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.

翻译：我们处理以下基本问题:人口和样本是什么? Frechet 意味着(或中位) 混合不相容 Erdos-Renyi 随机图?我们证明,如果我们使用 Hamming 距离来计算图表之间的距离,那么Frechet 表示(或中位) 指(或中位) 混合不相容随机图的组合图就是通过将预期合用物的匹配矩阵阈值来获得的。我们表明,当人口预期以样本替换相近矩阵时,结果也保留在样本平均值(或中位) 中。因此, Frechet 表示(或中位) 混合不相近 Erdos-Renyi 随机图显示了一个尖锐的门槛:要么是空图,要么是完整的图。这个新的理论结果产生了一些重大的实际后果;例如,混合混合随机图的混合值始终是空图。