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Multiple-scale theory of topology-driven patterns on directed networks.
Silvia Contemori1, Francesca Di Patti2, Duccio Fanelli2
1Dipartimento di Matematica e Informatica, Università degli Studi di Firenze, viale Morgagni 67/a, 50134 Firenze, Italy.
Physical Review. E
|April 15, 2016
Summary
Network topology can destabilize reaction-diffusion systems, leading to pattern formation. This study derives a nonlinear equation explaining these dynamics on directed graphs, validated by simulations.
Area of Science:
- Complex Systems
- Network Science
- Mathematical Biology
Background:
- Dynamical processes on networks are crucial across disciplines, with reaction-diffusion systems on directed graphs having broad applications in computer and traffic networks.
- The unique spectrum of the discrete Laplacian on directed graphs can destabilize homogeneous fixed points, a phenomenon not observed in undirected graphs.
- Linear analysis identifies instability conditions, but nonlinear treatment is needed for pattern characterization beyond exponential growth.
Purpose of the Study:
- To derive an effective nonlinear equation for the amplitude evolution of unstable modes near criticality in reaction-diffusion systems on directed graphs.
- To investigate the influence of network topology on pattern formation dynamics.
- To validate the derived theoretical model against numerical simulations.
Main Methods:
- Multiple time scale perturbative calculation to derive the effective nonlinear equation.
- Linear stability analysis to identify conditions for instability.
- Numerical simulations of a paradigmatic reaction-diffusion model on directed graphs.
Main Results:
- An effective nonlinear evolution equation, specifically a Stuart-Landau equation, was derived for the amplitude of the most unstable mode.
- The complex coefficients of the Stuart-Landau equation were found to depend on the topological features of the directed graph.
- The derived theory accurately predicted the behavior observed in numerical simulations.
Conclusions:
- Network topology plays a critical role in the stability and pattern formation of reaction-diffusion systems.
- The derived Stuart-Landau equation provides a powerful tool for understanding nonlinear dynamics near criticality in such systems.
- The findings are applicable to various real-world systems modeled by reaction-diffusion processes on directed networks.
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