Spectral fluctuations and crossovers in multilayer network.

Abstract

Spectral fluctuation analysis within the random matrix theory (RMT) framework is a powerful probe of complexity in networked systems, yet its extension to multilayer architectures remains unresolved. In general multilayer networks, the adjacency matrix possesses a heterogeneous block structure with unequal variances across layers, causing eigenvalue spacing statistics to deviate from RMT predictions even when each individual layer is perfectly random. We demonstrate that this variance mismatch is the central obstacle to observing spectral universality in multilayer systems and introduce a general blockwise normalization scheme that restores the correct variance structure across all blocks. Using higher-order spacing ratios, we show that once properly normalized, multilayer networks exhibit universal spectral fluctuations consistent with RMT across a broad class of configurations, including purely intralayer, interlayer, and multiplex structures. Focusing on the bilayer case, we introduce a crossover model parametrized by the relative interlayer to intralayer coupling strength, which captures the continuous transition from two independent Gaussian orthogonal ensembles (GOEs) to a single GOE. We find that this crossover sharpens with increasing system size, suggesting that in the large-system limit, arbitrarily weak interlayer coupling may be sufficient to induce global spectral correlations. Applying the framework to empirical multilayer networks derived from protein-crystal structures, we demonstrate that structural coupling drives analogous spectral transitions, directly linking the emergence of universality to physically meaningful organization. These results establish spectral universality as a robust feature of multilayer networks and provide a quantitative framework for understanding how structure and coupling govern collective behavior in complex interconnected systems.