Network science has a representation problem. Complex systems involve nonlinear interactions — epidemic spreading, opinion dynamics, neural computation — but the mathematics is cleaner when dynamics are linear. For decades, the field has handled this by studying nonlinear update rules on simple graphs: nodes have states, edges define who interacts with whom, and the nonlinearity lives entirely in the dynamics.
Recently, a different approach has emerged: explicitly model higher-order interactions using hypergraphs, where edges can connect three or more nodes simultaneously. This shifts some of the complexity from the dynamics to the structure. But the question of when these representations are equivalent — when you can trade dynamical nonlinearity for structural complexity — has been open.
Lacasa provides a precise answer. Multilinear dynamics on the vertices of a graph admit exact, finite-dimensional representations as linear dynamics on a hypergraph. The nonlinearity doesn't disappear — it is absorbed into the interaction structure. A product x_i · x_j in the update rule becomes a single hyperedge state evolving linearly.
The construction uses Carleman linearization, a classical technique that lifts nonlinear systems into higher-dimensional linear ones by treating products of variables as new independent coordinates. For multilinear terms (products where each variable appears at most once), the lifting terminates — the hypergraph has finitely many hyperedges and the equivalence is exact. For general polynomial nonlinearities of degree d, the lifting still terminates, but requires a richer structure: hyper-bag-graphs (hb-graphs), where the same vertex can appear multiple times in a single hyperedge, accommodating terms like x_i^2 · x_j.
For analytic nonlinearities (exponentials, trigonometric functions), the Taylor expansion never terminates, and the lifting is infinite-dimensional — an approximation, not an equivalence. This establishes a sharp boundary: polynomial nonlinearity can be exactly traded for structural complexity; transcendental nonlinearity cannot.
The practical implication is that the debate over whether to model a system with nonlinear graph dynamics or linear hypergraph dynamics is not a modeling choice. For the multilinear case, the two descriptions are the same mathematical object viewed from different coordinates. Which representation is more useful depends on the question, not the system.
The result also constrains the explanatory power of higher-order interactions. If a phenomenon can be fully captured by nonlinear pairwise dynamics, then invoking hypergraph structure adds no new physics — only a different notation for the same dynamics. Genuine higher-order effects would require dynamics that are linear on hypergraphs but cannot be reduced to any nonlinear graph dynamics, which the paper's framework makes testable.