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

We show that the modelocking dynamics in parametric frequency combs is equivalent to synchronization phenomena that occur in many physical systems as described by the Kuramoto model for coupled oscillators.

Synchronization Phenomena in Modelocked Parametric Frequency Combs

Consider a knotted loop of wire with charge distributed uniformly on it. The electric ﬁeld around such a loop was recently proven to have at least 2 t + 1 equilibrium points, where t is a topological invariant known as the knot’s tunnel number. Here we show numerically that this lower bound is not sharp, and we oﬀer conjectures (but not proofs) for the minimum number of equilibrium points around each type of knot with ﬁve or fewer crossings. We also visualize the equipotential surfaces of charged knots. These are the ﬁrst steps in the study of “electrostatic knot theory.”

Zeros of the electric ﬁeld around a charged knot

Superconducting systems built of underdamped Josephson junctions are of interest for ballistic and quantum vortex experiments. The dissipation in a Josephson tunnel junction can be extremely small because its subgap resistance can easily be made of the order of 1 MΩ which for a typical junction translates into a McCumber parameter as high as 107. In a Josephson system with such little dissipation, the vortex velocity can be high enough so that the energy stored in the electric field generated by the moving vortex has to be taken into account. This electric field contribution to the energy acts like a kinetic-energy term and the vortex mass is found to be proportional to the junction capacitance [1, 2, 3]. Measurements have indicated the existence of a mass term in the equation of motion for a vortex [4, 5]. The concept of a massive vortex has also been tested in quantum experiments: there is evidence for quantum tunneling of vortices [4, 5], for a collective, quantum phase transition triggered by a Bose-condensation of vortices [6], and for vortex interference (Aharonov-Casher effect) [7].

Vortex propagation in discrete josephson rings

We revisit a textbook example of a singularly perturbed nonlinear boundary-value problem. Unexpectedly, it shows a wealth of phenomena that seem to have been overlooked previously, including a pitchfork bifurcation in the number of solutions as one varies the small parameter, and transcendentally small terms in the initial conditions that can be calculated by elementary means. Based on our own classroom experience, we believe this problem could provide an enjoyable workout for students in courses on perturbation methods, applied dynamical systems, or numerical analysis.

Surprises in a classic boundary-layer problem

A tabletop waterwheel, designed and built by Prof. Willem Malkus (Math. Dept., MIT), is used to demonstrate chaos in a mechanical analog of the Lorenz equations. The waterwheel’s rotational damping rate can be adjusted by tightening or loosening a brake. When the brake is not too tight, the wheel settles into a steady rotation. Either direction of rotation is possible. When the brake was tightened in an attempt to make the wheel go chaotic, unfortunately it was tightened too much. Instead of settling into the desired pattern of irregular reversals, the wheel fell into a simple pendulum-like motion, turning once to the left, then back to the right, and so on. After the brake was loosened slightly, the motion became chaotic. (And the crowd went wild...) The waterwheel is an exact mechanical analog of the Lorenz equations (see Exercise 9.1.3 in Strogatz (1994)). The corresponding Lorenz parameters are b = 1, σ ∝ ν, and r ∝ ν, where ν is the rotational damping rate. So as the brake is tightened, the parameters are constrained to move along a hyperbola rσ = C in the (r, σ) parameter space.

https://core.ac.uk/download/pdf/4900597.pdf

Steven H. Strogatz

Papers

Synchronization Phenomena in Modelocked Parametric Frequency Combs

Zeros of the electric ﬁeld around a charged knot

Vortex propagation in discrete josephson rings

Surprises in a classic boundary-layer problem

Notes on the videotape Nonlinear Dynamics and Chaos: Lab Demonstrations