Medidas de centralidad en grafos. Aplicación a ligas deportivas

Abstract

Starting off with some basic concepts of graph theory and spectral theory, we will focus our study in the application of mathematics to the problem of classification in competitive leagues. Given that this problem is not completely solved, we want to offer a new view on this subject, proving that we have really powerful tools to develop a satisfactory solution to it with really interesting and desirable properties. This study will show the great applicability of functional analysis, giving an abstraction of a subject apparently not related to it, and obtaining new results helpful in the field of knowledge that we are working on. In the first Chapter, we give some basic definitions and results, already known, about graph theory, so that we can define what centrality measures are, and giving the expressions of the most commonly used in order to apply them in the next chapter. The second Chapter is dedicated to the main matter of this study: the classification of competitive leagues. It presents two main problems to solve. The first one is the problem of classification of different leagues or the same league along different seasons attending to its competitive balance which is a measure of concentration of wins, this is, how the victories are distributed among the teams participating in the competition. The second problem that is addressed in this chapter is the classification of the teams themselves in the league. This is answering these questions: who is the winner of the league? Who is next in the classification? Who is the last one? In other words, we are interested in giving a ranking method. We will solve these two problems thanks to functional analysis. The first one will part from an article written by Ávila-Cano and Triguero ([38]), and we will develop new results applying norms and distances in the vector space Rn. This will grant us with plenty of tools that can be used to give nice properties to the measures that we are looking for. We will define a new generalization of the competitive balance measures, calling them competitive balance normed measures and we will analyse and explain them in depth. The second one is a well known problem with plenty of literature related to it (view [12] as an example), so we are going to compile this literature and explain how we can apply graph theory to a competitive league, in order to obtain centrality measures of the graphs that explain the situations given in these leagues, so we can compare and conclude which centrality measure give us the best ranking possible. We will see that this measure is the eigenvector centrality, so an important result to apply this measure is the Perron-Frobenius Theorem, as it guarantees that this measure always finds a ranking of the teams, this is, we can always find a solution to the expression of the measure. As the Perron-Frobenius becomes a fundamental result in our study, we dedicate the third and final chapter of this assignment to, firstly, give an overview of the history of the theorem, so we can understand in which context it appeared. Afterwards, we give the definition of the Perron’s Theorem, which can be considered as a first approximation of the result that we are willing to prove. After giving all the pre-requisites that we need form the theory of functional analysis, we finally give the demonstration of the Perron-Frobenius Theorem, and, in order to provide the reader with a wider view on this subject, we also give an alternative proof of the theorem, more based on a geometric approach to the problem. In order to show the real importance of this result, we also enunciate the generalization of the theorem to Banach spaces, and we give some of the many applications that this fundamental result has in the field of scientific research.

El objetivo de este Trabajo Fin de Grado es aplicar técnicas y conocimientos propios del Análisis Funcional y, en concreto, de la Teoría Espectral, a ciertos problemas de la vida real, por lo que el enfoque de esta Memoria tiene un determinado componente interdisciplinar. Como motivación e hilo conductor del trabajo, hemos elegido varios problemas del mundo deportivo, que resultan extrapolables al contexto empresarial, dando una visión matemática abstracta a cuestiones que ya han sido estudiadas desde otros campos como puede ser la economía. La propuesta de Trabajo Fin de Grado, que en un primer momento se formalizó tenía como objetivo único el estudio de la centralidad del vector propio y, por tanto, del Teorema de Perron-Frobenius. Buscando alguna aplicación vistosa que sirviese de motivación llegamos al problema de la clasificación de equipos en una liga deportiva, en el que la centralidad del vector propio había sido una herramienta propuesta en multitud de trabajos, como por ejemplo [12], siendo referencia central en este enfoque el conocido artículo de J. P. Keener [18] (que dispone de 349 citas) si bien el primero de todos ellos, que hace uso del Teorema de Perrón Frobenius para clasificar equipos de fútbol, data de 1915 y se debe a E. Landau [21].