Weak-Correlation Universality and Macroscopic Fluctuations in Power-Law Recurrent Networks

Balanced network theory describes the dynamic state of randomly connected systems with intrinsic negative feedback, predicting the suppression of correlations as connectivity increases. We show that this weak-correlation universality holds broadly but breaks down in power-law networks where the largest out-degrees scale with system size. In such networks, residual correlations cross over to a regime that retains macroscopic fluctuations in the thermodynamic limit. Unlike Erdös-Rényi networks, the degree heterogeneity of power-law topologies amplifies coherent population-level fluctuations. Our results provide a theoretical foundation for the strong correlations observed in real-world networks with sparse and heterogeneous connectivity.