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.

/pdf/the-structure-and-function-of-complex-networks-34deompc9l.pdf

The Structure and Function of Complex Networks

. The Kuramoto model is a prototypical model used for rigorous mathematical analysis in the ﬁeld of synchronisation and nonlinear dynamics. A realisation of this model consists of a collection of identical oscillators with interactions given by a network, which we identify respectively with vertices and edges of a graph. In this paper, we show that a graph with suﬃcient expansion must be globally synchronising, meaning that the Kuramoto model on such a graph will converge to the fully synchronised state with all the oscillators with same phase, for every initial state up to a set of measure zero. In particular, we show that for any ε > 0 and p (cid:62) (1+ ε )(log n ) /n , the Kuramoto model on the Erdős–Rényi graph G ( n, p ) is globally synchronising with probability tending to one as n goes to inﬁnity. This improves on a previous result of Kassabov, Strogatz and Townsend and solves a conjecture of Ling, Xu and Bandeira. We also show that the Kuramoto model is globally synchronising on any d -regular Ramanujan graph with d (cid:62) 600 and that, for the same range of degrees, a d -regular random graph is typically globally synchronising.

Expander graphs are globally synchronising

Synchronization is studied in a population of phase oscillators with mean-field coupling—a special case of the more general Winfree model. Each oscillator is coupled to the mean-field with a strength dependent on its phase. The uncoupled frequencies of the oscillators are assumed to be randomly distributed according to a specified population density. The response of this system is considered as a function of two parameters. The first describes the strength of the coupling between each oscillator while the second characterizes the distribution of uncoupled frequencies in the population. As these parameters are varied, the synchronous solution can disappear as oscillators near the edge of the population (large deviation of their natural frequency from the population mean) no longer remain locked to the mean frequency. Using a PoincareLindstedt analysis, the bifurcation describing the loss of synchrony is characterized for a general oscillator population. The bifurcation is then calculated for several different frequency distributions, including even polynomial distributions and populations that correspond to a discrete system ofN oscillators. By formulating the discrete system in a continuum framework, the analysis can be carried through and the bifurcation curve can be calculated for arbitrarily large discrete populations. Results of the discrete system will be presented from N = 2 to N = 10 6 oscillators. As the number of discrete oscillators increases, approaching the continuum uniform distribution, the coefficients of the bifurcation curve increase and become singular as N grows without bound.

/pdf/synchronization-in-the-winfree-model-of-coupled-nonlinear-d3q4qwhn9p.pdf

Synchronization in the winfree model of coupled nonlinear oscillators

Einstein’s First Proof

We study the electric ﬁeld around a continuous one-dimensional loop of static charge, under the assumption that the charge is distributed uniformly along the loop. For rectangular or stadium-shaped loops in the plane, we ﬁnd that the electric ﬁeld can undergo a symmetry-breaking pitchfork bifurcation as the loop is elongated; the ﬁeld can have either one or three zeros, depending on the loop’s aspect ratio. For knotted charge distributions in three-dimensional space, we compute the electric ﬁeld numerically and compare our results to previously published theoretical bounds on the number of equilibrium points around charged knots. Our computations reveal that the previous bounds are far from sharp. The numerics also suggest conjectures for the actual minimum number of equilibrium points for all charged knots with ﬁve or fewer crossings. In addition, we provide the ﬁrst images of the equipotential surfaces around charged knots, and visualize their topological transitions as the level of the potential is varied.

/pdf/exploring-the-electric-field-around-a-loop-of-static-charge-1mj95vlk.pdf

Steven H. Strogatz

Papers

Expander graphs are globally synchronising

Synchronization in the winfree model of coupled nonlinear oscillators

Einstein’s First Proof

Exploring the electric field around a loop of static charge: Rectangles, stadiums, ellipses, and knots

Chaotic Mappings and Probability Distributions