Complex analyses involving multiple, dependent random quantities often lead to graphical models—a set of nodes denoting variables of interest, and corresponding edges denoting statistical interactions between nodes. To develop statistical analyses for graphical data, especially towards generative modeling, one needs mathematical representations and metrics for matching and comparing graphs, and subsequent tools, such as geodesics, means, and covariances. This paper utilizes a quotient structure to develop efficient algorithms for computing these quantities, leading to useful statistical tools, including principal component analysis, statistical testing, and modeling. We demonstrate the efficacy of this framework using datasets taken from several problem areas, including letters, biochemical structures, and social networks.

涉及多个相互依赖的随机量的复杂分析通常会导致图形化模型-一组表示关注变量的节点，以及相应的边表示节点之间的统计交互作用。为了开发图形数据的统计分析，尤其是针对生成模型的统计分析，需要数学表示和度量来匹配和比较图形，以及后续的工具，例如测地线，均值和协方差。本文利用商结构来开发用于计算这些数量的有效算法，从而得到有用的统计工具，包括主成分分析，统计测试和建模。我们使用来自几个问题领域的数据集（包括字母，生化结构和社交网络）证明了该框架的有效性。