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

Arrays of identical limit-cycle oscillators have been used to model a wide variety of pattern-forming systems, such as neural networks, convecting fluids, laser arrays, and coupled biochemical oscillators. These systems are known to exhibit rich collective behavior, from synchrony and traveling waves to spatiotemporal chaos and incoherence. Recently, Kuramoto and his colleagues reported a strange new mode of organization--here called the chimera state--in which coherence and incoherence exist side by side in the same system of oscillators. Such states have never been seen in systems with either local or global coupling; they are apparently peculiar to the intermediate case of nonlocal coupling. Here we give an exact solution for the chimera state, for a one-dimensional ring of phase oscillators coupled nonlocally by a cosine kernel. The analysis reveals that the chimera is born in a continuous bifurcation from a spatially modulated drift state, and dies in a saddle-node collision with an unstable version of itself.

Chimera States in a Ring of Nonlocally Coupled Oscillators

We analyze the dynamics of a solid-state laser driven by an injected sinusoidal field. For this type of laser, the cavity round-trip time is much shorter than its fluorescence time, yielding a dimensionless ratio of time scales $\sigma \ll 1$. Analytical criteria are derived for the existence, stability, and bifurcations of phase-locked states. We find three distinct unlocking mechanisms. First, if the dimensionless detuning $\Delta$ and injection strength $k$ are small in the sense that $k = O(\Delta) \ll \sigma^{1/2}$, unlocking occurs by a saddle-node infinite-period bifurcation. This is the classic unlocking mechanism governed by the Adler equation: after unlocking occurs, the phases of the drive and the laser drift apart monotonically. The second mechanism occurs if the detuning and the drive strength are large: $k =O(\Delta) \gg \sigma^{1/2}$. In this regime, unlocking is caused instead by a supercritical Hopf bifurcation, leading first to phase trapping and only then to phase drift as the drive is decreased. The third and most interesting mechanism occurs in the distinguished intermediate regime $k, \Delta = O(\sigma^{1/2})$. Here the system exhibits complicated, but nonchaotic, behavior. Furthermore, as the drive decreases below the unlocking threshold, numerical simulations predict a novel self-similar sequence of bifurcations whose details are not yet understood.

Nonlinear dynamics of a solid-state laser with injection

When one child goes to school with a throat infection, many of his or her classmates will often start to come down with a sore throat after two or three days A few of the children will get sick sooner, the very next day, while others may take about a week As such, there is a distribution of incubation periods – the time from exposure to illness – across the children in the class When plotted on a graph, the distribution of incubation periods is not the normal bell curve Rather the curve looks lopsided, with a long tail on the right Plotting the logarithms of the incubation periods, however, rather than the incubation periods themselves, does give a normal distribution As such, statisticians refer to this kind of curve as a “lognormal distribution" Remarkably, many other, completely unrelated, diseases – like typhoid fever or bladder cancer – also have approximately lognormal distributions of incubation periods This raised the question: why do such different diseases show such a similar curve? Working with a simple mathematical model in which chance plays a key role, Ottino-Loffler et al calculate how long it takes for a bacterial infection or cancer cell to take over a network of healthy cells The model explains why a lognormal-like distribution of incubation periods, modeled as takeover times, is so ubiquitous It emerges from the random dynamics of the incubation process itself, as the disease-causing microbe or mutant cancer cell competes with the cells of the host Intuitively, this new analysis builds on insights from the “coupon collector’s problem”: a classical problem in mathematics that describes the situation where a person collects items like baseball cards, stamps, or cartoon monsters in a videogame If a random item arrives every day, and the collector’s luck is bad, they may have to wait a long time to collect those last few items Similarly, in the model of Ottino-Loffler et al, the takeover time is dominated by dramatic slowdowns near the start or end of the infection process These effects lead to an approximately lognormal distribution, with long waits, as seen in so many diseases Ottino-Loffler et al do not anticipate that their findings will have direct benefits for medicine or public health Instead, they believe their results could help to advance basic research in the fields of epidemiology, evolutionary biology and cancer research The findings might also make an impact outside biology The term “contagion” has now become a familiar metaphor for the spread of everything from computer viruses to bank failures This model sheds light on how long it takes for a contagion to take over a network, for a variety of idealized networks and spreading processes

/pdf/evolutionary-dynamics-of-incubation-periods-2a0kzvk2uc.pdf

Evolutionary dynamics of incubation periods

Mutual synchronization of oscillators is ubiquitous in biology. The author reviews two examples in some detail. Everyone has heard of the phenomenon of synchronized menstrual cycles among women friends or roommates (Anonymous 1977). The first scientific study of menstrual synchrony was carried out by Martha McClintock (1971) while she was an undergraduate psychology major at Radcliffe in the late 60's. She studied 135 women undergraduates and had them keep records of their periods throughout the school year. In October, the cycles of close friends and roommates started an average of 8.5 days apart, but by March, the average spacing was down to 5 days, a statistically significant change. Randomly matched pairs of women showed no such change. In the animal world, groups of Southeast Asian fireflies provide a spectacular example of synchronization. Along the tidal rivers of Malaysia, Thailand and New Guinea, thousands of fireflies congregate in trees at night and flash on and off in unison. When they first arrive, their flickerings are uncoordinated. But as the night goes on, they build up the rhythm until eventually whole treefuls pulsate in silent concert. You can see this display on David Attenborough's (1992) television show The Trials of Life, in the episode called "Talking to Strangers". As he explains, "All those that are flashing are males, and their message, of course, is directed to the females, and it's a very simple one: 'Come hither-mate with me'".

Spontaneous synchronization in nature

We show that self-organization occurs in the phase dynamics of soliton modelocking in paramet- ric frequency combs. Reduction of the Lugiato-Lefever equation (LLE) to a simpler set of phase equations reveals that this self-organization arises via mechanisms akin to those in the Kuramoto model for synchronization of coupled oscillators. In addition, our simulations show that the phase equations evolve to a broadband phase-locked state, analogous to the soliton formation process in the LLE. Our simplified equations intuitively explain the origin of the pump phase offset in soliton- modelocked parametric frequency combs. They also predict that the phase of the intracavity field undergoes an anti-symmetrization that precedes phase synchronization, and they clarify the role of chaotic states in soliton formation in parametric combs.

Steven H. Strogatz

Papers

Chimera States in a Ring of Nonlocally Coupled Oscillators

Nonlinear dynamics of a solid-state laser with injection

Evolutionary dynamics of incubation periods

Spontaneous synchronization in nature

Self-organization in Kerr-cavity-soliton formation in parametric frequency combs