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Variance and Covariance of Distributions on Graphs
SIAM Review ( IF 10.8 ) Pub Date : 2022-05-05 , DOI: 10.1137/20m1361328 Karel Devriendt , Samuel Martin-Gutierrez , Renaud Lambiotte
SIAM Review ( IF 10.8 ) Pub Date : 2022-05-05 , DOI: 10.1137/20m1361328 Karel Devriendt , Samuel Martin-Gutierrez , Renaud Lambiotte
SIAM Review, Volume 64, Issue 2, Page 343-359, May 2022.
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 we 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.
中文翻译：
图上分布的方差和协方差
SIAM 评论，第 64 卷，第 2 期，第 343-359 页，2022 年 5 月。
我们开发了一种理论来测量在图的节点上定义的概率分布的方差和协方差，其中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，并保留了它的许多直观和期望的属性。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，我们在图上定义了关于有效阻力距离的最大方差问题，并在数值和理论上描述了该问题的解决方案。我们展示了最大方差分布如何集中在图的边界上，并在随机几何图的情况下说明这一点。我们的理论结果得到了数学概念网络上的一些实验的支持，我们使用方差和协方差作为分析工具来研究科学论文中关于这些概念之间（网络）关系的概念（共）现.
更新日期：2022-05-06
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 we 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.
中文翻译：
图上分布的方差和协方差
SIAM 评论，第 64 卷，第 2 期，第 343-359 页，2022 年 5 月。
我们开发了一种理论来测量在图的节点上定义的概率分布的方差和协方差，其中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，并保留了它的许多直观和期望的属性。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，我们在图上定义了关于有效阻力距离的最大方差问题，并在数值和理论上描述了该问题的解决方案。我们展示了最大方差分布如何集中在图的边界上，并在随机几何图的情况下说明这一点。我们的理论结果得到了数学概念网络上的一些实验的支持，我们使用方差和协方差作为分析工具来研究科学论文中关于这些概念之间（网络）关系的概念（共）现.
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