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

A number of recent studies have focused on the statistical properties of networked systems such as social networks and the Worldwide Web. Researchers have concentrated particularly on a few properties that seem to be common to many networks: the small-world property, power-law degree distributions, and network transitivity. In this article, we highlight another property that is found in many networks, the property of community structure, in which network nodes are joined together in tightly knit groups, between which there are only looser connections. We propose a method for detecting such communities, built around the idea of using centrality indices to find community boundaries. We test our method on computer-generated and real-world graphs whose community structure is already known and find that the method detects this known structure with high sensitivity and reliability. We also apply the method to two networks whose community structure is not well known—a collaboration network and a food web—and find that it detects significant and informative community divisions in both cases.

/pdf/community-structure-in-social-and-biological-networks-1jamu4rcfv.pdf

Community structure in social and biological networks

Complexity-defined in terms of the number of components and the nature of the interdependencies between them-is clearly a relevant feature of all tasks that groups perform. Yet the role that task complexity plays in determining group performance remains poorly understood, in part because no clear language exists to express complexity in a way that allows for straightforward comparisons across tasks. Here we avoid this analytical difficulty by identifying a class of tasks for which complexity can be varied systematically while keeping all other elements of the task unchanged. We then test the effects of task complexity in a preregistered two-phase experiment in which 1,200 individuals were evaluated on a series of tasks of varying complexity (phase 1) and then randomly assigned to solve similar tasks either in interacting groups or as independent individuals (phase 2). We find that interacting groups are as fast as the fastest individual and more efficient than the most efficient individual for complex tasks but not for simpler ones. Leveraging our highly granular digital data, we define and precisely measure group process losses and synergistic gains and show that the balance between the two switches signs at intermediate values of task complexity. Finally, we find that interacting groups generate more solutions more rapidly and explore the solution space more broadly than independent problem solvers, finding higher-quality solutions than all but the highest-scoring individuals.

https://www.pnas.org/content/pnas/118/36/e2101062118.full.pdf

Task complexity moderates group synergy.

Recent work on the structure of social networks and the internet has focussed attention on graphs with distributions of vertex degree that are significantly different from the Poisson degree distributions which 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 also apply our theory to some real-world graphs, including the world-wide web and collaboration graphs of scientists and business-people. 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 which is not captured by the random graph.

Random Graphs with Arbitrary Degree Distribution and Their Applications

Networks, Dynamics, and the Smafl-World

Social influence: the puzzling nature of success in cultural markets

Cooperation in repeated games has been widely studied in experimental settings; however, the duration over which players participate in such experiments is typically confined to at most hours, and often to a single game. Given that in real world settings people may have years of experience, it is natural to ask how behavior in cooperative games evolves over the long run. Here we analyze behavioral data from three distinct games involving 571 individual experiments conducted over a two-year interval. First, in the case of a standard linear public goods game we show that as players gain experience, they become less generous both on average and in particular towards the end of each game. Second, we analyze a multiplayer prisoner's dilemma where players are also allowed to make and break ties with their neighbors, finding that experienced players show an increase in cooperativeness early on in the game, but exhibit sharper "endgame" effects. Third, and finally, we analyze a collaborative search game in which players can choose to act selfishly or cooperatively, finding again that experienced players exhibit more cooperative behavior as well as sharper endgame effects. Together these results show consistent evidence of long-run learning, but also highlight directions for future theoretical work that may account for the observed direction and magnitude of the effects.

/pdf/long-run-learning-in-games-of-cooperation-249ncjxjq4.pdf

Duncan J. Watts

Papers

Task complexity moderates group synergy.

Random Graphs with Arbitrary Degree Distribution and Their Applications

Networks, Dynamics, and the Smafl-World

Social influence: the puzzling nature of success in cultural markets

Long-run learning in games of cooperation