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This paper challenges the traditional distinction between graphs and hypergraphs by proposing that graphs should be viewed as collections of focal neighborhoods centered on individual vertices rather than simply as pairwise connections. The authors demonstrate that while graph-based models can mathematically reproduce the dynamics of higher-order interactions found in hypergraphs, they lose important structural information about groupings, roles, and symmetries that hyperedges explicitly encode. This work establishes that dynamical equivalence between models does not imply structural equivalence, arguing that hypergraphs provide a more natural framework for representing complex interactions even when graph models can emulate their behavior.
Why it matters
Understanding the relationship between graph and hypergraph representations has practical implications for modeling complex systems in biology, social networks, and neural dynamics where group interactions are fundamental. This theoretical clarification could guide researchers in choosing appropriate mathematical frameworks that preserve not just the dynamics but also the interpretable structure of multi-way interactions in real-world systems.
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⚠️ Preprint – Noch nicht peer-reviewed
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Abstract: Graphs and hypergraphs are commonly distinguished by the order of the relations
they represent, with graphs encoding pairwise relations and hypergraphs encoding
interactions among arbitrary groups. We show that, for node-update dynamics, a more
natural viewpoint is to regard a graph as a collection of focal neighbourhoods, each centered
on a vertex and consisting of the vertex and its neighbours. From this
perspective, hypergraphs naturally generalise graph neighbourhoods by allowing
arbitrary interaction domains without requiring a distinguished focal node. This
viewpoint also clarifies the relationship between graph and hypergraph dynamical
models. Although sufficiently general graph-based node functions can reproduce the
dynamics of higher-order interactions, such dynamical equivalence does not imply
equivalence of the underlying interaction structures. Graph representations may
encode higher-order organisation implicitly in their functions while losing the
explicit grouping, roles, and symmetries carried by hyperedges. We therefore
distinguish dynamical expressivity from structural representation and show
that hypergraphs provide a generalisation of graph-based interaction
domains even when their dynamics can be emulated by graph models.
Source: Graphs as focal interaction systems in higher-order dynamics