The Non-commutative Spaces of Higher-Order Networks of Bach's Solo Violin Compositions: Dimension, Curvature, and Distance through the Spectral Triplet

Abstract: Our work is concerned with simplicial complexes that describe higher-order interactions in real complex systems. This description allows to go beyond the pairwise node-to-node representation that simple networks provide and to capture a hierarchy of interactions of different orders. The prime contribution of this work is the introduction of geometric measures for these simplicial complexes. We do so by noting the non-commutativity of the algebra associated with their matrix representations and consequently we bring to bear the spectral triplet formalism of Connes on these structures and then notions of associated dimensions, curvature, and distance can be computed to serve as characterizing features in addition to known topological metrics.