A Dual-Scale Framework for Nonlinear Spectral Graph Theory Analysis

Thesis Abstract

This study addresses the limitations of classical linear spectral methods in capturing nonlinear geometric and functional properties of large-scale graphs, where traditional eigen-based analyses fail to reveal intrinsic community structure, robustness, and dynamic diffusion patterns under irregular sampling and nonlinearity. The aim is to develop a dual-scale framework that integrates a nonlinear spectral embedding at the local scale with a macro-level diffusion-robust aggregation mechanism to enhance interpretability and predictive performance across diverse graph domains. Specific objectives are (i) to formulate a nonlinear spectral operator that combines p-Laplacian-inspired energy with scale-adaptive weighting to produce two complementary scales of graph representations, (ii) to derive theoretical guarantees on stability and convergence under perturbations and heterogeneous degree distributions, (iii) to construct an algorithmic pipeline for joint local-global spectral analysis, including efficient solvers for large sparse graphs, (iv) to evaluate the framework on multiple benchmark datasets with varying sparsity and noise levels, and (v) to demonstrate applicability to real-world problems in social networks and biological interaction graphs. The methodology adopts a deductive, design-science-inspired research design using a mixed-methods data-driven approach. The population encompasses large-scale graphs from social media platforms, biological networks, and citation networks, with sample sizes drawn from 20 real-world graphs ranging from 10k to 2M nodes to test scalability. Data collection utilizes publicly available graph datasets (e.g., TWITTER-like networks,STRING, and SNAP collections) accompanied by ground-truth community annotations where available. Instruments include a nonlinear spectral operator toolkit implemented in Python and C++, integrating a discretized p-Laplacian variant, adaptive edge-weight learning, and a diffusion-aware aggregation module. Validity and reliability are addressed through cross-validation on held-out subgraphs, synthetic benchmarks with known spectral properties, and sensitivity analyses over the nonlinearity parameter p and scale thresholds. Data analysis employs a combination of spectral analysis, clustering validity indices (Adjusted Rand Index, Normalized Mutual Information), and diffusion metrics (hitting times, commute times) to gauge local and global coherence. The analytical framework incorporates spectral clustering on the local scale followed by a macro diffusion-robust consensus stage, with theoretical support from nonlinear operator theory and variational principles. Analytical techniques include optimization-based solvers for the nonlinear eigenproblem, leveraging proximal gradient methods and alternating direction method of multipliers (ADMM) to ensure convergence in large graphs. The study integrates stability analysis via perturbation theory for nonlinear operators and network perturbations, and it employs regression analysis to relate spectral features to downstream tasks such as node classification and link prediction. The framework is grounded in two relevant theories the nonlinear spectral theory of p-Laplacians and diffusion geometry, complemented by network topology theory and robust statistics to handle noise and outliers. Expected findings anticipate that the dual-scale framework yields superior community detection quality and clustering stability relative to linear spectral baselines and single-scale nonlinear methods, particularly in graphs with heterogeneous degree distributions and nonuniform sampling. It is anticipated that the local nonlinear embedding will capture intricate community boundaries, while the macro diffusion-robust aggregation will improve resilience to perturbations and enhance generalization across domains. The study also expects improved performance in downstream tasks such as semi-supervised node classification and link prediction, with statistically significant gains demonstrated via paired t-tests and nonparametric equivalents across datasets. Contributions to knowledge include a novel dual-scale nonlinear spectral framework that unifies local nonlinear embedding with global diffusion-consistent aggregation, providing theoretical guarantees on convergence and stability, and offering scalable algorithms for massive graphs. The work extends existing nonlinear spectral theory by integrating scale-aware operators with practical diffusion considerations, delivering a transferable methodology applicable to social, biological, and information networks. Potential limitations are discussed in relation to parameter sensitivity and interpretability of dual-scale representations, with recommendations for adaptive parameter tuning and visualization tools. The study ultimately concludes that the proposed framework enhances both interpretability and predictive performance for complex networks, and it recommends further exploration into dynamic graphs, higher-order network representations, and integration with graph neural networks to further exploit the nonlinear spectral foundations.