Functional modularity induced by delayed interactions

Abstract

We investigate how modular and hierarchical delay structures shape synchronization dynamics in networks of uniformly connected phase oscillators. Using a Kuramoto model adapted to incorporate modular interaction delays, we isolate the effects of temporal heterogeneity from structural modularity. Our analysis identifies distinct synchronization regimes dependent on oscillator frequency: global coherence at low frequencies, modular synchronization at intermediate frequencies, and incoherence dominates at high frequencies, though localized coherence can reemerge under specific delay configurations. Under hierarchical delay arrangements, we further observe a reemergence of localized coherence within sub-modules at very high frequencies, indicating multi-scale synchronization facilitated purely by temporal delays. These findings highlight the fundamental role modular and hierarchical delays play in shaping functional network dynamics, offering insights relevant to the adaptability and multi-scale processing capabilities observed in neural and technological systems. © 2025 Elsevier Ltd