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.

/pdf/collective-dynamics-of-small-world-networks-205zhem3gm.pdf

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.

/pdf/statistical-mechanics-of-complex-networks-1zfjpvxipu.pdf

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 study of networks pervades all of science, from neurobiology to statistical physics. The most basic issues are structural: how does one characterize the wiring diagram of a food web or the Internet or the metabolic network of the bacterium Escherichia coli? Are there any unifying principles underlying their topology? From the perspective of nonlinear dynamics, we would also like to understand how an enormous network of interacting dynamical systems-be they neurons, power stations or lasers-will behave collectively, given their individual dynamics and coupling architecture. Researchers are only now beginning to unravel the structure and dynamics of complex networks.

/pdf/exploring-complex-networks-2pdzihykyi.pdf

Exploring complex networks

We explore several basic aspects of chaos, chaotic systems, and non-linear dynamics through three different setups. The logistic map, a canonical one-dimensional system exhibiting surprisingly complex and aperiodic behavior, is modeled as a function of its chaotic parameter. We consider maps of its phase space, and the progression through period-doubling bifurcations to the onset of chaos. The Feigenbaum ratio of successive bifurcation periods is estimated at 4.674, in good agreement with the accepted value. The Liapunov exponent, governing the exponential growth of small perturbations in chaotic systems, is calculated and its fractal structure compared to the corresponding bifurcation diagram for the logistic map. Using a non-linear p-n junction circuit we analyze the return maps and power spectrums of the resulting time series at various types of system behavior. Similarly, an electronic analog to a ball bouncing on a vertically driven table provides insight into real-world applications of chaotic motion. For both systems we calculate the fractal information dimension and compare with theoretical behavior for dissipative versus Hamiltonian systems. Subject headings: non-linear dynamics; non-linear dynamical systems; fractal dimension; chaos; strange attractors; logistic map

Nonlinear dynamics and Chaos

Recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the Poisson degree distributions that have been widely studied in the past. In this paper we develop in detail the theory of random graphs with arbitrary degree distributions. In addition to simple undirected, unipartite graphs, we examine the properties of directed and bipartite graphs. Among other results, we derive exact expressions for the position of the phase transition at which a giant component first forms, the mean component size, the size of the giant component if there is one, the mean number of vertices a certain distance away from a randomly chosen vertex, and the average vertex-vertex distance within a graph. We apply our theory to some real-world graphs, including the worldwide web and collaboration graphs of scientists and Fortune 1000 company directors. We demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world, while in others there is a measurable discrepancy between theory and reality, perhaps indicating the presence of additional social structure in the network that is not captured by the random graph.

Random graphs with arbitrary degree distributions and their applications.

Preface 1. Overview 1.0 Chaos, Fractals, and Dynamics 1.1 Capsule History of Dynamics 1.2 The Importance of Being Nonlinear 1.3 A Dynamical View of the World PART I. ONE-DIMENSIONAL FLOWS 2. Flows on the Line 2.0 Introduction 2.1 A Geometric Way of Thinking 2.2 Fixed Points and Stability 2.3 Population Growth 2.4 Linear Stability Analysis 2.5 Existence and Uniqueness 2.6 Impossibility of Oscillations 2.7 Potentials 2.8 Solving Equations on the Computer Exercises 3. Bifurcations 3.0 Introduction 3.1 Saddle-Node Bifurcation 3.2 Transcritical Bifurcation 3.3 Laser Threshold 3.4 Pitchfork Bifurcation 3.5 Overdamped Bead on a Rotating Hoop 3.6 Imperfect Bifurcations and Catastrophes 3.7 Insect Outbreak Exercises 4. Flows on the Circle 4.0 Introduction 4.1 Examples and Definitions 4.2 Uniform Oscillator 4.3 Nonuniform Oscillator 4.4 Overdamped Pendulum 4.5 Fireflies 4.6 Superconducting Josephson Junctions Exercises PART II. TWO-DIMENSIONAL FLOWS 5. Linear Systems 5.0 Introduction 5.1 Definitions and Examples 5.2 Classification of Linear Systems 5.3 Love Affairs Exercises 6. Phase Plane 6.0 Introduction 6.1 Phase Portraits 6.2 Existence, Uniqueness, and Topological Consequences 6.3 Fixed Points and Linearization 6.4 Rabbits versus Sheep 6.5 Conservative Systems 6.6 Reversible Systems 6.7 Pendulum 6.8 Index Theory Exercises 7. Limit Cycles 7.0 Introduction 7.1 Examples 7.2 Ruling Out Closed Orbits 7.3 Poincare-Bendixson Theorem 7.4 Lienard Systems 7.5 Relaxation Oscillators 7.6 Weakly Nonlinear Oscillators Exercises 8. Bifurcations Revisited 8.0 Introduction 8.1 Saddle-Node, Transcritical, and Pitchfork Bifurcations 8.2 Hopf Bifurcations 8.3 Oscillating Chemical Reactions 8.4 Global Bifurcations of Cycles 8.5 Hysteresis in the Driven Pendulum and Josephson Junction 8.6 Coupled Oscillators and Quasiperiodicity 8.7 Poincare Maps Exercises PART III. CHAOS 9. Lorenz Equations 9.0 Introduction 9.1 A Chaotic Waterwheel 9.2 Simple Properties of the Lorenz Equations 9.3 Chaos on a Strange Attractor 9.4 Lorenz Map 9.5 Exploring Parameter Space 9.6 Using Chaos to Send Secret Messages Exercises 10. One-Dimensional Maps 10.0 Introduction 10.1 Fixed Points and Cobwebs 10.2 Logistic Map: Numerics 10.3 Logistic Map: Analysis 10.4 Periodic Windows 10.5 Liapunov Exponent 10.6 Universality and Experiments 10.7 Renormalization Exercises 11. Fractals 11.0 Introduction 11.1 Countable and Uncountable Sets 11.2 Cantor Set 11.3 Dimension of Self-Similar Fractals 11.4 Box Dimension 11.5 Pointwise and Correlation Dimensions Exercises 12. Strange Attractors 12.0 Introductions 12.1 The Simplest Examples 12.2 Henon Map 12.3 Rossler System 12.4 Chemical Chaos and Attractor Reconstruction 12.5 Forced Double-Well Oscillator Exercises Answers to Selected Exercises References Author Index Subject Index

https://www.fefox.com/ARTICLES/Strogatz.pdf

Steven H. Strogatz

Papers

Collective dynamics of small-world networks

Exploring complex networks

Nonlinear dynamics and Chaos

Random graphs with arbitrary degree distributions and their applications.

Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering