Quantifying the Structural Stability of Simplicial Homology

Abstract Simplicial complexes are generalizations of classical graphs. Their homology groups widely used to characterize the structure and topology data in e.g. chemistry, neuroscience, transportation networks. In this work we assume given a simplicial complex that can act on its underlying graph, formed by set 1-simplices, investigate stability with respect perturbations edges such graph. Precisely, exploiting isomorphism between higher-order Laplacian operators, propose numerical method compute smallest graph perturbation sufficient change dimension simplex’s Hodge homology. Our approach is based matrix nearness problem formulated as differential equation, which requires an appropriate weighting normalizing procedure for boundary operators acting algebra’s groups. We develop bilevel optimization suitable illustrate method’s performance variety synthetic quasi-triangulation datasets real-world