Measuring Hierarchy

This paper presents a novel axiomatic approach to measuring and comparing hierarchical structures. Hierarchies are fundamental across a range of disciplines – from ecology to organizational science – yet existing measures of hierarchical degree often lack systematic criteria for comparison. We introduce a mathematically rigorous framework based on a simple partial pre-order over hierarchies, denoted as ≽H, and demonstrate its equivalence to intuitively appealing axioms for hierarchy comparisons. Our analysis yields three key results. First, we establish that for fixed-size hierarchies, one hierarchy is strictly more hierarchical than another according to ≽H if the latter can be derived from the former through a series of subordination removals. Second, we fully characterize the hierarchical pre-orders that align with ≽H using two fundamental axioms: Anonymity and Subordination Removal. Finally, we extend our framework to varying-size hierarchies through the introduction of a Replication Principle, which enables consistent comparisons across different scales.