We study optimal synchronization in networks of heterogeneous phase oscillators. Our main result is the derivation of a synchrony alignment function that encodes the interplay between network structure and oscillators' frequencies and can be readily optimized. We highlight its utility in two general problems: constrained frequency allocation and network design. In general, we find that synchronization is promoted by strong alignments between frequencies and the dominant Laplacian eigenvectors, as well as a matching between the heterogeneity of frequencies and network structure.
We review some of the recent literature, including Refs. [1-3], on the effects of non-normality on the synchronization of networks of oscillators, and provide numerical evidence that the basin of attraction about the synchronous solution is typically smaller for networks with non-normal Laplacian matrix, compared to networks with a normal Laplacian matrix.
In this Rapid Communication we investigate spatially constrained networks that realize optimal synchronization properties. After arguing that spatial constraints can be imposed by limiting the amount of `wire' available to connect nodes distributed in space, we use numerical optimization methods to construct networks that realize different trade-offs between optimal synchronization and spatial constraints. Over a large range of parameters such optimal networks are found to have a link length distribution characterized by power law tails $P(l)\propto l^{-α}$, with exponents $α$ increasing as th…
We consider optimization of phase response curves for stochastic synchronization of non-interacting limit-cycle oscillators by common Poisson impulsive signals. The optimal functional shape for sufficiently weak signals is sinusoidal, but can differ for stronger signals. By solving the Euler-Lagrange equation associated with the minimization of the Lyapunov exponent characterizing synchronization efficiency, the optimal phase response curve is obtained. We show that the optimal shape mutates from a sinusoid to a sawtooth as the constraint on its squared amplitude is varied.
We present an analytical scheme to achieve optimal synchronization in multiplex networks of frustrated and non-frustrated phase oscillators. We derive a multiplex synchrony alignment function (MSAF) for that purpose, the expression of which consists of structural as well as dynamical information of the layers of the multiplex network. Analyzing the MSAF, a set of frequencies (optimal frequencies) is determined to achieve optimal synchronization in the network. Further, using the scheme, we show that perfect synchronization can be achieved in a layer of the multiplex network for given coupling …
We provide a theoretical framework for quantifying the expected level of synchronization in a network of noisy oscillators. Through linearization around the synchronized state, we derive the following quantities as functions of the eigenvalues and eigenfunctions of the network Laplacian using a standard technique for dealing with multivariate Ornstein-Uhlenbeck processes: the magnitude of the fluctuations around a synchronized state and the disturbance coefficients $α_i$ that represent how strongly node $i$ disturbs the synchronization. With this approach, we can quantify the effect of individ…
We investigate the existence of an optimal interplay between the natural frequencies of a group chaotic oscillators and the topological properties of the network they are embedded in. We identify the conditions for achieving phase synchronization in the most effective way, i.e., with the lowest possible coupling strength. Specifically, we show by means of numerical and experimental results that it is possible to define a synchrony alignment function linking the natural frequencies of a set of non-identical phase-coherent chaotic oscillators with the topology of the Laplacian matrix $L$, the la…
We reply to the recent note "Comment on Synchronization dynamics in non-normal networks: the trade-off for optimality", showing that the authors base their claims mainly on general theoretical arguments that do not necessarily invalidate the adequacy of our previous study. In particular, they do not specifically tackle the correctness of our analysis but instead limit their discussion on the interpretation of our results and conclusions, particularly related to the concept of optimality of network structure related to synchronization dynamics. Nevertheless, their idea of optimal networks is st…
In this paper, we discuss distributed adaptive algorithms for synchronization of complex networks, consensus of multi-agents with or without pinning controller. The dynamics of individual node is governed by generalized QUAD condition. We design new algorithms, which can keep the left eigenvector of the adaptive coupling matrix corresponding to the zero eigenvalue invariant. Based on this invariance, various distributive adaptive synchronization, consensus, anti-synchronization models are given.
A recently proposed dimensional reduction approach for studying synchronization in the Kuramoto model is employed to build optimal network topologies to favor or to suppress synchronization. The approach is based in the introduction of a collective coordinate for the time evolution of the phase locked oscillators, in the spirit of the Ott-Antonsen ansatz. We show that the optimal synchronization of a Kuramoto network demands the maximization of the quadratic function $ω^T L ω$, where $ω$ stands for the vector of the natural frequencies of the oscillators, and $L$ for the network Laplacian matr…
We study optimal synchronization of networks of coupled phase oscillators. We extend previous theory for optimizing the synchronization properties of undirected networks to the important case of directed networks. We derive a generalized synchrony alignment function that encodes the interplay between network structure and the oscillators' natural frequencies and serves as an objective measure for the network's degree of synchronization. Using the generalized synchrony alignment function, we show that a network's synchronization properties can be systematically optimized. This framework also al…
From the flashes of fireflies to Josephson junctions and power infrastructure, networks of coupled phase oscillators provide a powerful framework to describe synchronization phenomena in many natural and engineered systems. Most real-world networks are under the influence of noisy, random inputs, potentially inhibiting synchronization. While noise is unavoidable, here we show that there exist optimal noise patterns which minimize desynchronizing effects and even enhance order. Specifically, using analytical arguments we show that in the case of a two-oscillator model, there exists a sharp tran…
Synchronizing phase frustrated Kuramoto oscillators, a challenge that has found applications from neuronal networks to the power grid, is an eluding problem, as even small phase-lags cause the oscillators to avoid synchronization. Here we show, constructively, how to strategically select the optimal frequency set, capturing the natural frequencies of all oscillators, for a given network and phase-lags, that will ensure perfect synchronization. We find that high levels of synchronization are sustained in the vicinity of the optimal set, allowing for some level of deviation in the frequencies wi…
We consider the problem of maximizing the synchronizability of oscillator networks by assigning weights and directions to the links of a given interaction topology. We first extend the well-known master stability formalism to the case of non-diagonalizable networks. We then show that, unless some oscillator is connected to all the others, networks of maximum synchronizability are necessarily non-diagonalizable and can always be obtained by imposing unidirectional information flow with normalized input strengths. The extension makes the formalism applicable to all possible network structures, w…
A recent paper by R. Muolo, T. Carletti, J. P. Gleeson, and M. Asllani [Entropy 23, 36 (2021)] presents a mainly numerical study on the role of non-normality in the synchronization of coupled periodic oscillators, deriving apparent contradictions with the existing literature. Here, we show that their conclusions are artifactual due to a misinterpretation of the master stability function formalism and confirm that the existing literature is correct. We also point to a broader existing literature in which research considered in this paper was correctly addressed numerically, analytically, and ex…
We develop an analytical framework to study the synchronization of a quantum self-sustained oscillator to an external signal. Our unified description allows us to identify the resource on which quantum synchronization relies, and to compare quantitatively the synchronization behavior of different limit cycles and signals. We focus on the most elementary quantum system that is able to host a self-sustained oscillation, namely a single spin 1. Despite the spin having no classical analogue, we first show that it can realize the van der Pol limit cycle deep in the quantum regime, which allows us t…
Synchronization of oscillations is a phenomenon prevalent in natural, social, and engineering systems. Controlling synchronization of oscillating systems is motivated by a wide range of applications from neurological treatment of Parkinson's disease to the design of neurocomputers. In this article, we study the control of an ensemble of uncoupled neuron oscillators described by phase models. We examine controllability of such a neuron ensemble for various phase models and, furthermore, study the related optimal control problems. In particular, by employing Pontryagin's maximum principle, we an…
We introduce a novel concept of generalized synchronization, able to encompass the setting of collective synchronized behavior for mutually coupled systems and networking systems featuring complex topologies in their connections. The onset of the synchronous regime is confirmed by the dependence of the system's Lyapunov exponents on the coupling parameter. The presence of a generalized synchronization regime is verified by means of the nearest neighbor method.
We consider two optimization problems on synchronization of oscillator networks: maximization of synchronizability and minimization of synchronization cost. We first develop an extension of the well-known master stability framework to the case of non-diagonalizable Laplacian matrices. We then show that the solution sets of the two optimization problems coincide and are simultaneously characterized by a simple condition on the Laplacian eigenvalues. Among the optimal networks, we identify a subclass of hierarchical networks, characterized by the absence of feedback loops and the normalization o…
This brief paper further investigates the locally and globally adaptive synchronization of an uncertain complex dynamical network. Several network synchronization criteria are deduced. Especially, our hypotheses and designed adaptive controllers for network synchronization are rather simple in form. It is very useful for future practical engineering design. Moreover, numerical simulations are also given to show the effectiveness of our synchronization approaches.
Today, complex networks have attracted increasing attention from various fields of science and engineering. It has been demonstrated that many complex networks display various synchronization phenomena. In this paper, we introduce a time-varying complex dynamical network model. We then further investigate its synchronization phenomenon and prove several network synchronization theorems. Especially, we show that synchronization of such a time-varying dynamical network is completely determined by the inner-coupling matrix, and the eigenvalues and the corresponding eigenvectors of the coupling co…
Due to time delays in signal transmission and processing, phase lags are inevitable in realistic complex oscillator networks. Conventional wisdom is that phase lags are detrimental to network synchronization. Here we show that judiciously chosen phase lag modulations can result in significantly enhanced network synchronization. We justify our strategy of phase modulation, demonstrate its power in facilitating and enhancing network synchronization with synthetic and empirical network models, and provide an analytic understanding of the underlying mechanism. Our work provides a new approach to s…
We propose a framework for achieving perfect synchronization in complex networks of Sakaguchi-Kuramoto oscillators in presence of higher order interactions (simplicial complexes) at a targeted point in the parameter space. It is achieved by using an analytically derived frequency set from the governing equations. The frequency set not only provides stable perfect synchronization in the network at a desired point, but also proves to be very effective in achieving high level of synchronization around it compared to the choice of any other frequency sets (Uniform, Normal etc.). The proposed frame…
We present an extended analysis, based on the dynamics towards synchronization of a system of coupled oscillators, of the hierarchy of communities in complex networks. In the synchronization process, different structures corresponding to well defined communities of nodes appear in a hierarchical way. The analysis also provides a useful connection between synchronization dynamics, complex networks topology and spectral graph analysis.
Understanding how the interplay between higher-order and multilayer structures of interconnections influences the synchronization behaviors of dynamical systems is a feasible problem of interest, with possible application in essential topics such as neuronal dynamics. Here, we provide a comprehensive approach for analyzing the stability of the complete synchronization state in simplicial complexes with numerous interaction layers. We show that the synchronization state exists as an invariant solution and derive the necessary condition for a stable synchronization state in presence of general c…
In this paper, we focus on the topic Synchronization and consensus of Complex Networks and their relationships. It is revealed that two topics are closely relating to each other and all results given in \cite{Li} can be obtained by the results in \cite{Lu2006}.
We study the relationship between topological scales and dynamic time scales in complex networks. The analysis is based on the full dynamics towards synchronization of a system of coupled oscillators. In the synchronization process, modular structures corresponding to well defined communities of nodes emerge in different time scales, ordered in a hierarchical way. The analysis also provides a useful connection between synchronization dynamics, complex networks topology and spectral graph analysis.
In this paper, we focus on the topic Synchronization and consensus of Complex Networks and their relationships. It is revealed that two topics are closely relating to each other and all results given in \cite{Li1,Li2} and many other papers can be obtained by the results in \cite{Chen1,Chen2}. It is pointed out that QUAD condition plays important role in discussing synchronization and consensus.
We consider the problem of global synchronization in a large random network of Kuramoto oscillators where some of them are subject to an external periodically driven force. We explore a recently proposed dimensional reduction approach and introduce an effective two-dimensional description for the problem. From the dimensionally reduced model, we obtain analytical predictions for some critical parameters necessary for the onset of a globally synchronized state in the system. Moreover, the low dimensional model also allows us to introduce an optimization scheme for the problem. Our main conclusi…
The phase-space of a simple synchronization model is thoroughly investigated. The model considers two-mode stochastic oscillators, coupled through a pulse-like interaction controlled by simple optimization rules. A complex phase space is uncovered as a function of two relevant model parameters that are related to the optimization threshold and the periods of the two oscillation modes. Several phases with different periodic global output signals are identified. It is shown that the system exhibits partial synchronization under unexpectedly general conditions.
Recent studies have shown that synchronizability of complex networks can be significantly improved by asymmetric couplings, and increase of coupling gradient is always in favor of network synchronization. Here we argue and demonstrate that, for typical complex networks, there usually exists an optimal coupling gradient under which the maximum network synchronizability is achieved. After this optimal value, increase of coupling gradient could deteriorate synchronization. We attribute the suppression of network synchronization at large gradient to the phenomenon of network breaking, and find tha…
Synchronization has been the subject of intense research during decades mainly focused on determining the structural and dynamical conditions driving a set of interacting units to a coherent state globally stable. However, little attention has been paid to the description of the dynamical development of each individual networked unit in the process towards the synchronization of the whole ensemble. In this paper, we show how in a network of identical dynamical systems, nodes belonging to the same degree class differentiate in the same manner visiting a sequence of states of diverse complexity …
The presence of higher-order interactions (simplicial complexes) in networks and certain types of multilayer networks has shown to lead to the abrupt first-order transition to synchronization. We discover that simplicial complexes on multilayer networks can yield multiple basins of attraction, leading to multiple routes to the abrupt first-order transition to synchronization. Using the Ott-Antonsen approach, we develop an analytical framework for simplicial complexes on multilayer networks, which thoroughly explains the origin and stability of all possible dynamical states, including multiple …
The emergence of explosive synchronization has been reported as an abrupt transition in complex networks of first-order Kuramoto oscillators. In this Letter, we demonstrate that the nodes in a second-order Kuramoto model, perform a cascade of transitions toward a synchronous macroscopic state, which is a novel phenomenon that we call \textit{cluster explosive synchronization}. We provide a rigorous analytical treatment using a mean-field analysis in uncorrelated networks. Our findings are in good agreement with numerical simulations and fundamentally deepen the understanding of microscopic mec…
Coherence resonance (CR), stochastic synchronization (SS), and spike-timing-dependent plasticity (STDP) are ubiquitous dynamical processes in biological neural networks. Whether there exists an optimal network and STDP configuration at which CR and SS are both pronounced is a fundamental question of interest that is still elusive. We expect such a configuration to enable the brain to make synergistic and optimal use of these phenomena to process information efficiently. This paper considers a small-world network of excitable Hodgkin-Huxley neurons driven by channel noise and STDP with an asymm…
A network of delay-coupled logistic maps exhibits two different synchronization regimes, depending on the distribution of the coupling delay times. When the delays are homogeneous throughout the network, the network synchronizes to a time-dependent state [Atay et al., Phys. Rev. Lett. 92, 144101 (2004)], which may be periodic or chaotic depending on the delay; when the delays are sufficiently heterogeneous, the synchronization proceeds to a steady-state, which is unstable for the uncoupled map [Masoller and Marti, Phys. Rev. Lett. 94, 134102 (2005)]. Here we characterize the transition from ti…
An incorporation of higher-order interactions is known to lead an abrupt first-order transition to synchronization in otherwise smooth second-order one for pair-wise coupled systems. Here, we show that adaptation in higher-order coupling strength may yield completely different phenomena, notably, second-order transition to synchronization and tiered synchronization. Using the Ott-Antonsen approach, we perform rigorous theoretical calculations illustrating the origin of these emerging phenomena along with a complete description of all (un)stable states. Numerical simulations for limited-size ne…
In this paper, the synchronization in a hyper-network of coupled dynamical systems is investigated for the first time. An evolving hyper-network model is proposed for better describing some complex systems. A concept of joint degree is introduced, and the evolving mechanism of hyper-network is given with respect to the joint degree. The hyper-degree distribution of the proposed evolving hyper-network is derived based on a rate equation method and obeys a power law distribution. Furthermore, the synchronization in a hyper-network of coupled dynamical systems is investigated for the first time. …
The concept of "remote synchronization" (RS) was introduced in [Phys. Rev. E 85, 026208 (2012)], where synchronization in a star network of Stuart-Landau oscillators was investigated. In the RS regime therein described, the central hub served as a transmitter of information between peripheral nodes, while maintaining independent dynamics that were asynchronous with the rest of the network. One of the key conclusions of that paper was that RS cannot occur in pure phase-oscillator networks. Here, we show that the RS regime can exist in networks of Kuramoto oscillators, and that hub nodes can act…
A novel regime of synchronization, called remote synchronization, where the peripheral nodes form a phase synchronized cluster not including the hub, was recently observed in star motifs. We show the existence of a more general dynamical state of remote synchronization in arbitrary networks of coupled oscillators. This state is characterized by the synchronization of pairs of nodes that are not directly connected via a physical link or any sequence of synchronized nodes. This phenomenon is almost negligible in networks of phase oscillators as its underlying mechanism is the modulation of the a…
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. For non identical elements, the lack of quantitative tools has hampered so far a systematic study of the mechanisms behind such a collective behavior. 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 "synchronization centrality", a measure which quantifies the role…
Synchronization processes play critical roles in the functionality of a wide range of both natural and man-made systems. Recent work in physics and neuroscience highlights the importance of higher-order interactions between dynamical units, i.e., three- and four-way interactions in addition to pairwise interactions, and their role in shaping collective behavior. Here we show that higher-order interactions between coupled phase oscillators, encoded microscopically in a simplicial complex, give rise to added nonlinearity in the macroscopic system dynamics that induces abrupt synchronization tran…
Realistic networks display not only a complex topological structure, but also a heterogeneous distribution of weights in the connection strengths. Here we study synchronization in weighted complex networks and show that the synchronizability of random networks with large minimum degree is determined by two leading parameters: the mean degree and the heterogeneity of the distribution of node's intensity, where the intensity of a node, defined as the total strength of input connections, is a natural combination of topology and weights. Our results provide a possibility for the control of synchro…
We propose a method for controlling synchronization patterns of limit-cycle oscillators by common noisy inputs, i.e., by utilizing noise-induced synchronization. Various synchronization patterns, including fully synchronized and clustered states, can be realized by using linear filters that generate appropriate common noisy signals from given noise. The optimal linear filter can be determined from the linear phase response property of the oscillators and the power spectrum of the given noise. The validity of the proposed method is confirmed by numerical simulations.
Complete Synchronization and its Transition in Higher Harmonic Sakaguchi--Kuramoto Oscillators
CSIR-Central Mechanical Engineering Research Institute（印度中央机械工程技术研究所）；Academy of Scientific and Innovative Research (AcSIR)（科学创新研究学院）；Department of Mathematics, National Institute of Technology（国家技术学院数学系）；Institute of Technical Physics and Materials Science, HUN-REN Centre for Energy Research（匈牙利科学院能源研究中心技术物理与材料科学研究所）；School of Computer Science, UPES（尤帕迪大学计算机科学学院）；Dhirubhai Ambani University（迪鲁拜·安巴尼大学）
Rhythmic activities that alternate between coherent and incoherent phases are ubiquitous in chemical, ecological, climate, or neural systems. Despite their importance, general mechanisms for their emergence are little understood. In order to fill this gap, we present a framework for describing the emergence of recurrent synchronization in complex networks with adaptive interactions. This phenomenon is manifested at the macroscopic level by temporal episodes of coherent and incoherent dynamics that alternate recurrently. At the same time, the dynamics of the individual nodes do not change quali…
Adaptation plays a fundamental role in shaping the structure of a complex network and improving its functional fitting. Even when increasing the level of synchronization in a biological system is considered as the main driving force for adaptation, there is evidence of negative effects induced by excessive synchronization. This indicates that coherence alone can not be enough to explain all the structural features observed in many real-world networks. In this work, we propose an adaptive network model where the dynamical evolution of the node states towards synchronization is coupled with an e…
Optimization of the stability of synchronized states between a pair of symmetrically coupled reaction-diffusion systems exhibiting rhythmic spatiotemporal patterns is studied in the framework of the phase reduction theory. The optimal linear filter that maximizes the linear stability of the in-phase synchronized state is derived for the case where the two systems are linearly coupled. The nonlinear optimal interaction function that theoretically gives the largest linear stability of the in-phase synchronized state is also derived. The theory is illustrated by using typical rhythmic patterns in…
We investigate the combined effects of distributed delay and the balance between excitatory and inhibitory nodes on the stability of synchronous oscillations in a network of coupled Stuart--Landau oscillators. To this end a network model is proposed for which the stability can be investigated analytically. It is found that beyond a critical inhibition ratio synchronization tends to be unstable. However, increasing distributional widths can counteract this trend leading to multiple resynchronization transitions at relatively high inhibition ratios. All studies are performed on two distribution …
