Tracking Structural Reorganisation in Complex Networks

Complex systems can undergo collective structural reorganisation as their internal interactions evolve between distinct configurations. We introduce a framework for characterising this evolution through complementary spectral and topological representations. Random-matrix methods provide statistical references for spectral properties induced by finite sampling, while network observables describe the geometry and collective connectivity of the system. We follow the temporal evolution of financial systems across multiple empirical periods, examining how their organisation changes between different configurations and around critical transitions. The empirical results show recurrent and non-monotonic changes in spectral and connectivity statistics, with different observables capturing complementary aspects of structural reorganisation. These results indicate that structural organisation can contain information about an evolving system before changes become apparent in conventional measures of fluctuations and covariance. The framework provides a general approach for tracking structural evolution in complex interconnected systems.