Abstract
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.
Highlights
The analysis of interdependence is crucial for understanding the staggering complexity of structures and behaviors manifested in biological, economic, and social systems
The total correlation (TC) accounts for the effect of collective constraints, which refer to regions of the phase space that the system explores less [33], while the dual total correlation (DTC) measures the amount of shared randomness between the variables, i.e., the amount of information that can be collected in one variable that refers to the activity of another [25]
While the previous section focused on the role of single intervals, our analyses focus on harmonic considerations
Summary
The analysis of interdependence is crucial for understanding the staggering complexity of structures and behaviors manifested in biological, economic, and social systems. The unprecedented amount of data available for scientific scrutiny provides unique opportunities to deepen our understanding of multivariate coevolving complex systems, including the orchestrated activity of multiple brain areas, the interactions between different genes, and the relationship between various econometric indices. What allows these systems to be more than the sum of their parts is not to be found in the material nature of their parts, but in the fine structure of their interdependencies [1].
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