  Graphical models for cardinal paired comparison data with and without
covariates are rigorously analyzed. Novel, graph--based, necessary and
sufficient conditions which guarantee strong consistency, asymptotic normality
and the exponential convergence of the estimated ranks are emphasized. A
complete theory for models with covariates is laid out. In particular
conditions under which covariates can be safely omitted from the model are
provided. The methodology is employed in the analysis of both finite and
infinite sets of ranked items specifically in the case of large sparse
comparison graphs. The proposed methods are explored by simulation and applied
to the ranking of teams in the National Basketball Association (NBA).
