Competitivity graphs analysis and structural comparison of rankings

Abstract

A complex networks based method is introduced for comparing different complete rankings of a finite family of elements. The concepts of competitivity graph and evolutive competitivity graph are introduced as the main tools for analyzing an (ordered) family of rankings of a fixed set of elements. It is shown how the structural properties of these competitivity graphs give deep information about the competitiveness of the elements according to the rankings considered. The relationships between competitivity graphs and some other well-known families of graphs, such as permutation graphs, comparability graphs and chordal graphs are also presented. Finally some applications are presented, including the analysis of sports rankings and, more precisely, the study of European soccer leagues.