Designing topological cluster synchronization patterns with the Dirac operator

Area of Science:

• Nonlinear dynamics
• Network science
• Computational neuroscience

Background:

• Cluster synchronization is crucial for understanding complex systems, particularly brain dynamics.
• Existing models exclusively use a node-based dynamical approach, limiting their scope.
• A new framework is needed to incorporate network topology more effectively.

Purpose of the Study:

• To propose a novel topological synchronization dynamics model.
• To design stable cluster synchronization patterns for both network nodes and edges.
• To leverage the topological Dirac operator for network dynamics analysis.

Main Methods:

• Developed a topological synchronization dynamics model using the topological Dirac operator.
• Constructed topological cluster synchronization patterns by modulating the ground state of free energy.
• Utilized linear stability analysis to predict pattern stability.
• Applied the model to real-world connectome data, random graphs, and stochastic block models.

Main Results:

• Successfully designed stable topological cluster synchronization patterns.
• Demonstrated the model's applicability to diverse network structures.
• Showcased the decomposition of dynamical states across nodes and edges.

Conclusions:

• The proposed topological synchronization model offers a powerful new approach to designing cluster synchronization patterns.
• This method extends synchronization dynamics beyond nodes to include network edges.
• The findings have significant implications for network science and understanding brain dynamics.