Bound the error from dense-to-sparse graph replacement

Derive an error bound for replacing the complete correlation-coupled XY interaction graph with the Ward-ordered interaction path, specifically relating the resulting partition function or observables to those of the dense graph.

Background

The portfolio model begins with a dense XY interaction graph whose couplings are given by pairwise empirical correlations. For computational tractability, the authors replace this graph with a path formed by connecting consecutive leaves in a Ward-clustering order. This reduces the tensor-network contraction cost to a path-like O(ND²) procedure but discards most direct interactions.

The paper explicitly notes that the Gromov hyperbolicity diagnostic does not determine the Ward path or quantify the effect of sparsification. Consequently, an unresolved problem is to establish a quantitative relationship between the dense model and the path-based surrogate, such as a bound on partition-function, marginal, magnetisation, or portfolio-weight errors.

References

The dense-to-sparse replacement changes the model and has no known error bound relative to the complete interaction graph.

— Tensor-Network Inference in a Field-Coupled XY Model for Portfolio Allocation  (2609.05045 - Chowdhry et al., 4 Sep 2026) in Section 4.5, subsection “Computational complexity”; also discussed in Section 3.4, subsection “Ward clustering, interaction paths, and approximation scope”