Applicability of hypergraph-based decomposition to Boolean, canalizing, and threshold node functions

Determine whether and how the hypergraph-based groupwise decomposition of node interaction functions, f_i(x_i, x_{∂i}) = g_i(x_i) + Σ_{c: i∈c} λ_c h_c(x_c), obtained by replacing an adjacency-matrix-based edgewise decomposition f_i(x_i, x_{∂i}) = g_i(x_i) + Σ_j W_{ij} h_{ij}(x_i, x_j) with a hypergraph parameterization, can be applied to models whose node functions are arbitrary Boolean functions, nested canalizing functions, or threshold functions. Clarify the conditions and constructions under which such a mapping is possible for these specific function classes.

Background

In contrasting graph-based and hypergraph-based parameterizations, the paper considers moving from an edgewise decomposition of node functions to a groupwise decomposition over hyperedges. While this replacement is straightforward for certain model classes, the authors note that extending it is not immediate in general.

Specifically, for node dynamics defined by general multivariate functions on neighborhoods (including historically important classes such as arbitrary Boolean functions, nested canalizing functions, and threshold functions), it is unclear how to realize the same functional form via a hypergraph summation over hyperedge terms. Establishing a concrete mapping would clarify the scope of hypergraph parameterizations and delineate when the hypergraph-based groupwise decomposition is applicable.

References

Despite this, one could try to make the case that for each parametrization based on an adjacency matrix \bm W, one could obtain a more general one by keeping everything else the same and replacing it by a hypergraph \bm \lambda, as Eq.~\ref{eq:group-wise} does to Eq.~\ref{eq:graph-wise}, to the extent that makes sense in a particular context (e.g.\ it is not clear how such construction can be applied to arbitrary Boolean functions, the nested canalizing, or threshold functions considered previously).

— Graphs are maximally expressive for higher-order interactions  (2602.16937 - Peixoto et al., 18 Feb 2026) in Section 3 (Hypergraphs constrain rather than generalize interactions), discussion comparing graph-wise and group-wise decompositions