Research Spotlights

Abstract

This issue features three Research Spotlight articles. The first of these is entitled “Variance and Covariance of Distributions on Graphs" and is coauthored by Karel Devriendt, Samuel Martin-Gutierrez, and Renaud Lambiotte. Given a distribution on the graph (that is, a function $p$ from the set of nodes to $[0,1]$ such that the sum of all values is one) the key ingredient in the authors' definition of the variance and covariance is a notion of distance between the nodes of a graph. Although their definitions of (co)variance are valid for different choices of distance, the authors focus on a metric called the effective resistance. The effective resistance resembles geodesic distance in that it reflects the length of the paths between a pair of nodes but differs in that it takes into account all paths, and their lengths, between a node pair. Conveniently, the resulting variance and covariance values can be calculated via evaluation of a quadratic product and matrix trace. In support of the newly introduced measures, the balance of the paper is devoted to the application of the new definitions in practice and to the explanation of the conceptual correspondence between their (co)variance measures and some known graph characteristics. Citing the benefits of the framework induced by their (co)variance measures, the authors leave the reader with suggestions for future application in fields such as neuroscience, economics, and social networks. Competitive tournaments are an integral part of daily life, from schoolyard games to professional sports, to politics and biology. Authors Alexander Strang, Karen C. Abbott, and Peter J. Thomas tackle the difficult problem of quantifying competition in tournament modeling. Their article, “The Network HHD: Quantifying Cyclic Competition in Trait-Performance Models of Tournaments,” outlines how to adapt the discrete Helmholtz--Hodge decomposition (HHD), first introduced in the literature as a method for ranking objects from incomplete and imbalanced data, to study competitive tournaments characterized by intransitivity, the presence of which provides a challenge in ranking. Specifically, an intransitive tournament is one in which there is no clear global ranking of all competitors, and it corresponds to a cycle in the model. The authors show that the HHD arises in the context of representing a generic tournament as a weighted sum of so-called perfectly transitive and perfectly cyclic components. Since a trait-performance model assumes that “the probability that one competitor beats another is a function of their traits” the goal becomes that of identifying which traits and performance functions influence the weights in the HHD. This is addressed with the theoretical results in section 4. Schematics and graphs complement the discussion. Using the code made available by the authors, the interested reader may wish to try an analysis of a tournament for themselves. The third article, “Bilinear Optimal Control of an Advection-Reaction-Diffusion System," authored by Roland Glowinski, Yongcun Song, Xiaoming Yuan, and Hangrui Yue, offers both theoretical and computational insights to solving the mathematical problem specified in the title. Readers may appreciate the space devoted to the motivation and introduction to the problem as well as to the description of the associated difficulties both in deriving existence results and in numerical computation of solutions. Following a few preliminaries, the authors derive a proof of existence of optimal controls for the problem, labeled in the introduction as BCP, without the special case assumptions used elsewhere in the literature. The balance of the article is devoted to issues associated with the numerical implementation. Although their proposed nested conjugate gradient method is straightforward to present on paper, several obstacles arise in the implementation, and the authors tackle each of these in turn. For computational efficiency, for example, an inexact line search is proposed to replace the computationally intractable exact line search; preconditioned CG is employed for two internal subproblems that arise. The choice of the time and space discretizations is explained as well. There are many technical details to consume, but a key point for the readers is well summarized by the authors just before the detailed experiments, namely, “despite its apparent complexity, the nested CG method” given here “is easy to implement" with aspects that are amenable to parallelizability. Misha E. Kilmer Section Editor Misha.Kilmer@tufts.edu