Thomas

Author(s): Kai Zhang, Ming Xu, Nana Li, Meng Xu, Qian Zhang, Eran Greenberg, Vitali B. Prakapenka, Yu-Sheng Chen, Matthias Wuttig, Ho-Kwang Mao, and Wenge Yang

Superconductivity and Anderson localization represent two extreme cases of electronic behavior in solids. Surprisingly, these two competing scenarios can occur in the same quantum system, e.g., in an amorphous superconductor. Although the disorder-driven quantum phase transition has attracted much a...

[Phys. Rev. Lett. 127, 127002] Published Wed Sep 15, 2021

Author(s): Dan Wilson and Bard Ermentrout

We use the theory of isostable reduction to incorporate higher order effects that are lost in the first order phase reduction of coupled oscillators. We apply this theory to weakly coupled complex Ginzburg-Landau equations, a pair of conductance-based neural models, and finally to a short derivation...

[Phys. Rev. Lett.] Published Wed Sep 18, 2019

The Kuramoto model, despite its popularity as a mean-field theory for many synchronization phenomenon of oscillatory systems, is limited to a first-order harmonic coupling of phases. For higher-order coupling, there only exists a low-dimensional theory in the thermodynamic limit. In this paper, we extend the formulation used by Watanabe and Strogatz to obtain a low-dimensional description of a system of arbitrary size of identical oscillators coupled all-to-all via their higher-order modes. To demonstrate an application of the formulation, we use a second harmonic globally coupled model, with a mean-field equal to the square of the Kuramoto mean-field. This model is known to exhibit asymmetrical clustering in previous numerical studies. We try to explain the phenomenon of asymmetrical clustering using the analytical theory developed here, as well as discuss certain phenomena not observed at the level of first-order harmonic coupling.

Author(s): A. A. Lushnikov

This paper reports the results on the study of cyclization kinetics in {} random graphs. The relationship between the generating function for the number of edges in the graph and the generating function of linked components applies for deriving the exact expression for the spectrum of linked compone...

[Phys. Rev. E] Published Thu Oct 04, 2018

A multilayer network dataset of interaction and influence spreading in a virtual world

Scientific Data, Published online: 10 October 2017; doi:10.1038/sdata.2017.144

We analyze the stability of the network's giant connected component under impact of adverse events, which we model through the link percolation. Specifically, we quantify the extent to which the largest connected component of a network consists of the same nodes, regardless of the specific set of deactivated links. Our results are intuitive in the case of single-layered systems: the presence of large degree nodes in a single-layered network ensures both its robustness and stability. In contrast, we find that interdependent networks that are robust to adverse events have unstable connected components. Our results bring novel insights to the design of resilient network topologies and the reinforcement of existing networked systems.

In the last two decades recurrence plots (RPs) were introduced in many different scientific disciplines. It turned out how powerful this method is. After introducing approaches of quantification of RPs and by the study of relationships between RPs and fundamental properties of dynamical systems, this method attracted even more attention. After 20 years of RPs it is time to summarise this development in a historical context.

Author(s): Xi-Wei Yao, Hengyan Wang, Zeyang Liao, Ming-Cheng Chen, Jian Pan, Jun Li, Kechao Zhang, Xingcheng Lin, Zhehui Wang, Zhihuang Luo, Wenqiang Zheng, Jianzhong Li, Meisheng Zhao, Xinhua Peng, and Dieter Suter

Analysis of the large amounts of image data requires increasingly expensive and time-consuming computational resources. Quantum computing may offer a shortcut. A new edge-detection algorithm based on a specific quantum image representation shows exponentially faster performance compared to classical methods.

[Phys. Rev. X 7, 031041] Published Mon Sep 11, 2017

Author(s): Sergei Maslov and Kim Sneppen

In this paper, the authors extend a model for the spreading of infectious diseases to explore the propagation of epidemics through more than one species. Their mathematical model highlights the importance of cross-species interactions, and shows how a significant imbalance in population could have far-reaching implications for the survival rates.

[Phys. Rev. E 96, 022412] Published Tue Aug 22, 2017

We present a Bayesian formulation of weighted stochastic block models that can be used to infer the large-scale modular structure of weighted networks, including their hierarchical organization. Our method is nonparametric, and thus does not require the prior knowledge of the number of groups or other dimensions of the model, which are instead inferred from data. We give a comprehensive treatment of different kinds of edge weights (i.e. continuous or discrete, signed or unsigned, bounded or unbounded), as well as arbitrary weight transformations, and describe an unsupervised model selection approach to choose the best network description. We illustrate the application of our method to a variety of empirical weighted networks, such as global migrations, voting patterns in congress, and neural connections in the human brain.

Kuramoto oscillator networks have the special property that their trajectories are constrained to lie on the (at most) 3D orbits of the Möbius group acting on the state space T N (the N -fold torus). This result has been used to explain the existence of the ##IMG## [http://ej.iop.org/images/1751-8121/50/35/355101/aaa7e39ieqn001.gif] {$N-3$} constants of motion discovered by Watanabe and Strogatz for Kuramoto oscillator networks. In this work we investigate geometric consequences of this Möbius group action. The dynamics of Kuramoto phase models can be further reduced to 2D reduced group orbits, which have a natural geometry equivalent to the unit disk ##IMG## [http://ej.iop.org/images/1751-8121/50/35/355101/aaa7e39ieqn002.gif] {$ \newcommand{\D}{\Delta} \D$} with the hyperbolic metric. We show that in this metric the original Kuramoto phase model (with order parameter Z 1 equal to th...

Spreading processes are ubiquitous in natural and artificial systems. They can be studied via a plethora of models, depending on the specific details of the phenomena under study. Disease contagion and rumour spreading are among the most important of these processes due to their practical relevance. However, despite the similarities between them, current models address both spreading dynamics separately. In this article, we propose a general spreading model that is based on discrete time Markov chains. The model includes all the transitions that are plausible for both a disease contagion process and rumour propagation. We show that our model not only covers the traditional spreading schemes but that it also contains some features relevant in social dynamics, such as apathy, forgetting, and lost/recovering of interest. The model is evaluated analytically to obtain the spreading thresholds and the early time dynamical behaviour for the contact and reactive processes in several scenarios. Comparison with Monte Carlo simulations shows that the Markov chain formalism is highly accurate while it excels in computational efficiency. We round off our work by showing how the proposed framework can be applied to the study of spreading processes occurring on social networks.

Author(s): Peng-Bi Cui (崔鹏碧), Francesca Colaiori, and Claudio Castellano

The spread of an infectious disease can, in some cases, promote the propagation of other pathogens favoring violent outbreaks, which cause a discontinuous transition to an endemic state. The topology of the contact network plays a crucial role in these cooperative dynamics. We consider a susceptible...

[Phys. Rev. E 96, 022301] Published Tue Aug 01, 2017

Nature Methods 14, 757 (2017). doi:10.1038/nmeth.4370

Authors: Martin Krzywinski & Naomi Altman

Decision trees are a simple but powerful prediction method.

Statistical techniques that analyse texts, referred to as text analytics, have departed from the use of simple word count statistics towards a new paradigm. Text mining now hinges on a more sophisticated set of methods, including the representations in terms of complex networks. While well-established word-adjacency (co-occurrence) methods successfully grasp syntactical features of written texts, they are unable to represent important aspects of textual data, such as its topical structure, that is the sequence of subjects developing at a mesoscopic level along the text. Such aspects are often overlooked by current methodologies. In order to grasp the mesoscopic characteristics of semantical content in written texts, we devised a network model which is able to analyse documents in a multi-scale fashion. In the proposed model, a limited amount of adjacent paragraphs are represented as nodes, which are connected whenever they share a minimum semantical content. To illustrate the capabilities of our model, we present, as a case example, a qualitative analysis of ‘Alice’s Adventures in Wonderland’. We show that the mesoscopic structure of a document, modelled as a network, reveals many semantic traits of texts. Such an approach paves the way to a myriad of semantic-based applications. In addition, our approach is illustrated in a machine learning context, in which texts are classified among real texts and randomized instances.

We study symmetric motifs in random geometric graphs. Symmetric motifs are subsets of nodes which have the same adjacencies. These subgraphs are particularly prevalent in random geometric graphs and appear in the Laplacian and adjacency spectrum as sharp, distinct peaks, a feature often found in real-world networks. We look at the probabilities of their appearance and compare these across parameter space and dimension. We then use the Chen–Stein method to derive the minimum separation distance in random geometric graphs which we apply to study symmetric motifs in both the intensive and thermodynamic limits. In the thermodynamic limit, the probability that the closest nodes are symmetric approaches one, while in the intensive limit this probability depends upon the dimension.

Author(s): Sho Shirasaka, Nobuhiro Watanabe, Yoji Kawamura, and Hiroya Nakao

We consider optimization of the linear stability of synchronized states between a pair of weakly coupled limit-cycle oscillators with cross coupling, where different components of state variables of the oscillators are allowed to interact. On the basis of the phase reduction theory, we derive the co...

[Phys. Rev. E 96, 012223] Published Wed Jul 26, 2017

We investigate different mean-field-like approximations for stochastic dynamics on graphs, within the framework of a cluster-variational approach. In analogy with its equilibrium counterpart, this approach allows one to give a unified view of various (previously known) approximation schemes, and suggests quite a systematic way to improve the level of accuracy. We compare the different approximations with Monte Carlo simulations on a reversible (susceptible-infected-susceptible) discrete-time epidemic-spreading model on random graphs.

We identify the breaking of time-translation invariance in a deterministic system of repulsively coupled Kuramoto oscillators, which are exposed to a distribution of natural frequencies. We analyze random and regular implementations of frequency distributions and consider grid sizes with different characteristics of the attractor space, which is by construction quite rich. This may cause long transients until the deterministic trajectories find their stationary orbits. The stationary orbits are limit cycles with periods that extend over orders of magnitude. It is the long transient times that cause the breaking of time-translation invariance in autocorrelation functions of oscillator phases. This feature disappears close to the transition to the monostable phase, where the phase trajectories are just irregular and no stationary behavior can be identified.

We investigate transition to synchrony in degree-frequency correlated Sakaguchi-Kuramoto (SK) model on complex networks both analytically and numerically. We analytically derive self-consistent equations for group angular velocity and order parameter for the model in the thermodynamic limit. Using the self-consistent equations we investigate transition to synchronization in SK model on uncorrelated scale-free (SF) and Erd\H{o}s-R\'enyi (ER) networks in detail. Depending on the degree distribution exponent ($\gamma$) of SF networks and phase-frustration parameter, the population undergoes from first order transition (explosive synchronization (ES)) to second order transition and vice versa. In ER networks transition is always second order irrespective of the phase-lag parameter. We observe that the critical coupling strength for the onset of synchronization is decreased by phase-frustration parameter in case of SF network where as in ER network, the phase-frustration delays the onset of synchronization. Extensive numerical simulations using SF and ER networks are performed to validate the analytical results. An analytical expression of critical coupling strength for the onset of synchronization is also derived from the self consistent equations considering the vanishing order parameter limit.

Author(s): James Burridge

A new model of language evolution assumes that changes in the spatial boundaries between dialects are controlled by a surface tension effect.

[Phys. Rev. X 7, 031008] Published Mon Jul 17, 2017

Author(s): Hyunsuk Hong

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. Kur...

[Phys. Rev. E 96, 012213] Published Fri Jul 14, 2017

Many real-world complex networks simultaneously exhibit topological features of scale-free behaviour and hierarchical organization. In this regard, deterministic scale-free [A.-L. Barab\'asi \etal, Physica A, 299, 3 (2001)] and pseudofractal scale-free [S. N. Dorogovtsev \etal, Phy. Rev. E, 65, 6 (2002)] networks constitute notable models which simultaneously incorporate the aforementioned properties. The rules governing the formation of such networks are completely deterministic. However, real-world networks are presumably neither completely deterministic, nor perfectly hierarchical. Therefore, we suggest here perfectly hierarchical scale-free networks with randomly rewired edges as better representatives of practical networked systems. In particular, we preserve the scale-free degree distribution of the various deterministic networks but successively relax the hierarchical structure while rewiring them. We utilize the framework of master stability function in investigating the synchronizability of dynamical systems coupled on such rewired networks. Interestingly, this reveals that the process of rewiring is capable of significantly enhancing, as well as, deteriorating the synchronizability of the resulting networks. We investigate the influence of rewiring edges on the topological properties of the rewired networks and, in turn, their relation to the synchronizability of the respective topologies. Finally, we compare the synchronizability of deterministic scale-free and pseudofractcal scale-free networks with that of random scale-free networks (generated using the classical Barab\'asi-Albert model of growth and preferential attachment) and find that the latter ones promote synchronizability better than their deterministic counterparts.

We investigate the dynamics of large, globally-coupled systems of Kuramoto oscillators with heterogeneous interaction delays. For the case of exponentially distributed time delays we derive the full stability diagram that describes the bifurcations in the system. Of particular interest is the onset of hysteresis where both the incoherent and partially synchronized states are stable -- this occurs at a codimension-two point at the intersection between a Hopf bifucration and saddle-node bifurcation of cycles. By studying this codimension-two point we find the full set of characteristic time delays and natural frequencies where bistability exists and identify the critical time delay and critical natural frequency below which bistability does not exist. Finally, we examine the dynamics of the more general system where time delays are drawn from a Gamma distribution, finding that more homogeneous time delay distributions tend to both promote the onset of synchronization and inhibit the presence of hysteresis.

Author(s): J. A. Méndez-Bermúdez, Guilherme Ferraz de Arruda, Francisco A. Rodrigues, and Yamir Moreno

Multilayer networks are widespread in natural and manmade systems. Key properties of these networks are their spectral and eigenfunction characteristics, as they determine the critical properties of many dynamics occurring on top of them. Here, we numerically demonstrate that the normalized localiza...

[Phys. Rev. E 96, 012307] Published Fri Jul 07, 2017

We investigate a non-Abelian generalization of the Kuramoto model proposed by Lohe and given by N quantum oscillators (‘nodes’) connected by a quantum network where the wavefunction at each node is distributed over quantum channels to all other connected nodes. It leads to a system of Schrödinger equations coupled by nonlinear self-interacting potentials given by their correlations. We give a complete picture of synchronization results, given on the relative size of the natural frequency and the coupling constant, for two non-identical oscillators and show complete phase synchronization for arbitrary ##IMG## [http://ej.iop.org/images/1751-8121/50/31/315101/aaa79c9ieqn001.gif] {$N>2$} identical oscillators. Our results are mainly based on the analysis of the ODE system satisfied by the correlations and on the introduction of a quantum order parameter, which is analogous to the one defined by Kuramoto in the classical model. As a consequence of the pr...

Kuramoto oscillator networks have the special property that their trajectories are constrained to lie on the (at most) 3D orbits of the M\"obius group acting on the state space $T^N$ (the $N$-fold torus). This result has been used to explain the existence of the $N-3$ constants of motion discovered by Watanabe and Strogatz for Kuramoto oscillator networks. In this work we investigate geometric consequences of this M\"obius group action. The dynamics of Kuramoto phase models can be further reduced to 2D reduced group orbits, which have a natural geometry equivalent to the unit disk $\Delta$ with the hyperbolic metric. We show that in this metric the original Kuramoto phase model (with order parameter $Z_1$ equal to the centroid of the oscillator configuration of points on the unit circle) is a gradient flow and the model with order parameter $iZ_1$ (corresponding to cosine phase coupling) is a completely integrable Hamiltonian flow. We give necessary and sufficient conditions for general Kuramoto phase models to be gradient or Hamiltonian flows in this metric. This allows us to identify several new infinite families of hyperbolic gradient or Hamiltonian Kuramoto oscillator networks which therefore have simple dynamics with respect to this geometry. We prove that for the $Z_1$ model, a generic 2D reduced group orbit has a unique fixed point corresponding to the hyperbolic barycenter of the oscillator configuration, and therefore the dynamics are equivalent on different generic reduced group orbits. This is not always the case for more general hyperbolic gradient or Hamiltonian flows; the reduced group orbits may have multiple fixed points, which also may bifurcate as the reduced group orbits vary.

We propose a new approximation framework that unifies and generalizes a number of existing mean-field approximation methods for the SIS epidemic model on complex networks. We derive the framework, which we call the Universal Mean-Field Framework (UMFF), as a set of approximations of the exact Markovian SIS equations. Our main novelty is that we describe the mean-field approximations from the perspective of the isoperimetric problem, an insight which results in bounds on the UMFF approximation error. These new bounds provide insight in the accuracy of existing mean-field methods, such as the widely-used N-Intertwined Mean-Field Approximation (NIMFA) and Heterogeneous Mean-Field method (HMF). Additionally, the geometric perspective of the isoperimetric problem enables the UMFF approximation accuracy to be related to the regularity notions of Szemer\'edi's regularity lemma, which yields a prediction about the behavior of the SIS process on large graphs.

Most information spreading models consider that all individuals are identical psychologically. They ignore, for instance, the curiosity level of people, which may indicate that they can be influenced to seek for information given their interest. For example, the game Pok\'emon GO spread rapidly because of the aroused curiosity among users. This paper proposes an information propagation model considering the curiosity level of each individual, which is a dynamical parameter that evolves over time. We evaluate the efficiency of our model in contrast to traditional information propagation models, like SIR or IC, and perform analysis on different types of artificial and real-world networks, like Google+, Facebook, and the United States roads map. We present a mean-field approach that reproduces with a good accuracy the evolution of macroscopic quantities, such as the density of stiflers, for the system's behavior with the curiosity. We also obtain an analytical solution of the mean-field equations that allows to predicts a transition from a phase where the information remains confined to a small number of users to a phase where it spreads over a large fraction of the population. The results indicate that the curiosity increases the information spreading in all networks as compared with the spreading without curiosity, and that this increase is larger in spatial networks than in social networks. When the curiosity is taken into account, the maximum number of informed individuals is reached close to the transition point. Since curious people are more open to a new product, concepts, and ideas, this is an important factor to be considered in propagation modeling. Our results contribute to the understanding of the interplay between diffusion process and dynamical heterogeneous transmission in social networks.

We formulate a linear phase and frequency response theory for hyperbolic flows, which generalizes phase response theory for autonomous limit cycle oscillators to hyperbolic chaotic dynamics. The theory is based on a shadowing conjecture, stating the existence of a perturbed trajectory shadowing every unperturbed trajectory on the system attractor for any small enough perturbation of arbitrary duration and a corresponding unique time isomorphism, which we identify as phase, such that phase shifts between the unperturbed trajectory and its perturbed shadow are well defined. The phase sensitivity function is the solution of an adjoint linear equation and can be used to estimate the average change of phase velocity to small time dependent or independent perturbations. These changes of frequency are experimentally accessible giving a convenient way to define and measure phase response curves for chaotic oscillators. The shadowing trajectory and the phase can be constructed explicitly in the tangent space of an unperturbed trajectory using co-variant Lyapunov vectors. It can also be used to identify the limits of the regime of linear response.