Higher-order interactions in complex networks of phase oscillators promote abrupt synchronization switching (1909.08057v2)

Abstract: Synchronization processes play critical roles in the functionality of a wide range of both natural and man-made systems. Recent work in physics and neuroscience highlights the importance of higher-order interactions between dynamical units, i.e., three- and four-way interactions in addition to pairwise interactions, and their role in shaping collective behavior. Here we show that higher-order interactions between coupled phase oscillators, encoded microscopically in a simplicial complex, give rise to added nonlinearity in the macroscopic system dynamics that induces abrupt synchronization transitions via hysteresis and bistability of synchronized and incoherent states. Moreover, these higher-order interactions can stabilize strongly synchronized states even when the pairwise coupling is repulsive. These findings reveal a self-organized phenomenon that may be responsible for the rapid switching to synchronization in many biological and other systems that exhibit synchronization without the need of particular correlation mechanisms between the oscillators and the topological structure.

• The paper demonstrates that higher-order interactions trigger abrupt synchronization transitions marked by hysteresis and bistability.
• The study shows that 2-simplex and 3-simplex couplings stabilize synchronization even under repulsive pairwise conditions.
• The authors employ analytical modeling with Ott-Antonsen reduction and numerical simulations on both real and synthetic networks to validate their findings.

Higher-order Interactions in Complex Networks of Phase Oscillators

The paper under review explores the dynamics of coupled phase oscillators within the framework of simplicial complexes and emphasizes the significance of higher-order interactions in shaping macroscopic system behaviors, particularly synchronization transitions. This research contributes to the existing body of knowledge on network-coupled dynamical systems by demonstrating how interactions beyond typical pairwise models, specifically three- and four-way interactions, introduce novel nonlinear dynamics that result in abrupt synchronization transitions via hysteresis and bistability.

Key Findings and Methodology

The paper extends the classic Kuramoto model to incorporate higher-order interactions by formulating a higher-order Kuramoto model that includes coupling terms for 2-simplex and 3-simplex interactions, alongside the standard pairwise 1-simplex interactions. The dynamics of such coupled systems are demonstrated using both real-world data, specifically the Macaque brain network, and synthetic datasets modeled as multiplex networks, with nodes and interactions structured into simplicial complexes.

Key observations include:

1. Abrupt Synchronization Transitions: The paper reveals that higher-order interactions, as encoded in the simplicial complex framework, induce abrupt transitions between incoherent and synchronized states. This is particularly evidenced in macroscopic systems via hysteresis, substantiating the concept of explosive synchronization that demands small systemic parameter changes to switch between states.
2. Stability of Synchronization in Repulsive Conditions: Remarkably, the presence of higher-order interactions stabilizes synchronized states even under repulsive pairwise coupling, which typically would not support synchronization. This insight suggests that higher-order interactions provide a robust synchronization mechanism beyond what pairwise coupling alone would predict.
3. Analytical and Computational Approach: Through analytical modeling using the dimensionality reduction techniques of the Ott-Antonsen framework, and extensive numerical simulations, the paper provides a comprehensive understanding of the stability and bifurcation behaviors in these systems. The authors validate theoretical predictions with simulation results, further illustrating the transition dynamics within an all-to-all coupled network scenario.

Implications and Future Directions

The implications of this research are manifold, affecting both theoretical and applied aspects of network synchronization. From a theoretical perspective, incorporating higher-order interactions brings complexity into modeling and analysis that better reflects systems like neural networks where such interactions naturally occur. Practically, understanding these interaction dynamics can enhance insights into biological systems cognition, particularly in brain networks that display rapid cognitive task switching through explosive synchronization.

Future developments could explore quantifying the impact of different simplicial complex configurations on synchronization and stability. Exploring other real-world systems, such as power grids and more generalized physical and chemical systems, for higher-order interaction effects on synchronization can broaden the applicability of this research. Moreover, extending the Kuramoto model's application to dynamic simplicial complexes where interactions can evolve over time could provide further insights into real-time network adaptability and resilience.

Overall, this paper underscores the transformative impact of considering higher-order interactions in dynamical systems, suggesting richer dynamics and more comprehensive models for understanding complex networks.