Torsion in higher-order networks: Application to music variations

Characterizing higher-order interactions, especially in simplicial complexes, provides metrics for comparing, classifying, and analyzing their structural complexity. Here, we introduce a framework based on spectral geometry that yields an interpretable descriptor, torsion. It provides a count of the higher-order spanning trees for each order of the simplicial complex, and allows to differentiate between topologically equivalent complexes that have the same Betti numbers but differ in the lengths of their fundamental cycles. We also recover the number of their closed primitive cycles through the Ihara-Zeta function, a closely related notion to torsion. These two measures provide complementary information about the adjacency matrices associated with all orders of the simplicial complex. We demonstrate these notions on a real data set of higher-order networks representing thirteen variations of a musical piece and reveal latent measures of similarity. This framework allowed us to compare them and cluster them into musicologically similar-sounding variations.