Geometrical Phase Transitions in Complex Networks

This paper presents a spectral-morphological framework to track the structural evolution of complex interconnected networks in real time, without relying on predictive historical time-series projections or autoregressive fitting. Traditional risk assessment models and data-driven methods are structurally constrained by retroactive computational biases and assumptions of stationarity, which can limit the detection of structural reorganizations prior to macroscopic failures. To address these limitations, the proposed approach characterizes network dynamics through complementary scales. Random-matrix methods, including the Marčenko–Pastur distribution and Wigner nearest-neighbour spacings, establish statistical baselines for finite sampling constraints. Concurrently, network observables—such as critical percolation fractions—delineate the topological geometry of the system. The methodology is empirically validated using multivariate financial networks restricted to N=20 core assets, spanning S&P 500 equities during the 2007–2009 crisis and two digital asset systems over the 2017–2019 and 2021–2023 periods. The empirical results show that systemic stress triggers a geometric phase transition toward topological saturation. By contrasting a conventional isolated-component baseline against a framework that incorporates these global topological constraints into a discrete binomial lattice, the resulting structural 'Safety Gap' operator measures the unregistered risk mass generated by collective network synchronization, showing a risk underestimation of up to 184.05% in conventional models during peak synchronization. Finally, the framework documents permanent morphological adaptation and structural hysteresis in post-crisis regimes.