图上分布的方差和协方差
Variance and covariance of distributions on graphs
摘要：
我们开发了一种理论来测量定义在图的节点上的概率分布的方差和协方差，该理论考虑了节点之间的距离。我们的方法将通常的（协）方差推广到加权图的设置，并保留了许多直观和理想的属性。有趣的是，我们发现图论和网络科学中的一些著名概念可以在这种情况下被重新解释为特定分布的方差和协方差。作为一个特殊的应用，我们定义了图上关于有效电阻距离的最大方差问题，并从数值和理论上刻画了这个问题的解。我们展示了最大方差分布是如何集中在图的边界上的，并在随机几何图的情况下说明了这一点。我们的理论结果得到了一些实验的支持，在网络上的数学概念，在那里我们使用的方差和协方差作为分析工具来研究（共）出现的概念在科学论文中的（网络）这些概念之间的关系。
英文摘要：
We develop a theory to measure the variance and covariance of probability distributions defined on the nodes of a graph, which takes into account the distance between nodes. Our approach generalizes the usual (co)variance to the setting of weighted graphs and retains many of its intuitive and desired properties. Interestingly, we find that a number of famous concepts in graph theory and network science can be reinterpreted in this setting as variances and covariances of particular distributions. As a particular application, we define the maximum variance problem on graphs with respect to the effective resistance distance, and characterize the solutions to this problem both numerically and theoretically. We show how the maximum variance distribution is concentrated on the boundary of the graph, and illustrate this in the case of random geometric graphs. Our theoretical results are supported by a number of experiments on a network of mathematical concepts, where we use the variance and covariance as analytical tools to study the (co-)occurrence of concepts in scientific papers with respect to the (network) relations between these concepts.
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