Pangraphs: Modeling Complex Higher-Order Interactions

• Pangraphs are quantitative generalizations of graphs that model complex higher-order interactions with distinct asymmetric vertex roles.
• They leverage real-valued incidence matrices and specialized Levi digraphs to encode both interaction strengths and topological details.
• Empirical applications, such as in coffee agroecosystems, demonstrate their capacity to refine centrality measures and elucidate network dynamics.

Pangraphs are a quantitative and robust generalization of traditional graphs designed to accurately model arbitrarily complex higher-order interactions (HOIs) within systems. While standard graphs capture pairwise relationships, and hypergraphs allow the representation of interactions among arbitrary subsets of vertices, pangraphs address the additional requirement of distinguishing the specific, potentially asymmetric roles of vertices in each higher-order interaction. This feature is critical for modeling systems, such as ecological networks or interaction modifications, where the involvement of entities in multi-way interactions is not symmetric or interchangeable (Iskrzyński et al., 14 Feb 2025).

1. Higher-Order Interactions and Existing Graph Models

Complex systems in nature frequently exhibit higher-order interactions, where more than two components concurrently affect each other or the system’s global properties. Traditional graphs represent only dyadic (pairwise) interactions, which limits their ability to capture the full structure of many real-world networks. Hypergraphs partially address this limitation by representing HOIs as hyperedges, i.e., arbitrary subsets of vertices. However, hypergraphs conflate all participating vertices into an unordered set, thereby failing to encode interaction asymmetry or role specificity. In practice, this means that information on directionality or individual roles within a multi-way relationship is lost, potentially altering the interpretation of the network’s dynamical properties (Iskrzyński et al., 14 Feb 2025).

2. Definition and Formalism of Pangraphs

Pangraphs are introduced to capture both the complexity and nuance of higher-order interactions. Pangraphs allow each vertex involved in a higher-order interaction to take on a distinct and possibly asymmetric role. This generality includes and extends many prior higher-order network representations. Several models proposed in the literature—such as some forms of hypergraphs and directed hypergraphs—are shown to be special cases or restrictions of the pangraph framework. Pangraphs are quantitative, enabling the assignment of real values to vertices’ participation in interactions, thus supporting both the topological and weighted analysis of complex systems (Iskrzyński et al., 14 Feb 2025).

3. Incidence Representation and Levi Digraphs

Pangraphs admit a flexible representation using incidence structures, including real-valued incidence matrices that encode the strength of association between vertices and their roles within interactions. To further conceptualize the structure, the Levi digraph is introduced: an incidence multilayer directed graph representation of a pangraph. This construction enables the translation of some problems in pangraph theory to the traditional field of multilayer (hyper)graph analysis. Despite this, the adaptation of recursively defined graph measures to pangraphs encounters fundamental obstacles. A consistent generalization of such measures cannot be reduced to analysis on the associated Levi digraph alone, suggesting that pangraphs possess unique structural features irreducible to classical incidence digraphs (Iskrzyński et al., 14 Feb 2025).

4. Generalization and Specialization of Higher-Order Representations

The pangraph formalism is demonstrated to encompass a range of higher-order network models developed previously. By specializing the way vertex roles and incidence information are encoded, pangraphs can recover conventional hypergraphs, dihypergraphs, and their variants as special cases. This generality positions pangraphs as a unifying framework for higher-order network theory, providing both the expressive power to encode complex HOI structure and a quantifiable basis for analysis (Iskrzyński et al., 14 Feb 2025).

5. Centrality Measures in Pangraphs

Traditional network centrality measures, such as degree and Katz centrality, quantify node importance within the context of pairwise graphs. Pangraphs require adaptation of these measures to account for generalized HOI participation and role asymmetry. Degree and Katz centrality are redefined within the pangraph context, quantifying both the frequency and quality (e.g., interaction strength) of a vertex’s involvement. Importantly, analytical and numerical comparisons between pangraph- and dihypergraph-based centralities reveal that the choice of higher-order representation significantly influences calculated values and the inferred rankings of vertices in empirical systems (Iskrzyński et al., 14 Feb 2025).

6. Application to Empirical Systems: The Coffee Agroecosystem

The utility of pangraphs is demonstrated through their application to a real-world ecological system: a coffee agroecosystem. In this case study, both analytic and computational assessments of Katz centrality are performed using pangraph and dihypergraph representations. The results show substantive differences in centrality values and vertex rankings between the two frameworks, underscoring the practical significance of faithful modeling of HOIs and participant roles. The adoption of real-valued incidence matrices enables precise quantification of both interaction strengths and vertex roles, further facilitating rigorous network analysis in ecological and other complex systems (Iskrzyński et al., 14 Feb 2025).

7. Quantitative Role Assignment and Interaction Strength

Pangraphs explicitly promote the use of real-valued incidence matrices to represent both the structure and quantitative strength of higher-order interactions, as well as the particular roles that vertices occupy within them. This quantification supports the analysis of not only network topology but also interaction heterogeneity, impact strength, and asymmetric participation. Such analytic tools are requisite for investigating stability and dynamical behavior in complex networks where differential roles and interaction strengths modulate system behavior (Iskrzyński et al., 14 Feb 2025).