An Adaptive Spectral Method for Dynamic Graphs in Enhancing Numerical Stability and Efficiency

Graph theory and numerical analysis meet in numerous real-world applications, including network evolution, dynamic data clustering, and multiscale modelling. Current spectral methods, however, often presume static graph structures, compromising their accuracy and efficiency for developing networks. This work presents an adaptive spectral technique that adjusts solver tolerance dynamically according to structural changes in the graph, quantified through the Frobenius norm of differences between consecutive Laplacians. We tested this on synthetic graphs and compared it with conventional fixed-tolerance techniques. The adaptive method exhibited runtime improvements of up to 50% and substantially reduced eigenvalue errors. These findings attest to its superior numerical stability and computational efficiency. We advocate the application of adaptive tolerance methods in real-time spectral analysis applications and propose follow-up work develop this framework for weighted, directed, or real-world dynamic networks.