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Abstract: Given a set of alternatives to be ranked, and some pairwise comparison data,ranking is a least squares computation on a graph. The vertices are thealternatives, and the edge values comprise the comparison data. The basic ideais very simple and old: come up with values on vertices such that theirdifferences match the given edge data. Since an exact match will usually beimpossible, one settles for matching in a least squares sense. This formulationwas first described by Leake in 1976 for rankingfootball teams and appears asan example in Professor Gilbert Strang-s classic linear algebra textbook. Ifone is willing to look into the residual a little further, then the problemreally comes alive, as shown effectively by the remarkable recent paper ofJiang et al. With or without this twist, the humble least squares problem ongraphs has far-reaching connections with many current areas ofresearch. Theseconnections are to theoretical computer science spectral graph theory, andmultilevel methods for graph Laplacian systems; numerical analysis algebraicmultigrid, and finite element exterior calculus; other mathematics Hodgedecomposition, and random clique complexes; and applications arbitrage, andranking of sports teams. Not all of these connections are explored in thispaper, but many are. The underlying ideas are easy to explain, requiring onlythe four fundamental subspaces from elementary linear algebra. One of our aimsis to explain these basic ideas and connections, to get researchers in manyfields interested in this topic. Another aim is to use our numericalexperiments for guidance on selecting methods and exposing the need for furtherdevelopment.

Autor: Anil N. Hirani, Kaushik Kalyanaraman, Seth Watts

Fuente: https://arxiv.org/

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