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

A few areas of human activity are examined here, using a number of different types of mathematical and computational models. First, we examine networks of five languages of the world, with their connectivity derived from the sounds of the words in these languages. We explore the graph-theoretic properties of these networks, finding that these phonological language networks have common properties, and are in turn topologically distinct from other types of complex networks observed in the literature. In addition, we discuss what these common properties imply for how we process language and why natural language is structured the way it is. In addition, by examining the networks of English and Spanish, we explain a surprising difference in processing that was uncovered in some recent experiments, and discuss some more general implications of competition or facilitation between different modes of cognition.
We next explore a more macro-scale area of human activity: cities. Superlinear scaling in cities, which appears in sociological quantities such as economic productivity and creative output relative to urban population size, has been observed but not been given a satisfactory theoretical explanation. We provide a model for the superlinear relationship between population size and innovation found in cities, with a reasonable range for the exponent.
Next, we examine collaboration and innovation in the scientific world. We attempt to understand how variations in ‘scientific distance’ among collaborators affect the degree to which that collaboration is a productive one. Using both mathematical models and empirical data, we explore the relationship between the scientific or social distance of collaborators and the fruitfulness of their output.
Last, we examine Joe DiMaggio's 56-game hitting streak and look at its probability, using a number of simple models. And it turns out that, contrary to many people's expectations, an extreme streak, while unlikely, is not unlikely to have occurred about once within the history of baseball. Surprisingly, however, such a record should have occurred far earlier in baseball history: back in the late 1800's or early 1900's. But not in 1941, when it actually happened.

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Complex dynamics of human activity: language, cities, collaboration, and baseball

Exact solution of percolation on random graphs with arbitrary degree distributions

The authors tackle the problem of infection spreading of mutant strains and how it is influenced by the structure of the network. The paper finds that the probability distributions that describe the impact of mutant contagion are similar if the spreading takes place on any infinite-dimensional network.

https://link.aps.org/pdf/10.1103/PhysRevResearch.2.033005

Descendant distributions for the impact of mutant contagion on networks

The incubation period of a disease is the time between an initiating pathologic event and the onset of symptoms. For typhoid fever, polio, measles, leukemia and many other diseases, the incubation period is highly variable. Some affected people take much longer than average to show symptoms, leading to a distribution of incubation periods that is right skewed and often approximately lognormal. Although this statistical pattern was discovered more than sixty years ago, it remains an open question to explain its ubiquity. Here we propose an explanation based on evolutionary dynamics on graphs. For simple models of a mutant or pathogen invading a network-structured population of healthy cells, we show that skewed distributions of incubation periods emerge for a wide range of assumptions about invader fitness, competition dynamics, and network structure. The skewness stems from stochastic mechanisms associated with two classic problems in probability theory: the coupon collector and the random walk. Unlike previous explanations that rely crucially on heterogeneity, our results hold even for homogeneous populations. Thus, we predict that two equally healthy individuals subjected to equal doses of equally pathogenic agents may, by chance alone, show remarkably different time courses of disease.

http://www.biorxiv.org/content/biorxiv/early/2017/05/30/144139.full.pdf

Steven H. Strogatz

Papers

Complex dynamics of human activity: language, cities, collaboration, and baseball

Exact solution of percolation on random graphs with arbitrary degree distributions

Descendant distributions for the impact of mutant contagion on networks

Evolutionary dynamics of incubation periods

Topology of zigzag chromatin.