What Is in a Simplicial Complex? A Metaplex-Based Approach to Its Structure and Dynamics

Area of Science:

• Complex Systems Analysis
• Mathematical Modeling
• Network Science

Background:

• Simplicial complexes offer unique representations of complex systems.
• Analyzing dynamical systems on these structures is challenging due to local continuous spaces and global discrete connectivity.

Purpose of the Study:

• To generalize the concept of metaplexes to geometric simplicial complexes.
• To define dynamical systems on these generalized structures.
• To address limitations of previous models for analyzing diffusion dynamics.

Main Methods:

• Generalization of metaplexes to geometric simplicial complexes.
• Definition of dynamical systems on simplicial metaplexes.
• Analysis of diffusion dynamics within the proposed framework.

Main Results:

• A novel framework, simplicial metaplexes, is presented for modeling complex systems.
• The model effectively describes diffusion dynamics, overcoming issues in prior approaches.
• The framework's applicability is demonstrated on a macaque visual cortex complex.

Conclusions:

• Simplicial metaplexes provide a robust model for complex systems with geometric and topological features.
• The model offers extensions to various complex structures like manifold complexes.
• This approach enhances the understanding of dynamical processes in real-world systems.