Explosive higher-order Kuramoto dynamics on simplicial complexes (1912.04405v3)

Abstract: The higher-order interactions of complex systems, such as the brain are captured by their simplicial complex structure and have a significant effect on dynamics. However, the existing dynamical models defined on simplicial complexes make the strong assumption that the dynamics resides exclusively on the nodes. Here we formulate the higher-order Kuramoto model which describes the interactions between oscillators placed not only on nodes but also on links, triangles, and so on. We show that higher-order Kuramoto dynamics can lead to an explosive synchronization transition by using an adaptive coupling dependent on the solenoidal and the irrotational component of the dynamics.

Overview of Higher-Order Kuramoto Dynamics on Simplicial Complexes

The paper by Ana P. Millán and colleagues introduces a novel approach to modeling synchronization phenomena in complex systems by leveraging the structure of simplicial complexes. Traditional Kuramoto models have typically focused on interactions at the node level within networks. However, this research extends the notion of synchronization dynamics to higher-order interactions, encompassing links, triangles, and higher-dimensional simplices.

Higher-Order Kuramoto Model

The authors present a higher-order Kuramoto dynamics that incorporates oscillators not only at individual nodes but also at higher-dimensional simplices such as links, triangles, and tetrahedra. This is achieved by employing a simplicial complex framework, which recognizes the role of multi-node interactions in the synchronization process. The model utilizes adaptive coupling influenced by the solenoidal and irrotational components of the dynamics, leading to insights into explosive synchronization transitions.

Key Findings

1. Continuous and Explosive Synchronization Transitions: The paper explores both simple and explosive higher-order Kuramoto dynamics. In simple dynamics, continuous synchronization transitions are observed, while explosive dynamics show discontinuous transitions. This dichotomy reveals intriguing behaviors in dynamical systems when higher-order interactions are considered.
2. Projection of Dynamics: The paper introduces a method to project dynamics defined on $n$-dimensional simplices to $(n+1)$- and $(n-1)$- dimensional simplices using higher-order incidence matrices. This approach shows how dynamics defined on higher-order structures can induce transitions on lower-order structures.
3. Implications of Hodge Decomposition: Utilizing Hodge decomposition, the model effectively separates the dynamics into solenoidal and irrotational components, providing a deeper understanding of the synchronization process in complex systems.

Implications

Practical Applications: The insights from this paper could be applied to various fields including neuroscience, social network analysis, and materials science, where higher-order interactions play a crucial role in system dynamics.

Theoretical Contributions: The research enriches the understanding of synchronization phenomena, moving beyond conventional node-based network models to encompass the geometric and topological aspects of simplicial complexes.

Future Developments: The framework sets a stage for exploring other coupling mechanisms and might inspire further developments in modeling synchronization across domains such as biological transport networks, neurological data analysis, and epidemic modeling on complex geometries.

Conclusion

The exploration of higher-order Kuramoto dynamics on simplicial complexes by Millán et al. represents a significant advancement in the modeling of complex systems. By challenging the prevailing assumptions in dynamical models and expanding the scope to higher-dimensional interactions, this research provides a foundation for future inquiries into the synchronization behavior across diverse scientific fields.