Augusto di Franco:
The study of complex networks was sparked at the end of the 90s with two seminal papers, describing their universality:
- small-worlds property: Strogatz S. H. and Watts D. J., 1998, Collective Dynamics of ‘Small-World’ Networks, Nature, 393, 440–442.
- scale-free nature: Albert R. and Barabasi A.-L., 1999, Emergence of Scaling in Random Networks, www.arXiv.org/abs/cond-mat/9910332.
Today, networks are ubiquitous: phenomena in the physical world (e.g., computer networks, transportation networks, power grids, spontaneous synchronization of systems of lasers), biological systems (e.g., neural networks, epidemiology, food webs, gene regulation), and social realms (e.g., trade networks, diffusion of innovation, trust networks, research collaborations, social affiliation) are best understood if characterized as networks.
The explosion of this field of research was and is coupled with the increasing availability of huge amounts of data, pouring in from neurobiology, genomics, ecology, finance and the Word-wide Web, …, computing power and storage facilities.
- Paradigm: The new paradigm states that it is best to understand a complex system, if it is mapped to a network. I.e., the links represent the some kind of interaction and the nodes are stripped of any intrinsic quality. So, as an example, you can forget about the complexity of the individual bird, if you model the flocks swarming behavior. (See these older posts: complex, fundamental, swarm theory, in a nutshell.)
- Weights: Only in the last years has the attention shifted from this topological level of analysis (either links are present or not) to incorporate weights of links, giving the strength relative to each other. Albeit being harder to tackle, these networks are closer to the real-world system it is modeling.
- Nodes: However, there is still one step missing: also the vertices of the network can be assigned with a value, which acts as a proxy for some real-world property that is coded into the network structure.
The two plots above illustrate the difference if the same network is visualized  using weights and values assigned to the vertices (left) or simply plotted as a binary (topological) network (right)… " (http://escoladeredes.net/group/bibliotecaalbertbarabsi)