Symmetries are ubiquitous in network systems and have profound impacts on the observable dynamics. At the most fundamental level, many synchronization patterns are induced by underlying network symmetry, and a high degree of symmetry is believed to enhance the stability of identical synchronization. Yet, here we show that the synchronizability of almost any symmetry cluster in a network of identical nodes can be enhanced precisely by breaking its structural symmetry. This counterintuitive effect holds for generic node dynamics and arbitrary network structure and is, moreover, robust against noise and imperfections typical of real systems, which we demonstrate by implementing a state-of-the-art optoelectronic experiment. These results lead to new possibilities for the topological control of synchronization patterns, which we substantiate by presenting an algorithm that optimizes the structure of individual clusters under various constraints.
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.
Synchronization of coupled oscillators is a ubiquitous phenomenon found throughout nature. Its robust realization is crucial to our understanding of various nonlinear systems, ranging from biological functions to electrical engineering. On another front, in condensed matter physics, topology is utilized to realize robust properties like topological edge modes, as demonstrated by celebrated topological insulators. Here, we integrate these two research avenues and propose a nonlinear topological phenomenon, namely topological synchronization, where only the edge oscillators synchronize while the…
Topological signals defined on nodes, links and higher dimensional simplices define the dynamical state of a network or of a simplicial complex. As such, topological signals are attracting increasing attention in network theory, dynamical systems, signal processing and machine learning. Topological signals defined on the nodes are typically studied in network dynamics, while topological signals defined on links are much less explored. Here we investigate Dirac synchronization, describing locally coupled topological signals defined on the nodes and on the links of a network, and treated using t…
The collective dynamics in populations of magnetic spin torque oscillators (STO) is an intensely studied topic in modern magnetism. Here, we show that arrays of STO coupled via dipolar fields can be modeled using a variant of the Kuramoto model, a well-known mathematical model in non-linear dynamics. By investigating the collective dynamics in arrays of STO we find that the synchronization in such systems is a finite size effect and show that the critical coupling-for a complete synchronized state-scales with the number of oscillators. Using realistic values of the dipolar coupling strength be…
Simplicial complexes capture the underlying network topology and geometry of complex systems ranging from the brain to social networks. Here we show that algebraic topology is a fundamental tool to capture the higher-order dynamics of simplicial complexes. In particular we consider topological signals, i.e., dynamical signals defined on simplices of different dimension, here taken to be nodes and links for simplicity. We show that coupling between signals defined on nodes and links leads to explosive topological synchronization in which phases defined on nodes synchronize simultaneously to pha…
We study synchronization dynamics of a population of pulse-coupled oscillators. In particular, we focus our attention in the interplay between networks topological disorder and its synchronization features. Firstly, we analyze synchronization time $T$ in random networks, and find a scaling law which relates $T$ to networks connectivity. Then, we carry on comparing synchronization time for several other topological configurations, characterized by a different degree of randomness. The analysis shows that regular lattices perform better than any other disordered network. The fact can be understo…
In this paper we study Algorithmic High-Frequency Financial Markets as dynamical networks. After an individual analysis of 24 stocks of the US market during a trading year of fully automated transactions by means of ordinal pattern series, we define an information-theoretic measure of pairwise synchronization for time series which allows us to study this subset of the US market as a dynamical network. We apply to the resulting network a couple of clustering algorithms in order to detect collective market states, characterized by their degree of centralized or descentralized synchronicity. This…
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…
Relay synchronization in complex networks is characterized by the synchronization of remote parts of the network due to their interaction via a relay. In multilayer networks, distant layers that are not connected directly can synchronize due to signal propagation via relay layers. In this work, we investigate relay synchronization of partial synchronization patterns like chimera states in three-layer networks of interacting FitzHugh-Nagumo oscillators. We demonstrate that the phenomenon of relay synchronization is robust to topological random inhomogeneities of small-world type in the layer ne…
Synchronization of Dirac-Bianconi driven oscillators
RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学研究所交叉理论与数学科学中心（iTHEMS））；Department of Systems and Control Engineering, Institute of Science Tokyo (former Tokyo Tech)（东京科学大学系统控制工程系（原东京工业大学））；Department of Applied Mathematics and Computer Science, Universidad de Cantabria（坎塔布里亚大学应用数学与计算机科学系）；Department of Complex and Intelligent Systems, Future University Hakodate（函馆未来大学复杂智能系统系）；International Research Frontiers Initiative, Institute of Science Tokyo (former Tokyo Tech)（东京科学大学国际研究前沿倡议项目（原东京工业大学））
Simplicial synchronization reveals the role that topology and geometry have in determining the dynamical properties of simplicial complexes. Simplicial network geometry and topology are naturally encoded in the spectral properties of the graph Laplacian and of the higher-order Laplacians of simplicial complexes. Here we show how the geometry of simplicial complexes induces spectral dimensions of the simplicial complex Laplacians that are responsible for changing the phase diagram of the Kuramoto model. In particular, simplicial complexes displaying a non-trivial simplicial network geometry can…
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…
In this work we study the synchronization of Kuramoto oscillators driven by external forces in complex modular networks. The motivation is the neuronal dynamics that takes place during information processing in the neural cortex. The neuron organization is modular, with clusters associated to different functions and brain structures, and need to constantly respond to external stimuli. We study the behavior of forced Kuramoto oscillators where only a fraction of them is subjected to a periodic external force. When all oscillators receive the external drive the system always synchronize with the…
Complexity of dynamical networks can arise not only from the complexity of the topological structure but also from the time evolution of the topology. In this paper, we study the synchronous motion of coupled maps in time-varying complex networks both analytically and numerically. The temporal variation is rather general and formalized as being driven by a metric dynamical system. Four network models are discussed in detail in which the interconnections between vertices vary through time randomly. These models are 1) i.i.d. sequences of random graphs with fixed wiring probability, 2) groups of…
We present a model of synchronization in networks of autonomous agents where the topology changes due to agents motion. We introduce two time scales, one for the topological change and another one for local synchronization. If the former scale is much shorter, an approximation that averages out the effect of motion is available. Here we show, however, that the time required for synchronization achievement increases with respect to that approximation in the opposite case. We find that this effect is more important close to the continuum percolation transition point. The simulation results are c…
We study the interplay between network topology and complex space-time patterns and introduce a concept to analytically predict complex patterns in networks of Stuart-Landau oscillators with linear symmetric and instantaneous coupling based solely on the network topology. These patterns consist of partial amplitude death and partial synchronization and are found to exist in large variety for all undirected networks of up to 5 nodes. The underlying concept is proved to be robust with respect to frequency mismatch and can also be extended to larger networks. In addition it directly links the sta…
Inspired by groups of animals and robots, we study the collective dynamics of large numbers of active particles, each one trying to get to its own randomly placed target, while avoiding collisions with each other. The particles we study are repulsive homing active Brownian particles (HABPs) - self-propelled particles whose orientation relaxes at a finite rate towards an absorbing target in $2d$ continuous space. For a wide range of parameters, these particles form synchronised system-wide chiral flocks, in spite of the absence of explicit alignment interactions. We show that this dramatic beha…
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…
We investigate the transition to synchronization in a two-layer network with time-switching inter-layer links. We focus on the role of the number of inter-layer links and the time-scale of topological changes. Initially, we observe a smooth transition to complete synchronization for the static inter-layer topology by increasing the number of inter-layer links. Next, for a dynamic topology with the existent inter-layer links randomly changing among identical units in the layers, we observe a significant improvement in the system synchronizability, i.e., the layers synchronize with lower inter-l…
Synchronization plays a key role in information processing in neuronal networks. Response of specific groups of neurons are triggered by external stimuli, such as visual, tactile or olfactory inputs. Neurons, however, can be divided into several categories, such as by physical location, functional role or topological clustering properties. Here we study the response of the electric junction C. elegans network to external stimuli using the partially forced Kuramoto model and applying the force to specific groups of neurons. Stimuli were applied to topological modules, obtained by the ModuLand p…
The Hegselmann-Krause (HK) model is a typical self-organizing system with local rule dynamics. In spite of its widespread use and numerous extensions, the underlying theory of its synchronization induced by noise still needs to be developed. In its original formulation, as a model first proposed to address opinion dynamics, its state-space was assumed to be bounded, and the theoretical analysis of noise-induced synchronization for this particular situation has been well established. However, when system states are allowed to exist in an unbounded space, mathematical difficulties arise whose th…
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…
Inter-layer synchronization is a dynamical state occurring in multi-layer networks composed of identical nodes. The state corresponds to have all layers synchronized, with nodes in each layer which do not necessarily evolve in unison. So far, the study of such a solution has been restricted to the case in which all layers had an identical connectivity structure. When layers are not identical, the inter-layer synchronous state is no longer a stable solution of the system. Nevertheless, when layers differ in just a few links, an approximate treatment is still feasible, and allows one to gather i…
We study synchronization dynamics in networks of coupled oscillators with bimodal distribution of natural frequencies. This setup can be interpreted as a simple model of frequency synchronization dynamics among generators and loads working in a power network. We derive the minimum coupling strength required to ensure global frequency synchronization. This threshold value can be efficiently found by solving a binary optimization problem, even for large networks. In order to validate our procedure, we compare its results with numerical simulations on a realistic network describing the European i…
Adaptive networks change their connectivity with time, depending on their dynamical state. While synchronization in structurally static networks has been studied extensively, this problem is much more challenging for adaptive networks. In this Letter, we develop the master stability approach for a large class of adaptive networks. This approach allows for reducing the synchronization problem for adaptive networks to a low-dimensional system, by decoupling topological and dynamical properties. We show how the interplay between adaptivity and network structure gives rise to the formation of stab…
University of Dschang（迪尚大学）；MoCLiS Research Group（MoCLiS研究组）；Quantum materials and Computing Group (QMaCG)（量子材料与计算小组（QMaCG））；Higher Teachers’ Training College, University of Bamenda（班达马师范高等专科学校，班达马大学）；Laboratory of Electrotechnics, Automatics and Energy, Higher Technical Teachers, Training College (ENSET) of Ebolowa, University of Ebolowa（埃博洛瓦高等技术教师培训学院（ENSET）电气技术、自动化与能源实验室，埃博洛瓦大学）；Centre for Audio, Acoustics and Vibration, Faculty of Engineering and IT, University of Technology Sydney（悉尼科技大学工程与信息学院音频、声学及振动中心）；IMT School for Advanced Studies Lucca（卢卡高等研究院（IMT））；Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, Departamento de Física（布宜诺斯艾利斯大学理学院物理系）；CONICET-Universidad de Buenos Aires, Instituto de Física Interdisciplinaria y Aplicada (INFINA)（阿根廷国家科学研究委员会-布宜诺斯艾利斯大学交叉与应用物理研究所（INFINA））
Symmetries naturally occur in real-world networks and can significantly influence the observed dynamics. For instance, many synchronization patterns result from the underlying network symmetries, and high symmetries are known to increase the stability of synchronization. Yet, here we find that general macroscopic features of network solutions such as regularity can be induced by breaking their symmetry of interactions. We demonstrate this effect in an ecological multilayer network where the topological asymmetries occur naturally. These asymmetries rescue the system from chaotic oscillations b…
We introduce an easily computable topological measure which locates the effective crossover between segregation and integration in a modular network. Segregation corresponds to the degree of network modularity, while integration is expressed in terms of the algebraic connectivity of an associated hyper-graph. The rigorous treatment of the simplified case of cliques of equal size that are gradually rewired until they become completely merged, allows us to show that this topological crossover can be made to coincide with a dynamical crossover from cluster to global synchronization of a system of…
The behavior at bifurcation from global synchronization to partial synchronization in finite networks of coupled oscillators is a complex phenomenon, involving the intricate dynamics of one or more oscillators with the remaining synchronized oscillators. This is not captured well by standard macroscopic model reduction techniques which capture only the collective behavior of synchronized oscillators in the thermodynamic limit. We introduce two mesoscopic model reductions for finite sparse networks of coupled oscillators to quantitatively capture the dynamics close to bifurcation from global to…
Dynamical properties of complex networks are related to the spectral properties of the Laplacian matrix that describes the pattern of connectivity of the network. In particular we compute the synchronization time for different types of networks and different dynamics. We show that the main dependence of the synchronization time is on the smallest nonzero eigenvalue of the Laplacian matrix, in contrast to other proposals in terms of the spectrum of the adjacency matrix. Then, this topological property becomes the most relevant for the dynamics.
Synchronization plays a fundamental role in healthy cognitive and motor function. However, how synchronization depends on the interplay between local dynamics, coupling and topology and how prone to synchronization a network with given topological organization is are still poorly understood issues. To investigate the synchronizability of both anatomical and functional brain networks various studies resorted to the Master Stability Function (MSF) formalism, an elegant tool which allows analysing the stability of synchronous states in a dynamical system consisting of many coupled oscillators. He…
Several complex systems can be modeled as large networks in which the state of the nodes continuously evolves through interactions among neighboring nodes, forming a high-dimensional nonlinear dynamical system. One of the main challenges of Network Science consists in predicting the impact of network topology and dynamics on the evolution of the states and, especially, on the emergence of collective phenomena, such as synchronization. We address this problem by proposing a Dynamics Approximate Reduction Technique (DART) that maps high-dimensional (complete) dynamics unto low-dimensional (reduc…
Natural and man-made networks often possess locally tree-like sub-structures. Taking such tree networks as our starting point, we show how the addition of links changes the synchronization properties of the network. We focus on two different methods of link addition. The first method adds single links that create cycles of a well-defined length. Following a topological approach we introduce cycles of varying length and analyze how this feature, as well as the position in the network, alters the synchronous behaviour. We show that in particular short cycles can lead to a maximum change of the L…
We found that a network-organized metapopulation of cooperators, defectors and destructive agents playing the public goods game with mutations, can collectively reach global synchronization or chimera states. Global synchronization is accompanied by a collective periodic burst of cooperation, whereas chimera states reflect the tendency of the networked metapopulation to be fragmented in clusters of synchronous and incoherent bursts of cooperation. Numerical simulations have shown that the system's dynamics alternates between these two steady states through a first order transition. Depending o…
The Kuramoto model is a system of nonlinear differential equations that models networks of coupled oscillators and is often used to study synchronization among the oscillators. In this paper we study steady state solutions of the Kuramoto model by assigning to each steady state a tuple of integers which records how the state twists around the cycles in the network. We then use this new classification of steady states to obtain a "Weyl" type of asymptotic estimate for the number of steady states as the number of oscillators becomes arbitrarily large while preserving the cycle structure. We furt…
We investigate synchronization in networks of Kuramoto oscillators with inertia. More specifically, we introduce a rewiring algorithm consisting basically in a {\em hill climb} scheme in which the edges of the network are swapped in order to enhance its synchronization capacity. We show that the the synchrony-optimized networks generated by our algorithm have some interesting topological and dynamical properties. In particular, they typically exhibit an anticipation of the synchronization onset and are more robust against certain types of perturbations. We consider synthetic random networks an…
We consider the two-dimensional (2D) noisy Kuramoto model of synchronization with short-range coupling and a Gaussian distribution of intrinsic frequencies, and investigate its ordering dynamics following a quench. We consider both underdamped (inertial) and over-damped dynamics, and show that the long-term properties of this intrinsically out-of-equilibrium system do not depend on the inertia of individual oscillators. The model does not exhibit any phase transition as its correlation length remains finite, scaling as the inverse of the standard deviation of the distribution of intrinsic freq…
The coupling of some types of oscillators requires the mediation of a physical link between them, rendering the distance between oscillators a critical factor to achieve synchronization. In this paper we propose and explore a greedy algorithm to grow spatially embedded oscillator networks. The algorithm is constructed in such a way that nodes are sequentially added seeking to minimize the cost of the added links' length and optimize the linear stability of the growing network. We show that, for appropriate parameters, the stability of the resulting network, measured in terms of the dynamics of…
