Complex systems often comprise many kinds of components which vary over many orders of magnitude in size: Populations of cities in countries, individual and corporate wealth in economies, species abundance in ecologies, word frequency in natural language, and node degree in complex networks. Here, we introduce ‘allotaxonometry’ along with ‘rank-turbulence divergence’ (RTD), a tunable instrument for comparing any two ranked lists of components. We analytically develop our rank-based divergence in a series of steps, and then establish a rank-based allotaxonograph which pairs a map-like histogram for rank-rank pairs with an ordered list of components according to divergence contribution. We explore the performance of rank-turbulence divergence, which we view as an instrument of ‘type calculus’, for a series of distinct settings including: Language use on Twitter and in books, species abundance, baby name popularity, market capitalization, performance in sports, mortality causes, and job titles. We provide a series of supplementary flipbooks which demonstrate the tunability and storytelling power of rank-based allotaxonometry.

复杂系统通常由多种组成部分组成，这些组成部分的规模有多个数量级：国家中的城市人口、经济中的个人和企业财富、生态中的物种丰富度、自然语言中的词频以及复杂网络中的节点度。在这里，我们介绍“allotaxonometry”和“rank-turbulence divergence”（RTD），这是一种用于比较任何两个排名成分列表的可调工具。我们通过一系列步骤分析开发基于等级的散度，然后建立一个基于等级的异位图，它将等级-等级对的类似地图的直方图与根据散度贡献的有序组件列表配对。我们探索了等级湍流散度的性能，我们将其视为“类型微积分”的工具，针对一系列不同的设置，包括：Twitter 和书籍中的语言使用、物种丰富度、婴儿名字流行度、市值、体育表现、死亡率和职位。我们提供了一系列补充活页夹，展示了基于等级的异种分类学的可调性和讲故事的能力。