Topology as a language for emergent organization in complex systems: Multiscale structure, higher-order interactions, and structural diagnostics.

Abstract

Complex systems are difficult to study not only because they are nonlinear, multiscale, and nonstationary, but because their scientifically relevant organization is often distributed across components, relations, and interaction orders. Topology provides a mathematical language for describing that organization through connectedness, recurrence, branching, closure, cavities, and persistence across scale. This review synthesizes persistent homology, Mapper, simplicial complexes, hypergraphs, and relation-level operator methods through a unified workflow from empirical data to representation, topological construction, output, and scientific interpretation. Across nonlinear dynamics, finance, neuroscience, biology, ecology, materials, and engineered systems, topological and topology-inspired methods make state-space organization, collective constraints, and structural reorganization available as observables that can be integrated with statistics, dynamics, mechanistic models, and machine learning. The review distinguishes the claim that a representation makes structure visible from the stronger claim that it improves detection or prediction, and it summarizes comparative evidence where such benchmarks exist. Prospective early-warning evidence remains uneven, but several studies demonstrate useful structural diagnostics, data-efficient classification, anomaly detection, and reductions in false alarms. The central conclusion is that topology is most valuable when representation is treated as a scientific hypothesis and topological descriptions are connected to domain-matched inference and mechanism.