Overview

Random networks arise when nodes are randomly deployed on the plane and randomly connected to each other. Depending on the specific rules used to construct them, they create structures that can resemble what is observed in real natural, as well as in artificial, complex systems. Thus, they provide simple models that allow us to use probability theory as a tool to explain the observable behaviour of real systems and to formally study and predict phenomena that are not amenable to analysis with a deterministic approach. This often leads to useful design guidelines for the development and optimal operation of real systems.

Historically, random networks has been a field of study in mathematics and statistical physics, although many models were inspired by practical questions of engineering interest. One of the early mathematical models appeared in a series of papers starting in 1959 by the two Hungarian mathematicians Paul Erd s and Alfr d R nyi. They investigated what a typical graph of n vertices and m edges looks like, by connecting nodes at random. They showed that many properties of these graphs are almost always predictable, as they suddenly arise with very high probability when the model parameters are chosen appropriately. This peculiar property generated much interest among mathematicians, and their papers marked the starting point of the field of random graph theory. The graphs they considered, however, were abstract mathematical objects and there was no notion of geometric position of vertices and edges.

Mathematical models inspired by more practical questions...