Least Squares Ranking on Graphs

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 the alternatives, and the edge values comprise the comparison data. The basic idea is very simple and old: come up with values on vertices such that their differences match the given edge data. Since an exact match will usually be impossible, one settles for matching in a least squares sense. This formulation was first described by Leake in 1976 for rankingfootball teams and appears as an example in Professor Gilbert Strang's classic linear algebra textbook. If one is willing to look into the residual a little further, then the problem really comes alive, as shown effectively by the remarkable recent paper of Jiang et al. With or without this twist, the humble least squares problem on graphs has far-reaching connections with many current areas ofresearch. These connections are to theoretical computer science (spectral graph theory, and multilevel methods for graph Laplacian systems); numerical analysis (algebraic multigrid, and finite element exterior calculus); other mathematics (Hodge decomposition, and random clique complexes); and applications (arbitrage, and ranking of sports teams). Not all of these connections are explored in this paper, but many are. The underlying ideas are easy to explain, requiring only the four fundamental subspaces from elementary linear algebra. One of our aims is to explain these basic ideas and connections, to get researchers in many fields interested in this topic. Another aim is to use our numerical experiments for guidance on selecting methods and exposing the need for further development.Comment: Added missing references, comparison of linear solvers overhauled, conclusion section added, some new figures adde

Hodge Laplacians on simplicial meshes and graphs

We present in this dissertation some developments in the discretizations of exterior calculus for problems posed on simplicial discretization (meshes) of geometric manifolds and analogous problems on abstract simplicial complexes. We are primarily interested in discretizations of elliptic type partial differential equations, and our model problem is the Hodge Laplacian Poisson problem on differential k-forms on n dimensional manifolds. One of our major contributions in this work is the computational quantification of the solution using the weak mixed formulation of this problem on simplicial meshes using discrete exterior calculus (DEC), and its comparisons with the solution due to a different discretization framework, namely, finite element exterior calculus (FEEC). Consequently, our important computational result is that the solution of the Poisson problem on different manifolds in two- and three-dimensions due to DEC recovers convergence properties on many sequences of refined meshes similar to that of FEEC. We also discuss some potential attempts for showing this convergence theoretically. In particular, we demonstrate that a certain formulation of a variational crimes approach that can be used for showing convergence for a generalized FEEC may not be directly applicable to DEC convergence in its current formulation. In order to perform computations using DEC, a key development that we present is exhibiting sign rules that allow for the computation of the discrete Hodge star operators in DEC on Delaunay meshes in a piecewise manner. Another aspect of computationally solving the Poisson problem using the mixed formulation with either DEC or FEEC requires knowing the solution to the corresponding Laplace's problem, namely, the harmonics. We present a least squares method for computing a basis for the space of such discrete harmonics via their isomorphism to cohomology. We also provide some numerics to quantify the efficiency of this solution in comparison with previously known methods. Finally, we demonstrate an application to obtain the ranking of pairwise comparison data. We model this data as edge weights on graphs with 3-cliques included and perform its Hodge decomposition by solving two least squares problems. An outcome of this exploration is also providing some computational evidence that algebraic multigrid linear solvers for the resulting linear systems on Erdős-Rényi random graphs and on Barabási-Albert graphs do not perform very well in comparison with iterative Krylov solvers

Laplacians Of Cellular Sheaves: Theory And Applications

Cellular sheaves are a discrete model for the theory of sheaves on cell complexes. They carry a canonical cochain complex computing their cohomology. This thesis develops the theory of the Hodge Laplacians of this complex, as well as avenues for their application to concrete engineering and data analysis problems. The sheaf Laplacians so developed are a vast generalization of the graph Laplacians studied in spectral graph theory. As such, they admit generalizations of many results from spectral graph theory and the spectral theory of discrete Hodge Laplacians. A theory of approximation of cellular sheaves is developed, and algorithms for producing spectrally good approximations are given, as well as a generalization of the notion of expander graphs. Sheaf Laplacians allow development of various dynamical systems associated with sheaves, and their behavior is studied. Finally, applications to opinion dynamics, extracting network structure from data, linear control systems, and distributed optimization are outlined

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal