Complex network

In network science, which touches scientific disciplines as varied as computer science, mathematics, physics, biology and sociology, the term "complex network" refers to a network (graph) that has non-trivial topological structure. Most real world networks can be considered complex on account of their having several topological features that do not exist in simple networks, e.g., a heavy-tail in the degree distribution, a high clustering coefficient, assortativity or disassortativity among vertices, community structure at many scales and evidence of a hierarchical structure. In contrast, simple networks have none of these properties, and are typically represented by graphs such as a lattice or a random graph, which look either exactly or roughly the same no matter what part is examined.

The two most well-known examples of complex networks are those of scale-free networks and small-world networks. Both are specific models of complex networks discovered in the late 1990s by physicists, and are canonical case-studies in the field. However, as network science has continued to grow in importance and popularity, other models of complex networks have been developed. Indeed, the field continues to develop at a brisk pace, and has brought together researchers from a variety of fields. Network science, and the study and use of complex networks in particular, shows some promise of helping to unravel the structure of the genetic regulatory network, to explain how to build robust and scalable communication networks both wired and wireless, to aid in the development of more efficient vaccination strategies, and to produce a near endless stream of attractive pictures.

Scale-free networks
A network is named scale-free if its degree distribution, i.e., the probability that a node selected uniformly at random has a certain number of links (degree), follows a particular mathematical function called a power law. The power law implies that the degree distribution of these networks has no characteristic scale. In contrast, network with a single well-defined scale are somewhat similar to a lattice in that every node has (roughly) the same degree. Examples of networks with a single scale include the Erdos- Renyi random graph and hypercubes. 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, then it wouldn't be a scale-free distribution!

Interest in scale-free networks began in the late 1990s with the apparent discovery of a power-law degree distribution in many real world networks such as the World Wide Web, the network of Autonomous systems (ASs), the network of Internet routers, protein interaction networks, email networks, etc. Although many of these distributions are not unambiguously power laws, their breadth, both in degree and in domain, shows that networks exhibiting such a distribution are clearly very different from what you would expect if edges existed independently and at random (a Poisson distribution). Indeed, 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.g., The Gibrat principle by Herbert Simon, the Matthew effect, cumulative advantage and, most recently, preferential attachment by  Barabasi and Albert for power-law degree distributions.

Networks with a power-law degree distribution can be highly resistant to the random deletion of vertices, i.e., the vast majority of vertices remain connected together in a giant component. Such networks can also be quite sensitive to targeted attacks aimed at fracturing the network quickly. When the graph is uniformly random except for the degree distribution, these critical vertices are the ones with the highest degree, and have thus been implicated in the spread of disease (natural and artificial) in social and communication networks, and in the spread of fads (both of which are modeled by a percolation or branching process).

Small-world networks
A network is called a small-world network by analogy with the small-world phenomenon (popularly known as six degrees of separation). The small world hypothesis, was first described by the Hungarian writer Frigyes Karinthy in 1929, and tested experimentally by Stanley Milgram (1967), is the idea that two arbitrary people are connected by only six degrees of separation, i.e. the diameter of the corresponding graph of social connections is not much larger than six. In 1998, Duncan J. Watts and Steve Strogatz published the first small-world network model, which through a single parameter smoothly interpolates between a random graph to a lattice. Their model demonstrated that with the addition of only a small number of long-range links, a regular graph, in which the diameter is proportional to the size of the network, can be transformed into a "small world" in which the average number of edges between any two vertices is very small (mathematically, it should grow as the logarithm of the size of the network). It is known that a wide variety of abstract graphs exhibit the small-world property, e.g., random graphs and  scale-free networks. Further, real world networks such as the World Wide Web and the metabolic network also exhibit this property.

In the scientific literature on networks, there is some ambiguity associated with the term "small world." In addition to referring to the size of the diameter of the network, it can also refer to the co-occurrence of a small diameter and a high clustering coefficient. The clustering coefficient is a metric that represents the density of triangles in the network. For instance, sparse random graphs have a vanishingly small clustering coefficient while real world networks often have a coefficient significantly larger. Scientists point to this difference as suggesting that edges are correlated in real world networks.

Researchers and scientists

 * L.A.N. Amaral
 * Albert-László Barabási
 * Yaneer Bar-Yam
 * Sergey Dorogovtsev
 * Jose Fernando F. Mendes
 * Mark Newman
 * Sidney Redner
 * Kim Sneppen
 * Ricard V. Solé
 * Steven Strogatz
 * Duncan J. Watts

Books

 * Barabási, Albert-László Linked: How Everything is Connected to Everything Else, 2004 ISBN 0452284392
 * S.N. Dorogovtsev and J.F.F. Mendes, Evolution of Networks: from biological networks to the Internet and WWW, Oxford University Press, 2003, ISBN 0198515901
 * Duncan J. Watts, Six Degrees: The Science of a Connected Age, W. W. Norton & Company, 2003, ISBN 0393041425
 * Duncan J. Watts, Small Worlds : The Dynamics of Networks between Order and Randomness, Princeton University Press, 2003, ISBN 0691117047