Graphs are focal hypergraphs: strict containment in higher-order interaction dynamics

Abstract

We introduce a taxonomy of interaction types and show that graphs are focal hypergraphs: every graph is canonically a focal hypergraph via its closed neighbourhood structure, and every graph dynamical model is a special case of the general hypergraph dynamical model. The central distinction is between \emph{focal} interactions, in which the interaction domain is defined relative to a designated reference node, and \emph{non-focal} interactions, in which all participants stand in equivalent structural relationship. Closed graph neighbourhoods are precisely focal hyperedges, so hyperedges generalise graph neighbourhoods by removing the focal constraint. This yields a strict three-level hierarchy: graph models $\subsetneq$ focal hypergraph models $\subsetneq$ general hypergraph models. Moreover, graph models do encode genuinely higher-order (many-body) interactions, in the sense that each node's update function may depend jointly on all members of its closed neighbourhood, but they remain a strict special case of the hypergraph dynamical model, not equivalent to it. We further show that universal encodings such as bipartite factor graphs are neutral with respect to this hierarchy, and that the symmetry condition of the hypergraph dynamical model -- often treated as an additional constraint relative to the graph model -- is in fact the dynamical definition of a non-focal interaction. The taxonomy is grounded in concrete phenomena from physics, biology, ecology, and social systems, and yields a principle of representational alignment: the choice between graph and hypergraph models should be governed by the type of interaction, not by a blanket preference for one formalism over the other.