Networks are pervasive in the real world. Nature, society, information, and technology are supported by ostensibly different networks that in fact share an amazing number of interesting structural properties.
The majority of these networks exist since many years, some of them (biological networks) are here since millions of years. For decades, we assumed that the components of such complex systems are randomly wired together. In the last years many researchers independently showed that such an assumption is wrong: real networks have similar architectures, regardless of their age, function, and scope, that elude the random world.
A random network is generated by laying down a number of nodes and adding a number of edges between them at random. In a random network the majority of nodes have the same typical degree (number of neighbors) and nodes deviating from the average are extremely rare. In other words, the distribution of degree is peaked at a characteristic scale.
However, most real networks are different from a random network. In real networks, most of the nodes (the trivial many) have low degree and a significant few of them (the vital few) have extremely large degree. The degree distribution recalls a Pareto law, similar to distribution of wealth among citizens in a nation. These networks are called scale-free, meaning that they miss a characteristic mean value for the node degree.
Why are real networks scale-free and not random? The given explanation is two-fold:
Network science is a rapidly growing filed devoted to the holistic analysis of complex systems through the study of the structure of networks that wire their components. The expansion of this new field of science was boosted by the availability of large databases on the topology of various real networks, mainly the Web and biological networks.