AI 中文总结
该研究针对几何随机图的社区边界估计问题，建立了一致估计边界位置的充要条件，设计了收敛的估计量并给出误差界，为随机块模型的几何变体提供了理论支撑。
AI 中文摘要
单位正方形$I=[0,1]^2$被竖直线$x=p$分为两个矩形，其中$p\in(0,1)$。考虑$N$个独立且均匀分布在$I$上的点，将其视为个体群体，直线$x=p$代表将群体划分为两个社区的社区边界。个体对是否相连取决于它们在$I$中的位置以及是否属于同一社区：同一社区内的两个个体若距离在$R_N$以内则相连，不同社区的两个个体若距离在$R_N'$以内则相连，由此产生了所谓随机块模型的几何变体。统计学家观察所得图的邻接矩阵及个体的几何位置，任务是估计边界位置$p$。根据$R_N$和$R_N'$随$N\to\infty$的缩放方式，我们建立了$p$的一致估计的必要条件和充分条件。只要一致估计可行，我们就设计了一个随$N\to\infty$收敛到$p$的估计量，并给出了其估计误差的明确界值。
英文摘要
The unit square $I=[0,1]^2$ is divided into two rectangles by the vertical line $x=p$, where $p\in(0,1)$. Consider $N$ independent uniformly distributed points on $I$, which we interpret as a population of individuals, with the line $x=p$ representing a community boundary that separates the population into two communities. Whether a pair of individuals share a connection depends on their locations in $I$ and on whether they belong to the same community. Specifically, two individuals in the same community are connected if they are within distance $R_N$ of each other, while two individuals in different communities are connected if they are within distance $R_N'$ of each other, giving rise to a geometric variant of the so-called \emph{stochastic block model}. A statistician observes the adjacency matrix of the resulting graph together with the geometric locations of the individuals and is tasked with estimating the boundary location $p$. Depending on how $R_N$ and $R_N'$ scale as $N\to\infty$, we establish necessary and sufficient conditions for consistent estimation of $p$. Whenever consistent estimation is possible, we devise an estimator that converges to $p$ as $N\to\infty$ and provide explicit bounds on its estimation error.
Comments19 pages, 2 figures; comments welcome
