Quantifying high-order interdependencies on individual patterns via the local O-information: Theory and applications to music analysis

High-order, beyond-pairwise interdependencies are at the core of biological, economic, and social complex systems, and their adequate analysis is paramount to understand, engineer, and control such systems. This paper presents a framework to measure high-order interdependence that disentangles their effect on each individual pattern exhibited by a multivariate system. The approach is centered on the local O-information, a new measure that assesses the balance between synergistic and redundant interdependencies at each pattern. To illustrate the potential of this framework, we present a detailed analysis of music scores from J. S. Bach, which reveals how high-order interdependence is deeply connected with highly nontrivial aspects of the musical discourse. Our results place the local O-information as a promising tool of wide applicability, which opens other perspectives for analyzing high-order relationships in the patterns exhibited by complex systems.