Explosive synchronization in mono and multilayer networks

A recent study has found an explosive synchronization in a Kurammoto model on scale-free networks when the natural frequencies of oscillators are equal to their degrees. In this work, we introduce a quantity to characterize the correlation between the structural and the dynamical properties and investigate the impacts of the correlation on the synchronization transition in the Kuramoto model on scale-free networks. We find that the synchronization transition may be either a continuous one or a discontinuous one depending on the correlation and that strong correlation always postpones both the transitions from the incoherent state to a synchronous one and the transition from a synchronous state to the incoherent one. We find that the dependence of the synchronization transition on the correlation is also valid for other types of distributions of natural frequency.

The neural networks of the human brain exhibit dual structural characteristics of smallworld and scale-free networks, and their electrical activity is readily modulated by electromagnetic fields. Constructing neural network models that mimic biological structures is crucial for elucidating the brain's information processing mechanisms and the pathological basis of neurological disorders. This paper constructs a Small-World Scale-Free Neural Network (SWSFNN) model under electromagnetic effects using discrete memristors. By optimizing network topology via graph theory, we systematically investigate how memristor initial values and electromagnetic induction intensity influence the network dynamics. Numerical simulations reveal that memristor initial values affect neuronal firing patterns and regulate network synchronization. We also discovered a spontaneous “synchronization – cluster synchronization – synchronization” transition under constant parameters. This demonstrates that even without parameter changes, the inherent nonlinear interactions within the neural network system can drive spontaneous state transitions, yielding rich dynamical behaviors. Furthermore, increasing electromagnetic induction intensity similarly enhances network synchronization. This study provides theoretical foundations for understanding the nonlinear dynamical mechanisms and synchronization control of neural networks in electromagnetic environments, offering insights for neural computation and information processing.

Networks of phase oscillators have become a widely established paradigmatic model for studying emergent collective behavior across several real-world systems, including neuronal networks, populations of chemical oscillators, and power grids. The Kuramoto model, involving one-dimensional or two-dimensional phase oscillators, demonstrates the potential for networks to showcase exceptional collective dynamics. This encompasses various outcomes such as full, partial, explosive, and asymmetry-induced synchronization, clusters, chimeras, solitary states, and generalized splay states. Notably, increasing all-to-all coupling in the classical Kuramoto model induces full synchronization as the most probable outcome and dominant rhythm. Kuramoto networks with repulsive coupling usually display splay, generalized, and cluster splay states, but the conditions under which a certain rhythm can arise and prevail are not entirely understood. Equally important for connecting Kuramoto networks to practical physical systems is understanding the function of higher-order coupling terms. These terms display a Fourier decomposition of a general 2π-periodic interaction function [1]. Previous studies have demonstrated that the inclusion of higher-order terms in the classical Kuramoto model of oscillators with all-to-all attractive coupling can lead to multiple synchronous states and switching between synchronization clusters. However, the impact of higher-order coupling modes on rhythm generation in repulsive networks remains unexplored. In this work, we present significant progress in addressing the critical issue related to repulsive Kuramoto–Sakaguchi networks of phase oscillators with phase-lagged first-order and higher-order coupling. We demonstrate that weakly repulsive networks of even and odd numbers of oscillators with first-order coupling are dominated by two-cluster and three-cluster splay states, respectively. The three-cluster splay states consist of two distinct coherent clusters and one solitary oscillator. These tripod states can be considered a fusion of a two-body chimera and a solitary state. We have dubbed these patterns of three oscillators as “Cyclops states” in reference to the Greek mythological giant with a single eye. The solitary oscillator and synchronous clusters respectively represent the Cyclops’ eye and shoulders. We present a remarkable discovery that the inclusion of higher-order coupling modes leads to worldwide stability of cyclops states across almost the entire range of the phase-lag parameter controlling repulsion [2]. Beyond the Kuramoto oscillators, we demonstrate the robust presence of this effect in networks of canonical theta-neurons with adaptive coupling. Furthermore, our results provide insight into identifying dominant rhythms within repulsive physical and biological networks.

The Kuramoto model consisting of large ensembles of coupled phase oscillators serves as an illustrative paradigm for studying the synchronization transitions and collective behaviors in various self-sustained systems. In recent years, the research of the high-order coupled phase oscillators has attracted extensive interest for the high-order coupled structure playing an essential role in modeling the dynamics of code and data storage. By studying the effects of high-order coupling, we extend the Kuramoto model of high-order structure by considering the correlations between frequency and coupling, which reflects the intrinsic properties of heterogeneity of interactions between oscillators. Several novel dynamic phenomena occur in the model, including clustering, extensive multistability, explosive synchronization and oscillatory state. The universal form of the critical coupling strength characterizing the transition from disorder to order is obtained via an analysis of the stability of the incoherent state. Furthermore, we present the self-consistent approach and find the multi-cluster with their expressions of order parameters. The stability analysis of multi-cluster is performed in the subspace getting stability condition together with the stable solutions of order parameters. The discussion of all the results summarizes the mechanism of the transition from hysteresis to oscillatory states. In addition, we emphasize that the combination of the Kuramoto order parameter capturing the asymmetry of the system and the Daido order parameter representing the clustering can give a complete description of the macroscopic dynamics of the system. The research of this paper can improve the understanding of the effects of the heterogeneity among populations and the explosive synchronization occurring in higher-order coupled phase oscillators.

Indicating the synchronization bifurcation points using the early warning signals in two case studies: Continuous and explosive synchronization

We investigate the phenomenon of first order transition (explosive\nsynchronization (ES)) in adaptively coupled phase-frustrated bi-layer multiplex\nnetwork. We consider Sakaguchi-Kuramoto (SK) dynamics over the top of multiplex\nnetworks and we establish that ES can emerge in all layers of a multiplex\nnetwork even when one of the layers may not exhibit ES in the absence of the\ninter layer connections. We clearly identify the regions of the parameter\nspace, in which the multiplexity wins over frustration parameter and network\nstructure for the emergence of ES. Based on the mean field analysis around the\ncoherent state and a perturbative approach around the incoherent state we\nanalytically derive the synchronization transition points (backward and\nforward) of all layers of multiplex network as well as its mono-layer\ncounterpart satisfying a close agreement with the numerical results.\n

Networks incorporating higher-order interactions are increasingly recognized for their ability to introduce novel dynamics into various processes, including synchronization. Previous studies on synchronization within multilayer networks have often been limited to specific models, such as the Kuramoto model, or have focused solely on higher-order interactions within individual layers. Here, we present a comprehensive framework for investigating synchronization, particularly global synchronization, in multilayer networks with higher-order interactions. Our framework considers interactions beyond pairwise connections, both within and across layers. We demonstrate the existence of a stable global synchronous state, with a condition resembling the master stability function, contingent on the choice of coupling functions. Our theoretical findings are supported by simulations using Hindmarsh-Rose neuronal and Rössler oscillators. These simulations illustrate how synchronization is facilitated by higher-order interactions, both within and across layers, highlighting the advantages over scenarios involving interactions within single layers. Published by the American Physical Society 2024

Continuous and explosive synchronization of phase oscillators on star network: Effect of degree-frequency correlations and time-delays

Explosive synchronization transition in a ring of coupled oscillators

In this work, we study the synchronization in a generalized Kuramoto model with frequency-weighted coupling. In particular, we focus on the situations in which the frequency distributions of oscillators are asymmetric. For typical unimodal frequency distributions, such as Lorentzian, Gaussian, triangle, and even special Rayleigh, we find that the synchronization transition in the model generally converts from the first order to the second order as the central frequency shifts toward positive direction. We characterize two interesting coherent states in the system: In the former, two phase-locking clusters are formed, rotating with the same frequency. They correspond to those oscillators with relatively high frequencies while the oscillators with relatively small frequencies are not entrained. In the latter, two phase-locking clusters rotate with different frequencies, leading to the oscillation of the order parameter. We further conduct theoretical analysis to reveal the relation between the asymmetric frequency distribution and the conversion of synchronization type, and justify the coherent states observed in the system.

The accurate construction of mono-, bi- and multi-layer networks has been an important challenge, especially for bi- and multi-layer networks. Monolayer, bilayer, sandwich bilayer, four-layer, and multi-layer two-dimensional pillararene-type metal-organic coordination networks have been constructed from functionalized pillar[5]arene and pillar[6]arene by utilizing the coordination interaction of cobalt and copper ions and combining with temperature control and guest induction. These two-dimensional coordination networks exhibit the excellent plasticity of pillararenes and structural variety, which are characterized by X-ray single crystal diffraction and PXRD, confirming that pillararenes units can function as excellent tunable scaffolds for structural regulation. Two-dimensional chiral double-layer structure products are also constructed from R- and S-pillar[6]arene, which are obtained by high-performance liquid chromatography. Atomic force microscopic imaging confirms the thicknesses of these networks. Moreover, these networks also exhibit high iodine adsorption capacity in aqueous environments at ambient temperature. The monolayer, bilayer, sandwich bilayer, four-layer and multi-layer structures of the pillararene-type networks represent a new facile supramolecular self-assembly strategy and platform for designing more mono-, bi- and multi-layer two-dimensional nanomaterials and chiral two-dimensional double-layer structures provide a new method for the construction of more two-dimensional chiral polymers.

Topology analysis of Lanzhou public transport network based on double-layer complex network theory

Synchronization of networked oscillators is known to depend fundamentally on the interplay between the dynamics of the graph's units and the microscopic arrangement of the network's structure. We here propose an effective network whose topological properties reflect the interplay between the topology and dynamics of the original network. On that basis, we are able to introduce the effective centrality, a measure that quantifies the role and importance of each network's node in the synchronization process. In particular, in the context of explosive synchronization, we use such a measure to assess the propensity of a graph to sustain an irreversible transition to synchronization. We furthermore discuss a strategy to induce the explosive behavior in a generic network, by acting only upon a fraction of its nodes.

We reveal that an introduction of frequency-weighted inter-layer coupling term in networks of frequency dipoles can induce explosive synchronization transitions. The reason for explosive synchronization is that the oscillators with synchronization superiority are moderately suppressed. The findings show that a super-linear correlation induces explosive synchronization in networks of frequency dipoles, while a linear or sub-linear correlation excites chimera-like states. Clearly, the synchronization transition mode of networks of frequency dipoles is controlled by the power-law exponent. In addition, by means of the mean-field approximation, we obtain the critical values of the coupling strength within and between layers in two limit cases. The results of theoretical analysis are in good agreement with those of numerical simulation. Compared with the previous models, the model proposed in this paper retains the topological structure of network and the intrinsic properties of oscillators, so it is easy to realize pinning control.

In this paper, we study cluster synchronization in networks of coupled nonidentical dynamical systems. The vertices in the same cluster have the same dynamics of uncoupled node system but the uncoupled node systems in different clusters are different. We present conditions guaranteeing cluster synchronization and investigate the relation between cluster synchronization and the unweighted graph topology. We indicate that two conditions play key roles for cluster synchronization: the common intercluster coupling condition and the intracluster communication. From the latter one, we interpret the two cluster synchronization schemes by whether the edges of communication paths lie in inter- or intracluster. By this way, we classify clusters according to whether the communications between pairs of vertices in the same cluster still hold if the set of edges inter- or intracluster edges is removed. Also, we propose adaptive feedback algorithms to adapting the weights of the underlying graph, which can synchronize any bi-directed networks satisfying the conditions of common intercluster coupling and intracluster communication. We also give several numerical examples to illustrate the theoretical results.