New Theory Explains Ultrametric Stability in Sparse Perturbations

The subdominant (minmax) ultrametric is a fundamental tree-structured summary derived from a dissimilarity matrix, often associated with single-linkage clustering. While its stability has traditionally been analyzed using L-infinity or Gromov-Hausdorff metrics, these approaches are not well-suited for understanding the effects of sparse perturbations, where only a few pairwise distances are altered. This limitation leaves a gap in understanding how minor changes can impact hierarchical structures. A new L0-type stability theory has been developed to address this. The analysis reveals that sparse edits to a dissimilarity matrix propagate changes exclusively through the minimum spanning tree (MST). Specifically, a pairwise ultrametric value can only change if its corresponding tree path intersects an edited edge or a cut newly exposed by an off-tree edited edge. This insight leads to sharp per-edit exposed-cut scores and a tree-only global envelope. These findings result in Hamming-Lipschitz bounds, which quantify the number of ultrametric entries that can change due to sparse perturbations. The researchers also proved sharpness results, demonstrating that this dependence on tree geometry is unavoidable. Under strict cut separation, the tree-edge bound is precisely attained, and for off-tree edits, explicit families exist where a single edited distance can alter a quadratic number of ultrametric entries. Additionally, a conditional near-additivity principle for multiple edits was proven, applicable when per-edit changed regions are large and aggregate overlap is negligible. Experiments on deep-embedding graphs show these structural scores are useful for diagnosing vulnerabilities in hierarchical representations.