A Quotient Space Formulation for Statistical Analysis of Graphical Data

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, one needs mathematical representations and metrics for matching and comparing graphs, and other geometrical tools, such as geodesics, means, and covariances, on representation spaces of graphs. This paper utilizes a quotient structure to develop efficient algorithms for computing these quantities, leading to useful statistical tools, including principal component analysis, linear dimension reduction, and analytical statistical modeling. The efficacy of this framework is demonstrated using datasets taken from several problem areas, including alphabets, video summaries, social networks, and biochemical structures.