  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.
