arXiv 2104.072592021-04math.PRmath.CO

基于图子的随机图中子图计数的涨落

Fluctuations of Subgraph Counts in Graphon Based Random Graphs

Bhaswar B. Bhattacharya, Anirban Chatterjee, Svante Janson

备注：: 36 pages, 6 figures. Major updates. Svante Janson joins as co-author. New results on higher-order limits and degeneracies added

摘要：

给定一个图$W$和一个有限简单图$H$，其顶点集为$V(H)$，用$X_n(H, W)$表示$W$ -随机图在$n$顶点上的$H$的拷贝数.在$H$是团的情况下，最近Hladkalski、Pelekis和Šileikis（2021）获得了$X_n(H, W)$的渐近分布。本文将这一结果推广到任何固定图$H$。为此，我们引入了一个概念的$H$ -正则性的graphons，并表明，如果graphon $W$是不$H$ -正则的，那么$X_n(H, W)$有高斯波动与缩放$n^{|V(H)|-\frac{1}{2}}$。另一方面，如果$W$是$H$ -正则的，则波动是$n^{|V(H)|-1}$阶的，并且$X_n(H, W)$的极限分布可以具有高斯和非高斯分量，其中非高斯分量是中心卡方随机变量的（可能）无限加权和，其权重由从$W$导出的图子的光谱性质确定。我们的证明使用了由Janson和Nowicki（1991）发展的广义$U$ -统计量的渐近理论。我们还研究了$H$ -正则图子的结构，其中高斯或非高斯分量的极限分布（但不是两者）是退化的。有趣的是，也有$H$ -正则图子$W$，其高斯或非高斯分量都是简并的，也就是说，即使在缩放$n^{|V(H)|-1}$下，$X_n(H, W)$也具有简并极限。我们给出了一个例子，这种简并与$H=K_{1, 3}$（3星级），也建立了非简并在几个例子。这自然会导致关于高阶简并的有趣的开放性问题。

英文摘要：

Given a graphon $W$ and a finite simple graph $H$, with vertex set $V(H)$, denote by $X_n(H, W)$ the number of copies of $H$ in a $W$-random graph on $n$ vertices. The asymptotic distribution of $X_n(H, W)$ was recently obtained by Hladký, Pelekis, and Šileikis (2021) in the case where $H$ is a clique. In this paper, we extend this result to any fixed graph $H$. Towards this we introduce a notion of $H$-regularity of graphons and show that if the graphon $W$ is not $H$-regular, then $X_n(H, W)$ has Gaussian fluctuations with scaling $n^{|V(H)|-\frac{1}{2}}$. On the other hand, if $W$ is $H$-regular, then the fluctuations are of order $n^{|V(H)|-1}$ and the limiting distribution of $X_n(H, W)$ can have both Gaussian and non-Gaussian components, where the non-Gaussian component is a (possibly) infinite weighted sum of centered chi-squared random variables with the weights determined by the spectral properties of a graphon derived from $W$. Our proofs use the asymptotic theory of generalized $U$-statistics developed by Janson and Nowicki (1991). We also investigate the structure of $H$-regular graphons for which either the Gaussian or the non-Gaussian component of the limiting distribution (but not both) is degenerate. Interestingly, there are also $H$-regular graphons $W$ for which both the Gaussian or the non-Gaussian components are degenerate, that is, $X_n(H, W)$ has a degenerate limit even under the scaling $n^{|V(H)|-1}$. We give an example of this degeneracy with $H=K_{1, 3}$ (the 3-star) and also establish non-degeneracy in a few examples. This naturally leads to interesting open questions on higher-order degeneracies.

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摘要：

英文摘要：

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