Bias in generation of random graphs

We study the statistical properties of the generation of random graphs according the configuration model, where one assigns randomly degrees to nodes. This model is often used, e.g., for the scale-free degree distribution ~d^gamma. For the efficient variant, where non-feasible edges are rejected and the construction of a graph continues, there exists a bias, which we calculate explicitly for a small sample ensemble. We find that this bias does not disappear with growing system size. This becomes also visible, e.g., for scale-free graphs when measuring quantities like the graph diameter. Hence, the efficient generation of general scale-free graphs with a very broad distribution (gamma <2) remains an open problem.