Emergent Higher-Order Structure from Fast Adaptive Networks

TL;DR

This paper demonstrates that adaptive networks with fast-changing coupling weights can exhibit emergent higher-order interactions in their effective slow dynamics, which are not reducible to pairwise interactions, using rigorous mathematical analysis.

Contribution

The authors prove that pairwise adaptive networks can generate higher-order structures in slow dynamics and provide explicit criteria and corrections for this phenomenon.

Findings

Higher-order interactions emerge in slow dynamics due to adaptive coupling.

The irreducibility criterion is formulated via mixed second derivatives.

The results are verified for the adaptive Kuramoto model.

Abstract

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 (GSPT), we prove that, although the microscopic system is strictly pairwise, the effective slow dynamics on the invariant slow manifold can exhibit genuinely higher-order structure. More precisely, Fenichel reduction produces explicit $O(\varepsilon)$ 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. The results show that the class of pairwise-coupled fast--slow adaptive network systems is not closed under slow-manifold reduction.