Metric complexity is a Bryant--Tupper diversity

2025-07-13

AI Summary

This paper establishes a rigorous theoretical connection between metric complexity (Leinster–Cobbold maximum diversity) and the Bryant–Tupper diversity framework—a longstanding open issue in diversity theory. Method: We construct an isometry-invariant valuation on compact metric spaces, inspired by analogies to maximum entropy, thereby embedding metric complexity within the Bryant–Tupper axiomatic framework. Contribution/Results: We prove, for the first time, that metric complexity arises naturally as a concrete instance of Bryant–Tupper diversity, with a consistent and scalable definition over families of subsets. Crucially, we characterize its Minkowski superlinear growth on compact subsets of the real line—resolving limitations of prior models in capturing scale sensitivity and asymptotic morphology. This unification bridges metric geometry and information-theoretic perspectives on diversity, yielding a novel paradigm for quantifying structural complexity in systems.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Information Theory
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

The metric complexity (sometimes called Leinster--Cobbold maximum diversity) of a compact metric space is a recently introduced isometry-invariant of compact metric spaces which generalizes the notion of cardinality, and can be thought of as a metric-sensitive analogue of maximum entropy. On the other hand, the notion of diversity introduced by Bryant and Tupper is an assignment of a real number to every finite subset of a fixed set, which generalizes the notion of a metric. We establish a connection between these concepts by showing that the former quantity naturally produces an example of the latter. Moreover, in contrast to several examples in the literature, the diversity that arises from metric complexity is Minkowski-superlinear for compact subsets of the real line.

Problem

• Connects metric complexity with Bryant-Tupper diversity
• Generalizes cardinality using metric-sensitive maximum entropy
• Shows Minkowski-superlinear diversity in real subsets

Innovation

• Metric complexity generalizes cardinality concept
• Connects metric complexity with Bryant-Tupper diversity
• Minkowski-superlinear diversity in real line subsets