Switching between pairwise and higher-order synchronization on simplicial complexes.

Resilience Beyond Pairwise Networks

We derive a one-dimensional reduction for nonlinear dynamics on simplicial complexes containing both pairwise and triangular (higher-order) interactions. The effective state is defined using a mixed weight determined by the pairwise and triangular degrees of each node. The resulting reduced equation retains two structural coefficients, associated separately with the pairwise and higher-order coupling channels. A fluctuation expansion identifies the closure assumptions underlying the reduction and shows how deviations of individual node states from the effective state contribute to the approximation error. We numerically validate the proposed framework on Gene-regulatory dynamics, the double-well system, and SIS spreading. The states of the reduced model are compared with full-network simulations through coupling-parameter sweeps, steady-state branch calculations, and progressive node-removal experiments on synthetic and real-world networks. The reduced model successfully reproduces the principal transitions and steady-state branches in all three dynamical systems considered. Agreement is strongest for relatively homogeneous networks and deteriorates when structural heterogeneity produces a broader distribution of node states. The closure diagnostics account for this loss of accuracy and indicate when a single effective state is no longer sufficient. The reduction therefore provides a tractable description of resilience in systems with coexisting pairwise and higher-order interactions.

Onset of synchronization in multiplex networks with higher order interactions

Synchronization of directed hypergraphs with heterogeneities via dynamic coupling

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.

Explosive death in interacting networks under mixed coupling

Explosive death - a discontinuous, first-order transition from oscillatory dynamics to a steady state - is investigated in a framework of interacting networks under mixed coupling. Unlike previous studies focused on monolayer topologies, this work explores the interplay between local diffusive coupling among peripheral nodes and their respective hubs, and conjugate (dissimilar variable) interactions between the hubs of different layers. We demonstrate that by tuning the intralayer and interlayer coupling strengths, the system exhibits a sudden collapse of oscillations into a steady state. The generality of this explosive transition is verified across a diverse range of dynamical systems, including the Stuart-Landau limit cycle oscillator, and the Hindmarsh-Rose bursting neuron model with star and scale-free networks. Our results suggest that the combination of star-type or scale-free network connectivity and mismatch in coupling variables provides a robust mechanism for inducing abrupt transitions to quenching in multilayered nonlinear systems within the explored parameter space.

Perturbation responses on topological synchrony in simplicial Kuramoto model

Synchronization is conventionally understood as the emergence of a phase-locked collective state among interacting dynamical units. However, extending this notion to systems whose dynamical variables are defined on higher-dimensional simplices introduces fundamentally new constraints arising from the topology of the underlying simplicial complex. In the simplicial Kuramoto model, nontrivial topological cycles give rise to a higher dimensional harmonic subspace that is unaffected by the coupling and can therefore drift indefinitely, preventing a globally synchronized state. To resolve this we employ Hodge decomposition on simplicial Kuramoto dynamics and investigated the components. Despite this topological obstruction, we show that the simplicial Kuramoto model admits fixed-point states in the exact and coexact sectors of the Hodge decomposition above critical coupling strengths, which we derive analytically for both sectors. This decomposition provides a generalized notion of synchronization in which the non-harmonic components converge to fixed states. We further investigate the robustness of these fixed-point states to external perturbations. Excluding the drifting and non-interacting harmonic component, the perturbation response is governed by the spectrum of weighted Laplacian and recovers a Kirchhoff index dependence analogous to standard Kuramoto case. In contrast, the harmonic sector is non-dissipative, and consequently the fragility of the system increases with the dimension of this topological subspace. We characterize this effect numerically using triangulated tori with varying first Betti number and find a superlinear scaling relation between system fragility and the dimension of the harmonic sector.

Impact of Higher-Order Interactions on Collective Motion

Collective motion in self-propelled particle systems has been widely studied using the Vicsek model, which relies on pairwise alignment interactions. We introduce a generalized Vicsek model that incorporates higher-order (triadic) alignment interactions. Using agent-based simulations and mean-field theory, we demonstrate that pure triadic alignment induces a discontinuous phase transition, evidenced by hysteresis, a double-well free-energy landscape, and a Binder cumulant minimum that deepens with system size, whereas the standard pairwise model exhibits a continuous transition at the same system sizes. We further show that higher-order interactions require higher particle densities to sustain collective order and produce sharper fluctuation peaks near the transition with lower critical noise. These results establish that the microscopic structure of the alignment interaction, whether pairwise or many-body, is an independent control parameter for the order of the phase transition in active matter, with implications for understanding collective behavior in biological and synthetic systems.