  We propose a number of techniques for obtaining a global ranking from data
that may be incomplete and imbalanced -- characteristics almost universal to
modern datasets coming from e-commerce and internet applications. We are
primarily interested in score or rating-based cardinal data. From raw ranking
data, we construct pairwise rankings, represented as edge flows on an
appropriate graph. Our statistical ranking method uses the graph Helmholtzian,
the graph theoretic analogue of the Helmholtz operator or vector Laplacian, in
much the same way the graph Laplacian is an analogue of the Laplace operator or
scalar Laplacian. We study the graph Helmholtzian using combinatorial Hodge
theory: we show that every edge flow representing pairwise ranking can be
resolved into two orthogonal components, a gradient flow that represents the
L2-optimal global ranking and a divergence-free flow (cyclic) that measures the
validity of the global ranking obtained -- if this is large, then the data does
not have a meaningful global ranking. This divergence-free flow can be further
decomposed orthogonally into a curl flow (locally cyclic) and a harmonic flow
(locally acyclic but globally cyclic); these provides information on whether
inconsistency arises locally or globally. An obvious advantage over the NP-hard
Kemeny optimization is that discrete Hodge decomposition may be computed via a
linear least squares regression. We also investigated the L1-projection of edge
flows, showing that this is dual to correlation maximization over bounded
divergence-free flows, and the L1-approximate sparse cyclic ranking, showing
that this is dual to correlation maximization over bounded curl-free flows. We
discuss relations with Kemeny optimization, Borda count, and Kendall-Smith
consistency index from social choice theory and statistics.
