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 number subgraphs $F$ each vertex in participates in, for some fixed small $F$. How many other graphs would look essentially same to us, i.e., have similar local structure? In this paper, derive upper and lower bounds whose lies close (in Kolmogorov-Smirnov distance) $G$. Our are given as solutions maximum entropy problem random size $k$ does not depend $n$, under $d$ global density constraints. The asymptotically close, with gap vanishes at rate depends concentration function center ball.