A Novel Multiscale Graph Signal Processing and Network Dynamics Approach to Vibration Analysis for Stone Size Discrimination via Nonlinear Manifold Embeddings and a Convolutional Self-Attention Model

Abstract

Understanding nonlinear dynamics is critical for analyzing the hidden complexities of vibrational behavior in real-world systems. This study introduces a graph-theoretic approach to analyze the complex nonlinear temporal patterns in vibrational signals, utilizing the Tri-Axial Vibro-Dynamic Stone Classification dataset. This dataset captures high-resolution acceleration signals from controlled stone-crushing experiments, providing a unique opportunity to investigate temporal dynamics associated with distinct stone sizes. A 12-level Maximal Overlap Discrete Wavelet Transform is employed to perform multiscale signal decomposition, enabling the construction of transition graphs that encode transient and stable structural characteristics. Conceptually, transition graphs are analyzed as dynamic networks to uncover the interactions and temporal patterns embedded within vibrational signals. These networks are studied using a comprehensive suite of complexity metrics derived from information theory, graph theory, network science, and dynamical systems analysis. Metrics such as Shannon and Von Neumann's entropy evaluate signal dynamics' stochasticity and information retention. At the same time, the spectral radius measures the network's stability and structural robustness. Lyapunov exponents and fractal dimensions, informed by chaos theory and fractal geometry, further capture the degree of nonlinearity and temporal complexity. Complementing these dynamic measures, static network metrics-including the clustering coefficient, modularity, and the static Kuramoto index-offer critical discernment into the network's community structures, synchronization phenomena, and connectivity efficiency. Manifold learning techniques address the high-dimensional feature space derived from complexity metrics, with UMAP outperforming ISOMAP, Spectral Embedding, and PCA in preserving critical data structures. The reduced features are input into a convolutional self-attention model, combining localized feature extraction with long-term sequence modeling, achieving 100% classification accuracy across stone-size categories. This study presents a comprehensive framework for vibrational signal analysis, integrating multiscale graph-based representations, nonlinear dynamics quantification, and UMAP-based dimensionality reduction with a convolutional self-attention classifier. The proposed approach supports accurate classification and contributes to the development of data-driven tools for automated diagnostics and predictive maintenance in industrial and engineering contexts.