  The homology groups of a simplicial complex reveal fundamental properties of
the topology of the data or the system and the notion of topological stability
naturally poses an important yet not fully investigated question. In the
current work, we study the stability in terms of the smallest perturbation
sufficient to change the dimensionality of the corresponding homology group.
Such definition requires an appropriate weighting and normalizing procedure for
the boundary operators acting on the Hodge algebra's homology groups. Using the
resulting boundary operators, we then formulate the question of structural
stability as a spectral matrix nearness problem for the corresponding
higher-order graph Laplacian. We develop a bilevel optimization procedure
suitable for the formulated matrix nearness problem and illustrate the method's
performance on a variety of synthetic quasi-triangulation datasets and
transportation networks.
