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:让$G$成为大(简单、未贴标签的)坚硬的坚硬图。 假设我们只知道或者能够估计每个G$顶点所参与的子集数量的实证分布情况, 对于某些固定的小方块来说,每个G$的顶点是$F$。 有多少其他图表看起来基本相同, 也就是说, 本地结构会相似? 在本文中, 我们从实验分布接近( 科尔莫戈罗夫- 斯米尔诺夫距离) $G$ 的图表数量上得出上下限。 我们的界限作为解决方案, 解决固定大小的随机图中不依赖于$美元、 低于美元的全球密度限制。 边距是微小的, 以取决于科尔莫戈洛夫- 斯米尔诺夫球中心的集中功能的速度以美元消失差距。