Contrarians Synchronize beyond the Limit of Pairwise Interactions

Exotic swarming dynamics of high-dimensional swarmalators

Swarmalators are oscillators that can swarm as well as sync via a dynamic balance between their spatial proximity and phase similarity. Swarmalator models employed so far in the literature comprise only one-dimensional phase variables to represent the intrinsic dynamics of the natural collectives. Nevertheless, the latter can indeed be represented more realistically by high-dimensional phase variables. For instance, the alignment of velocity vectors in a school of fish or a flock of birds can be more realistically set up in three-dimensional space, while the alignment of opinion formation in population dynamics could be multidimensional, in general. We present a generalized D-dimensional swarmalator model, which more accurately captures self-organizing behaviors of a plethora of real-world collectives by self-adaptation of high-dimensional spatial and phase variables. For a more sensible visualization and interpretation of the results, we restrict our simulations to three-dimensional spatial and phase variables. Our model provides a framework for modeling complicated processes such as flocking, schooling of fish, cell sorting during embryonic development, residential segregation, and opinion dynamics in social groups. We demonstrate its versatility by capturing the maneuvers of a school of fish, qualitatively and quantitatively, by a suitable extension of the original model to incorporate appropriate features besides a gallery of its intrinsic self-organizations for various interactions. We expect the proposed high-dimensional swarmalator model to be potentially useful in describing swarming systems and programmable and reconfigurable collectives in a wide range of disciplines, including the physics of active matter, developmental biology, sociology, and engineering.

Solvable dynamics of the three-dimensional Kuramoto model with frequency-weighted coupling

The presence of coupling heterogeneity is deemed to be a natural attribute in realistic systems comprised of many interacting agents. In this work, we study dynamics of the 3D Kuramoto model with heterogeneous couplings, where the strength of coupling for each agent is weighted by its intrinsic rotation frequency. The critical coupling strength for the instability of incoherence is rigorously derived to be zero by carrying out a linear stability analysis of an incoherent state. For positive values of the coupling strength, at which the incoherence turns out to be unstable, a self-consistency approach is developed to theoretically predict the degree of global coherence of the model. Our theoretical analyses match well with numerical simulations, which helps us to deepen the understanding of collective behaviors spontaneously emerged in heterogeneously coupled high-dimensional dynamical networks.

Higher-Order Components Dictate Higher-Order Contagion Dynamics in Hypergraphs

The presence of the giant component is a necessary condition for the emergence of collective behavior in complex networked systems. Unlike networks, hypergraphs have an important native feature that components of hypergraphs might be of higher order, which could be defined in terms of the number of common nodes shared between hyperedges. Although the extensive higher-order component (HOC) could be witnessed ubiquitously in real-world hypergraphs, the role of the giant HOC in collective behavior on hypergraphs has yet to be elucidated. In this Letter, we demonstrate that the presence of the giant HOC fundamentally alters the outbreak patterns of higher-order contagion dynamics on real-world hypergraphs. Most crucially, the giant HOC is required for the higher-order contagion to invade globally from a single seed. We confirm it by using synthetic random hypergraphs containing adjustable and analytically calculable giant HOC.

Effect of higher-order interactions on chimera states in two populations of Kuramoto oscillators

We investigate the effect of the fraction of pairwise and higher-order interactions on the emergent dynamics of the two populations of globally coupled Kuramoto oscillators with phase-lag parameters. We find that the stable chimera exists between saddle-node and Hopf bifurcations, while the breathing chimera lives between Hopf and homoclinic bifurcations in the two-parameter phase diagrams. The higher-order interaction facilitates the onset of the bifurcation transitions at a much lower disparity between the inter- and intra-population coupling strengths. Furthermore, the higher-order interaction facilitates the spread of breathing chimera in a large region of the parameter space while suppressing the spread of the stable chimera. A low degree of heterogeneity among the phase-lag parameters promotes the spread of both stable chimera and breathing chimera to a large region of the parameter space for a large fraction of the higher-order coupling. In contrast, a large degree of heterogeneity is found to decrease the spread of both chimera states for a large fraction of the higher-order coupling. A global synchronized state is observed above a critical value of heterogeneity among the phase-lag parameters. We have deduced the low-dimensional evolution equations for the macroscopic order parameters using the Ott-Antonsen Ansatz. We have also deduced the analytical saddle-node and Hopf bifurcation curves from the evolution equations for the macroscopic order parameters and found them to match with the bifurcation curves obtained using the software XPPAUT and with the simulation results.

Order parameter dynamics in complex systems: From models to data

Collective ordering behaviors are typical macroscopic manifestations embedded in complex systems and can be ubiquitously observed across various physical backgrounds. Elements in complex systems may self-organize via mutual or external couplings to achieve diverse spatiotemporal coordinations. The order parameter, as a powerful quantity in describing the transition to collective states, may emerge spontaneously from large numbers of degrees of freedom through competitions. In this minireview, we extensively discussed the collective dynamics of complex systems from the viewpoint of order-parameter dynamics. A synergetic theory is adopted as the foundation of order-parameter dynamics, and it focuses on the self-organization and collective behaviors of complex systems. At the onset of macroscopic transitions, slow modes are distinguished from fast modes and act as order parameters, whose evolution can be established in terms of the slaving principle. We explore order-parameter dynamics in both model-based and data-based scenarios. For situations where microscopic dynamics modeling is available, as prototype examples, synchronization of coupled phase oscillators, chimera states, and neuron network dynamics are analytically studied, and the order-parameter dynamics is constructed in terms of reduction procedures such as the Ott-Antonsen ansatz, the Lorentz ansatz, and so on. For complicated systems highly challenging to be well modeled, we proposed the eigen-microstate approach (EMP) to reconstruct the macroscopic order-parameter dynamics, where the spatiotemporal evolution brought by big data can be well decomposed into eigenmodes, and the macroscopic collective behavior can be traced by Bose-Einstein condensation-like transitions and the emergence of dominant eigenmodes. The EMP is successfully applied to some typical examples, such as phase transitions in the Ising model, climate dynamics in earth systems, fluctuation patterns in stock markets, and collective motion in living systems.

Contrarian role of phase and phase velocity coupling in synchrony of second-order phase oscillators

Positive phase coupling plays an attractive role in inducing in-phase synchrony in an ensemble of phase oscillators. Positive coupling involving both amplitude and phase continues to be attractive, leading to complete synchrony in identical oscillators (limit cycle or chaotic) or phase coherence in oscillators with heterogeneity of parameters. In contrast, purely positive phase velocity coupling may originate a repulsive effect on pendulumlike oscillators (with rotational motion) to bring them into a state of diametrically opposite phases or a splay state. Negative phase velocity coupling is necessary to induce synchrony or coherence in the general sense. The contrarian roles of phase coupling and phase velocity coupling on the synchrony of networks of second-order phase oscillators have been explored here. We explain our proposition using networks of two model systems, a second-order phase oscillator representing the pendulum or the superconducting Josephson junction dynamics, and a voltage-controlled oscillations in neurons model. Numerical as well as semianalytical approaches are used to confirm our results.

Synchronization onset for contrarians with higher-order interactions in multilayer systems

We investigate the impact of contrarians (via negative coupling) in multilayer networks of phase oscillators having higher-order interactions. We report that the multilayer framework facilitates synchronization onset in the negative pairwise coupling regime. The multilayering strength governs the onset of synchronization and the nature of the phase transition, whereas the higher-order interactions dictate the backward critical coupling. Specifically, the system does not synchronize below a critical value of the multilayering strength. The analytical calculations using the mean-field Ott-Antonsen approach agree with the simulations. The results presented here may be useful for understanding emergent behaviors in real-world complex systems with contrarians and higher-order interactions, such as the brain and social system.

Impact of phase lag on synchronization in frustrated Kuramoto model with higher-order interactions

The study of first order transition (explosive synchronization) in an ensemble (network) of coupled oscillators has been the topic of paramount interest among the researchers for more than one decade. Several frameworks have been proposed to induce explosive synchronization in a network and it has been reported that phase frustration in a network usually suppresses first order transition in the presence of pairwise interactions among the oscillators. However, on the contrary, by considering networks of phase frustrated coupled oscillators in the presence of higher-order interactions (up to 2-simplexes) we show here, under certain conditions, phase frustration can promote explosive synchronization in a network. A low-dimensional model of the network in the thermodynamic limit is derived using the Ott-Antonsen ansatz to explain this surprising result. Analytical treatment of the low-dimensional model, including bifurcation analysis, explains the apparent counter intuitive result quite clearly.

Higher-order interactions in Kuramoto oscillators with inertia

How do higher-order interactions influence the dynamical landscape of a network of the second-order phase oscillators? We address this question using three coupled Kuramoto phase oscillators with inertia under pairwise and higher-order interactions, in search of various collective states, and new states, if any, that show marginal presence or absence under pairwise interactions. We explore this small network for varying phase lag in the coupling and over a range of negative to positive coupling strength of pairwise as well as higher-order or group interactions. In the extended coupling parameter plane of the network we record several well-known states such as synchronization, frequency chimera states, and rotating waves that appear with distinct boundaries. In the parameter space, we also find states generated by the influence of higher-order interactions: The 2+1 antipodal point and the 2+1 phase-locked states. Our results demonstrate the importantance of the choices of the phase lag and the sign of the higher-order coupling strength for the emergent dynamics of the network. We provide analytical support to our numerical results.

Asymmetry in the Kuramoto model with nonidentical coupling

Synchronization and desynchronization of coupled oscillators appear to be the key property of many physical systems. It is believed that to predict a synchronization (or desynchronization) event, the knowledge on the exact structure of the oscillatory network is required. However, natural sciences often deal with observations where the coupling coefficients are not available. In the present paper we suggest a way to characterize synchronization of two oscillators without the reconstruction of coupling. Our method is based on the Kuramoto chain with three oscillators with constant but nonidentical coupling. We characterize coupling in this chain by two parameters: the coupling strength s and disparity σ. We give an analytical expression of the boundary s_{max} of synchronization occurred when s>s_{max}. We propose asymmetry A of the generalized order parameter induced by the coupling disparity as a new characteristic of the synchronization between two oscillators. For the chain model with three oscillators we present the self-consistent inverse problem. We explore scaling properties of the asymmetry A constructed for the inverse problem. We demonstrate that the asymmetry A in the chain model is maximal when the coupling strength in the model reaches the boundary of synchronization s_{max}. We suggest that the asymmetry A may be derived from the phase difference of any two oscillators if one pretends that they are edges of an abstract chain with three oscillators. Performing such a derivation with the general three-oscillator Kuramoto model, we show that the crossover from the chain to general network of oscillators keeps the interrelation between the asymmetry A and synchronization. Finally, we apply the asymmetry A to describe synchronization of the solar magnetic field proxies and discuss its potential use for the forecast of solar cycle anomalies.

Higher-order interactions shape collective dynamics differently in hypergraphs and simplicial complexes

Higher-order networks have emerged as a powerful framework to model complex systems and their collective behavior. Going beyond pairwise interactions, they encode structured relations among arbitrary numbers of units through representations such as simplicial complexes and hypergraphs. So far, the choice between simplicial complexes and hypergraphs has often been motivated by technical convenience. Here, using synchronization as an example, we demonstrate that the effects of higher-order interactions are highly representation-dependent. In particular, higher-order interactions typically enhance synchronization in hypergraphs but have the opposite effect in simplicial complexes. We provide theoretical insight by linking the synchronizability of different hypergraph structures to (generalized) degree heterogeneity and cross-order degree correlation, which in turn influence a wide range of dynamical processes from contagion to diffusion. Our findings reveal the hidden impact of higher-order representations on collective dynamics, highlighting the importance of choosing appropriate representations when studying systems with nonpairwise interactions.

Solvable Dynamics of Coupled High-Dimensional Generalized Limit-Cycle Oscillators

We introduce a new model consisting of globally coupled high-dimensional generalized limit-cycle oscillators, which explicitly incorporates the role of amplitude dynamics of individual units in the collective dynamics. In the limit of weak coupling, our model reduces to the D-dimensional Kuramoto phase model, akin to a similar classic construction of the well-known Kuramoto phase model from weakly coupled two-dimensional limit-cycle oscillators. For the practically important case of D=3, the incoherence of the model is rigorously proved to be stable for negative coupling (K<0) but unstable for positive coupling (K>0); the locked states are shown to exist if K>0; in particular, the onset of amplitude death is theoretically predicted. For D≥2, the discrete and continuous spectra for both locked states and amplitude death are governed by two general formulas. Our proposed D-dimensional model is physically more reasonable, because it is no longer constrained by fixed amplitude dynamics, which puts the recent studies of the D-dimensional Kuramoto phase model on a stronger footing by providing a more general framework for D-dimensional limit-cycle oscillators.

An optimization-based algorithm for obtaining an optimal synchronizable network after link addition or reduction

Achieving a network structure with optimal synchronization is essential in many applications. This paper proposes an optimization algorithm for constructing a network with optimal synchronization. The introduced algorithm is based on the eigenvalues of the connectivity matrix. The performance of the proposed algorithm is compared with random link addition and a method based on the eigenvector centrality. It is shown that the proposed algorithm has a better synchronization ability than the other methods and also the scale-free and small-world networks with the same number of nodes and links. The proposed algorithm can also be applied for link reduction while less disturbing its synchronization. The effectiveness of the algorithm is compared with four other link reduction methods. The results represent that the proposed algorithm is the most appropriate method for preserving synchronization.

Synchronization in repulsively coupled oscillators

A long-standing expectation is that two repulsively coupled oscillators tend to oscillate in opposite directions. It has been difficult to achieve complete synchrony in coupled identical oscillators with purely repulsive coupling. Here, we introduce a general coupling condition based on the linear matrix of dynamical systems for the emergence of the complete synchronization in pure repulsively coupled oscillators. The proposed coupling profiles (coupling matrices) define a bidirectional cross-coupling link that plays the role of indicator for the onset of complete synchrony between identical oscillators. We illustrate the proposed coupling scheme on several paradigmatic two-coupled chaotic oscillators and validate its effectiveness through the linear stability analysis of the synchronous solution based on the master stability function approach. We further demonstrate that the proposed general condition for the selection of coupling profiles to achieve synchronization even works perfectly for a large ensemble of oscillators.

Diffusion-driven instability of topological signals coupled by the Dirac operator

The study of reaction-diffusion systems on networks is of paramount relevance for the understanding of nonlinear processes in systems where the topology is intrinsically discrete, such as the brain. Until now, reaction-diffusion systems have been studied only when species are defined on the nodes of a network. However, in a number of real systems including, e.g., the brain and the climate, dynamical variables are not only defined on nodes but also on links, faces, and higher-dimensional cells of simplicial or cell complexes, leading to topological signals. In this work, we study reaction-diffusion processes of topological signals coupled through the Dirac operator. The Dirac operator allows topological signals of different dimension to interact or cross-diffuse as it projects the topological signals defined on simplices or cells of a given dimension to simplices or cells of one dimension up or one dimension down. By focusing on the framework involving nodes and links, we establish the conditions for the emergence of Turing patterns and we show that the latter are never localized only on nodes or only on links of the network. Moreover, when the topological signals display a Turing pattern their projection does as well. We validate the theory hereby developed on a benchmark network model and on square lattices with periodic boundary conditions.

Dynamics on higher-order networks: a review

Network science has evolved into an indispensable platform for studying complex systems. But recent research has identified limits of classical networks, where links connect pairs of nodes, to comprehensively describe group interactions. Higher-order networks, where a link can connect more than two nodes, have therefore emerged as a new frontier in network science. Since group interactions are common in social, biological and technological systems, higher-order networks have recently led to important new discoveries across many fields of research. Here, we review these works, focusing in particular on the novel aspects of the dynamics that emerges on higher-order networks. We cover a variety of dynamical processes that have thus far been studied, including different synchronization phenomena, contagion processes, the evolution of cooperation and consensus formation. We also outline open challenges and promising directions for future research.