Nature of synchronization transitions in random networks of coupled oscillators

Validity of annealed approximation in a high-dimensional system

This study investigates the suitability of the annealed approximation in high-dimensional systems characterized by dense networks with quenched link disorder, employing models of coupled oscillators. We demonstrate that dynamic equations governing dense-network systems converge to those of the complete-graph version in the thermodynamic limit, where link disorder fluctuations vanish entirely. Consequently, the annealed-network systems, where fluctuations are attenuated, also exhibit the same dynamic behavior in the thermodynamic limit. However, a significant discrepancy arises in the incoherent (disordered) phase wherein the finite-size behavior becomes critical in determining the steady-state pattern. To explicitly elucidate this discrepancy, we focus on identical oscillators subject to competitive attractive and repulsive couplings. In the incoherent phase of dense networks, we observe the manifestation of random irregular states. In contrast, the annealed approximation yields a symmetric (regular) incoherent state where two oppositely coherent clusters of oscillators coexist, accompanied by the vanishing order parameter. Our findings imply that the annealed approximation should be employed with caution even in dense-network systems, particularly in the disordered phase.

Exploring nonlinear dynamics and network structures in Kuramoto systems using machine learning approaches

Recent advances in machine learning (ML) have facilitated its application to a wide range of systems, from complex to quantum. Reservoir computing algorithms have proven particularly effective for studying nonlinear dynamical systems that exhibit collective behaviors, such as synchronizations and chaotic phenomena, some of which still remain unclear. Here, we apply ML approaches to the Kuramoto model to address several intriguing problems, including identifying the transition point and criticality of a hybrid synchronization transition, predicting future chaotic behaviors, and understanding network structures from chaotic patterns. Our proposed method also has further implications, such as inferring the structure of neural networks from electroencephalogram signals. This study, finally, highlights the potential of ML approaches for advancing our understanding of complex systems.

Collective excitability in highly diluted random networks of oscillators

We report on collective excitable events in a highly diluted random network of non-excitable nodes. Excitability arises thanks to a self-sustained local adaptation mechanism that drives the system on a slow timescale across a hysteretic phase transition involving states with different degrees of synchronization. These phenomena have been investigated for the Kuramoto model with bimodal distribution of the natural frequencies and for the Kuramoto model with inertia and a unimodal frequency distribution. We consider global and partial stimulation protocols and characterize the system response for different levels of dilution. We compare the results with those obtained in the fully coupled case showing that such collective phenomena are remarkably robust against network diluteness.

First-order like phase transition induced by quenched coupling disorder

We investigate the collective dynamics of a population of X Y model-type oscillators, globally coupled via non-separable interactions that are randomly chosen from a positive or negative value and subject to thermal noise controlled by temperature T. We find that the system at T = 0 exhibits a discontinuous, first-order like phase transition from the incoherent to the fully coherent state; when thermal noise is present ( T > 0 ), the transition from incoherence to the partial coherence is continuous and the critical threshold is now larger compared to the deterministic case ( T = 0 ). We derive an exact formula for the critical transition from incoherent to coherent oscillations for the deterministic and stochastic case based on both stability analysis for finite oscillators as well as for the thermodynamic limit ( N → ∞) based on a rigorous mean-field theory using graphons, valid for heterogeneous graph structures. Our theoretical results are supported by extensive numerical simulations. Remarkably, the synchronization threshold induced by the type of random coupling considered here is identical to the one found in studies, which consider uniform input or output strengths for each oscillator node [H. Hong and S. H. Strogatz, Phys. Rev. E 84(4), 046202 (2011); Phys. Rev. Lett. 106(5), 054102 (2011)], which suggests that these systems display a "universal" character for the onset of synchronization.

Extended mean-field approach for chimera states in random complex networks

Identical oscillators in the chimera state exhibit a mixture of coherent and incoherent patterns simultaneously. Nonlocal interactions and phase lag are critical factors in forming a chimera state within the Kuramoto model in Euclidean space. Here, we investigate the contributions of nonlocal interactions and phase lag to the formation of the chimera state in random networks. By developing an extended mean-field approximation and using a numerical approach, we find that the emergence of a chimera state in the Erdös-Rényi network is due mainly to degree heterogeneity with nonzero phase lag. For a regularly random network, although all nodes have the same degree, we find that disordered connections may yield the chimera state in the presence of long-range interactions. Furthermore, we show a nontrivial dynamic state in which all the oscillators drift more slowly than a defined frequency due to connectivity disorder at large phase lags beyond the mean-field solutions.

A Model for Evolutionary Structural Plasticity and Synchronization of a Network of Neurons

A model of time-dependent structural plasticity for the synchronization of neuron networks is presented. It is known that synchronized oscillations reproduce structured communities, and this synchronization is transient since it can be enhanced or suppressed, and the proposed model reproduces this characteristic. The evolutionary behavior of the couplings is comparable to those of a network of biological neurons. In the structural network, the physical connections of axons and dendrites between neurons are modeled, and the evolution in the connections depends on the neurons' potential. Moreover, it is shown that the coupling force's function behaves as an adaptive controller that leads the neurons in the network to synchronization. The change in the node's degree shows that the network exhibits time-dependent structural plasticity, achieved through the evolutionary or adaptive change of the coupling force between the nodes. The coupling force function is based on the computed magnitude of the membrane potential deviations with its neighbors and a threshold that determines the neuron's connections. These rule the functional network structure along the time.

Phase synchronization in the two-dimensional Kuramoto model: Vortices and duality

We study a system of Kuramoto oscillators arranged on a two-dimensional periodic lattice where the oscillators interact with their nearest neighbors, and all oscillators have the same natural frequency. The initial phases of the oscillators are chosen to be distributed uniformly between (-π,π]. During the relaxation process to the final stationary phase, we observe different features in the phase field of the oscillators: initially, the state is randomly oriented, then clusters form. As time evolves, the size of the clusters increases and vortices that constitute topological defects in the phase field form in the system. These defects, being topological, annihilate in pairs; i.e., a given defect annihilates if it encounters another defect with opposite polarity. Finally, the system ends up either in a completely phase synchronized state in case of complete annihilation or a metastable phase locked state characterized by presence of vortices and antivortices. The basin volumes of the two scenarios are estimated. Finally, we carry out a duality transformation similar to that carried out for the XY model of planar spins on the Hamiltonian version of the Kuramoto model to expose the underlying vortex structure.

Onset of synchronization of Kuramoto oscillators in scale-free networks

Despite the great attention devoted to the study of phase oscillators on complex networks in the last two decades, it remains unclear whether scale-free networks exhibit a nonzero critical coupling strength for the onset of synchronization in the thermodynamic limit. Here, we systematically compare predictions from the heterogeneous degree mean-field (HMF) and the quenched mean-field (QMF) approaches to extensive numerical simulations on large networks. We provide compelling evidence that the critical coupling vanishes as the number of oscillators increases for scale-free networks characterized by a power-law degree distribution with an exponent 2<γ≤3, in line with what has been observed for other dynamical processes in such networks. For γ>3, we show that the critical coupling remains finite, in agreement with HMF calculations and highlight phenomenological differences between critical properties of phase oscillators and epidemic models on scale-free networks. Finally, we also discuss at length a key choice when studying synchronization phenomena in complex networks, namely, how to normalize the coupling between oscillators.

Finite-size scaling in the system of coupled oscillators with heterogeneity in coupling strength

We consider a mean-field model of coupled phase oscillators with random heterogeneity in the coupling strength. The system that we investigate here is a minimal model that contains randomness in diverse values of the coupling strength, and it is found to return to the original Kuramoto model [Y. Kuramoto, Prog. Theor. Phys. Suppl. 79, 223 (1984)10.1143/PTPS.79.223] when the coupling heterogeneity disappears. According to one recent paper [H. Hong, H. Chaté, L.-H. Tang, and H. Park, Phys. Rev. E 92, 022122 (2015)10.1103/PhysRevE.92.022122], when the natural frequency of the oscillator in the system is "deterministically" chosen, with no randomness in it, the system is found to exhibit the finite-size scaling exponent ν[over ¯]=5/4. Also, the critical exponent for the dynamic fluctuation of the order parameter is found to be given by γ=1/4, which is different from the critical exponents for the Kuramoto model with the natural frequencies randomly chosen. Originally, the unusual finite-size scaling behavior of the Kuramoto model was reported by Hong et al. [H. Hong, H. Chaté, H. Park, and L.-H. Tang, Phys. Rev. Lett. 99, 184101 (2007)10.1103/PhysRevLett.99.184101], where the scaling behavior is found to be characterized by the unusual exponent ν[over ¯]=5/2. On the other hand, if the randomness in the natural frequency is removed, it is found that the finite-size scaling behavior is characterized by a different exponent, ν[over ¯]=5/4 [H. Hong, H. Chaté, L.-H. Tang, and H. Park, Phys. Rev. E 92, 022122 (2015)10.1103/PhysRevE.92.022122]. Those findings brought about our curiosity and led us to explore the effects of the randomness on the finite-size scaling behavior. In this paper, we pay particular attention to investigating the finite-size scaling and dynamic fluctuation when the randomness in the coupling strength is considered.

Finite-size scaling, dynamic fluctuations, and hyperscaling relation in the Kuramoto model

We revisit the Kuramoto model to explore the finite-size scaling (FSS) of the order parameter and its dynamic fluctuations near the onset of the synchronization transition, paying particular attention to effects induced by the randomness of the intrinsic frequencies of oscillators. For a population of size N, we study two ways of sampling the intrinsic frequencies according to the same given unimodal distribution g(ω). In the "random" case, frequencies are generated independently in accordance with g(ω), which gives rise to oscillator number fluctuation within any given frequency interval. In the "regular" case, the N frequencies are generated in a deterministic manner that minimizes the oscillator number fluctuations, leading to quasiuniformly spaced frequencies in the population. We find that the two samplings yield substantially different finite-size properties with clearly distinct scaling exponents. Moreover, the hyperscaling relation between the order parameter and its fluctuations is valid in the regular case, but it is violated in the random case. In this last case, a self-consistent mean-field theory that completely ignores dynamic fluctuations correctly predicts the FSS exponent of the order parameter but not its critical amplitude.

Driven synchronization in random networks of oscillators

Synchronization is a universal phenomenon found in many non-equilibrium systems. Much recent interest in this area has overlapped with the study of complex networks, where a major focus is determining how a system's connectivity patterns affect the types of behavior that it can produce. Thus far, modeling efforts have focused on the tendency of networks of oscillators to mutually synchronize themselves, with less emphasis on the effects of external driving. In this work, we discuss the interplay between mutual and driven synchronization in networks of phase oscillators of the Kuramoto type, and explore how the structure and emergence of such states depend on the underlying network topology for simple random networks with a given degree distribution. We find a variety of interesting dynamical behaviors, including bifurcations and bistability patterns that are qualitatively different for heterogeneous and homogeneous networks, and which are separated by a Takens-Bogdanov-Cusp singularity in the parameter region where the coupling strength between oscillators is weak. Our analysis is connected to the underlying dynamics of oscillator clusters for important states and transitions.

Model reduction for networks of coupled oscillators

We present a collective coordinate approach to describe coupled phase oscillators. We apply the method to study synchronisation in a Kuramoto model. In our approach, an N-dimensional Kuramoto model is reduced to an n-dimensional ordinary differential equation with n≪N, constituting an immense reduction in complexity. The onset of both local and global synchronisation is reproduced to good numerical accuracy, and we are able to describe both soft and hard transitions. By introducing two collective coordinates, the approach is able to describe the interaction of two partially synchronised clusters in the case of bimodally distributed native frequencies. Furthermore, our approach allows us to accurately describe finite size scalings of the critical coupling strength. We corroborate our analytical results by comparing with numerical simulations of the Kuramoto model with all-to-all coupling networks for several distributions of the native frequencies.