Breathing synchronization in interconnected networks

Abstract: Global synchronization in a complex network of oscillators emerges from the interplay between its topology and the dynamics of the pairwise interactions among its numerous components. When oscillators are spatially separated, however, a time delay appears in the interaction which might obstruct synchronization. Here we study the synchronization properties of interconnected networks of oscillators with a time delay between networks and analyze the dynamics as a function of the couplings and communication lag. We discover a new breathing synchronization regime, where two groups appear in each network synchronized at different frequencies. Each group has a counterpart in the opposite network, one group is in phase and the other in anti-phase with their counterpart. For strong couplings, instead, networks are internally synchronized but a phase shift between them might occur. The implications of our findings on several socio-technical and biological systems are discussed.

• The paper identifies breathing synchronization, where oscillators split into two frequency-locked communities due to delayed inter-network interactions.
• It employs an extended Kuramoto model with time-delay to reveal transitions from global synchrony to competing and supernode states as intra-network coupling increases.
• The research provides analytical and numerical insights with significant implications for biological and technological systems involving modular interactions and signal delays.

Breathing Synchronization in Interconnected Oscillator Networks

Introduction

The study addresses synchronization phenomena in complex systems of coupled oscillators, focusing on the scenario of two interconnected networks with heterogeneous coupling strengths and explicit communication delays in inter-network interactions. The research utilizes the Kuramoto model framework, augmenting it with a time delay parameter for inter-network links, and explores the emergent dynamical regimes as functions of intra- and inter-network coupling strengths and the delay value. The principal result is the identification and characterization of a breathing synchronization regime, which fundamentally contrasts with the classically expected collective behaviors in isolated or instantaneously coupled networks.

Model Formulation and Dynamical Regimes

The networks considered comprise $n$ Kuramoto oscillators with identical natural frequency $\omega_0$, each network implemented on a random graph topology. The intra-network (within-network) interactions are assumed instantaneous, governed by coupling $\sigma_{\text{IN}}$, while inter-network (between-network) couplings $\sigma_{\text{EX}}$ are subject to a delay $\tau$. The system is governed by a set of coupled differential equations extending the traditional Kuramoto approach to include delayed arguments for inter-network interactions.

This architecture allows the investigation of the interplay between local rapid synchronization mechanisms and delayed cross-network feedback, a setting relevant for a broad class of real-world systems, from neuronal structures to large-scale socio-technical infrastructures.

Emergent Breathing Synchronization

A distinct dynamical regime, termed breathing synchronization, emerges in the limit of weak intra-network coupling and finite, significant inter-network delay. In this regime, oscillators within each individual network split into two frequency-adapted communities. Each community maintains synchronization with a counterpart in the opposing network: one in-phase, the other in anti-phase. The resulting global state is not static but characterized by temporal variations in the network order parameters, reflecting the "breathing" alternation between alignment and anti-alignment of these subgroups.

Figure 1: Steady-state population snapshots and temporal order parameter evolution demonstrating the segregation of two intra-network frequency communities and the ensuing breathing behavior.

The analytic insight for the existence of two synchronized frequencies derives from bifurcations permissible in the underlying delay-coupled pairwise interactions, according to the Schuster-Wagner formulation. Notably, neither of the emergent frequencies need coincide with the intrinsic frequency $\omega_0$ of the isolated oscillators; rather, the possible locked values are functions of $\omega_0$, $\tau$, and $\sigma_{\text{EX}}$.

Intra-Network Coupling and Transition to Phase-Locked Regimes

As $\sigma_{\text{IN}}$ increases, intra-network coupling exerts increasing dominance. The breathing regime is destabilized beyond a threshold: one of the two frequency groups becomes preponderant, leading to global synchronization on a unique frequency within each network. If $\sigma_{\text{IN}}$ is further increased, the networks behave effectively as supernodes, each phase-locked internally, but exhibiting inter-network phase relationships that can be either in-phase or anti-phase, dependent on initial conditions. These transitions are abrupt with clear demarcating boundaries in the coupling parameter space.

Figure 2: Frequency pair scatter plots for intra-network neighbors across coupling regimes, illustrating the dominance and coalescence of frequency groups as intra-network coupling is increased.

Systematic analysis reveals that the average phase displacement between inter-network pairs transitions from $\Delta = \pi$ (anti-phase) through intermediary values back toward zero (in-phase) as the nature of global synchronization shifts. The robustness of this phenomenology is affirmed across network sizes and topologies within those tested, and the regime boundaries display strong dependence on the value of the imposed delay $\tau$.

Figure 3: Phase diagram depicting regions of breathing, Kuramoto-type, competing, and supernode synchronization states as a function of intra- and inter-network couplings.

Theoretical and Numerical Implications

The results elucidate how networked oscillatory systems can exhibit nontrivial multistable collective behaviors when internal versus cross-network interactions are hierarchically disparate and subject to non-negligible signal latency. The breathing synchronization regime specifically demonstrates that even homogeneous populations can spontaneously segregate into subcommunities with distinct collective dynamics, directly shaped by systemic delays. This challenges prevailing assumptions about the sufficiency of coupling strength enhancements for achieving global synchrony: increasing intra-network coupling can, paradoxically, foster regime transitions to states with more separated collective frequencies (supernodes) rather than robust unification.

The transitions between these regimes are numerically sharp and the present framework could be further formalized with master stability function approaches or linear stability analysis to analytically determine the transition boundaries.

Relevance to Biological and Technological Systems

The findings have direct implications for systems where time-lagged communication and modular organization are central features. For example, in biological contexts such as Physarum polycephalum and large-scale neural networks, the results predict the possibility of spontaneous emergence of intra-module frequency groups resulting from cross-module delays—results that can be experimentally interrogated by varying spatial separation or modulating inter-module communication channels.

Technologically, the dynamics are highly pertinent to engineered interdependent networks such as interconnected power grids, or multi-site computational clusters, where physical transmission constraints and decentralized internal coupling cannot be idealized away. The presence or absence of breathing-like regimes could have implications for predicting and controlling the collective response of such systems under perturbation or attack.

Conclusions

This research establishes that interconnected oscillator networks with delayed cross-network coupling do not, in general, converge to a unique synchronized state. Instead, they can exhibit breathing synchronization—an internally organized segregation into two phase-locked frequency communities—if intra-network coupling is sufficiently weak. Increasing intra-network connections induces sharp, qualitative changes in the collective state, sometimes to competing or supernode regimes, and not always to the classically predicted global synchrony. Practical systems involving modular organization and intrinsic communication delays must therefore account for the nontrivial spectrum of possible synchronized states and transition thresholds revealed by this work.

Future extensions should articulate analytical stability boundaries via group synchronization theory and validate the presented phenomenology in experimentally accessible biological and engineered oscillator systems.