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Emergent Higher-Order Structure from Fast Adaptive Networks

Christian Kuehn, Fergal Murphy

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.

Emergent Higher-Order Structure from Fast Adaptive Networks

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 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.
Paper Structure (11 sections, 8 theorems, 67 equations)

This paper contains 11 sections, 8 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Pairwise and higher-order vector fields
  3. Geometric Singular Perturbation Theory
  4. Limiting problems and normal hyperbolicity
  5. Graph case and first-order expansion
  6. Adaptive network models and emergent triplet interactions
  7. Model class
  8. Critical manifold and slow manifold
  9. First-order correction and reduced phase dynamics
  10. Genuinely nonpairwise structure in the reduced flow
  11. Discussion and Outlook

Key Result

Proposition 2.4

If $F\in \mathcal{V}_{\mathrm{pair}}$ and $\Phi$ is node-respecting, then the pushforward vector field $\Phi_*F$ also belongs to $\mathcal{V}_{\mathrm{pair}}$.

Theorems & Definitions (24)

  • Definition 2.1: Pairwise vector field
  • Definition 2.2: Genuinely higher-order vector field
  • Definition 2.3: Node-respecting diffeomorphism
  • Proposition 2.4: Invariance of the pairwise class
  • proof
  • Remark 2.5: Coordinate artefacts versus emergence
  • Proposition 2.6: Mixed-derivative vanishing for pairwise fields
  • proof
  • Corollary 2.7: Mixed-derivative certificate
  • Theorem 3.1: Fenichel Fenichel1979Kuehn2015
  • ...and 14 more