Least Squares Ranking on Graphs, Hodge Laplacians, Time Optimality, and Iterative Methods

Given a set of alternatives and some pairwise comparison values, ranking is a least squares computation on a graph. The graph vertices are the alternatives, with a weighted oriented edge between each pair for which there is a pairwise score. The orientations are arbitrary. The set of edges may be sparse or dense. The basic idea of the computation is very simple and old – come up with a vertex potential such that the potential difference matches the given edge data. Since an exact match will usually be impossible, one settles for matching the edge data in a least squares sense. This formulation was first described by Leake in 1976 for ranking football teams [21]. The residual can be further analyzed for discovering inconsistencies in the given pairwise comparison data, and this leads to a second least squares problem. This whole process was formulated recently by Jiang et al. as a Hodge decomposition of the edge values [19]. The second problem, besides being an important refinement of the basic least squares ranking, has other potential applications, such as in economics [19]. In a recent breakthrough paper, Koutis et al. [20] showed that symmetric diagonally dominant (SDD) linear systems can be solved in time approaching optimality (we’ll refer to their algorithm as the KMP solver). We first show as an easy consequence of their result that for an arbitrary graph, the first least squares problem of ranking can be solved in time approaching optimality using the KMP solver. We show that the second least squares problem involves the Hodge 2-Laplacian, which is different from the graph Laplacian. It has not been studied in the theoretical computer science literature. We show that if a graph is the 1-skeleton of a cell complex on a compact surface the second least squares system matrix is also SDD, which implies optimality via KMP. For all the above cases we also give bounds on the number of conjugate gradient iterations required to achieve a given error bound. For the surface graphs we do this first for boundaryless surface. We show that the system matrix is the same as the graph Laplacian for the dual graph. If the embedding surface does have boundary, then we use Cauchy’s interlacing theorem to give bounds on conjugate gradient iterations required to solve the second problem by patching the holes and using bounds for the boundaryless case. These special cases are important in computational topology, and we show that the least squares problems of ranking are a 2-norm version of the optimal homologous chain problem of computational topology [10]. For a general graph with cells filled in, we show that the second least squares system matrix is, in general, not diagonally dominant. Thus KMP does not apply directly and nothing is known about its spectrum in general. In this case the best approach is to use an iterative Krylov method and we show numerical results for several choices. Krylov methods are also useful for large graphs where the loss of sparsity in forming the system matrix might be a storage issue. The second problem’s system matrix will be in general singular and have a high dimensional kernel equal to the dimension of the second homology which is the number of independent spheres such as tetrahedra amongst the cliques. Krylov methods work in a space orthogonal to the kernel and no kernel modding is required. ∗Author for correspondence. Department of Computer Science, University of Illinois at Urbana-Champaign, [email protected] http://www.cs.illinois.edu/hirani †Department of Computer Science, University of Illinois at Urbana-Champaign, [email protected] ‡Department of Mech. Sci. & Eng., University of Illinois at Urbana-Champaign, [email protected] ar X iv :1 01 1. 17 16 v1 [ cs .N A ] 8 N ov 2 01 0