Randomness and Complexity in Networks

I start by reviewing some basic properties of random graphs. I then consider the role of random walks in complex networks and show how they may be used to explain why so many long tailed distributions are found in real data sets. The key idea is that in many cases the process involves copying of properties of near neighbours in the network and this is a type of short random walk which in turn produce a natural preferential attachment mechanism. Applying this to networks of fixed size I show that copying and innovation are processes with special mathematical properties which include the ability to solve a simple model exactly for any parameter values and at any time. I finish by looking at variations of this basic model.Comment: Survey paper based on talk given at the workshop on ``Stochastic Networks and Internet Technology'', Centro di Ricerca Matematica Ennio De Giorgi, Matematica nelle Scienze Naturali e Sociali, Pisa, 17th - 21st September 2007. To appear in proceeding

Exact Solutions for Network Rewiring Models

Evolving networks with a constant number of edges may be modelled using a rewiring process. These models are used to describe many real-world processes including the evolution of cultural artifacts such as family names, the evolution of gene variations, and the popularity of strategies in simple econophysics models such as the minority game. The model is closely related to Urn models used for glasses, quantum gravity and wealth distributions. The full mean field equation for the degree distribution is found and its exact solution and generating solution are given.Comment: 7 pages, 7 figures. Minor changes and corrections made for publicatio

Thermal Bosonic Green Functions Near Zero Energy (6 characters removed for email)

The properties of the various types of bosonic Green functions at finite temperature in the zero energy limit are considered in the light of recent work.Comment: 14, LaTeX (article style), Imperial/TP/91-92/3

Complex Networks

An outline of recent work on complex networks is given from the point of view of a physicist. Motivation, achievements and goals are discussed with some of the typical applications from a wide range of academic fields. An introduction to the relevant literature and useful resources is also given.Comment: Review for Contemporary Physics, 31 page

Thermal Bubble Diagrams Near Zero Energy

The zero four-momentum and equal mass limits are taken for the bubble diagram of scalar fields. It is seen that RTF and ITF are in complete agreement. However contributions from this diagram to both retarded and time-ordered functions do depend on the order of the limits and can be infinite in some cases. This shows explicitly that the relation between the free energy and a derivative expansion of a thermal effective action is generally much more complicated that is the case at zero temperature.Comment: 13 pages, LaTeX (3 PS figures appended), Imperial/TP/92-93/4

Clique Graphs and Overlapping Communities

It is shown how to construct a clique graph in which properties of cliques of a fixed order in a given graph are represented by vertices in a weighted graph. Various definitions and motivations for these weights are given. The detection of communities or clusters is used to illustrate how a clique graph may be exploited. In particular a benchmark network is shown where clique graphs find the overlapping communities accurately while vertex partition methods fail.Comment: 23 pages plus 16 additional pages in appendice