Matching topological formalisms to complex-system representations

Determine which topological formalism—such as one-parameter persistence, multiparameter persistence, or zigzag persistence—appropriately matches a given complex-system representation characterized by multiple notions of scale, non-monotone evolution, or higher-order relational structure, in order to capture organization without misrepresentation.

Background

Many complex systems evolve with multiple scales and non-monotone changes that are not naturally captured by a single filtration parameter.

The authors discuss richer frameworks (e.g., zigzag and multiparameter persistence) but note their practical challenges and limited adoption in applications.

They identify the need to decide, per representation, which formalism best preserves the relevant organizational structure.

References

The open problem here is not simply to generalize for the sake of generality. It is to determine which topological formalism matches which kind of complex-system representation.

— Topology as a Language for Emergent Organization in Complex Systems: Multiscale Structure, Higher-Order Interactions, and Early Warning Signals  (2603.25760 - Bailey, 25 Mar 2026) in Section 8.5: The one-parameter paradigm is too narrow for many systems

More broadly, metric discovery remains an open problem: identifying which geometric spaces best capture relational structure between concepts.

— Are You Thinking What I am Thinking? : Examining Conceptual Separation in Neural Architectures  (2609.00764 - Ponde et al., 1 Sep 2026) in Section 6, Limitations and Future Work

Our PH framework requires many choices, which influence both our qualitative and quantitative results. The ability to make such choices is both a benefit and a drawback of our approach. The flexibility of our approach is very beneficial, but how should one make these choices?

— Using Persistent Homology to Analyze Access to Heterogeneous-Quality Resources and Heterogeneous-Severity Nuisances  (2609.34090 - Tymochko et al., 28 Sep 2026) in Section Discussion