Scientifically matched null models for TDA in complex-systems data

Develop domain-appropriate null models for topological data analysis that are scientifically matched to weighted, temporal, correlation-derived, higher-order, and spatially constrained datasets arising in complex-systems applications, so that statistical significance of persistence-based summaries can be assessed against meaningful alternatives rather than generic random baselines.

Background

The paper emphasizes that statistical significance for persistence-based summaries cannot rely solely on generic random-point-cloud baselines because many complex-systems datasets are weighted, temporal, correlation-derived, higher-order, or spatially constrained.

While recent advances provide universal null distributions in limited settings, the authors argue that complex-systems applications require null models that preserve key structural constraints (e.g., marginal distributions, degree/strength sequences, spatial embedding, autocorrelation) to avoid misleading inferences.

They identify the lack of such scientifically tailored null models as a central barrier to rigorous inference with TDA in real-world systems.

References

For complex-systems science, this is a central open problem. The field needs null models that are not merely mathematically convenient, but scientifically matched to weighted, temporal, correlation-derived, higher-order, and spatially constrained data.

— 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 8.2: Stability is not significance: inference and null models

Although persistence diagrams provide a robust multiscale representation of the O-information field, a complete statistical assessment of the resulting structures requires the specification of suitable null models. In the present framework, the construction of such models is nontrivial because municipalities are represented by multivariate empirical risk distributions and the associated O-information depends on both marginal probabilities and higher-order dependency structures. Consequently, different randomization schemes, including label shuffling, bootstrap resampling, subsampling, or correlation-preserving surrogates, correspond to distinct null hypotheses. The development and comparison of these null-model families represents an important direction for future work. Accordingly, the present study should be interpreted as an exploratory methodological investigation aimed at introducing and evaluating the proposed information-topological framework, while a systematic analysis of statistical significance and uncertainty remains a topic for subsequent research.

— Topological Uncertainty and Higher-Order Interactions in Spatial Networks  (2609.18439 - Pomarico et al., 16 Sep 2026) in Discussion, paragraphs beginning “Although persistence diagrams provide a robust multiscale representation of the O-information field” and “Accordingly, the present study should be interpreted...”