A Survey on Spectral Energy Measures under Structural Perturbations

Structural perturbations provide a powerful framework for understanding how localized topological modifications influence global network behaviour. Spectral graph theory addresses this problem by mapping discrete changes in network structure to continuous variations in eigenvalues through energy-based descriptors such as graph energy and Laplacian energy. These measures offer computationally efficient alternatives to large scale simulations by enabling analytical assessment of network robustness, resilience, and structural integrity. However, a fundamental challenge remains in determining whether global spectral energy measures can adequately capture localized structural and functional dynamics, particularly in complex and modular systems. This issue is especially relevant in computational neuroscience, where spectral variations are often used as biomarkers of neurological disorders. This survey reviews the development of spectral energy measures under structural perturbations, examining their theoretical foundations, perturbation sensitivity, and applications in brain connectivity networks. The review highlights inconsistencies among different energy measures, the lack of standardized benchmarking frameworks, and the limited investigation of weighted, directed, and time-varying networks. Finally, it identifies key research gaps and outlines future directions for developing unified, scalable, and application-oriented spectral frameworks.