Higher-Order Interactions in Complex Systems: Mechanisms, Behaviour, Representation and Reducibility

Abstract: Higher-order interactions couple three or more elements in ways that cannot be decomposed into pairwise contributions. In diverse complex systems they produce striking collective behaviours such as explosive synchronization, discontinuous transitions and altered stability. Neither such behaviour nor a description written in many-body terms establishes a many-body mechanism, since both depend on how the system is represented. This review distinguishes three questions: what mechanism governs the system, what behaviour does it produce, and how is either represented? Representation covers both which variables are included and how relations among them are encoded. We examine direct many-body mechanisms, nonlinear but pairwise mechanisms that mimic them, and indirect routes through shared environments or eliminated variables, together with temporal aggregation, an artefact of observation rather than a mechanism. The same behaviour can arise from different mechanisms, and a given higher-order mechanism has no universal behavioural consequence. We then formulate reducibility, the question of whether a higher-order model can be replaced by a lower-order one, as a comparison between a source model and an admissible class of lower-order models, relative to a target and a tolerance. The target may be a microscopic rule, an observable or a qualitative feature. Information-theoretic and inferential methods are reviewed as complementary tools for detecting higher-order dependence, testing lower-order alternatives and reconstructing interaction structure. However, observational data alone generally cannot establish the mechanism. We close by identifying open problems in linking mechanisms to measurable signatures, coarse-graining, identifiability and causality, and in extending reductions beyond mean field. (Abridged)