Higher-order Laplacian Renormalization

TL;DR

This paper introduces a novel renormalization group scheme for higher-order networks using diffusion dynamics, enabling analysis of scale-invariance and structure across different interaction orders in complex systems.

Contribution

It presents the first general RG scheme for higher-order networks, extending diffusion-based methods to polyadic interactions and providing tools for multilevel structural analysis.

Findings

Effective detection of higher-order scale-invariance in synthetic systems

Application to real-world systems reveals order-specific profiles

New coarse-graining scheme for complex higher-order networks

Abstract

We propose a cross-order Laplacian renormalization group (X-LRG) scheme for arbitrary higher-order networks. The renormalization group is a pillar of the theory of scaling, scale-invariance, and universality in physics. An RG scheme based on diffusion dynamics was recently introduced for complex networks with dyadic interactions. Despite mounting evidence of the importance of polyadic interactions, we still lack a general RG scheme for higher-order networks. Our approach uses a diffusion process to group nodes or simplices, where information can flow between nodes and between simplices (higher-order interactions). This approach allows us (i) to probe higher-order structures, defining scale-invariance at various orders, and (ii) to propose a coarse-graining scheme. We demonstrate our approach on controlled synthetic higher-order systems and then use it to detect the presence of order-specific scale-invariant profiles of real-world complex systems from multiple domains.