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 center of the Kolmogorov-Smirnov ball.

翻译：设$G$是一个包含$n$个顶点的大规模（简单、无标记）稠密图。假设我们仅知道或能够估计$G$中每个顶点参与的某个固定小图$F$的子图数量的经验分布。那么，有多少其他图会呈现出与我们观察到的本质上相同的局部结构呢？本文中，我们推导了经验分布（在 Kolmogorov-Smirnov 距离下）接近$G$的经验分布的图的数量上界和下界。这些界被表示为在$d$个全局密度约束下，一个关于固定规模$k$（与$n$无关）的随机图的最大熵问题的解。这些界是渐近接近的，其间隙随着$d$的增加而消失，消失速率取决于 Kolmogorov-Smirnov 球心的集中函数。