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Spectral coarse-graining scheme inspired by Laplacian renormalization group for higher-order network
Linhe Zhu1, Wenlong Zhang1, Shuling Shen2
1Jiangsu University, School of Mathematical Sciences, Zhenjiang 212013, China.
Physical Review. E
|May 16, 2026
Summary
This study introduces a new spectral coarse-graining method for complex network dynamics, preserving epidemic model behaviors. The approach simplifies systems while maintaining essential dynamic properties and Turing instability conditions.
Area of Science:
- Complex Systems Science
- Network Science
- Mathematical Biology
Background:
- Coarse-graining simplifies complex systems but its dynamic preservation in network reaction-diffusion (RD) systems is understudied.
- Renormalization group (RG) inspired methods are used for structural compression in networks, preserving static diffusion properties.
Purpose of the Study:
- To investigate the preservation of dynamics in network RD systems, specifically epidemic models, using spectral coarse-graining.
- To propose and validate a novel Laplacian RG-inspired spectral coarse-graining scheme for complex networks.
Main Methods:
- Developed a spectral coarse-graining scheme based on a multiorder Laplacian matrix, inspired by renormalization group concepts.
- Constructed a susceptible-infected-recovered-dead (SIRD) epidemic model on simplicial complexes.
- Utilized Turing patterns on random and empirical networks to evaluate dynamic behavior preservation under coarse-graining.
Main Results:
- The proposed method effectively preserves key dynamic behaviors of network RD systems, including infected density distributions.
- Demonstrated preservation of the time to stable pattern formation and conditions for Turing instability.
- Achieved significant reduction in system complexity while maintaining dynamic fidelity.
Conclusions:
- The spectral coarse-graining method successfully preserves essential dynamic properties of network reaction-diffusion systems, particularly epidemic models.
- This approach offers a powerful tool for analyzing complex network dynamics with reduced computational cost.
- The method is validated on both random and empirical networks, showing broad applicability.
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