Functional reducibility of higher-order networks

Abstract: Empirical complex systems are widely assumed to be characterized not only by pairwise interactions, but also by higher-order (group) interactions that affect collective phenomena, from metabolic reactions to epidemics. Nevertheless, higher-order networks' superior descriptive power -- compared to classical pairwise networks -- comes with a much increased model complexity and computational cost. Consequently, it is of paramount importance to establish a quantitative method to determine when such a modeling framework is advantageous with respect to pairwise models, and to which extent it provides a parsimonious description of empirical systems. Here, we propose a principled method, based on information compression, to analyze the reducibility of higher-order networks to lower-order interactions, by identifying redundancies in diffusion processes while preserving the relevant functional information. The analysis of a broad spectrum of empirical systems shows that, although some networks contain non-compressible group interactions, others can be effectively approximated by lower-order interactions -- some technological and biological systems even just by pairwise interactions. More generally, our findings mark a significant step towards minimizing the dimensionality of models for complex systems

• The paper introduces a framework that uses a generalized density matrix with a multiorder Laplacian to assess when higher-order network interactions can be reduced.
• It quantifies reducibility by using the modified message length—combining Kullback-Leibler divergence and model complexity—to determine the optimal interaction order.
• Empirical analysis on synthetic and real-world datasets shows that optimal order identification varies across network types, guiding the simplification of complex systems.

Functional Reducibility of Higher-Order Networks: An Overview

This paper presents an in-depth exploration of the functional reducibility of higher-order networks, a topic that explores when and how such complex network models can be simplified without significant loss of information. The authors propose a method grounded in information compression principles to assess the necessity and efficiency of higher-order interaction representations in networks, compared to more conventional pairwise models.

Main Contributions

The core contribution of this work is the introduction of a framework to analyze the reducibility of higher-order networks. This framework is based on a generalization of the density matrix used in network science, which incorporates higher-order diffusive dynamics through the use of a multiorder Laplacian. This approach leverages the formalism of quantum statistical physics, bridging it with higher-order network analysis. The primary objective is to determine the optimal order of interactions that maintains an accurate yet parsimonious description of the network’s functional characteristics.

Methodological Insights

1. Density Matrix for Higher-Order Networks:
   • The authors extend the notion of pairwise network density matrices to hypergraphs by using a multiorder Laplacian that operates over varying interaction orders. This generalized density matrix serves as a tool for capturing complex interactions among network nodes beyond simple dyadic connections.
2. Quantification of Reducibility:
   • A pivotal aspect of the paper is the development of a measure termed the "modified message length," which is a sum of the Kullback-Leibler divergence (quantifying information loss) and the model complexity. This measure allows for determining the order up to which network interactions should be considered to ensure a functionally optimal model.
3. Optimal Order Identification:
   • The authors propose finding a balance between model accuracy and complexity through minimization of the modified message length. The result is an optimal order that indicates the degree to which a network’s higher-order structure can be compressed.

Empirical Examination

The paper conducts an extensive empirical analysis on both synthetic and real-world datasets. The synthetic analyses confirm the theoretical framework, showing that random hypergraphs (lacking inter-order correlation) are generally non-reducible, while simplicial complexes exhibit reducibility due to their inherent nested structures. The real-world analyses reveal variability in reducibility across various datasets, including coauthorship networks, contact networks, and biological systems, indicating the method's applicability and utility in diverse settings.

Implications and Future Directions

This study's implications are twofold: First, it highlights scenarios where higher-order interactions provide indispensable insights into network functioning, pushing the boundaries of current modeling techniques in complex systems. Second, it facilitates the identification of situations where simpler, pairwise models suffice, providing a basis for reducing computational and analytical complexity in network studies.

Looking ahead, this research opens avenues for further exploration into the conditions under which higher-order modeling significantly enhances the understanding of complex systems. Future research may explore more comprehensive empirical validations across varying domains, enhance computational tools for higher-order network analysis, and integrate this framework into dynamic systems modeling.

The paper's approach represents a substantial step forward in network science, marrying concepts from quantum mechanics, statistical physics, and information theory to offer a robust tool for network analysis. It encourages a re-evaluation of modeling assumptions in complex systems by systematically exploring the necessity and implications of higher-order interactions.