Most social, biological, and technological networks display substantial non-trivial topological features, with patterns of connection between networks an introduction mark newman pdf elements that are neither purely regular nor purely random. Such features include a heavy tail in the degree distribution, a high clustering coefficient, assortativity or disassortativity among vertices, community structure, and hierarchical structure. In the case of directed networks these features also include reciprocity, triad significance profile and other features. In contrast, many of the mathematical models of networks that have been studied in the past, such as lattices and random graphs, do not show these features.

The most complex structures can be realized by networks with a medium number of interactions. Two well-known and much studied classes of complex networks are scale-free networks and small-world networks, whose discovery and definition are canonical case-studies in the field. Both are characterized by specific structural featuresâ€”power-law degree distributions for the former and short path lengths and high clustering for the latter.

However, as the study of complex networks has continued to grow in importance and popularity, many other aspects of network structure have attracted attention as well. Recently, the study of complex networks has been expanded to networks of networks.

If those networks are interdependent, they become significantly more vulnerable to random failures and targeted attacks and exhibit cascading failures and first-order percolation transitions. Furthermore, the collective behavior of a network in the presence of nodes failure and recovery has been studied.

The field continues to develop at a brisk pace, and has brought together researchers from many areas including mathematics, physics, biology, climate, computer science, sociology, epidemiology, and others. Research on networks are regularly published in the most visible scientific journals and obtain vigorous funding in many countries.

Network theory was found recently useful to identify bottlenecks in city traffic. Network science is the topic of many conferences in a variety of different fields, and has been the subject of numerous books both for the lay person and for the expert. An example of complex scale-free network.

A network is named scale-free if its degree distribution, i. The power law implies that the degree distribution of these networks has no characteristic scale.

In a network with a scale-free degree distribution, some vertices have a degree that is orders of magnitude larger than the average – these vertices are often called “hubs”, although this is a bit misleading as there is no inherent threshold above which a node can be viewed as a hub. If there were such a threshold, the network would not be scale-free. Internet routers, protein interaction networks, email networks, etc. There are many different ways to build a network with a power-law degree distribution.

The Yule process is a canonical generative process for power laws, and has been known since 1925. However, it is known by many other names due to its frequent reinvention, e. The Gibrat principle by Herbert A.