We live in a world of networks, and, moreover, networks are the core of our civilization. Nowadays, in the epoch of economic, political, and cultural globalization, one can see this with perfect clarity. Thus, the world of networks is our inevitable future, for better or worse.

We are in a period of revolution in network science. Communication networks, the WWW, the Internet, peer-to-peer networks, genome, chemical reaction networks, catalytic networks, nets of metabolic reactions, protein networks, idiotypic networks, neural networks, signalling networks, artery nets, transportation networks, river networks, ecological and food webs, social networks, networks of collaborations, terrorist networks, nets of citations in scientific literature, telephone call graphs, mail networks, power grids, relations between enterprises, nets of ownership, networks of influence, the Word Web, electronic circuits, nets of software components, landscape networks, ‘geometric’ networks …—what will be next?

The progress is so immediate and astounding that we actually face a new science based on a new concept, and, one may even say, on a new philosophy: the natural philosophy of a small world. Old ideas from mathematics, statistical physics, biology, computer science, and so on take quite a new form in application to real evolving networks. Let us list some of basic points of this new conception:

• • Evolving networks are one of the fundamental objects of statistical physics.

• • Evolving networks self-organize into complex structures.

• • The result of this self-organization is nets with the crucial role of highly connected vertices. The entire spectrum of connections is important in networks with such an architecture, but just the highly connected vertices determine the topology and basic properties of the networks, stability, transmission of interactions, spread of diseases, and so on.

• • These networks have several levels of structural organization, several distinct scales: the local structure of connections of a vertex, the structure of connections in its environment, and the long-range structure of a network. Each of these levels determines a distinct set of properties of the networks.

• • Networks are extremely ‘compact’ objects without well defined metric structure. Their organization varies between a tree-like form and a highly correlated structure, rich in loops, which are the two faces of a network.

More than 40 years ago, in 1960, Erdös and Rényi wrote their seminal paper ‘On the evolution of random graphs’, which is one of the starting points (p. 220 ) of mathematical random graph theory. What Erdös and Rényi called ‘random graphs’ were simple equilibrium networks with the Poisson distribution of connections. What they called ‘evolution’ was, actually, their construction procedure for these equilibrium graphs. Major recent achievements in network science are related to the transition to the study of the evolving, self-organizing networks with fat-tailed, non-Poisson, distributions of connections.

Most basic networks, both natural and artificial, belong to this class: the Internet, the WWW, networks of protein interactions, and many, many others. As such, the impressive recent progress in this field represents a significant step towards understanding the most exciting networks of our world: the Internet and WWW, and basic biological networks. It is a step towards understanding one of the few fundamental objects of the Universe: a network.