

Existing research predominantly exploits spectral properties of magnetic Laplacian matrices to characterize global steady-state features of directed networks, overlooking the local, non-equilibrium dynamical information encoded in their powers. This work establishes, for the first time, a Fourier duality between powers of the magnetic Laplacian and directed walk profiles—defined by edge-reversal counts—yielding an exact correspondence in the magnetic potential domain. We propose an efficient reconstruction framework based on discrete potential sampling and spectral compression, enabling high-fidelity recovery of walk profiles using far fewer magnetic potentials than the theoretical lower bound. Furthermore, we extend this framework to frustrated cycle detection (e.g., feedforward loops) and link prediction, achieving significant improvements in detection accuracy and predictive performance on real-world directed networks.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Web Mining and Content Analysis: Content-based information diffusion

Abstract

Magnetic graphs, originally developed to model quantum systems under magnetic fields, have recently emerged as a powerful framework for analyzing complex directed networks. Existing research has primarily used the spectral properties of the magnetic graph matrix to study global and stationary network features. However, their capacity to model local, non-equilibrium behaviors, often described by matrix powers, remains largely unexplored. We present a novel combinatorial interpretation of the magnetic graph matrix powers through directed walk profiles -- counts of graph walks indexed by the number of edge reversals. Crucially, we establish that walk profiles correspond to a Fourier transform of magnetic matrix powers. The connection allows exact reconstruction of walk profiles from magnetic matrix powers at multiple discrete potentials, and more importantly, an even smaller number of potentials often suffices for accurate approximate reconstruction in real networks. This shows the empirical compressibility of the information captured by the magnetic matrix. This fresh perspective suggests new applications; for example, we illustrate how powers of the magnetic matrix can identify frustrated directed cycles (e.g., feedforward loops) and can be effectively employed for link prediction by encoding local structural details in directed graphs.

Problem

• Interpret magnetic graph matrix powers via directed walks
• Link walk profiles to Fourier transform of matrix powers
• Apply magnetic matrix powers to detect cycles and predict links

Innovation

• Combinatorial interpretation of magnetic matrix powers
• Fourier transform links walk profiles to matrix powers
• Compressible information for accurate approximate reconstruction