Author(s): Sarbendu Rakshit, Zahra Faghani, Fatemeh Parastesh, Shirin Panahi, Sajad Jafari, Dibakar Ghosh, and Matjaž Perc
Chimera states have been a vibrant subject of research in the recent past, but the analytical treatment of transitions from chimeras to coherent states remains a challenge. Here we analytically derive the necessary conditions for this transition by means of the coherent stability function approach, ...
[Phys. Rev. E 100, 012315] Published Tue Jul 30, 2019
The integrity and functionality of many real-world complex systems hinge on a small set of pivotal nodes, or influencers. In different contexts, these influencers are defined as either structurally important nodes that maintain the connectivity of networks, or dynamically crucial units that can disproportionately impact certain dynamical processes. In practice, identification of the optimal set of influencers in a given system has profound implications in a variety of disciplines. In this review, we survey recent advances in the study of influencer identification developed from different perspectives, and present state-of-the-art solutions designed for different objectives. In particular, we first discuss the problem of finding the minimal number of nodes whose removal would breakdown the network (i.e., the optimal percolation or network dismantle problem), and then survey methods to locate the essential nodes that are capable of shaping global dynamics with either continuous (e.g., independent cascading models) or discontinuous phase transitions (e.g., threshold models). We conclude the review with a summary and an outlook.
The information theoretic quantity known as mutual information finds wide use in classification and community detection analyses to compare two classifications of the same set of objects into groups. In the context of classification algorithms, for instance, it is often used to compare discovered classes to known ground truth and hence to quantify algorithm performance. Here we argue that the standard mutual information, as commonly defined, omits a crucial term which can become large under real-world conditions, producing results that can be substantially in error. We demonstrate how to correct this error and define a mutual information that works in all cases. We discuss practical implementation of the new measure and give some example applications.
We propose a Bayesian nonparametric model based on Markov Chain Monte Carlo (MCMC) methods for unveiling the structure of the invariant global stable manifold from observed time-series data. The underlying unknown dynamical process is possibly contaminated by additive noise. We introduce the Stable Manifold Geometric Stick Breaking Reconstruction (SM-GSBR) model with which we reconstruct the unknown dynamic equations and in parallel we estimate the global structure of the perturbed stable manifold. Our method works for noninvertible maps without modifications. The stable manifold estimation procedure is demonstrated specifically in the case of polynomial maps. Simulations based on synthetic time series are presented.
For nonlinear time-delay systems, domains of attraction are rarely studied despite their importance for technological applications. The present paper provides methodological hints for the determination of an upper bound on the radius of attraction by numerical means. Thereby, the respective Banach space for initial functions has to be selected and primary initial functions have to be chosen. The latter are used in time-forward simulations to determine a first upper bound on the radius of attraction. Thereafter, this upper bound is refined by secondary initial functions, which result a posteriori from the preceding simulations. Additionally, a bifurcation analysis should be undertaken. This analysis results in a possible improvement of the previous estimation. An example of a time-delayed swing equation demonstrates the various aspects.
Janus oscillators have been recently introduced as a remarkably simple phase oscillator model that exhibits non-trivial dynamical patterns -- such as chimeras, explosive transitions, and asymmetry-induced synchronization -- that once were only observed in specifically tailored models. Here we study ensembles of Janus oscillators coupled on large homogeneous and heterogeneous networks. By virtue of the Ott-Antonsen reduction scheme, we find that the rich dynamics of Janus oscillators persists in the thermodynamic limit of random regular, Erd\H{o}s-R\'enyi and scale-free random networks. We uncover for all these networks the coexistence between partially synchronized state and a multitude of states displaying global oscillations. Furthermore, abrupt transitions of the global and local order parameters are observed for all topologies considered. Interestingly, only for scale-free networks, it is found that states displaying global oscillations vanish in the thermodynamic limit.
Author(s): Yang Liu and Jürgen Kurths
Current studies have shown that there is a positive correlation between the network assortativity and robustness and that the assortativity also plays an important role in explosive synchronization. In this paper, taking the network robustness as a global property, we investigate its significance as...
[Phys. Rev. E 100, 012312] Published Thu Jul 25, 2019
In the study of discrete dynamical systems, we typically start with a function from a space into itself, and ask questions about the properties of sequences of iterates of the function. In this paper we reverse the direction of this study. In particular, restricting to the complex plane, we start with a sequence of complex numbers and study the functions (if any) for which this sequence is an orbit under iteration. This gives rise to questions of existence and of uniqueness. We resolve some questions, and show that these issues can be quite delicate.
We study a convex optimization framework for bounding extreme events in nonlinear dynamical systems governed by ordinary or partial differential equations (ODEs or PDEs). This framework bounds from above the largest value of an observable along trajectories that start from a chosen set and evolve over a finite or infinite time interval. The approach needs no explicit trajectories. Instead, it requires constructing suitably constrained auxiliary functions that depend on the state variables and possibly on time. Minimizing bounds over auxiliary functions is a convex problem dual to the non-convex maximization of the observable along trajectories. This duality is strong, meaning that auxiliary functions give arbitrarily sharp bounds, for sufficiently regular ODEs evolving over a finite time on a compact domain. When these conditions fail, strong duality may or may not hold; both situations are illustrated by examples. We also show that near-optimal auxiliary functions can be used to construct spacetime sets that localize trajectories leading to extreme events. Finally, in the case of polynomial ODEs and observables, we describe how polynomial auxiliary functions of fixed degree can be optimized numerically using polynomial optimization. The corresponding bounds become sharp as the polynomial degree is raised if strong duality and mild compactness assumptions hold. Analytical and computational ODE examples illustrate the construction of bounds and the identification of extreme trajectories, along with some limitations. As an analytical PDE example, we bound the maximum fractional enstrophy of solutions to the Burgers equation with fractional diffusion.
We develop a linear response theory by computing the asymptotic value of the order parameter from the linearized equation of continuity around the nonsynchronized reference state using the Laplace transform in time. The proposed theory is applicable to a wide class of coupled phase oscillator systems and allows for any coupling functions, any natural frequency distributions, any phase-lag parameters, and any values for the time-delay parameter. This generality is in contrast to the limitation of the previous methods of the Ott--Antonsen ansatz and the self-consistent equation for an order parameter, which are restricted to a model family whose coupling function consists of only a single sinusoidal function. The theory is verified by numerical simulations.
Author(s): M. E. J. Newman
The spectrum of the adjacency matrix plays several important roles in the mathematical theory of networks and network data analysis, for example in percolation theory, community detection, centrality measures, and the theory of dynamical systems on networks. A number of methods have been developed f...
[Phys. Rev. E 100, 012314] Published Fri Jul 26, 2019
Super-critical Kuramoto oscillators with distributed frequencies separate into two disjoint groups: an ordered one locked to the mean field, and a disordered one consisting of effectively decoupled oscillators -- at least so in the thermodynamic limit. In finite ensembles, in contrast, such clear separation fails: The mean field fluctuates due to finite-size effects and thereby induces order in the disordered group. To our best knowledge, this publication is the first to reveal such an effect, similar to noise-induced synchronization, in a purely deterministic system. We start by modeling the situation as a stationary mean field with additional white noise acting on a pair of unlocked Kuramoto oscillators. An analytical expression shows that the cross-correlation between the two increases with decreasing ratio of natural frequency difference and noise intensity. In a deterministic finite Kuramoto model, the strength of the mean field fluctuations is inextricably linked to the typical natural frequency difference. Therefore, we let a fluctuating mean field, generated by a finite ensemble of active oscillators, act on pairs of passive oscillators with a microscopic natural frequency difference between which we then measure the cross-correlation, at both super- and sub-critical coupling.
This article refines the classical notion of a stochastic D-bifurcation to the respective family of n-point motions for homogeneous Markovian stochastic semiflows, such as stochastic Brownian flows of homeomorphisms, and their generalizations. This notion essentially detects at which level $k\leq n$ the support of the invariant measure of the k-point bifurcation has more than one connected component. Stochastic Brownian flows and their invariant measures which were shown by Kunita (1990) to be rigid, in the sense of being uniquely determined by the $1$-and $2$-point motions, and hence only stochastic n-point bifurcation of level $n=1$ or $n=2$ can occur. For general homogeneous stochastic Markov semiflows this turns out to be false. This article constructs minimal examples of where this rigidity is false in general on finite space and studies the complexity of the resulting n-point bifurcations.
For a large class of dynamical systems, the optimally time-dependent (OTD) modes, a set of deformable orthonormal tangent vectors that track directions of instabilities along any trajectory, are known to depend "pointwise" on the state of the system on the attractor, and not on the history of the trajectory. We leverage the power of neural networks to learn this "pointwise" mapping from phase space to OTD space directly from data. The result of the learning process is a cartography of directions associated with strongest instabilities in phase space. Implications for data-driven prediction and control of dynamical instabilities are discussed.
Author(s): Emanuele Cozzo, Guilherme Ferraz de Arruda, Francisco A. Rodrigues, and Yamir Moreno
Network robustness is a central point in network science, both from a theoretical and a practical point of view. In this paper, we show that layer degradation, understood as the continuous or discrete loss of links' weight, triggers a structural transition revealed by an abrupt change in the algebra...
[Phys. Rev. E 100, 012313] Published Thu Jul 25, 2019
Author(s): Wei Zou, Meng Zhan, and Jürgen Kurths
A second-order continuous synchronization has been well documented for the classic Kuramoto model. Here we generalize the classic Kuramoto model by incorporating a low-pass filter (LPF) in the coupling, which serves as a simple form of indirect coupling through a common external dynamic environment....
[Phys. Rev. E 100, 012209] Published Mon Jul 15, 2019
Author(s): Can Xu, Jian Gao, Stefano Boccaletti, Zhigang Zheng, and Shuguang Guan
We fully describe the mechanisms underlying synchronization in starlike networks of phase oscillators. In particular, the routes to synchronization and the critical points for the associated phase transitions are determined analytically. In contrast to the classical Kuramoto theory, we unveil that r...
[Phys. Rev. E 100, 012212] Published Tue Jul 23, 2019
Chimera referring to a coexistence of coherent and incoherent states, is traditionally very difficult to control due to its peculiar nature. Here, we provide a recipe to construct chimera states in the multiplex networks with the aid of multiplexing-delays. The chimera state in multiplex networks is produced by introducing heterogeneous delays in a fraction of inter-layer links, referred as multiplexing-delay, in a sequence. Additionally, the emergence of the incoherence in the chimera state can be regulated by making appropriate choice of both inter- and intra-layer coupling strengths, whereas the extent and the position of the incoherence regime can be regulated by appropriate placing and {strength} of the multiplexing delays. The proposed technique to construct such {engineered} chimera equips us with multiplex network's structural parameters as tools in gaining both qualitative- and quantitative-control over the incoherent section of the chimera states and, in turn, the chimera. Our investigation can be of worth in controlling dynamics of multi-level delayed systems and attain desired chimeric patterns.
The emergence of large-scale connectivity and synchronization are crucial to the structure, function and failure of many complex socio-technical networks. Thus, there is great interest in analyzing phase transitions to large-scale connectivity and to global synchronization, including how to enhance or delay the onset. These phenomena are traditionally studied as second-order phase transitions where, at the critical threshold, the order parameter increases rapidly but continuously. In 2009, an extremely abrupt transition was found for a network growth process where links compete for addition in attempt to delay percolation. This observation of "explosive percolation" was ultimately revealed to be a continuous transition in the thermodynamic limit, yet with very atypical finite-size scaling, and it started a surge of work on explosive phenomena and their consequences. Many related models are now shown to yield discontinuous percolation transitions and even hybrid transitions. Explosive percolation enables many other features such as multiple giant components, modular structures, discrete scale invariance and non-self-averaging, relating to properties found in many real phenomena such as explosive epidemics, electric breakdowns and the emergence of molecular life. Models of explosive synchronization provide an analytic framework for the dynamics of abrupt transitions and reveal the interplay between the distribution in natural frequencies and the network structure, with applications ranging from epileptic seizures to waking from anesthesia. Here we review the vast literature on explosive phenomena and synthesize the fundamental connections between models and survey the application areas. We attempt to classify explosive phenomena based on underlying mechanisms and to provide a coherent overview and perspective for future research to address the many vital questions that remained unanswered.
The emergence of large-scale connectivity and synchronization are crucial to the structure, function and failure of many complex socio-technical networks. Thus, there is great interest in analyzing phase transitions to large-scale connectivity and to global synchronization, including how to enhance or delay the onset. These phenomena are traditionally studied as second-order phase transitions where, at the critical threshold, the order parameter increases rapidly but continuously. In 2009, an extremely abrupt transition was found for a network growth process where links compete for addition in attempt to delay percolation. This observation of "explosive percolation" was ultimately revealed to be a continuous transition in the thermodynamic limit, yet with very atypical finite-size scaling, and it started a surge of work on explosive phenomena and their consequences. Many related models are now shown to yield discontinuous percolation transitions and even hybrid transitions. Explosive percolation enables many other features such as multiple giant components, modular structures, discrete scale invariance and non-self-averaging, relating to properties found in many real phenomena such as explosive epidemics, electric breakdowns and the emergence of molecular life. Models of explosive synchronization provide an analytic framework for the dynamics of abrupt transitions and reveal the interplay between the distribution in natural frequencies and the network structure, with applications ranging from epileptic seizures to waking from anesthesia. Here we review the vast literature on explosive phenomena and synthesize the fundamental connections between models and survey the application areas. We attempt to classify explosive phenomena based on underlying mechanisms and to provide a coherent overview and perspective for future research to address the many vital questions that remained unanswered.
The Phase Response Curve (PRC) is a tool used in neuroscience that measures the phase shift experienced by an oscillator due to a perturbation applied at different phases of the limit cycle. In this paper we present a new approach to PRCs based on the parameterization method. The underlying idea relies on the construction of a periodic system whose corresponding stroboscopic map has an invariant curve. We demonstrate the relationship between the internal dynamics of this invariant curve and the PRC, which yields a method to numerically compute the PRCs. Moreover, we link the existence properties of this invariant curve as the amplitude of the perturbation is increased with changes in the PRC waveform and with the geometry of isochrons. The invariant curve and its dynamics will be computed by means of the parameterization method consisting of solving an invariance equation. We show that the method to compute the PRC can be extended beyond the breakdown of the curve by means of introducing a modified invariance equation. The method also computes the amplitude response functions (ARCs) which provide information on the displacement away from the oscillator due to the effects of the perturbation. Finally, we apply the method to several classical models in neuroscience to illustrate how the results herein extend the framework of computation and interpretation of the PRC and ARC for perturbations of large amplitude and not necessarily pulsatile.
This paper investigates how models of spatiotemporal dynamics in the form of nonlinear partial differential equations can be identified directly from noisy data using a combination of sparse regression and weak formulation. Using the 4th-order Kuramoto-Sivashinsky equation for illustration, we show how this approach can be optimized in the limits of low and high noise, achieving accuracy that is orders of magnitude better than what existing techniques allow. In particular, we derive the scaling relation between the accuracy of the model, the parameters of the weak formulation, and the properties of the data, such as its spatial and temporal resolution and the level of noise.
In the last 15 years, statistical physics has been a very successful framework to model complex networks. On the theoretical side, this approach has brought novel insights into a variety of physical phenomena, such as self-organisation, scale invariance, emergence of mixed distributions and ensemble non-equivalence, that display unconventional features on heterogeneous networks. At the same time, thanks to their deep connection with information theory, statistical physics and the principle of maximum entropy have led to the definition of null models for networks reproducing some features of real-world systems, but otherwise as random as possible. We review here the statistical physics approach and the various null models for complex networks, focusing in particular on the analytic frameworks reproducing the local network features. We then show how these models have been used to detect statistically significant and predictive structural patterns in real-world networks, as well as to reconstruct the network structure in case of incomplete information. We further survey the statistical physics models that reproduce more complex, semi-local network features using Markov chain Monte Carlo sampling, as well as the models of generalised network structures such as multiplex networks, interacting networks and simplicial complexes.
As network research becomes more sophisticated, it is more common than ever for researchers to find themselves not studying a single network but needing to analyze sets of networks. An important task when working with sets of networks is network comparison, developing a similarity or distance measure between networks so that meaningful comparisons can be drawn. The best means to accomplish this task remains an open area of research. Here we introduce a new measure to compare networks, the Network Portrait Divergence, that is mathematically principled, incorporates the topological characteristics of networks at all structural scales, and is general-purpose and applicable to all types of networks. An important feature of our measure that enables many of its useful properties is that it is based on a graph invariant, the network portrait. We test our measure on both synthetic graphs and real world networks taken from protein interaction data, neuroscience, and computational social science applications. The Network Portrait Divergence reveals important characteristics of multilayer and temporal networks extracted from data.
Author(s): Iván León and Diego Pazó
Phase reduction is a powerful technique that makes possible to describe the dynamics of a weakly perturbed limit-cycle oscillator in terms of its phase. For ensembles of oscillators, a classical example of phase reduction is the derivation of the Kuramoto model from the mean-field complex Ginzburg-L...
[Phys. Rev. E 100, 012211] Published Fri Jul 19, 2019
A reservoir computer is a dynamical system that may be used to perform computations. A reservoir computer usually consists of a set of nonlinear nodes coupled together in a network so that there are feedback paths. Training the reservoir computer consists of inputing a signal of interest and fitting the time series signals of the reservoir computer nodes to a training signal that is related to the input signal. It is believed that dynamical systems function most efficiently as computers at the "edge of chaos", the point at which the largest Lyapunov exponent of the dynamical system transitions from negative to positive. In this work I simulate several different reservoir computers and ask if the best performance really does come at this edge of chaos. I find that while it is possible to get optimum performance at the edge of chaos, there may also be parameter values where the edge of chaos regime produces poor performance. This ambiguous parameter dependance has implications for building reservoir computers from analog physical systems, where the parameter range is restricted.
Complex networks have abundant and extensive applications in real life. Recently, researchers have proposed a number of complex networks, in which some are deterministic and others are random. Compared with deterministic networks, random network is not only interesting and typical but also practical to illustrate and study many real-world complex networks, especially for random scale-free networks. Here, we introduce three types of operations, i.e., type-A operation, type-B operation and type-C operation, for generating random scale-free networks $N(p,q,r,t)$. On the basis of our operations, we put forward the concrete process of producing networks, which constitute the network space $\mathcal{N}(p,q,r,t)$, and then discuss their topological properties. Firstly, we calculate the range of the average degree of each member in our network space and discover that each member is a sparse network. Secondly, we prove that each member in our space obeys the power-law distribution with degree exponent $\gamma=1+\frac{\ln(4-r)}{\ln2}$, which implies that each member is scale-free. Next, we analyze the diameter, and find that the diameter may abruptly transform from small to large due to type-B operation. Afterwards, we study the clustering coefficient of network and discover that its value is only determined by type-C operation. Ultimately, we make an elaborate conclusion. \\ \textbf{Keywords:} Random network; degree distribution; diameter; clustering coefficient.
We numerically investigate the onset of multi-chimera states in a linear array of coupled oscillators. As the phase delay $\alpha$ is increased, they exhibit a continuous transition from the globally synchronized state to the multichimera state consisting of asynchronous and synchronous domains. Large-scale simulations show that the fraction of asynchronous sites $\rho_a$ obeys the power law $\rho_a \sim (\alpha - \alpha_c)^{\beta_a}$, and that the spatio-temporal gaps between asynchronous sites show power-law distributions at the critical point. The critical exponents are distinct from those of the (1+1)-dimensional directed percolation and other absorbing-state phase transitions, indicating that this transition belongs to a new class of non-equilibrium critical phenomena. Crucial roles are played by traveling waves that rejuvenate asynchronous clusters by mediating non-local interactions between them.