On the Number of Graphs with a Given Histogram

Let G be a large (simple, unlabeled) dense graph on n vertices. Suppose that we only know, or can estimate, the empirical distribution of the number of subgraphs F that each vertex in G participates in, for some fixed small graph F. How many other graphs would look essentially the same to us, i.e., would have a similar local structure? In this paper, we derive upper and lower bounds on the number of graphs whose empirical distribution lies close (in the Kolmogorov-Smirnov distance) to that of G. Our bounds are given as solutions to a maximum entropy problem on random graphs of a fixed size k that does not depend on n , under d global density constraints. The bounds are asymptotically close, with a gap that vanishes with d at a rate that depends on the concentration function of the distribution at the center of the Kolmogorov-Smirnov ball.