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

Manuel Miranda1, Gissell Estrada-Rodriguez2, Ernesto Estrada1

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Summary
This summary is machine-generated.

This study introduces simplicial metaplexes, a novel model for analyzing complex systems. This framework enhances the study of dynamical systems on geometric simplicial complexes, improving upon previous models.

Keywords:
brain networksdiffusiongeometric realizationhigher-order networksmetaplexessimplicial complex

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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.