Complex network

In contrast, many of the mathematical models of networks that have been studied in the past, such as lattices and random graphs, do not show these features.

The field continues to develop at a brisk pace, and has brought together researchers from many areas including mathematics, physics, electric power systems,[9] biology, climate, computer science, sociology, epidemiology, and others.

Network science is the topic of many conferences in a variety of different fields, and has been the subject of numerous books both for the lay person and for the expert.

In contrast, networks with a single well-defined scale are somewhat similar to a lattice in that every node has (roughly) the same degree.

Most of these reported "power laws" fail when challenged with rigorous statistical testing, but the more general idea of heavy-tailed degree distributions—which many of these networks do genuinely exhibit (before finite-size effects occur) -- are very different from what one would expect if edges existed independently and at random (i.e., if they followed a Poisson distribution).

However, it is known by many other names due to its frequent reinvention, e.g., The Gibrat principle by Herbert A. Simon, the Matthew effect, cumulative advantage and, preferential attachment by Barabási and Albert for power-law degree distributions.

Some networks with a power-law degree distribution (and specific other types of structure) can be highly resistant to the random deletion of vertices—i.e., the vast majority of vertices remain connected together in a giant component.

In 1998, Duncan J. Watts and Steven Strogatz published the first small-world network model, which through a single parameter smoothly interpolates between a random graph and a lattice.

Approaches have been developed to generate network models that exhibit high correlations, while preserving the desired degree distribution and small-world properties.