AVERAGE DISTANCE IN G(w), PART II

with probability 1 − o(1). Last time it was shown that almost surely d ≥ (1 + o(1))(log n/ log d̃), so we now concentrate on proving that this is also an upper bound. We will need three lemmas before attacking the theorem itself. We also mention that some technical details will be omitted, and we refer the reader to [1] for the full treatment. In particular, it is worth mentioning that this theorem will not be true unless we restrict our attention to the subset of G(w) having admissible expected degree sequences. Of course, the property of being admissible depends only upon the weights wi and not on the outcome of any random variables, so that we may still treat these graphs as random graphs.