均值网络何时能捕捉网络样本的拓扑结构？
When does the mean network capture the topology of a sample of networks?
AI总结：
本文首次给出随机块模型下样本Fréchet均值的解析估计，证明汉明距离无法捕捉网络拓扑，而有效电阻距离能恢复正确划分与边密度，为网络值机器学习中的度量选择提供理论依据。
摘要：
Fréchet均值（又称“重心”）网络的概念是大多数需要估计“位置”参数以分析网络值数据的机器学习算法的核心工具。在此背景下，网络重心能否继承训练数据集中网络的拓扑结构至关重要。度量——用于衡量网络之间的接近程度——控制着重心的结构性质。这项工作的意义在于，它首次为随机块模型提供了样本Fréchet均值的解析估计，而随机块模型正处于随机网络严格概率分析的前沿。我们证明，使用汉明距离计算的均值网络无法捕捉训练样本中网络的拓扑结构，而使用有效电阻距离计算的均值网络能够恢复正确的划分及相关的边密度。从实际角度看，我们的工作为在利用样本Fréchet均值网络刻画网络拓扑以进行网络值机器学习的场景中选择度量提供了依据。
英文摘要：
The notion of Fréchet mean (also known as "barycenter") network is the workhorse of most machine learning algorithms that require the estimation of a "location" parameter to analyse network-valued data. In this context, it is critical that the network barycenter inherits the topological structure of the networks in the training dataset. The metric - which measures the proximity between networks - controls the structural properties of the barycenter. This work is significant because it provides for the first time analytical estimates of the sample Fréchet mean for the stochastic blockmodel, which is at the cutting edge of rigorous probabilistic analysis of random networks. We show that the mean network computed with the Hamming distance is unable to capture the topology of the networks in the training sample, whereas the mean network computed using the effective resistance distance recovers the correct partitions and associated edge density. From a practical standpoint, our work informs the choice of metrics in the context where the sample Fréchet mean network is used to characterise the topology of networks for network-valued machine learning
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