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The theory of pattern formation on directed networks
Malbor Asllani1, Joseph D Challenger2, Francesco Saverio Pavone3
11] Dipartimento di Scienza e Alta Tecnologia, University of Insubria, via Valleggio 11, 22100 Como, Italy [2] Dipartimento di Fisica e Astronomia, University of Florence, INFN and CSDC, Via Sansone 1, Sesto Fiorentino, 50019 Florence, Italy.
Nature Communications
|August 1, 2014
Summary
Pattern formation on directed networks is now understood. New instabilities arise from network topology, leading to traveling waves or stable patterns in reaction-diffusion systems.
Area of Science:
- Mathematical modeling
- Network theory
- Dynamical systems
Background:
- Pattern formation in reaction-diffusion systems is well-studied on symmetric networks.
- Applications span neuroscience, computer networks, and traffic systems.
- Previous theories primarily focused on undirected graphs.
Purpose of the Study:
- Extend pattern formation theory to directed networks.
- Investigate novel instabilities arising from network topology.
- Analyze the behavior of reaction-diffusion systems on asymmetric graphs.
Main Methods:
- Linear stability analysis to identify instability regions.
- Mathematical analysis of the network Laplacian in directed graphs.
- Numerical simulations to observe pattern evolution.
Main Results:
- The dispersion relation for directed networks possesses both real and imaginary components.
- Network topology can destabilize homogeneous fixed points, creating new instability types.
- Analytical tracing of instability regions is feasible.
- Numerical simulations reveal traveling waves and quasi-stationary patterns.
Conclusions:
- Directed network topology introduces unique instabilities in reaction-diffusion systems.
- These findings expand the understanding of pattern formation beyond symmetric networks.
- The study provides a framework for analyzing complex network dynamics.
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