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

Recent papers have demonstrated that synchronization in the Lorenz system is highly robust to additive perturbation of the drive signal. This property has led to a concept known as chaotic signal masking and recovery. This paper presents experiments and an approximate analytical model that quantify and explain the observed robustness of synchronization in the Lorenz system. In particular, we explain why speech and other narrowband perturbations can be recovered faithfully, even though the synchronization error is comparable in power to the message itself.

Robustness and signal recovery in a synchronized chaotic system

Everyone is familiar with the small-world phenomenon: soon after meeting a stranger, we are often suprised to discover that we have a mutual friend, or that we are somehow linked by a short chain of friends. In this talk, I’ll present evidence that the small-world phenomenon is more than a curiosity of social networks — it is actually a general property of large, sparse networks whose topology is neither completely regular nor completely random. To check this idea, Duncan Watts and I have analyzed three networks of scientific interest: the neural network of the nematode worm C. elegans, the electrical power grid of the western United States, and the collaboration graph of actors in feature films. All three are small worlds, in the sense that the average number of “handshakes” separating any two members is extremely small (close to the theoretical lower limit set by a random graph). Yet at the same time, all three networks exhibit much more local clustering than a random net, demonstrating that they are not random. I’ll also discuss a class of model networks that interpolate between regular lattices and random graphs. Previous theoretical research on complex systems in a wide range of disciplines has focused almost exclusively on networks that are either regular or random. Real networks often lie somewhere in between. Our mathematical model shows that networks in this middle ground tend to exhibit the small-world phenomenon, thanks to the presence of a few long-range edges that link parts of the graph that would otherwise be far apart. Furthermore, we find that when various dynamical systems are coupled in a small-world fashion, they exhibit much greater propagation speed, computational power, and synchronizability than their locally connected, regular counterparts. We explore the implications of these results for simple models of disease spreading, global computation in cellular automata, and collective locking of biological oscillators.

Small-world networks

Pattern Formation in Continuous and Coupled Systems

Collective dynamics of coupled oscillators with random pinning

Some of the most pivotal moments in intellectual history occur when a new ideology sweeps through a society, supplanting an established system of beliefs in a rapid revolution of thought. Yet in many cases the new ideology is as extreme as the old. Why is it then that moderate positions so rarely prevail? Here, in the context of a simple model of opinion spreading, we test seven plausible strategies for deradicalizing a society and find that only one of them significantly expands the moderate subpopulation without risking its extinction in the process.

Steven H. Strogatz

Papers

Robustness and signal recovery in a synchronized chaotic system

Small-world networks

Pattern Formation in Continuous and Coupled Systems

Collective dynamics of coupled oscillators with random pinning

Encouraging moderation: clues from a simple model of ideological conflict