Emergent higher-order structure from fast adaptive networks

We study adaptive network models in which coupling weights evolve on a fast time scale relative to the phase dynamics of the nodes. Using Geometric Singular Perturbation Theory, we prove that, although the microscopic node–edge system contains only pairwise node interactions, the effective slow dynamics on the invariant slow manifold can exhibit genuinely higher-order structure. More precisely, Fenichel reduction produces explicit O ( ε ) triplet terms in the reduced phase dynamics. In addition, we give a rigorous criterion ensuring that these terms are irreducible, in the sense that the reduced vector field does not admit a pairwise decomposition in node coordinates. We derive the first-order slow-manifold correction explicitly, formulate the irreducibility criterion via mixed second derivatives, and verify it for the adaptive Kuramoto phase oscillator model. More generally, the coefficient at order ε n involves at most n + 2 nodes. For adaptive Kuramoto coupling this bound is attained through order N − 2 , so the exact reduced field has interaction order N for all sufficiently small ε > 0 . These sharpness conclusions also hold for an open dense set of sufficiently regular model data, including when the natural frequencies are fixed. Nontruncation of the slow-manifold expansion alone, however, does not imply increasing interaction order.