Higher-order Laplacian Renormalization (2401.11298v2)

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.

• The paper introduces a cross-order diffusion process that captures scale-invariant properties in higher-order network structures.
• It defines novel cross-order Laplacians to extend traditional adjacency notions beyond pairwise interactions.
• Empirical tests on synthetic and real-world datasets validate its renormalization scheme for uncovering hidden structural dynamics.

An Overview of Higher-order Laplacian Renormalization

The paper "Higher-order Laplacian Renormalization" offers a significant extension to the renormalization group (RG) framework by introducing the cross-order Laplacian renormalization group (X-LRG) scheme tailored for higher-order networks. Higher-order networks encompass interactions that involve more than two entities, which are typically represented as simplices in structures such as simplicial complexes or hypergraphs. This formalism enhances our ability to model complex systems across various disciplines where interaction orders exceed pairwise relationships.

Context and Motivation

The renormalization group is foundational in physics for understanding how properties of systems change with scale, characterized by universal behavior and scale invariance. Traditional applications have been predominantly restricted to dyadic or pairwise interactions as reflected in graphs. However, many real-world systems exhibit polyadic interactions, demanding an RG approach that accounts for higher orders. Prior attempts, such as the Laplacian RG for network structures, primarily considered node-centric diffusion processes and lacked the comprehensive framework necessary for arbitrariness in the order of interactions.

Key Contributions

• Cross-order Diffusion Process: The authors introduce a sophisticated diffusion process that allows information flow not only between nodes but also between simplices of differing orders through the defined cross-order Laplacians. This process is instrumental in observing and defining scale-invariance within these higher-order constructs.
• Cross-order Laplacians: These Laplacians extend adjacency notions beyond conventional vertex-edge relationships, facilitating the capture of diffusion dynamics across different simplex orders. The paper defines these matrices in a way analogous to classical graph Laplacians while encapsulating adjacency of $k$-simplices via $m$-simplices.
• Informational Notion of Scale-invariance: The research applies the concept of von Neumann entropy in conjunction with entropic susceptibility to discern informational scale invariance, translating these principles to handle diffusion in arbitrary higher-order networks.
• Renormalization Scheme: The work develops a robust renormalization scheme predicated on cross-order Laplacians. By iteratively partitioning and coarse-graining the simplicial complexes based on diffusion times and scales, this method reveals and exploits scale-invariance properties efficiently.

Empirical and Theoretical Insights

The authors exemplify their approach on both synthetic and real-world datasets:

• Synthetic Data: The paper employs pseudofractal simplicial complexes to demonstrate the X-LRG's ability to recover self-similar structures and validate the method's accuracy in capturing higher-order scale invariances.
• Real-world Applications: Applying their method to datasets like network science co-authorship networks, the authors observe varying degrees of higher-order scale-invariance not previously detectable through traditional node-centric analyses. This insight extends the potential for understanding structural organization in complex systems through a higher-order lens.

Implications and Future Directions

The implications of this research are manifold and cross-disciplinary. Practically, the X-LRG method suggests improved approaches for detecting underlying organizational principles in systems modeled by higher-order networks. This can lead to better models in neuroscience, social sciences, and wherever complex polyadic interactions prevail. Theoretically, it stimulates discourse on extending RG concepts to multifaceted network structures, paving the way for future work on simultaneous multicross-order RG frameworks.

In conclusion, this paper charts a significant step in generalizing the RG framework to higher-order networks. Its novel methodology provides a foundation for unveiling the scale-invariant properties of complex structures, offering promising avenues for further exploration in the field of network science and beyond.