论文标题
不均匀的erdos-renyi随机图的特征平均值（或中位数）的尖锐阈值
Sharp Threshold for the Frechet Mean (or Median) of Inhomogeneous Erdos-Renyi Random Graphs
论文作者
论文摘要
我们解决以下基础问题：不均匀的Erdos-Renyi随机图的集合的人群和样本，特征的平均值（或中位数）图是什么？我们证明，如果我们使用锤子距离来计算图之间的距离，则通过阈值集团的预期邻接矩阵获得了不均匀随机图的集合的均值（或中位数）图。我们表明，当预期的邻接矩阵被样品平均邻接矩阵替换时，样本平均值（或中位数）也适用于样本平均值（或中位数）。因此，不均匀的Erdos-Renyi随机图的特雷切特均值（或中位数）图显示出尖锐的阈值：它是空图，或完整的图。这个新颖的理论结果产生了一些重大的实际后果。例如，稀疏不均匀随机图的集合的特征平均值始终是空图。
We address the following foundational question: what is the population, and sample, Frechet mean (or median) graph of an ensemble of inhomogeneous Erdos-Renyi random graphs? We prove that if we use the Hamming distance to compute distances between graphs, then the Frechet mean (or median) graph of an ensemble of inhomogeneous random graphs is obtained by thresholding the expected adjacency matrix of the ensemble. We show that the result also holds for the sample mean (or median) when the population expected adjacency matrix is replaced with the sample mean adjacency matrix. Consequently, the Frechet mean (or median) graph of inhomogeneous Erdos-Renyi random graphs exhibits a sharp threshold: it is either the empty graph, or the complete graph. This novel theoretical result has some significant practical consequences; for instance, the Frechet mean of an ensemble of sparse inhomogeneous random graphs is always the empty graph.
