Foundations of complex networks
M. Ángeles Serrano and Marián Boguñá
A wide class of real systems of many interacting elements can be mapped into graphs or networks. Under this approach, vertices or nodes of the network represent the elements of the system whereas edges or links among them stand for interactions between different elements. This mapping has triggered a huge number of works and a surge of interest in the field of complex networks that has lead to a general framework within which to analyze their topology as well as the dynamical processes running on top of them.
In many cases, these dynamical processes are directly related to functionality and involve some kind of transport or traffic flow. Furthermore, the very existence of those networks could be naturally explained as a direct consequence of the communication need among its constituents. The Internet or the World Wide Web are clear examples. In order to preserve functionality, networks characterized by transport processes must be connected, that is, a path must exist between any pair of nodes, or, at least, there must exist a macroscopic portion of vertices --or giant component-- able to communicate. In this context, percolation theory appears as an indispensable tool to analyze the conditions under which such connected structures emerge in large networks.
Our research in this field is focused towards the study of structural properties of networks and new percolation phenomena, such as percolation in random directed networks with one and two points degree correlations, bidirectional connections and clustering. These are ubiquitous properties in the networks of the real life which have strong implications in their percolation properties.