Principled tractable models for higher-order interactions

Develop tractable probabilistic models that capture the diverse effects of higher-order interactions in complex systems in a principled manner, overcoming the combinatorial explosion inherent to exhaustive high-order representations while retaining interpretability and analytic tractability.

Background

Higher-order interactions (HOIs) are pervasive across biological and artificial neural systems and are linked to phenomena such as bistability, hysteresis, and explosive phase transitions. However, modeling HOIs typically leads to an exponential increase in parameters when higher-order terms are considered exhaustively. This challenge has limited prior studies to highly homogeneous scenarios or to relatively low-order models.

The paper introduces curved neural networks derived from a generalized maximum entropy principle as one approach to address these challenges. While this framework provides a tractable avenue, the broader problem of constructing principled, tractable HOI models remains a central motivation and is explicitly noted as unresolved in the introduction.

References

In fact, it is currently unclear how to construct tractable models to address the diverse effects of HOIs in a principled manner.

— Explosive neural networks via higher-order interactions in curved statistical manifolds  (2408.02326 - Aguilera et al., 2024) in Introduction

Conceptually, GraphK addresses the open problem articulated by , reconciling symmetry principles with sparse network properties by permitting limited dependencies between edges, and advances a simple, scalable mechanism that empirically produces realistic community structure while keeping the model permutation-agnostic in its latent parametrization.

— GraphK: Variable-Size Graph Generation with Efficient Edge Construction  (2608.18777 - Tugay et al., 19 Aug 2026) in Section Methodology, paragraph 'Exchangeability, projectivity, and sparsity'

The agenda is ambitious because its hardest questions remain open. Identity must itself become probabilistic: uncertain coreference and owl:sameAs cannot simply be assumed away. Tractability is equally fundamental. Weighted model counting is #P-hard, higher-order conditional tensors grow combinatorially, and practical systems will depend on sparsity, factorisation, and lifted inference over ontology symmetries. The numbers also require scrutiny: logit-derived confidences vary across models and phrasings, so calibration methods and source reliability must travel with the beliefs they produce.

— Semantic Bayesian World Models  (2609.03834 - Soru, 3 Sep 2026) in Section 4, What Must Be Built