Random Graphs and Branching Processes.- Percolation, Connectivity, Coverage and Colouring of Random Geometric Graphs.- Scaling Properties of Complex Networks and Spanning Trees.- Random Tree Growth with Branching Processes - A Survey.- Reaction-diffusion Processes in Scale-free Networks.- Toward Understanding the Structure and Function of Cellular Interaction Networks.- Scale-Free Cortical Planar Networks.- Reconstructing Cortical Networks: Case of Directed Graphs with High Level of Reciprocity.- k-Clique Percolation and Clustering.- The Inverse Problem of Evolving Networks - with Application to Social Nets.- Learning and Representation: From Compressive Sampling to the 'Symbol Learning Problem'.- Telephone Call Network Data Mining: A Survey with Experiments.
"Chapter 1 Random Graphs and Branching Processes (p. 15-16)
B´ELA BOLLOB´AS and OLIVER RIORDAN
During the past decade or so, there has been much interest in generating and analyzing graphs resembling large-scale real-world networks such as the world wide web, neural networks, and social networks. As these large-scale networks seem to be ‘random’, in the sense that they do not have a transparent, well-de?ned structure, it does not seem too unreasonable to hope to ?nd classical models of random graphs that share their basic properties.
Such hopes are quickly dashed, however, since the classical random graphs are all homogeneous, in the sense that all vertices (or indeed all k-sets of vertices) are a priori equivalent in the model. Most real-world networks are not at all like this, as seen most easily from their often unbalanced (power-law) degree sequences. Thus, in order to model such graphs, a host of inhomogeneous random graph models have been constructed and studied.
In this paper we shall survey a number of these models and the basic results proved about the inhomogeneous sparse (bounded average degree) random graphs they give rise to. We shall focus on mathematically tractable models, which often means models with independence between edges, and in particular on the very general sparse inhomogeneous models of Bollob´as, Janson and Riordan. The ?rst of these encompasses a great range of earlier models of this type; the second, the inhomogeneous clustering model, goes much further, allowing for the presence of clustering while retaining tractability.
We are not only interested in our inhomogeneous random graphs themselves, but also in the random subgraphs obtained by keeping their edges with a certain probability p. Our main interest is in the phase transition that takes place around a certain critical value p0 of p, when the component structure of the random subgraph undergoes a sudden change. The quintessential phase transition occurs in the classical binomial random graph G(n, c/n) as c grows from less than 1 to greater than 1 and, as shown by Erd?os and R´enyi, a unique largest component, the giant component, is born.
A ubiquitous theme of our paper is the use of branching processes in the study of random graphs. This ‘modern’ approach to random graphs is crucial in the study of the very general models of inhomogeneous random graphs mentioned above. To illustrate the power of branching processes, we show how they can be used to reprove sharp results about the classical random graph G(n, c/n), ?rst proved by Bollob´as and Luczak over twenty years ago. When it comes to inhomogeneous models, we shall have time only to sketch the connection to branching processes. Finally, we close by discussing the question of how to tell whether a given model is appropriate in a given situation. This leads to many fascinating questions about metrics for sparse graphs, and their relationship to existing models and potential new models."
