Pangraphs as models of higher-order interactions
Mateusz Iskrzyński, Aleksandra Puchalska, Aleksandra Grzelik, Gökhan Mutlu
TL;DR
This work formalises the representation of higher-order ecological interactions with ubergraphs and their incidence Levi-graph representations, introducing ecographs to capture flows and functional dependencies. It extends key graph measures—degree, centrality, paths, clustering, and information-theoretic metrics—to hypergraphs and ubergraphs, including recursive constructs like Katz centrality and trophic levels, while accounting for edge depth and classes. A central contribution is demonstrating that the choice of representation (hypergraph vs ubergraph vs ecograph) materially alters centrality rankings in real ecosystems (e.g., a coffee agroecosystem), highlighting the importance of preserving interaction roles and flow quantities. The work also establishes a duality between dynamical ecological models (ODEs) and weighted ubergraph structures, and provides a mapping framework to translate between equations and ubergraph representations, enabling principled analysis of higher-order ecological dynamics. Collectively, these tools offer a flexible, quantitative framework for studying HOIs in complex ecosystems with explicit directionality and flow quantities, with potential applications to trait-mediated interactions and dynamic robustness.
Abstract
Graphs depict pairwise relationships between objects within a system. Higher-order interactions (HOIs), which involve more than two objects simultaneously, are common in nature. Such interactions can change the stability of a complex system. Hypergraphs can represent an HOI as an arbitrary subset of vertices. However, they fail to capture the specific roles of the vertices involved, which can be highly asymmetric, particularly in the case of interaction modifications. We introduce pangraphs, a robust and quantitative generalisation of graphs that accurately captures arbitrarily complex higher-order interactions. We demonstrate that several higher-order representations proposed in the literature are specific instances of pangraphs. Additionally, we introduce an incidence multilayer digraph representation of a pangraph, referred to as Levi digraph. We adapt degree and Katz centrality measures to the pangraph framework and show that a consistent generalisation of recursive graph measures cannot be simplified to a Levi digraph of a pangraph. We construct a pangraph for a real-world coffee agroecosystem and compare Katz centrality between its dihypergraph and pangraph representations, both analytically and numerically. The choice of representation significantly affects centrality values and alters vertex ranks. Additionally, we emphasise the use of real-valued incidence matrices to quantify interaction strengths and the roles of vertices within the system.