  We give sufficient conditions under which a random graph with a specified
degree sequence is symmetric or asymmetric. In the case of bounded degree
sequences, our characterisation captures the phase transition of the symmetry
of the random graphs. This phase transition coincides with that of the graph
connectivity. However, when the maximum degree is a growing function as the
number of vertices tends to infinity, our results suggest that these two
thresholds do not coincide any more
