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Synchronization of directed hypergraphs with heterogeneities via dynamic coupling
Aiwin Thomas Vadakkan, Pietro De Lellis, Mario di Bernardo
TL;DR
Higher-order interactions and parameter mismatches complicate complete synchronization, while existing MSF frameworks do not jointly address mismatches, dynamic coupling, and directed higher-order interactions. This paper derives conditions for synchronization and develops an MSF framework for directed hypergraphs with dynamic coupling. Its simulations validate the predictions and demonstrate complete leader-follower consensus in a heterogeneous opinion-dynamics pinning problem.
Problem
Existing MSF frameworks do not jointly address parameter mismatch, dynamic coupling, and directed higher-order interactions, while heterogeneous agents under diffusive coupling generally achieve only bounded synchronization.
Source: §1 · pp. 1–2Method
The paper derives synchronization-manifold invariance and local-stability conditions, then develops an MSF framework for directed hypergraphs with proportional-integral coupling.
Source: §4 · pp. 6, 9, §6 · p. 14Results
Simulations on directed Lorenz hypergraphs validated the MSF predictions, and opinion-dynamics pinning simulations demonstrated complete leader-follower consensus despite nodal heterogeneity.
Source: §6 · p. 15Takeaways & Limitations
Complete synchronization is possible only when parameter mismatches enter the transverse dynamics as constant additive disturbances that integral action can compensate.
Source: §6 · p. 14The MSF analysis relies on integral action; without it, the synchronization manifold is no longer invariant and the MSF is not meaningful.
Source: §4 · p. 9
Abstract
from arXiv · showhide
Many real-world networks involve interactions among three or more agents that cannot be reduced to pairwise coupling, making hypergraphs a natural modeling framework. In this work, we study complete synchronization in directed hypergraphs of nonlinear agents with parameter mismatches under dynamic diffusive coupling. Although proportional-integral coupling schemes are known to achieve consensus in heterogeneous linear networks, their ability to enforce complete synchronization in nonlinear systems is more limited, generally yielding only bounded synchronization. We show that complete synchronization can be attained only when the effect of parameter mismatches is structurally equivalent, in the transverse dynamics, to a constant disturbance. Under this condition, we derive invariance and local stability conditions for the synchronization manifold and develop a Master Stability Function framework for directed hypergraphs with dynamic coupling. The proposed approach accounts for distinct proportional and integral hypergraph layers and provides spectral criteria for predicting synchronization regions. Numerical simulations on directed hypergraphs of Lorenz oscillators validate the theoretical predictions and show how the integral action can compensate destabilizing effects induced by the proportional layer. We further demonstrate the applicability of the framework to a pinning-control problem in nonlinear opinion dynamics, where dynamic diffusive coupling achieves complete leader-follower consensus in the presence of heterogeneity.
1. Introduction
Higher-order interactions and heterogeneous agent dynamics limit what pairwise, identical-agent synchronization frameworks can represent. This work combines directed hypergraphs, parameter mismatches, dynamic coupling, and an MSF framework to study when complete synchronization is possible.
1. Introduction
- Interactions among three or more agents can be irreducible to pairwise couplings, motivating hypergraphs as a model for higher-order interactions.The passage also notes sensing and actuation settings where only an aggregated group state is measurable.Source: §1 · p. 1
- With parameter mismatches, diffusive coupling generally yields bounded synchronization rather than vanishing synchronization error.The introduction describes a similar bounded steady-state error in consensus under nodal heterogeneity and disturbances.Source: §1 · p. 2
- PI coupling adds integral action to diffusive coupling, but in heterogeneous nonlinear networks it generally achieves only bounded synchronization.Prior PI results cited here establish consensus for heterogeneous linear systems under constant perturbations, not general complete synchronization for nonlinear systems.Source: §1 · p. 2
- The paper derives invariance and local-stability conditions and an MSF framework for predicting synchronization regions in directed hypergraphs with parameter mismatches and dynamic coupling.The framework addresses a combination not covered by existing MSF approaches, according to the authors.Source: §1 · p. 2
- The paper tests its approach on directed hypergraphs of Lorenz systems and a nonlinear opinion-dynamics pinning problem.The authors report that these analyses identify conditions for complete synchronization and underline the relevance of integral action.Source: §1 · p. 2
2. Preliminaries
The preliminaries establish mathematical notation and define the hypergraph structures used in the paper. They distinguish undirected hyperedges from directed hyperedges with ordered tail and head sets.
2. Preliminaries
- The preliminaries define matrix and vector notation, concatenation, real and imaginary parts, spectra, eigenvalue conventions, Kronecker products, and Jacobians.Source: §2 · pp. 2–3
2.1. Hypergraphs
- An undirected hypergraph consists of nodes and nonempty node subsets called hyperedges; subsets larger than two encode higher-order interactions.Rank and co-rank are the maximum and minimum hyperedge cardinalities; equal values define a k-uniform hypergraph.Source: §2.1 · p. 3
- A directed hyperedge has disjoint ordered tail and head sets, and E_j denotes the hyperedges whose head set contains node j.Source: §2.1 · p. 3
3. Problem Statement
The problem statement models heterogeneous nonlinear agents interacting through directed hyperedges and introduces dynamic diffusive coupling to seek complete synchronization. The closed loop has proportional and integral hypergraph layers, with the coupling signals formed from weighted differences between tail and head states.
3. Problem Statement
- The model uses a proportional hyperdiffusive coupling over a directed hypergraph, with node states and parameters entering each agent's nonlinear dynamics.Source: §3 · p. 3
- Each hyperedge sends its heads a signal based on the difference between convex combinations of its tail and head states.The condition g(0) = 0 makes this protocol noninvasive, but parameter mismatches can still prevent synchronization-manifold invariance.Source: §3 · p. 3
- Dynamic diffusive coupling adds an integral action over a directed hypergraph layer, creating a two-layer closed-loop network alongside the proportional layer.The proportional and integral layers are represented by their respective hyperedge sets.Source: §3 · pp. 3–4
4. Results
The results characterize when parameter-mismatched agents can maintain an invariant synchronization manifold under dynamic coupling, then give conditions for its local transverse stability. The stability criteria use a Master Stability Function (MSF) for distinct proportional and integral hypergraph layers, with a separate result for a complete uniform integral layer.
Invariance and mismatch compensation
- The synchronization manifold is invariant only when each node’s vector field plus control input agrees on the manifold.For first-order transverse dynamics, mismatch cancellation requires δu = −(I_N ⊗ J_μf)δμ; persistence requires the mismatch contribution to be constant in time, which is ensured when J_μf is constant.Source: §4 · p. 6
Local stability
- Local transverse stability follows when the integral-layer Laplacian has a simple zero eigenvalue and all transverse dynamic blocks have negative maximum Lyapunov exponents.The theorem also assumes bounded ‖J_xg(0)‖, constant J_μf, and an initially matched integral state.Source: §4 · pp. 7–8
Master Stability Function
- The MSF tests transverse stability by checking whether the proportional- and integral-layer eigenvalue pairs fall in its negative region.The MSF is the maximum Lyapunov exponent of the master stability equation as a function of the two complex coupling parameters.Source: §4 · pp. 8–9
Scope
- The MSF framework depends on integral action; without it, the synchronization manifold is not invariant and only bounded synchronization error is established.The MSF is not meaningful when the integral-layer parameter is zero.Source: §4 · p. 9
Special integral-layer case
- A complete k-uniform integral layer with equal coupling strengths supports an MSF stability criterion even when the general layer assumption does not hold.Under the theorem’s other hypotheses, stability follows when the maximum MSF over the nonzero proportional-layer eigenvalues and the integral-layer parameter is negative.Source: §4 · p. 9
5. Numerical Applications
The numerical applications test the MSF predictions on directed-hypergraph Lorenz systems and demonstrate dynamic coupling in a nonlinear opinion-leader problem. They show synchronization despite destabilizing proportional-layer eigenvalues and complete leader-follower consensus under the stated equilibrium conditions.
5.1.1. The integral action alone can lead to complete synchronization
- A proportional-layer eigenvalue of −0.216 is compatible with synchronization: the MSF predicts the stability boundary, and stronger proportional coupling requires stronger integral action.The example uses an ER-like directed hypergraph with triadic interactions and a complete-graph integral layer.Source: §5.1.1 · p. 11
- For complex layer spectra, a minimal ℜ(𝜚) exists beyond which complete synchronization is attained; this threshold increases with ℑ(𝜚), and similar behavior occurs with proportional control.Source: §5.1.1 · p. 11
5.1.2. Validation of Theorem 2
- In 6-node and 100-node Lorenz networks, the observed transition to synchronization agrees with the MSF prediction.The larger-network example uses coincident ER-like directed-hypergraph layers with triadic interactions.Source: §5.1.2 · pp. 11–12
5.2. An opinion dynamics application
- Dynamic coupling eliminates the persistent opinion mismatch and brings all 150 agents asymptotically to the pinner’s opinion, unlike purely diffusive coupling, which leaves bounded error.The demonstration uses both coupling layers on a real-world social-network topology.Source: §5.2 · pp. 12–13
- The opinion-model Jacobian with respect to its heterogeneous parameter is not constant, but at the limiting consensus equilibrium the mismatch contribution is constant and can be compensated by integral action.Thus the example concerns convergence to a constant leader opinion, not a generic time-varying synchronous trajectory.Source: §5.2 · p. 13
6. Conclusion
Dynamic diffusive coupling can achieve complete synchronization in heterogeneous directed hypergraphs when parameter mismatches act as constant additive disturbances in the transverse dynamics. The work derives invariance and stability criteria and validates its predictions in Lorenz-oscillator and opinion-dynamics simulations.
6. Conclusion
- Complete synchronization is possible only when parameter mismatches enter the transverse dynamics as constant additive disturbances that the integral coupling can compensate.Dynamic diffusive coupling can restore synchronization under these precise structural conditions.Source: §6 · p. 14
- The signed-graph correspondence yields necessary and sufficient local-invariance conditions, and the MSF assesses transverse stability using spectra from both hypergraph layers.The layers are the proportional and integral Laplacians.Source: §6 · pp. 14–15
- The integral layer can enlarge the synchronizability region and sometimes overcome destabilizing effects of negative-real-part eigenvalues in the hypergraph Laplacian.Source: §6 · p. 15
- Lorenz-oscillator simulations validate the MSF-predicted transition between synchronized and nonsynchronized regimes, while opinion-dynamics simulations show complete leader-follower consensus without persistent errors under diffusive coupling.The opinion-dynamics result is demonstrated in the presence of nodal heterogeneity.Source: §6 · p. 15
- Future extensions include broader mismatch classes, time-varying or adaptive topologies, delays, stochastic perturbations, general nonlinear coupling, and less restrictive MSF spectral conditions.Source: §6 · p. 15