Networks of coupled dynamical systems have been used to model biological oscillators, Josephson junction arrays, excitable media, neural networks, spatial games, genetic control networks and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks 'rewired' to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them 'small-world' networks, by analogy with the small-world phenomenon (popularly known as six degrees of separation. The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.

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Collective dynamics of small-world networks

There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

抗原变异可使得多种致病微生物易于逃避宿主免疫应答。表达在感染红细胞表面的恶性疟原虫红细胞表面蛋白1（PfPMP1）与感染红细胞、内皮细胞、树突状细胞以及胎盘的单个或多个受体作用，在黏附及免疫逃避中起关键的作用。每个单倍体基因组var基因家族编码约60种成员，通过启动转录不同的var基因变异体为抗原变异提供了分子基础。

疟原虫var基因转换速率变化导致抗原变异[英]／Paul H, Robert P, Christodoulou Z, et al//Proc Natl Acad Sci U S A

Controlling chaos

The synchronization of chaotic systems

We enhance the response of a ``stochastic resonator'' by coupling it into a chain of identical resonators. Specifically, we show via numerical simulation that local linear coupling of overdamped nonlinear oscillators significantly enhances the signal-to-noise ratio of the response of a single oscillator to a time-periodic signal and noise. We relate this array enhanced stochastic resonance to the global spatiotemporal dynamics of the array and show how noise, coupling, and bistable potential cooperate to organize spatial order, temporal periodicity, and peak signal-to-noise ratio.

Array enhanced stochastic resonance and spatiotemporal synchronization.

DISORDERand noise in physical systems usually tend to destroy spatial and temporal regularity, but recent research into nonlinear systems provides intriguing counterexamples. In the phenomenon of stochastic resonance1, for example, the presence of noise improves the ability of some nonlinear systems to transfer information reliably. Noise can also remove chaos in a model oscillator2, and facilitate synchronization in an extended array of bistable elements3. Here we explore the use of disorder as a means to control spatiotemporal chaos4–8 in coupled arrays of forced, damped, nonlinear oscillators. Chaotic behaviour in spatially extended systems, especially in biology and physiology9,10, might be amenable to control, as occurs in lowdimensional temporally chaotic systems11,12. In our numerical experiments, one and twodimensional arrays of identical oscillators behave chaotically, but the introduction of slight, uncorrelated differences between the oscillators induces ordered motion characterized by complex but regular spatiotemporal patterns.

Taming spatiotemporal chaos with disorder

Recently, the synchronization and signal processing ability of a locally and linearly coupled array of bistable elements was enhanced by the addition of uncorrelated noise [J. F. Lindner, B. K. Meadows, W. L. Ditto, M. E. Inchiosa, and A. R. Bulsara, Phys. Rev. Lett. 75, 3 (1995)]. Here, we detail the performance of such an array as a function of both coupling and noise. Simple theoretical arguments, grounded in extensive numerical studies, suggest how to ``tune'' the array for best synchronization and signal-to-noise ratio. Specifically, we propose that, for large array size N, the optimal coupling scales like ${\mathit{N}}^{2}$ and the optimal noise variance scales like N. This scaling matches the coupling-induced correlation length to the array length and the noise-generated mean hopping time to the modulation period, thereby creating a stochastic resonance in space and time. \textcopyright{} 1996 The American Physical Society.

Scaling laws for spatiotemporal synchronization and array enhanced stochastic resonance.

We use noise to extend signal propagation in one- and two-dimensional arrays of two-way coupled bistable oscillators. In a numerical model, we sinusoidally force one end of a chain of noisy oscillators. We record a signal-to-noise ratio at each oscillator. We demonstrate that moderate noise significantly extends the propagation of the sinusoidal input. Oscillators far from the input, where noise extends the signal, exhibit a classical stochastic resonance. We obtain similar results with two-dimensional arrays. The simplicity of the model suggests the generality of the phenomenon.

Noise Enhanced Propagation

The unprecedented light curves of the Kepler space telescope document how the brightness of some stars pulsates at primary and secondary frequencies whose ratios are near the golden mean, the most irrational number. A nonlinear dynamical system driven by an irrational ratio of frequencies generically exhibits a strange but nonchaotic attractor. For Kepler's "golden" stars, we present evidence of the first observation of strange nonchaotic dynamics in nature outside the laboratory. This discovery could aid the classification and detailed modeling of variable stars.

John F. Lindner

Papers

Array enhanced stochastic resonance and spatiotemporal synchronization.

Taming spatiotemporal chaos with disorder

Scaling laws for spatiotemporal synchronization and array enhanced stochastic resonance.

Noise Enhanced Propagation

Strange Nonchaotic Stars