Thomas

Several recent studies have tackled the issue of optimal network immunization by providing efficient criteria to identify key nodes to be removed in order to break apart a network, thus preventing the occurrence of extensive epidemic outbreaks. Yet, although the efficiency of those criteria has been demonstrated also in empirical networks, preventive immunization is rarely applied to real-world scenarios, where the usual approach is the a posteriori attempt to contain epidemic outbreaks using quarantine measures. Here we compare the efficiency of prevention with that of quarantine in terms of the tradeoff between the number of removed and saved nodes on both synthetic and empirical topologies. We show how, consistent with common sense, but contrary to common practice, in many cases preventing is better than curing: depending on network structure, rescuing an infected network by quarantine could become inefficient soon after the first infection.

A large variety of dynamical systems, such as chemical and biomolecular systems, can be seen as networks of nonlinear entities. Prediction, control, and identification of such nonlinear networks require knowledge of the state of the system. However, network states are usually unknown, and only a fraction of the state variables are directly measurable. The observability problem concerns reconstructing the network state from this limited information. Here, we propose a general optimization-based approach for observing the states of nonlinear networks and for optimally selecting the observed variables. Our results reveal several fundamental limitations in network observability, such as the trade-off between the fraction of observed variables and the observation length on one side, and the estimation error on the other side. We also show that owing to the crucial role played by the dynamics, purely graph- theoretic observability approaches cannot provide conclusions about one's practical ability to estimate the states. We demonstrate the effectiveness of our methods by finding the key components in biological and combustion reaction networks from which we determine the full system state. Our results can lead to the design of novel sensing principles that can greatly advance prediction and control of the dynamics of such networks.

Current understanding of the critical outbreak condition on temporal networks relies on approximations (time scale separation, discretization) that may bias the results. We propose a theoretical framework to compute the epidemic threshold in continuous time through the infection propagator approach. We introduce the {\em weak commutation} condition allowing the interpretation of annealed networks, activity-driven networks, and time scale separation into one formalism. Our work provides a coherent connection between discrete and continuous time representations applicable to realistic scenarios.

Author(s): Yuanzhao Zhang, Takashi Nishikawa, and Adilson E. Motter

A scenario has recently been reported in which in order to stabilize complete synchronization of an oscillator network—a symmetric state—the symmetry of the system itself has to be broken by making the oscillators nonidentical. But how often does such behavior—which we term asymmetry-induced synchro...

[Phys. Rev. E 95, 062215] Published Fri Jun 16, 2017

Author(s): Niels Lörch, Simon E. Nigg, Andreas Nunnenkamp, Rakesh P. Tiwari, and Christoph Bruder

Classically, the tendency towards spontaneous synchronization is strongest if the natural frequencies of the self-oscillators are as close as possible. We show that this wisdom fails in the deep quantum regime, where the uncertainty of amplitude narrows down to the level of single quanta. Under thes...

[Phys. Rev. Lett. 118, 243602] Published Thu Jun 15, 2017

We present a generalization of the Kuramoto phase oscillator model in which phases advance in discrete phase increments through Poisson processes, rendering both intrinsic oscillations and coupling inherently stochastic. We study the effects of phase discretization on the synchronization and precision properties of the coupled system both analytically and numerically. Remarkably, many key observables such as the steady-state synchrony and the quality of oscillations show distinct extrema while converging to the classical Kuramoto model in the limit of a continuous phase. The phase-discretized model provides a general framework for coupled oscillations in a Markov chain setting.

Author(s): Suman Saha, Arindam Mishra, E. Padmanaban, Sourav K. Bhowmick, Prodyot K. Roy, Bivas Dam, and Syamal K. Dana

We propose a set of general coupling conditions to select a coupling profile (a set of coupling matrices) from the linear flow matrix of dynamical systems for realizing global stability of complete synchronization (CS) in identical systems and robustness to parameter perturbation. The coupling matri...

[Phys. Rev. E 95, 062204] Published Fri Jun 09, 2017

Author(s): Benjamin Schäfer, Moritz Matthiae, Xiaozhu Zhang, Martin Rohden, Marc Timme, and Dirk Witthaut

Shifting our electricity generation from fossil fuel to renewable energy sources introduces large fluctuations to the power system. Here, we demonstrate how increased fluctuations, reduced damping, and reduced intertia may undermine the dynamical robustness of power grid networks. Focusing on fundam...

[Phys. Rev. E 95, 060203(R)] Published Fri Jun 09, 2017

In the last decade it has been shown that a large class of phase oscillator models admit low dimensional descriptions for the macroscopic system dynamics in the limit of an infinite number N of oscillators. The question of whether the macroscopic dynamics of other similar systems also have a low dimensional description in the infinite N limit has, however, remained elusive. In this paper we show how techniques originally designed to analyze noisy experimental chaotic time series can be used to identify effective low dimensional macroscopic descriptions from simulations with a finite number of elements. We illustrate and verify the effectiveness of our approach by applying it to the dynamics of an ensemble of globally coupled Landau-Stuart oscillators for which we demonstrate low dimensional macroscopic chaotic behavior with an effective 4-dimensional description. By using this description we show that one can calculate dynamical invariants such as Lyapunov exponents and attractor dimensions. One could also use the reconstruction to generate short-term predictions of the macroscopic dynamics.

While there exist a wide range of effective methods for community detection in networks, most of them require one to know in advance how many communities one is looking for. Here we present a method for estimating the number of communities in a network using a combination of Bayesian inference with a novel prior and an efficient Monte Carlo sampling scheme. We test the method extensively on both real and computer-generated networks, showing that it performs accurately and consistently, even in cases where groups are widely varying in size or structure.

Author(s): Xiaowen Chen, Takashi Nishikawa, and Adilson E. Motter

Fractal geometry is inherent to chaotic systems, but this is not well understood in undriven dissipative processes, in which chaos necessarily vanishes over time. A new analysis leads to a novel framework for studying these systems and shows, for the first time, that they too can generically exhibit fractal behavior.

[Phys. Rev. X 7, 021040] Published Thu Jun 08, 2017

This paper studies the problem of designing networks of nonidentical coupled oscillators in order to achieve a desired level of phase cohesiveness, defined as the maximum asymptotic phase difference across the edges of the network. In particular, we consider the following two design problems: (i) the nodal-frequency design problem, in which we tune the natural frequencies of the oscillators given the topology of the network, and (ii) the (robust) edge-weight design problem, in which we design the edge weights assuming that the natural frequencies are given (or belong to a given convex uncertainty set). For both problems, we optimize an objective function of the design variables while considering a desired level of phase cohesiveness as our design constraint. This constraint defines a convex set in the nodal-frequency design problem. In contrast, in the edge-weight design problem, the phase cohesiveness constraint yields a non-convex set, unless the underlying network is either a tree or an arbitrary graph with identical edge weights. We then propose a convex semidefinite relaxation to approximately solve the (non-convex) edge-weight design problem for general (possibly cyclic) networks with nonidentical edge weights. We illustrate the applicability of our results by analyzing several network design problems of practical interest, such as power re-dispatch in power grids, sparse network design, (robust) network design for distributed wireless analog clocks, and the detection of edges leading to the Braess' paradox in power grids.

The dynamical systems found in Nature are rarely isolated. Instead they interact and influence each other. The coupling functions that connect them contain detailed information about the functional mechanisms underlying the interactions and prescribe the physical rule specifying how an interaction occurs. Here, we aim to present a coherent and comprehensive review encompassing the rapid progress made recently in the analysis, understanding and applications of coupling functions. The basic concepts and characteristics of coupling functions are presented through demonstrative examples of different domains, revealing the mechanisms and emphasizing their multivariate nature. The theory of coupling functions is discussed through gradually increasing complexity from strong and weak interactions to globally-coupled systems and networks. A variety of methods that have been developed for the detection and reconstruction of coupling functions from measured data is described. These methods are based on different statistical techniques for dynamical inference. Stemming from physics, such methods are being applied in diverse areas of science and technology, including chemistry, biology, physiology, neuroscience, social sciences, mechanics and secure communications. This breadth of application illustrates the universality of coupling functions for studying the interaction mechanisms of coupled dynamical systems.

Real-world networks often have power-law degrees and scale-free properties such as ultra-small distances and ultra-fast information spreading. In this paper, we study a third universal property: three-point correlations that suppress the creation of triangles and signal the presence of hierarchy. We quantify this property in terms of $\bar c(k)$, the probability that two neighbors of a degree-$k$ node are neighbors themselves. We investigate how the clustering spectrum $k\mapsto\bar c(k)$ scales with $k$ in the hidden variable model and show that $c(k)$ follows a {\it universal curve} that consists of three $k$-ranges where $\bar c(k)$ remains flat, starts declining, and eventually settles on a power law $\bar c(k)\sim k^{-\alpha}$ with $\alpha$ depending on the power law of the degree distribution. We test these results against ten contemporary real-world networks and explain analytically why the universal curve properties only reveal themselves in large networks.

We implement a scale-free version of the pivot algorithm and use it to sample pairs of three-dimensional self-avoiding walks, for the purpose of efficiently calculating an observable that corresponds to the probability that pairs of self-avoiding walks remain self-avoiding when they are concatenated. We study the properties of this Markov chain, and then use it to find the critical exponent γ for self-avoiding walks to unprecedented accuracy. Our final estimate for γ is ##IMG## [http://ej.iop.org/images/1751-8121/50/26/264003/aaa7231ieqn001.gif] {$1.156\, 953\, 00(95)$} .

In dynamical systems, the full stability of fixed point solutions is determined by their basin of attraction. Characterizing the structure of these basins is, in general, a complicated task, especially in high dimensionality. Recent works have advocated to quantify the non-linear stability of fixed points of dynamical systems through the relative volumes of the associated basins of attraction [D. A. Wiley {\it et al.} Chaos {\bf 16}, 015103 (2006), P. J. Menck {\it et al.} Nat. Phys. {\bf 9}, 89 (2013)]. Here we revisit this issue and propose an efficient numerical method to estimate these volumes. The algorithm first identifies stable fixed points. Second, a set of initial conditions is considered that are randomly distributed at the surface of hypercubes centered on each fixed point. These initial conditions are dynamically evolved. The linear size of each basin of attraction is finally determined by the proportion of initial conditions which converge back to the fixed point. Armed with this algorithm, we revisit the problem considered by Wiley et al. in a seminal paper [D. A. Wiley {\it et al.} Chaos {\bf 16}, 015103 (2006)] that inspired the title of the present manuscript, and consider the equal-frequency Kuramoto model on a cycle. Fixed points of this model are characterized by an integer winding number $q$ and the number $n$ of oscillators. We find that the basin volumes scale as $(1-4q/n)^n$, contrasting with the Gaussian behavior postulated in Wiley et al.'s paper. Finally, we show the applicability of our method to complex models of coupled oscillators with different natural frequencies and on meshed networks.

The emergence of collective synchronization was reproduced long ago by Winfree in a classical model consisting of an ensemble of pulse-coupled phase oscillators. By means of the Ott-Antonsen ansatz, we derive an exact low-dimensional representation which is exhaustively investigated for a variety of pulse types and phase response curves (PRCs). Two structurally different synchronization scenarios are found, which are linked via the mutation of a Bogdanov-Takens point. From our results, we infer a general rule of thumb relating pulse shape and PRC offset with each scenario. Finally, we compare the exact synchronization threshold with the prediction of the averaging approximation given by the Kuramoto-Sakaguchi model. At the leading order, the discrepancy appears to behave as an odd function of the PRC offset.

Recurrently coupled networks of inhibitory neurons robustly generate oscillations in the gamma band. Nonetheless, the corresponding Wilson-Cowan type firing rate equation for such an inhibitory population does not generate such oscillations without an explicit time delay. We show that this discrepancy is due to a voltage-dependent spike-synchronization mechanism inherent in networks of spiking neurons which is not captured by standard firing rate equations. Here we investigate an exact low-dimensional description for a network of heterogeneous canonical type-I inhibitory neurons which includes the sub-threshold dynamics crucial for generating synchronous states. In the limit of slow synaptic kinetics the spike-synchrony mechanism is suppressed and the standard Wilson-Cowan equations are formally recovered as long as external inputs are also slow. However, even in this limit synchronous spiking can be elicited by inputs which fluctuate on a time-scale of the membrane time-constant of the neurons. Our meanfield equations therefore represent an extension of the standard Wilson-Cowan equations in which spike synchrony is also correctly described.

Publication date: 8 May 2017
Source:Physics Reports, Volume 687
Author(s): Matjaž Perc, Jillian J. Jordan, David G. Rand, Zhen Wang, Stefano Boccaletti, Attila Szolnoki
Extensive cooperation among unrelated individuals is unique to humans, who often sacrifice personal benefits for the common good and work together to achieve what they are unable to execute alone. The evolutionary success of our species is indeed due, to a large degree, to our unparalleled other-regarding abilities. Yet, a comprehensive understanding of human cooperation remains a formidable challenge. Recent research in the social sciences indicates that it is important to focus on the collective behavior that emerges as the result of the interactions among individuals, groups, and even societies. Non-equilibrium statistical physics, in particular Monte Carlo methods and the theory of collective behavior of interacting particles near phase transition points, has proven to be very valuable for understanding counterintuitive evolutionary outcomes. By treating models of human cooperation as classical spin models, a physicist can draw on familiar settings from statistical physics. However, unlike pairwise interactions among particles that typically govern solid-state physics systems, interactions among humans often involve group interactions, and they also involve a larger number of possible states even for the most simplified description of reality. The complexity of solutions therefore often surpasses that observed in physical systems. Here we review experimental and theoretical research that advances our understanding of human cooperation, focusing on spatial pattern formation, on the spatiotemporal dynamics of observed solutions, and on self-organization that may either promote or hinder socially favorable states.

It is well known that the addition of noise in a multistable system can induce random transitions between stable states. The rate of transition can be characterised in terms of the noise-free system's dynamics and the added noise: for potential systems in the presence of asymptotically low noise the well-known Kramers' escape time gives an expression for the mean escape time. This paper examines some general properties and examples of transitions between local steady and oscillatory attractors within networks: the transition rates at each node may be affected by the dynamics at other nodes. We use first passage time theory to explain some properties of scalings noted in the literature for an idealised model of initiation of epileptic seizures in small systems of coupled bistable systems with both steady and oscillatory attractors. We focus on the case of sequential escapes where a steady attractor is only marginally stable but all nodes start in this state. As the nodes escape to the oscillatory regime, we assume that the transitions back are very infrequent in comparison. We quantify and characterise the resulting sequences of noise-induced escapes. For weak enough coupling we show that a master equation approach gives a good quantitative understanding of sequential escapes, but for strong coupling this description breaks down.

A scenario has recently been reported in which in order to stabilize complete synchronization of an oscillator network---a symmetric state---the symmetry of the system itself has to be broken by making the oscillators nonidentical. But how often does such behavior---which we term asymmetry-induced synchronization (AISync)---occur in oscillator networks? Here we present the first general scheme for constructing AISync systems and demonstrate that this behavior is the norm rather than the exception in a wide class of physical systems that can be seen as multilayer networks. Since a symmetric network in complete synchrony is the basic building block of cluster synchronization in more general networks, AISync should be common also in facilitating cluster synchronization by breaking the symmetry of the cluster subnetworks.

Complex network formalism allows to explain the behavior of systems composed by interacting units. Several prototypical network models have been proposed thus far. The small-world model has been introduced to mimic two important features observed in real-world systems: i) local clustering and ii) the possibility to move across a network by means of long-range links that significantly reduce the characteristic path length. A natural question would be whether there exist several "types" of small-world architectures, giving rise to a continuum of models with properties (partially) shared with other models belonging to different network families. Here, we take advantage of the interplay between network theory and time series analysis and propose to investigate small-world signatures in complex networks by analyzing multifractal characteristics of time series generated from such networks. In particular, we suggest that the degree of right-sided asymmetry of multifractal spectra is linked with the degree of small-worldness present in networks. This claim is supported by numerical simulations performed on several parametric models, including prototypical small-world networks, scale-free, fractal and also real-world networks describing protein molecules. Our results also indicate that right-sided asymmetry emerges with the presence of the following topological properties: low edge density, low average shortest path, and high clustering coefficient.

In this paper we investigate the topological and spatial features of public transport networks (PTN) within the UK. Networks investigated include London, Manchester, West Midlands, Bristol, national rail and coach networks during 2011. Using methods in complex network theory and statistical physics we are able to discriminate PTNs with respect to their stability; which is the first of this kind for national networks. Moreover, taking advantage of various fractal properties we gain useful insights into the serviceable area of stations. These features can be employed as key performance indicators in aid of further developing efficient and stable PTNs.

Extensive cooperation among unrelated individuals is unique to humans, who often sacrifice personal benefits for the common good and work together to achieve what they are unable to execute alone. The evolutionary success of our species is indeed due, to a large degree, to our unparalleled other-regarding abilities. Yet, a comprehensive understanding of human cooperation remains a formidable challenge. Recent research in social science indicates that it is important to focus on the collective behavior that emerges as the result of the interactions among individuals, groups, and even societies. Non-equilibrium statistical physics, in particular Monte Carlo methods and the theory of collective behavior of interacting particles near phase transition points, has proven to be very valuable for understanding counterintuitive evolutionary outcomes. By studying models of human cooperation as classical spin models, a physicist can draw on familiar settings from statistical physics. However, unlike pairwise interactions among particles that typically govern solid-state physics systems, interactions among humans often involve group interactions, and they also involve a larger number of possible states even for the most simplified description of reality. The complexity of solutions therefore often surpasses that observed in physical systems. Here we review experimental and theoretical research that advances our understanding of human cooperation, focusing on spatial pattern formation, on the spatiotemporal dynamics of observed solutions, and on self-organization that may either promote or hinder socially favorable states.

Collective behavior among coupled dynamical units can emerge in various forms as a result of different coupling topologies as well as different types of coupling functions. Chimera states have recently received ample attention as a fascinating manifestation of collective behavior, in particular describing a symmetry breaking spatiotemporal pattern where synchronized and desynchronized states coexist in a network of coupled oscillators. In this perspective, we review the emergence of different chimera states, focusing on the effects of different coupling topologies that describe the interaction network connecting the oscillators. We cover chimera states that emerge in local, nonlocal and global coupling topologies, as well as in modular, temporal and multilayer networks. We also provide an outline of challenges and directions for future research.

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.

Author(s): Marcello A. Budroni, Andrea Baronchelli, and Romualdo Pastor-Satorras

Methods connecting dynamical systems and graph theory have attracted increasing interest in the past few years, with applications ranging from a detailed comparison of different kinds of dynamics to the characterization of empirical data. Here we investigate the effects of the (multi)fractal propert…

[Phys. Rev. E 95, 052311] Published Tue May 16, 2017

Author(s): Manuel Jiménez-Martín, Javier Rodríguez-Laguna, Otti D'Huys, Javier de la Rubia, and Elka Korutcheva

We study the synchronization of chaotic units connected through time-delayed fluctuating interactions. Focusing on small-world networks of Bernoulli and Logistic units with a fixed chiral backbone, we compare the synchronization properties of static and fluctuating networks in the regime of large de…

[Phys. Rev. E 95, 052210] Published Tue May 16, 2017

We present an approach to generate chimera dynamics (localized frequency synchrony) in oscillator networks with two groups of (at least) two elements using a general method based on delayed interaction with linear and quadratic terms. The coupling design yields robust chimeras through a phase-model based design of the delay and the ratio of linear and quadratic components of the interactions. We demonstrate the method in the Brusselator model and experiments with electrochemical oscillators. The technique opens the way to directly bridge theory for phase models and the dynamics of real-world oscillator networks.

The Kuramoto model (KM) of coupled phase oscillators on scale free graphs is analyzed in this work. The W-random graph model is used to define a convergent family of sparse graphs with power law degree distribution. For the KM on this family of graphs, we derive the mean field description of the system's dynamics in the limit as the size of the network tends to infinity. The mean field equation is used to study two problems: synchronization in the coupled system with randomly distributed intrinsic frequencies and existence and bifurcations of chimera states in the KM with repulsive coupling. The analysis of both problems highlights the role of the scale free network organization in shaping dynamics of the coupled system. The analytical results are complemented with the results of numerical simulations.