A Switching Approach to Random Graphs with a Fixed Degree Sequence

For a fixed degree sequence D=(d_1,...,d_n), let G(D) be a uniformly chosen (simple) graph on {1,...,n} where the vertex i has degree d_i. The study of G(D) is of special interest in order to model real-world networks that can be described by their degree sequence, such as scale-free networks. While many aspects of G(D) have been extensively studied, most of the obtained results only hold provided that the degree sequence D satisfies some technical conditions. In this talk we will introduce a new approach (based on the switching method) that allows us to study the random graph G(D) imposing no conditions on D. Most notably, this approach provides a new criterion on the existence of a giant component in G(D). Moreover, this method is also useful to determine whether there exists a percolation threshold in G(D). The first part of this talk is joint work with F. Joos, D. Rautenbach and B. Reed, and the second part, with N. Fountoulakis and F. Joos.