A Unified Framework for Multiscale Spectral Graph Signal Theory

Thesis Abstract

This study addresses the fragmentation between discrete and continuous perspectives in spectral graph signal theory and proposes a unified multiscale framework that integrates wavelet-like hierarchies with graph Fourier analysis to enhance signal representation, denoising, and inference on complex networks. The central aim is to develop a theoretical and computational framework that unifies multiscale spectral methods with kernel-based and variational perspectives to enable robust analysis of signals defined on graphs with heterogeneous topology. Specific objectives include (1) formulating a unified multiscale spectral decomposition that couples graph Laplacian eigenmodes with scalable diffusion operators, (2) deriving a multiresolution embedding that preserves both local geometric structure and global spectral information, (3) establishing theoretical guarantees on approximation error, stability under graph perturbations, and convergence rates, (4) designing efficient algorithms for multiscale signal processing tasks such as denoising, interpolation, and semi-supervised learning, and (5) validating the framework on synthetic benchmarks and real-world networks. The methodology adopts a theory-driven, mixed-methods research design grounded in harmonic analysis on graphs, spectral graph theory, and variational optimization. The population consists of synthetic networks (Erd?s–Rényi, Barabási–Albert, stochastic block models) and real-world networks from social, biological, and information domains (e.g., citation networks, protein–protein interaction networks, and transportation networks). A stratified sampling approach selects networks with varied size (n ? 1,000–100,000 nodes) and average degree to test scalability and robustness. Data collection employs both benchmark graph data and generated signals with controlled noise and masking patterns to simulate measurement error and missing data. Instrumentation includes software implementations in Python using libraries such as NetworkX, PyTorch Geometric, and SciPy, along with custom modules for multiscale graph wavelets and diffusion-operator constructions. Analytical techniques combine theoretical derivations with empirical evaluation. The core analytical framework builds on (i) spectral graph theory, (ii) multiscale diffusion processes and graph wavelets, and (iii) variational and kernel methods, with explicit incorporation of the following named theories the graph Fourier transform, diffusion geometry, and reproducing kernel Hilbert spaces. The study derives a unified multiscale operator that interpolates between classical Laplacian-based spectra and diffusion-based embeddings, providing finite-energy guarantees and stability bounds under perturbations in either the adjacency or Laplacian structure. Algorithmic development includes fast multiscale decompositions via truncated spectral expansions and hierarchical diffusion schemes, with complexity analyzed as O(m log n) for sparse graphs. Data analysis comprises quantitative assessments using mean squared error for denoising, accuracy for semi-supervised classification, and spectral gap sensitivity analysis, complemented by qualitative evaluation of signal interpretability on illustrative networks. Regression-based and information-theoretic metrics assess the alignment between multiscale representations and ground-truth signals, while ablation studies quantify the contribution of each framework component. The key expected findings include (a) a principled, provably stable multiscale decomposition that unifies spectral and diffusion-based representations; (b) improved denoising performance and semi-supervised learning accuracy relative to single-scale and purely spectral methods, especially on heterogeneous networks; (c) theoretical guarantees on approximation error bounds and perturbation resilience under edge addition/deletion and weight perturbations; and (d) scalable algorithms with practical runtimes on networks with up to 100,000 nodes. The study anticipates that the unified framework will reveal intrinsic multiscale structure in signals on graphs, enabling more accurate inference of node attributes and more robust recovery of missing values. Contributions to knowledge include (i) a novel, unified theory integrating multiscale spectral graph signal processing with diffusion-based and kernel approaches, (ii) a mathematically rigorous framework offering stability and convergence guarantees, and (iii) practical algorithms and empirical evidence demonstrating superior performance on benchmark and real-world networks. The main conclusion is that a harmonized multiscale spectral framework improves interpretability and predictive accuracy for graph signals, with broad applicability to network data analysis, sensor networks, and biological systems. Recommendations for future work include extending the framework to dynamic graphs, exploring adaptive scale selection guided by network topology, and integrating the approach with graph neural networks to enhance task-specific performance while preserving interpretability.