Efficacy and safety of minimally invasive intramedullary calcaneal nail "nail-in-nail" technique versus extensile lateral approach plate internal fixation for Sanders type II-III calcaneal fractures:

Quantifying the relative strength of competitors is a fundamental aspect of any sport, traditionally accomplished through linear ranking systems. However, these one-dimensional lists often oversimplify the complex web of relationships and performance dynamics that arise from direct, pairwise competition. Network science offers a robust framework for modelling complex systems by representing entities as nodes and their interactions as edges. Applying this paradigm to the world of professional sports, where athletes are nodes and match outcomes are directed edges, provides a novel lens to analyse competitive hierarchies and predict future performance. By employing advanced analytical tools and methodologies such as centrality measures, motif analysis, and temporal network analysis, one can uncover the hidden features of complex networks such as the networks of matches. Understanding these delicate aspects of networks not only enriches our comprehension of the network’s structure and function but also enhances our ability to model, predict, and optimize the behaviour of complex systems, whether in sports tournaments, social interactions, or other competitive environments. The present research examines key features of directed networks related to pairwise sports contests within tournament frameworks. We will use the terms“contest”and“match”, as well as“players”and “contestants”, interchangeably. The topic related to rankings and networks of competitors have recently become popular for interpreting results in the realm of sports tournaments. As for rankings in general, the recent book by P. Érdi has an extensive overview of the many facets and applications of the field. While rankings are usually created by comparing the relevance of the entities considered, distilling the information gained into an order that faithfully reflects the results of these comparisons is a complex theoretical challenge. A common situation is when the more“important”of the two entities is given a higher score or rank, where importance can correspond to various features. For example, in sports, a contestant’s winning potential is considered a key feature. The task of ranking from pairwise comparisons has received significant attention in social choice theory. This body of literature highlights that the transition from individual matches or comparisons to a global hierarchy is rarely straightforward, as evidenced by several impossibility theorems that illustrate the fundamental challenges and logical constraints of such systems. For a broader perspective on these mechanics, Langville and Meyer provide a comprehensive overview of the mathematical algorithms used to rate and rank diverse subjects-ranging from sports teams and political candidates to products and Web pages-further emphasising that the methodology behind a ranking system is just as critical as the competitive data it seeks to organise. In addition to linear (one dimensional) rankings, in the present work we also analyse the competitive relationships as a hierarchy, which provides a more nuanced and more complex ordering of players, with elite performers at the top of the hierarchy. In general, hierarchical organisation is a ubiquitous feature of complex networks, observed in a remarkable range of systems, from intricate regulatory networks within cells and social structures of animal groups, to the organisation of online news content, scientific journals and scientific fields, and even the grand scale of ecological systems and evolution. In such networks, nodes positioned higher in the hierarchy typically have a greater influence than those at lower levels. Identifying and quantifying this hierarchical structure is a non-trivial challenge, with various approaches ranging from statistical inference based on network topology to the development of specific hierarchy measures. Hierarchies are often represented as directed acyclic graphs that capture asymmetric relationships, such as parent-child or leader-follower, that define the hierarchical ordering of nodes.