Spectral coarse-graining scheme inspired by Laplacian renormalization group for higher-order network reaction-diffusion systems.

Abstract

Coarse-graining methods are crucial for characterizing complex systems across different levels of description. Inspired by renormalization group ideas, such approaches have been applied to structural compression studies in complex networks, where a Laplacian renormalization group based on spectral coarse graining has been shown to preserve the static properties of diffusion systems. However, their capacity to preserve dynamics in network reaction-diffusion (RD) systems remains underexplored. This paper addresses this gap by investigating network RD systems, particularly epidemic models. We propose a Laplacian renormalization group-inspired spectral coarse-graining scheme based on multiorder Laplacian matrix and validate its advantages in both random and empirical networks. Based on simplicial complexes, we construct a network susceptible-infected-recovered-dead system and employ Turing patterns as a testing platform to evaluate the preservation of dynamic behavior under coarse graining. The results demonstrate that the proposed method effectively preserves key dynamic behaviors, including the infected density distributions and the time required for stable pattern formation, while maintaining the necessary conditions for Turing instability. This preservation is achieved alongside a significant reduction in system complexity.