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

Systems as diverse as genetic networks or the World Wide Web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature was found to be a consequence of two generic mechanisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach preferentially to sites that are already well connected. A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.

https://arxiv.org/pdf/cond-mat/9910332.pdf

Emergence of Scaling in Random Networks

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

The emergence of order in natural systems is a constant source of inspiration for both physical and biological sciences. While the spatial order characterizing for example the crystals has been the basis of many advances in contemporary physics, most complex systems in nature do not offer such high degree of order. Many of these systems form complex networks whose nodes are the elements of the system and edges represent the interactions between them. 
Traditionally complex networks have been described by the random graph theory founded in 1959 by Paul Erdohs and Alfred Renyi. One of the defining features of random graphs is that they are statistically homogeneous, and their degree distribution (characterizing the spread in the number of edges starting from a node) is a Poisson distribution. In contrast, recent empirical studies, including the work of our group, indicate that the topology of real networks is much richer than that of random graphs. In particular, the degree distribution of real networks is a power-law, indicating a heterogeneous topology in which the majority of the nodes have a small degree, but there is a significant fraction of highly connected nodes that play an important role in the connectivity of the network. 
The scale-free topology of real networks has very important consequences on their functioning. For example, we have discovered that scale-free networks are extremely resilient to the random disruption of their nodes. On the other hand, the selective removal of the nodes with highest degree induces a rapid breakdown of the network to isolated subparts that cannot communicate with each other. 
The non-trivial scaling of the degree distribution of real networks is also an indication of their assembly and evolution. Indeed, our modeling studies have shown us that there are general principles governing the evolution of networks. Most networks start from a small seed and grow by the addition of new nodes which attach to the nodes already in the system. This process obeys preferential attachment: the new nodes are more likely to connect to nodes with already high degree. We have proposed a simple model based on these two principles wich was able to reproduce the power-law degree distribution of real networks. Perhaps even more importantly, this model paved the way to a new paradigm of network modeling, trying to capture the evolution of networks, not just their static topology.

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Statistical mechanics of complex networks

Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.

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The Structure and Function of Complex Networks

In recent numerical experiments on series arrays of overdamped Josephson junctions, Nichols and Wiesenfeld [Phys. Rev. A 45, 8430 (1992)] discovered that the periodic states known as splay states are neutrally stable in all but four directions in phase space. We present a theory that accounts for this enormous degree of neutral stability. The theory also predicts the four non-neutral Floquet multipliers to within 0.1% of their numerically computed values. The analytical approach used here may be appli- cable to other globally coupled systems of oscillators, such as multimode lasers, electronic oscillator circuits, and solid-state laser arrays.

Splay states in globally coupled Josephson arrays: Analytical prediction of Floquet multipliers.

The network characteristics based on the phonological similarities in the lexicons of several languages were examined. These languages differed widely in their history and linguistic structure, but commonalities in the network characteristics were observed. These networks were also found to be different from other networks studied in the literature. The properties of these networks suggest explanations for various aspects of linguistic processing and hint at deeper organization within human language.

The Structure of Phonological Networks Across Multiple Languages

Dynamics of circular arrays of Josephson junctions and the discrete sine-Gordon equation

During prolonged temporal isolation in caves or windowless rooms, human subjects often develop complicated sleep-wake patterns. Seeking lawful structure in these patterns, we have reanalyzed the spontaneous timing of 359 sleep-wake cycles recorded from 15 internally desynchronized human subjects. The observed sleep-wake patterns obey a simple rule: The phase of the circadian temperature rhythm at bedtime determines the lengths of both prior wake (alpha) and subsequent sleep (rho). From this rule we derive an average alpha:rho relationship that depends on circadian phase. The relationship reconciles the established negative alpha:rho correlation observed in synchronized subjects with the positive alpha:rho correlation found in desynchronized subjects. Our most surprising result concerns the residual deviations of alpha and rho from their circadian phase-adjusted mean values. We report that there is no significant positive correlation between the residuals of alpha and rho, contrary to the prediction of restorative models of sleep duration. Our findings illuminate the mechanisms underlying sleep regulation and provide much-needed tests of mathematical models of the sleep-wake cycle.

/pdf/circadian-regulation-dominates-homeostatic-control-of-sleep-14oa0zuo72.pdf

Circadian regulation dominates homeostatic control of sleep length and prior wake length in humans.

The authors study mutual synchronisation in a model of interacting limit cycle oscillators with random intrinsic frequencies. It is shown rigorously that the model exhibits no long-range order in one dimension, and that in higher-dimensional lattices, large clusters of synchronised oscillators necessarily have a sponge-like structure. Surprisingly, the phase-locking behaviour of the mean-field model is completely different from that of any finite-dimensional lattice, indicating that d= infinity is the upper critical dimension for phase locking.

https://iopscience.iop.org/article/10.1088/0305-4470/21/13/005/pdf

Steven H. Strogatz

Papers

Splay states in globally coupled Josephson arrays: Analytical prediction of Floquet multipliers.

The Structure of Phonological Networks Across Multiple Languages

Dynamics of circular arrays of Josephson junctions and the discrete sine-Gordon equation

Circadian regulation dominates homeostatic control of sleep length and prior wake length in humans.

Collective synchronisation in lattices of nonlinear oscillators with randomness