Constraints Define Coherence: A Hierarchical Theory of Dynamical States in Complex Systems

ABSTRACT Coherent dynamical states are not generic outcomes of complex systems, but arise only within constrained regions of parameter space. Despite extensive research on synchronization, network topology, and critical dynamics, these mechanisms are typically studied in isolation. This work introduces a unified, data-driven framework that integrates structural connectivity, network topology, and dynamical coupling into a hierarchical model governing the existence, activation, and stability of globally coherent states. Using networks of coupled oscillators with systematically varied connectivity, topology, and coupling strength, we identify a set of interacting constraints. First, a structural threshold exists at a critical connectivity ratio k/N approx 0.3, below which global coherence cannot emerge. Second, network topology determines the ability of the system to integrate: clustered networks support stable coherence, while randomly organized networks with comparable density fail to do so. Third, dynamical coupling acts as an activation mechanism, enabling coherent states only within structurally admissible systems. Beyond these conditions, we identify a critical dynamical regime characterized by maximal temporal variability at intermediate levels of coherence. This regime is distinct from both rigid high-coherence states and incoherent disordered states, demonstrating that maximal dynamical complexity does not coincide with maximal synchronization. To assess the relevance of the framework to real-world systems, a preliminary empirical analysis was performed using EEG data. The results reveal a consistent pattern, with maximal variability occurring at intermediate coherence levels (R approx 0.71), in agreement with the predicted critical regime. Together, these findings establish that coherent dynamical states are constrained phenomena defined by a hierarchy of structural and dynamical conditions. The proposed framework unifies previously disconnected approaches and provides measurable criteria for the emergence, stability, and collapse of integrated yet dynamically sensitive states in complex systems.