In network science, collective dynamics of complex systems are typically modeled as (nonlinear, often including many-body) vertex-level update rules evolving over a graph interaction structure. In recent years, frameworks that explicitly model such higher-order interactions in the interaction backbone (e.g., hypergraphs) have been advanced, somehow shifting the imputation of the effective nonlinearity from the dynamics to the interaction structure. In this work, we discuss such structural-dynamical representation duality, and we investigate how and when a nonlinear dynamics defined on the vertex set of a graph allows an equivalent representation in terms of a linear dynamics defined on the state space of a sufficiently richer, higher-order interaction structure. Using Carleman linearization arguments, we show that finite polynomial dynamics defined in the $\left|V\right|$ vertices of a graph admit an exact representation as linear dynamics on the state space of an hb-graph of order $\left|V\right|$, a combinatorial structure that extends hypergraphs by allowing vertex multiplicity, where the specific shape of the nonlinearity indicates whether the hb-graph is either finite or infinite (in terms of the number of hb-edges). For more general analytic nonlinearities, exact linear representation always require an hb-graph of infinite size, and its finite-size truncation provides an approximate representation of the original nonlinear graph-based dynamics.

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• Collective behavior in networks