Abstract
The degree sequence of an n -vertex graph is d 0 ,…, d n −1 , where each d i is the number of vertices of degree i in the graph. A random graph with degree sequence d 0 ,…, d n −1 is a randomly selected member of the set of graphs on {1,…, n } with that degree sequence, all choices being equally likely. Let λ 0 ,λ 1 ,… be a sequence of nonnegative reals summing to 1. A class of finite graphs has degree sequences approximated by λ 0 ,λ 1 ,… if, for every i and n , the members of the class of size n have λ i n + o(n) vertices of degree i . Our main result is a convergence law for random graphs with degree sequences approximated by some sequence λ 0 ,λ 1 ,…. With certain conditions on the sequence λ 0 ,λ 1 ,…, the probability of any first-order sentence on random graphs of size n converges to a limit as n grows.
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