SIAM Review, Volume 64, Issue 2, Page 360-391, May 2022.
Competitive tournaments appear in sports, politics, population ecology, and animal behavior. All of these fields have developed methods for rating competitors and ranking them accordingly. A tournament is intransitive if it is not consistent with any ranking. Intransitive tournaments contain rock-paper-scissors type cycles. The discrete Helmholtz--Hodge decomposition (HHD) is well adapted to describing intransitive tournaments. It separates a tournament into perfectly transitive and perfectly cyclic components, where the perfectly transitive component is associated with a set of ratings. The size of the cyclic component can be used as a measure of intransitivity. Here we show that the HHD arises naturally from two classes of tournaments with simple statistical interpretations. We then discuss six different sets of assumptions that define equivalent decompositions. This analysis motivates the choice to use the HHD among other existing methods. Success in competition is often mediated by the traits of the competitors. A trait-performance model assumes that the probability that one competitor beats another is a function of their traits. We show that if the traits of each competitor are drawn independently and identically from a trait distribution, then the expected degree of intransitivity in the network can be computed explicitly. We show that increasing the number of pairs of competitors who could compete promotes cyclic competition, and that correlation in the performance of $A$ against $B$ with the performance of $A$ against $C$ promotes transitive competition. The expected size of cyclic competition can thus be understood by analyzing this correlation.

中文翻译：

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网络 HHD：量化锦标赛特征性能模型中的循环竞争

SIAM 评论，第 64 卷，第 2 期，第 360-391 页，2022 年 5 月。
竞技比赛出现在体育、政治、人口生态和动物行为中。所有这些领域都开发了对竞争对手进行评级和相应排名的方法。如果锦标赛与任何排名不一致，则锦标赛是不及物的。不及物锦标赛包含石头剪刀布类型的循环。离散亥姆霍兹-霍奇分解 (HHD) 非常适合描述不及物锦标赛。它将锦标赛分为完美传递和完美循环组件，其中完美传递组件与一组评级相关联。循环分量的大小可以用作不传递性的度量。在这里，我们展示了 HHD 是由具有简单统计解释的两类锦标赛自然产生的。然后，我们讨论了六组不同的定义等效分解的假设。这种分析促使人们选择在其他现有方法中使用 HHD。竞争的成功往往取决于竞争对手的特质。特质绩效模型假设一个竞争对手击败另一个竞争对手的概率是他们的特质的函数。我们表明，如果每个竞争者的特征都是从特征分布中独立且相同地提取的，则可以明确计算网络中预期的不传递性程度。我们表明，增加可以竞争的竞争者对的数量会促进循环竞争，并且 $A$ 对 $B$ 的表现与 $A$ 对 $C$ 的表现之间的相关性促进了传递竞争。