ABSTRACT
This paper presents a comprehensive spectral-based framework for analyzing time-evolving networks, such as social networks, transportation systems, and biological networks. While traditional spectral graph theory has been extensively applied to static graphs, dynamic networks remain an underexplored domain. We extend the Laplacian and adjacency matrices to time-dependent graphs, providing a mathematical foundation for spectral analysis of dynamic systems. By examining the eigenvalue behaviors over time, we uncover insights into network stability, anomaly detection, and community evolution. Our approach leverages advanced spectral methods, including time-series spectral embeddings and spectral clustering, to analyze structural changes in evolving graphs. We demonstrate our framework's effectiveness through experiments on real-world datasets, offering new perspectives for understanding dynamic networks. The findings contribute to the advancement of spectral techniques for analyzing time-evolving graph structures and their applications in diverse fields.
