Averages of unlabeled networks: Geometric characterization and asymptotic behavior
Averages of unlabeled networks: Geometric characterization and asymptotic behavior
复制标题
DOI：
10.1214/19-aos1820
复制
发表时间：
2017-09
期刊：
影响因子：
--
通讯作者：
E. Kolaczyk;Lizhen Lin;S. Rosenberg;Jie Xu;Jackson Walters
中科院分区：
文献类型：
--
作者：
E. Kolaczyk;Lizhen Lin;S. Rosenberg;Jie Xu;Jackson Walters
大量网络数据对象的集合变得越来越普遍，即网络被视为基本观察单位的数据集。因此，迫切需要开发基于网络的类似工具，甚至是许多已经成为标量和矢量数据标准的最基本工具。在本文中，我们的重点是具有边缘权重的未标记、无向网络的平均值。具体来说，我们（i）表征所有此类网络的空间的特定概念，（ii）描述与进行概率和统计相关的该空间的关键拓扑和几何属性，以及（iii）使用这些属性来建立从该空间支持的分布采样下的经验平均值的广义概念的渐近行为。我们的结果依赖于几何、概率论和统计形状分析工具的组合。特别是，缺乏顶点标签需要使用商空间来修改标签的排列。这导致了未标记网络空间的重要几何形状，进而发现这对可能获得的概率和统计结果的类型以及获得它们所需的技术具有重要影响。
It is becoming increasingly common to see large collections of network data objects -- that is, data sets in which a network is viewed as a fundamental unit of observation. As a result, there is a pressing need to develop network-based analogues of even many of the most basic tools already standard for scalar and vector data. In this paper, our focus is on averages of unlabeled, undirected networks with edge weights. Specifically, we (i) characterize a certain notion of the space of all such networks, (ii) describe key topological and geometric properties of this space relevant to doing probability and statistics thereupon, and (iii) use these properties to establish the asymptotic behavior of a generalized notion of an empirical mean under sampling from a distribution supported on this space. Our results rely on a combination of tools from geometry, probability theory, and statistical shape analysis. In particular, the lack of vertex labeling necessitates working with a quotient space modding out permutations of labels. This results in a nontrivial geometry for the space of unlabeled networks, which in turn is found to have important implications on the types of probabilistic and statistical results that may be obtained and the techniques needed to obtain them.