Any collection can be ranked. Sports and games are common examples of ranked systems: players and teams are constantly ranked using different methods. The statistical properties of rankings have been studied for almost a century in a variety of fields. More recently, data availability has allowed us to study rank dynamics: how elements of a ranking change in time. Here, we study the rank distributions and rank dynamics of 12 datasets from different sports and games. To study rank dynamics, we consider measures that we have defined previously: rank diversity, change probability, rank entropy, and rank complexity. We also introduce a new measure that we call “system closure” that reflects how many elements enter or leave the rankings in time. We use a random walk model to reproduce the observed rank dynamics, showing that a simple mechanism can generate similar statistical properties as the ones observed in the datasets. Our results show that while rank distributions vary considerably for different rankings, rank dynamics have similar behaviors, independently of the nature and competitiveness of the sport or game and its ranking method. Our results also suggest that our measures of rank dynamics are general and applicable for complex systems of different natures.

中文翻译：

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体育和游戏排名的统计特性

任何集合都可以排名。体育和游戏是排名系统的常见示例：球员和球队不断地使用不同的方法进行排名。近一个世纪以来，各个领域都对排名的统计特性进行了研究。最近，数据可用性使我们能够研究排名动态：排名元素如何随时间变化。在这里，我们研究了来自不同运动和游戏的 12 个数据集的排名分布和排名动态。为了研究排名动态，我们考虑了我们之前定义的度量：排名多样性、变化概率、排名熵和排名复杂性。我们还引入了一种称为“系统关闭”的新度量，它反映了有多少元素及时进入或离开排名。我们使用随机游走模型来重现观察到的排名动态，表明一个简单的机制可以产生与数据集中观察到的相似的统计特性。我们的结果表明，虽然不同排名的排名分布差异很大，但排名动态具有相似的行为，与运动或游戏的性质和竞争力及其排名方法无关。我们的结果还表明，我们对等级动态的测量是通用的，适用于不同性质的复杂系统。