Edmilson Roque

We generalize the concept of basin of attraction of a stable state in order to facilitate the analysis of dynamical systems with noise and to assess stability properties of metastable states and long transients. To this end we examine the notions of mean sojourn times and absorption probabilities for Markov chains and study their relation to the basins of attraction. Our approach is applicable to a large variety of problems since in most cases the transfer operator associated to a dynamical system can be approximated by a Markov chain.

We illustrate the geometric phase associated with the cyclic dynamics of a classical system of coupled oscillators. We use an analogy between a classical coupled oscillator and a two-state quantum mechanical system to represent the evolution of the oscillator on an equivalent Hilbert space, which may be represented as a trajectory on the surface of a sphere. The cyclic evolution of the system leads to a change in phase, which consists of a dynamic phase along with an additional phase shift dependent on the geometry of the evolution. A simple experiment suitable for advanced undergraduate students is designed to study the geometric phase incurred during cyclic evolution of a coupled oscillator.

We present and analyze a nonabelian version of the Kuramoto system, which we call the quantum Kuramoto system. We study the stability of several classes of special solutions to this system, and show that for certain connection topologies the system supports multiple attractors. We also present estimates on the maximal possible heterogeneity in this system that can support an attractor, and study the effect of modifications analogous to phase-lag.

Author(s): Andreas Brechtel, Philipp Gramlich, Daniel Ritterskamp, Barbara Drossel, and Thilo Gross

We study diffusion-driven pattern formation in networks of networks, a class of multilayer systems, where different layers have the same topology, but different internal dynamics. Agents are assumed to disperse within a layer by undergoing random walks, while they can be created or destroyed by reac...

[Phys. Rev. E 97, 032307] Published Mon Mar 19, 2018

Author(s): David J. Jörg, Luis G. Morelli, and Frank Jülicher

The dynamics of oscillators in biological systems can involve chains of events with many noisy steps. In this work the authors investigate how precision and synchronization can be increased by coupling two such genetic oscillators. Their findings may shed light on the operating regime of cellular genetic oscillator systems in which precise timing is vital, such as circadian clocks.

[Phys. Rev. E 97, 032409] Published Mon Mar 19, 2018

Author(s): David Darmon

In the absence of mechanistic or phenomenological models of real-world systems, data-driven models become necessary. The discovery of various embedding theorems in the 1980s and 1990s motivated a powerful set of tools for analyzing deterministic dynamical systems via delay-coordinate embeddings of o...

[Phys. Rev. E 97, 032206] Published Fri Mar 16, 2018

A model-based approach to forecasting chaotic dynamical systems utilizes knowledge of the physical processes governing the dynamics to build an approximate mathematical model of the system. In contrast, machine learning techniques have demonstrated promising results for forecasting chaotic systems purely from past time series measurements of system state variables (training data), without prior knowledge of the system dynamics. The motivation for this paper is the potential of machine learning for filling in the gaps in our underlying mechanistic knowledge that cause widely-used knowledge-based models to be inaccurate. Thus we here propose a general method that leverages the advantages of these two approaches by combining a knowledge-based model and a machine learning technique to build a hybrid forecasting scheme. Potential applications for such an approach are numerous (e.g., improving weather forecasting). We demonstrate and test the utility of this approach using a particular illustrative version of a machine learning known as reservoir computing, and we apply the resulting hybrid forecaster to a low-dimensional chaotic system, as well as to a high-dimensional spatiotemporal chaotic system. These tests yield extremely promising results in that our hybrid technique is able to accurately predict for a much longer period of time than either its machine-learning component or its model-based component alone.

Symmetry broken states arise naturally in oscillatory networks. In this Letter, we investigate chaotic attractors in an ensemble of four mean-coupled Stuart-Landau oscillators with two oscillators being synchronized. We report that these states with partially broken symmetry, so-called chimera states, have different set-wise symmetries in the incoherent oscillators, and in particular some are and some are not invariant under a permutation symmetry on average. This allows for a classification of different chimera states in small networks. We conclude our report with a discussion of related states in spatially extended systems, which seem to inherit the symmetry properties of their counterparts in small networks.

In this paper, time delay effect and distributed shear are considered in the Kuramoto model. On the Ott-Antonsen's manifold, through analyzing the associated characteristic equation of the reduced functional differential equation, the stability boundary of the incoherent state is derived in multiple-parameter space. Moreover, very rich dynamical behavior such as stability switches inducing synchronization switches can occur in this equation. With the loss of stability, Hopf bifurcating coherent states arise, and the criticality of Hopf bifurcations is determined by applying the normal form theory and the center manifold theorem. On one hand, theoretical analysis indicates that the width of shear distribution and time delay can both eliminate the synchronization then lead the Kuramoto model to incoherence. On the other, time delay can induce several coexisting coherent states. Finally, some numerical simulations are given to support the obtained results where several bifurcation diagrams are drawn, and the effect of time delay and shear is discussed.

The Braess paradox can be observed in road networks used by selfish users. It describes the counterintuitive situation in which adding a new, per se faster, origin-destination connection to a road network results in increased travel times for all network users. We study the network as originally proposed by Braess but introduce microscopic particle dynamics based on the totally asymmetric exclusion processes. In contrast to our previous work [10.1103/PhysRevE.94.062312], where routes were chosen randomly according to turning rates, here we study the case of drivers with fixed route choices. We find that travel time reduction due to the new road only happens at really low densities and Braess' paradox dominates the largest part of the phase diagram. Furthermore, the domain wall phase observed in [10.1103/PhysRevE.94.062312] vanishes. In the present model gridlock states are observed in a large part of phase space. We conclude that the construcion of a new road can often be very critical and should be considered carefully.

We study for the first time linear response for random compositions of maps, chosen independently according to a distribution $\PP$. We are interested in the following question: how does an absolutely continuous stationary measure (acsm) of a random system change when $\PP$ changes smoothly to $\PP_{\eps}$? For a wide class of one dimensional random maps, we prove differentiability of acsm with respect to $\eps$; moreover, we obtain a linear response formula. We apply our results to iid compositions, with respect to various distributions $\PP_{\eps}$, of uniformly expanding circle maps, Gauss-R\'enyi maps (random continued fractions) and Pomeau-Manneville maps. Our results yield an exact formula for the invariant density of random continued fractions; while for Pomeau-Manneville maps our results provide a precise relation between their linear response under certain random perturbations and their linear response under deterministic perturbations.

Author(s): Julian Lee

Time-reversal symmetry of the microscopic laws dictates that the equilibrium distribution of a stochastic process must obey the condition of detailed balance. However, cyclic Markov processes that do not admit equilibrium distributions with detailed balance are often used to model systems driven out...

[Phys. Rev. E 97, 032110] Published Wed Mar 14, 2018

The construction of numerical schemes for the Kuramoto model is challenging due to the structural properties of the system which are essential in order to capture the correct physical behavior, like the description of stationary states and phase transitions. Additional difficulties are represented by the high dimensionality of the problem in presence of multiple frequencies. In this paper, we develop numerical methods which are capable to preserve these structural properties of the Kuramoto equation in the presence of diffusion and to solve efficiently the multiple frequencies case. The novel schemes are then used to numerically investigate the phase transitions in the case of identical and non identical oscillators.

Many physical and biological systems exhibit intrinsic cyclic dynamics that are altered by random external perturbations. We examine continuous-time autonomous dynamical systems exhibiting a stable limit cycle, perturbed by additive Gaussian white noise. We derive a formal approximation for the dynamics of sample paths that stay close to the limit cycle, in terms of a phase coordinate and a deviation perpendicular to the limit cycle. To leading order in the deviation, the phase advances at the deterministic speed superimposed by a Brownian-motion-like drift. The deviation itself takes the form of an (n-1)-dimensional Ornstein-Uhlenbeck process. We apply these results to the case of limit cycles emerging through a supercritical Hopf bifurcation, which is widespread in ecological and epidemiological models. We derive approximation formulas for the system's stationary autocovariance and power spectral density. The latter two reflect the effects of perturbations on the temporal coherence and spectral bandwidth of perturbed limit cycles. We verify our results using numerical simulations and exemplify their application to the El Ni\~no Southern Oscillation.

Dynamical systems with high intrinsic dimensionality are often characterized by extreme events having the form of rare transitions several standard deviations away from the mean. For such systems, order-reduction methods through projection of the governing equations have limited applicability due to the large intrinsic dimensionality of the underlying attractor but also the complexity of the transient events. An alternative approach is data-driven techniques that aim to quantify the dynamics of specific modes utilizing data-streams. Several of these approaches have improved performance by expanding the state representation using delayed coordinates. However, such strategies are limited in regions of the phase space where there is a small amount of data available, as is the case for extreme events. In this work, we develop a blended framework that integrates an imperfect model, obtained from projecting equations into a subspace that still contains crucial dynamical information, with data-streams through a recurrent neural network (RNN) architecture. In particular, we employ the long-short-term memory (LSTM), to model portions of the dynamics which cannot be accounted by the equations. The RNN is trained by analyzing the mismatch between the imperfect model and the data-streams, projected in the reduced-order space. In this way, the data-driven model improves the imperfect model in regions where data is available, while for locations where data is sparse the imperfect model still provides a baseline for the prediction of the system dynamics. We assess the developed framework on two challenging prototype systems exhibiting extreme events and show that the blended approach has improved performance compared with methods that use either data streams or the imperfect model alone. The improvement is more significant in regions associated with extreme events, where data is sparse.

Network structure from rich but noisy data

Network structure from rich but noisy data, Published online: 12 March 2018; doi:10.1038/s41567-018-0076-1

A technique allows optimal inference of the structure of a network when the available observed data are rich but noisy, incomplete or otherwise unreliable.

The phenomenon of self-synchronization in populations of oscillatory units appears naturally in neurosciences. However, in some situations, the formation of a coherent state is damaging. In this article we study a repulsive mean-field Kuramoto model that describes the time evolution of n points on the unit circle, which are transformed into incoherent phase-locked states. It has been recently shown that such systems can be reduced to a three-dimensional system of ordinary differential equations, whose mathematical structure is strongly related to hyperbolic geometry. The orbits of the Kuramoto dynamical system are then described by a ow of M\"obius transformations. We show this underlying dynamic performs statistical inference by computing dynamically M-estimates of scatter matrices. We also describe the limiting phase-locked states for random initial conditions using Tyler's transformation matrix. Moreover, we show the repulsive Kuramoto model performs dynamically not only robust covariance matrix estimation, but also data processing: the initial configuration of the n points is transformed by the dynamic into a limiting phase-locked state that surprisingly equals the spatial signs from nonparametric statistics. That makes the sign empirical covariance matrix to equal 1 2 id2, the variance-covariance matrix of a random vector that is uniformly distributed on the unit circle.

Most empirical studies of networks assume that the network data we are given represent a complete and accurate picture of the nodes and edges in the system of interest, but in real-world situations this is rarely the case. More often the data only specify the network structure imperfectly -- like data in essentially every other area of empirical science, network data are prone to measurement error and noise. At the same time, the data may be richer than simple network measurements, incorporating multiple measurements, weights, lengths or strengths of edges, node or edge labels, or annotations of various kinds. Here we develop a general method for making estimates of network structure and properties from any form of network data, simple or complex, when the data are unreliable, and give example applications to a selection of social and biological networks.

Relational inference leverages relationships between entities and links in a network to infer information about the network from a small sample. This method is often used when global information about the network is not available or difficult to obtain. However, how reliable is inference from a small labelled sample? How should the network be sampled, and what effect does it have on inference error? How does the structure of the network impact the sampling strategy? We address these questions by systematically examining how network sampling strategy and sample size affect accuracy of relational inference in networks. To this end, we generate a family of synthetic networks where nodes have a binary attribute and a tunable level of homophily. As expected, we find that in heterophilic networks, we can obtain good accuracy when only small samples of the network are initially labelled, regardless of the sampling strategy. Surprisingly, this is not the case for homophilic networks, and sampling strategies that work well in heterophilic networks lead to large inference errors. These findings suggest that the impact of network structure on relational classification is more complex than previously thought.

The phenomenon of self-synchronization in populations of oscillatory units appears naturally in neurosciences. However, in some situations, the formation of a coherent state is damaging. In this article we study a repulsive mean-field Kuramoto model that describes the time evolution of n points on the unit circle, which are transformed into incoherent phase-locked states. It has been recently shown that such systems can be reduced to a three-dimensional system of ordinary differential equations, whose mathematical structure is strongly related to hyperbolic geometry. The orbits of the Kuramoto dynamical system are then described by a ow of M\"obius transformations. We show this underlying dynamic performs statistical inference by computing dynamically M-estimates of scatter matrices. We also describe the limiting phase-locked states for random initial conditions using Tyler's transformation matrix. Moreover, we show the repulsive Kuramoto model performs dynamically not only robust covariance matrix estimation, but also data processing: the initial configuration of the n points is transformed by the dynamic into a limiting phase-locked state that surprisingly equals the spatial signs from nonparametric statistics. That makes the sign empirical covariance matrix to equal 1 2 id2, the variance-covariance matrix of a random vector that is uniformly distributed on the unit circle.

A faithful description of the state of a complex dynamical network would require, in principle, the measurement of all its $d$ variables, an infeasible task for systems with practical limited access and composed of many nodes with high dimensional dynamics. However, even if the network dynamics is observable from a reduced set of measured variables, how to reliably identifying such a minimum set of variables providing full observability remains an unsolved problem. From the Jacobian matrix of the governing equations of nonlinear systems, we construct a {\it pruned fluence graph} in which the nodes are the state variables and the links represent {\it only the linear} dynamical interdependences encoded in the Jacobian matrix after ignoring nonlinear relationships. From this graph, we identify the largest connected sub-graphs where there is a path from every node to every other node and there are not outcoming links. In each one of those sub-graphs, at least one node must be measured to correctly monitor the state of the system in a $d$-dimensional reconstructed space. Our procedure is here validated by investigating large-dimensional reaction networks for which the determinant of the observability matrix can be rigorously computed.

We consider the problem of identifying the most influential nodes for a spreading process on a network when prior knowledge about structure and dynamics of the system is incomplete or erroneous. Specifically, we perform a numerical analysis where the set of top spreaders is determined on the basis of prior information that is artificially altered by a certain level of noise. We then measure the optimality of the chosen set by measuring its spreading impact in the true system. Whereas we find that the identification of top spreaders is optimal when prior knowledge is complete and free of mistakes, we also find that the quality of the top spreaders identified using noisy information does not necessarily decrease as the noise level increases. For instance, we show that it is generally possible to compensate for erroneous information about dynamical parameters by adding synthetic errors in the structure of the network. Further, we show that, in some dynamical regimes, even completely losing prior knowledge on network structure may be better than relying on certain but incomplete information.

Language can be described as a network of interacting objects with different qualitative properties and complexity. These networks include semantic, syntactic, or phonological levels and have been found to provide a new picture of language complexity and its evolution. A general approach considers language from an information theory perspective that incorporates a speaker, a hearer, and a noisy channel. The later is often encoded in a matrix connecting the signals used for communication with meanings to be found in the real world. Most studies of language evolution deal in a way or another with such theoretical contraption and explore the outcome of diverse forms of selection on the communication matrix that somewhat optimizes communication. This framework naturally introduces networks mediating the communicating agents, but no systematic analysis of the underlying landscape of possible language graphs has been developed. Here we present a detailed analysis of network properties on a generic model of a communication code, which reveals a rather complex and heterogeneous morphospace of language networks. Additionally, we use curated data of English words to locate and evaluate real languages within this language morphospace. Our findings indicate a surprisingly simple structure in human language unless particles are introduced in the vocabulary, with the ability of naming any other concept. These results refine and for the first time complement with empirical data a lasting theoretical tradition around the framework of \emph{least effort language}.

An excitatory-inhibitory recurrent neuronal network is established to numerically study the effect of inhibitory neurons on the synchronization degree of neuronal systems. The obtained results show that, with the number of inhibitory neurons and the coupling strength from an inhibitory neuron to an excitatory neuron increasing, inhibitory neurons can not only reduce the synchronization degree when the synchronization degree of the excitatory population is initially higher, but also enhance it when it is initially lower. Meanwhile, inhibitory neurons could also help the neuronal networks to maintain moderate synchronized states. In this paper, we call this effect as modulation effect of inhibitory neurons. With the obtained results, it is further revealed that the ratio of excitatory neurons to inhibitory neurons being nearly 4 : 1 is an economic and affordable choice for inhibitory neurons to realize this modulation effect.

Asynchrony between Antarctic temperature and CO₂ associated with obliquity over the past 720,000 years

Asynchrony between Antarctic temperature and CO₂ associated with obliquity over the past 720,000 years, Published online: 06 March 2018; doi:10.1038/s41467-018-03328-3

The Antarctic temperature record displays a puzzling asynchrony with changes in CO2 through glacial cycles. Here, the authors show that a 720,000-year Antarctic temperature record is affected by variations in obliquity-induced local insolation that are associated with phase modulation of eccentricity cycle.

We consider social networks of competing agents that evolve dynamically over time. Such dynamic competition networks are directed, where a directed edge from nodes $u$ to $v$ corresponds a negative social interaction. We present a novel hypothesis that serves as a predictive tool to uncover alliances and leaders within dynamic competition networks. Our focus is in the present study is to validate it on competitive networks arising from social game shows such as Survivor and Big Brother.

According to a recent information-theoretical proposal, the problem of defining and identifying communities in networks can be interpreted as a classical communication task over a noisy channel: memberships of nodes are information bits erased by the channel, edges and non-edges in the network are parity bits introduced by the encoder but degraded through the channel, and a community identification algorithm is a decoder. The interpretation is perfectly equivalent to the one at the basis of well-known statistical inference algorithms for community detection. The only difference in the interpretation is that a noisy channel replaces a stochastic network model. However, the different perspective gives the opportunity to take advantage of the rich set of tools of coding theory to generate novel insights on the problem of community detection. In this paper, we illustrate two main applications of standard coding-theoretical methods to community detection. First, we leverage a state-of-the-art decoding technique to generate a family of quasi-optimal community detection algorithms. Second and more important, we show that the Shannon's noisy-channel coding theorem can be invoked to establish a lower bound, here named as decodability bound, for the maximum amount of noise tolerable by an ideal decoder to achieve perfect detection of communities. When computed for well-established synthetic benchmarks, the decodability bound explains accurately the performance achieved by the best community detection algorithms existing on the market, telling us that only little room for their improvement is still potentially left.

Phase response curves are important for analysis and modeling of oscillatory dynamics in various applications, particularly in neuroscience. Standard experimental technique for determining them requires isolation of the system and application of a specifically designed input. However, isolation is not always feasible and we are compelled to observe the system in its natural environment under free-running conditions. To that end we propose an approach relying only on passive observations of the system and its input. We illustrate it with simulation results of an oscillator driven by a stochastic force.

Author(s): Han-Xin Yang, Tao Zhou, and Zhi-Xi Wu

Recently, Antonioni and Cardillo proposed a coevolutionary model based on the intertwining of oscillator synchronization and evolutionary game theory [Phys. Rev. Lett. {118}, 238301 (2017)], in which each Kuramoto oscillator can decide whether to interact-or not-with its neighbors, and all oscillato...

[Phys. Rev. E] Published Fri Mar 02, 2018

We investigate the dependence of the largest Lyapunov exponent (LLE) of an N -particle self-gravitating ring model at equilibrium with respect to the number of particles and its dependence on energy. This model has a continuous phase-transition from a ferromagnetic to homogeneous phase, and we numerically confirm with large scale simulations the existence of a critical exponent associated to the LLE, although at variance with the theoretical estimate. The existence of strong chaos in the magnetized state evidenced by a positive Lyapunov exponent is explained by the coupling of individual particle oscillations to the diffusive motion of the center of mass of the system and also results in a change of the scaling of the LLE with the number of particles. We also discuss thoroughly for the model the validity and limits of the approximations made by a geometrical model for their analytic estimate.