Thomas

Author(s): Diego Pazó and Ernest Montbrió

The Winfree model, a well-known mathematical model for describing collective synchronization in living systems, such as flashing fireflies, has been under-utilized because of its daunting technical complexity. Now scientists have found a way to dramatically reduce it to a technically tractable form and demonstrate the power of the reduction with findings of new “chimera” states in populations of pulse-coupled oscillators.

[Phys. Rev. X 4, 011009] Published Wed Jan 29, 2014

Author(s): Gerald F. Frasco, Jie Sun, Hernán D. Rozenfeld, and Daniel ben-Avraham

How does the geographic distribution of human populations correlate with their networks of social connections? In a simple mathematical model, theorists, for the first time, tie these two phenomena together and show that the model reproduces several interesting features found in real populations.

[Phys. Rev. X 4, 011008] Published Tue Jan 28, 2014

Author(s): Jaegon Um, Hyunsuk Hong, and Hyunggyu Park

We consider a system of phase oscillators with random intrinsic frequencies coupled through sparse random networks and investigate how the connectivity disorder affects the nature of collective synchronization transitions. Various distribution types of intrinsic frequencies are considered: uniform, ...

[Phys. Rev. E 89, 012810] Published Wed Jan 22, 2014

Phase transition in its strict sense can only be observed in an infinite system, for which equilibration takes an infinitely long time at criticality. In numerical simulations, we are often limited both by the finiteness of the system size and by the finiteness of the observation time scale. We prop...

Author(s): C. Schmeltzer, J. Soriano, I. M. Sokolov, and S. Rüdiger

Motivated by experiments on activity in neuronal cultures [J. Soriano, M. Rodríguez Martínez, T. Tlusty, and E. Moses, Proc. Natl. Acad. Sci. 105, 13758 (2008)PNASA610.1073/pnas.0707492105], we investigate the percolation transition and critical exponents of spatially embedded Erdős-Rényi networks w...

[Phys. Rev. E 89, 012116] Published Mon Jan 13, 2014

Author(s): Di Yuan, Mei Zhang, and Junzhong Yang

As a paradigmatic model, the Kuramoto model has provided a platform for investigating synchronization among nonidentical oscillators. In this work, we consider the Kuramoto model consisting of conformists with positive coupling strength and contrarians with negative coupling strength. We introduce t...

[Phys. Rev. E 89, 012910] Published Mon Jan 13, 2014

Author(s): Yoji Kawamura

We derive the Kuramoto-Sivashinsky-type phase equation from the Kuramoto-Sakaguchi-type phase model via the Ott-Antonsen-type complex amplitude equation and demonstrate heterogeneity-induced collective-phase turbulence in nonlocally coupled individual-phase oscillators.

[Phys. Rev. E 89, 010901] Published Mon Jan 13, 2014

We use a simple dynamical model and explore coherent dynamics of wavepackets in complex networks of optical fibers. We start from a symmetric lattice and through the application of a Monte-Carlo criterion we introduce structural disorder and deform the lattice into a small-world network regime. We investigate in the latter both structural (correlation length) as well as dynamical (diffusion exponent) properties and find that both exhibit a rapid crossover from the ordered to the fully random regime. For a critical value of the structural disorder parameter $\rho \approx 0.25$ transport changes from ballistic to sub-diffusive due to the creation strongly connected local clusters and channels of preferential transport in the small world regime.

Author(s): Isabel M. Kloumann, Ian M. Lizarraga, and Steven H. Strogatz

We consider a Kuramoto model of coupled oscillators that includes quenched random interactions of the type used by van Hemmen in his model of spin glasses. The phase diagram is obtained analytically for the case of zero noise and a Lorentzian distribution of the oscillators' natural frequencies. Dep...

[Phys. Rev. E 89, 012904] Published Mon Jan 06, 2014

We investigate common-noise-induced phase synchronization between uncoupled identical Hele-Shaw cells exhibiting oscillatory convection. Using the phase description method for oscillatory convection, we demonstrate that the uncoupled systems of oscillatory Hele-Shaw convection can exhibit in-phase s...

We derive the Kuramoto-Sivashinsky-type phase equation from the Kuramoto-Sakaguchi-type phase model via the Ott-Antonsen-type complex amplitude equation and demonstrate heterogeneity-induced collective-phase turbulence in nonlocally-coupled individual-phase oscillators.

Author(s): Laurent Hébert-Dufresne, Antoine Allard, Jean-Gabriel Young, and Louis J. Dubé

The k-core decomposition of a network has thus far mainly served as a powerful tool for the empirical study of complex networks. We now propose its explicit integration in a theoretical model. We introduce a hard-core random network (HRN) model that generates maximally random networks with arbitrary...

[Phys. Rev. E 88, 062820] Published Mon Dec 30, 2013

Author(s): Hideyuki Kato, Miguel C. Soriano, Ernesto Pereda, Ingo Fischer, and Claudio R. Mirasso

We study how reliably generalized synchronization can be detected and characterized from time-series analysis. To that end, we analyze synchronization in a generalized sense of delay-coupled chaotic oscillators in unidirectional ring configurations. The generalized synchronization condition can be v...

[Phys. Rev. E 88, 062924] Published Thu Dec 26, 2013

Nature Physics 10, 34 (2014). doi:10.1038/nphys2819

Authors: Antonio Majdandzic, Boris Podobnik, Sergey V. Buldyrev, Dror Y. Kenett, Shlomo Havlin & H. Eugene Stanley

Much research has been carried out to explore the structural properties and vulnerability of complex networks. Of particular interest are abrupt dynamic events that cause networks to irreversibly fail. However, in many real-world phenomena, such as brain seizures in neuroscience or sudden market crashes in finance, after an inactive period of time a significant part of the damaged network is capable of spontaneously becoming active again. The process often occurs repeatedly. To model this marked network recovery, we examine the effect of local node recoveries and stochastic contiguous spreading, and find that they can lead to the spontaneous emergence of macroscopic ‘phase-flipping’ phenomena. As the network is of finite size and is stochastic, the fraction of active nodes z switches back and forth between the two network collective modes characterized by high network activity and low network activity. Furthermore, the system exhibits a strong hysteresis behaviour analogous to phase transitions near a critical point. We present real-world network data exhibiting phase switching behaviour in accord with the predictions of the model.

Synchronization of coupled oscillators on networks has been investigated in a wide range of topologies. One of the latest findings is the explosive synchronization in the scale-free network with a positive frequency-degree correlation (Gómez G. J. et al. , Phys. Rev. Lett. , 106 (2011) 128701). In this letter, we generalize this study and explore the effect of mixing parts on the Kuramoto model with positive correlation between frequencies and degrees. It is shown that small or weak mixing parts on module networks may accelerate the synchronization of the whole network while large and strong mixing parts may hinder synchronization. In particular, by altering the mixing part of a joint-star network, a two-step shaped transition of synchronization is observed with theoretical analysis on the critical points. Our findings indicate that mesoscopic structures should be of importance to affect network explosive synchronization.

The collective dynamics of a network of excitable nodes changes dramatically when inhibitory nodes are introduced. We consider inhibitory nodes which may be activated just like excitatory nodes but, upon activating, decrease the probability of activation of network neighbors. We show that, although the direct effect of inhibitory nodes is to decrease activity, the collective dynamics becomes self-sustaining. We explain this counterintuitive result by defining and analyzing a "branching function" which may be thought of as an activity-dependent branching ratio. The shape of the branching function implies that for a range of global coupling parameters dynamics are self-sustaining. Within the self-sustaining region of parameter space lies a critical line along which dynamics take the form of avalanches with universal scaling of size and duration, embedded in ceaseless timeseries of activity. Our analyses, confirmed by numerical simulation, suggest that inhibition may play a counterintuitive role in excitable networks.

We develop a methodology for analyzing the percolation phenomena of two mutually coupled (interdependent) networks based on the cavity method of statistical mechanics. In particular, we take into account the influence of degree--degree correlations inside and between the networks on the network robu...

The availability of data from many different sources and fields of science has made it possible to map out an increasing number of networks of contacts and interactions. However, quantifying how reliable these data are remains an open problem. From Biology to Sociology and Economics, the identification of false and missing positives has become a problem that calls for a solution. In this work we extend one of the newest, best performing models—due to Guimerá and Sales-Pardo in 2009—to directed networks. The new methodology is able to identify missing and spurious directed interactions with more precision than previous approaches, which renders it particularly useful for analyzing data reliability in systems like trophic webs, gene regulatory networks, communication patterns and several social systems. We also show, using real-world networks, how the method can be employed to help search for new interactions in an efficient way.

Author(s): Jianxi Gao, Sergey V. Buldyrev, H. Eugene Stanley, Xiaoming Xu, and Shlomo Havlin

Percolation theory is an approach to study the vulnerability of a system. We develop an analytical framework and analyze the percolation properties of a network composed of interdependent networks (NetONet). Typically, percolation of a single network shows that the damage in the network due to a fai...

[Phys. Rev. E 88, 062816] Published Fri Dec 20, 2013

We consider a system of phase oscillators with random intrinsic frequencies coupled through sparse random networks, and investigate how the connectivity disorder affects the nature of collective synchronization transitions. Various distribution types of intrinsic frequencies are considered: uniform,...

As a paradigmatic model, the Kuramoto model has provided a platform for investigating synchronization among non-identical oscillators. In this work, we consider the Kuramoto model consisting of conformists with positive coupling strength and contrarians with negative coupling strength. We introduce ...

We study the temporal percolation properties of temporal networks by taking as a representative example the recently proposed activity driven network model [N. Perra et al., Sci. Rep. 2, 469 (2012)]. Building upon an analytical framework based on a mapping to hidden variables networks, we provide expressions for the percolation time marking the onset of a giant connected component in the integrated network. In particular, we consider both the generating function formalism, valid for degree uncorrelated networks, and the general case of networks with degree correlations. We discuss the different limits of the two approach, indicating the parameter regions where the correlated threshold collapses onto the uncorrelated case. Our analytical prediction are confirmed by numerical simulations of the model. The temporal percolation concept can be fruitfully applied to study epidemic spreading on temporal networks. We show in particular how the susceptible-infected- removed model on an activity driven network can be mapped to the percolation problem up to a time given by the spreading rate of the epidemic process. This mapping allows to obtain addition information on this process, not available for previous approaches.

We consider a Kuramoto model of coupled oscillators that includes quenched random interactions of the type used by van Hemmen in his model of spin glasses. The phase diagram is obtained analytically for the case of zero noise and a Lorentzian distribution of the oscillators' natural frequencies. Dep...

Different types of interactions coexist and coevolve to shape the structure and function of a multiplex network. We propose here a general class of growth models in which the various layers of a multiplex network coevolve through a set of non-linear preferential attachment rules. We show, both numerically and analytically, that by tuning the level of non-linearity these models allow to reproduce either homogeneous or heterogeneous degree distributions, together with positive or negative degree correlations across layers. In particular, we derive the condition for the appearance of a condensed state in which one node in each layer attracts an extensive fraction of all the edges.

Author(s): J. Platig, E. Ott, and M. Girvan

In various applications involving complex networks, network measures are employed to assess the relative importance of network nodes. However, the robustness of such measures in the presence of link inaccuracies has not been well characterized. Here we present two simple stochastic models of false a...

[Phys. Rev. E 88, 062812] Published Wed Dec 11, 2013

Stimulus information is encoded in the spatial-temporal structures of external inputs to the neural system. The ability to extract the temporal information of inputs is fundamental to brain function. It has been found that the neural system can memorize temporal intervals of visual inputs in the order of seconds. Here...

The ability to understand and eventually predict the emergence of information and activation cascades in social networks is core to complex socio-technical systems research. However, the complexity of social interactions makes this a challenging enterprise. Previous works on cascade models assume that the emergence of this collective phenomenon is related to the activity observed in the local neighborhood of individuals, but do not consider what determines the willingness to spread information in a time-varying process. Here we present a mechanistic model that accounts for the temporal evolution of the individual state in a simplified setup. We model the activity of the individuals as a complex network of interacting integrate-and-fire oscillators. The model reproduces the statistical characteristics of the cascades in real systems, and provides a framework to study the time evolution of cascades in a state-dependent activity scenario.

The k-core decomposition of a network has thus far mainly served as a powerful tool for the empirical study of complex networks. We now propose its explicit integration in a theoretical model. We introduce a Hard-core Random Network model that generates maximally random networks with arbitrary degre...