The total information that can be stored in the system () is decomposed by a given state of knowledge (i.e., a probability distribution) into two parts: what is determined by the constraints [the negentropy, and what is not instantiated until an actual measurement takes place [the entropy, . Both terms can be further decomposed into their individual and collective components, yielding different perspectives on interdependency seen as either collective constraints [measured by the total correlation or shared randomness [corresponding to the dual total correlation .

Double diamond diagram with the possible sequences of binary partitions of three variables. Every path from the source node () to the two sink nodes and corresponds to a decomposition of either or .

Diagram of how values of the O-information impose limits on the strength of interactions—as measured by —at different scales. Positive (negative) values of put lower (upper) bounds on subsets of , and higher absolute values of put bounds on subsystems of smaller sizes.

Mean value and confidence intervals of ensembles of systems of spins with randomly generated Hamiltonians. By including high-order interaction terms, net synergy increases and decreases.

The sum of the TC and DTC (denoted by ) is an accurate approximation of the TSE complexity. Each dot corresponds to probability distribution over binary variables, which are sampled uniformly at random from the corresponding probability simplex.

(upper line) and (lower line) evaluated on a distribution resulting from a linear mixture between a copy (left) and an (right), showing that the TSE complexity conflates synergy and redundancy. The figure shows the case , but results are qualitatively similar for larger systems.

Top: Entropy of the frequencies of appearance of each note in the studied pieces of Bach and Corelli, measured in muts (logarithm to base 13); standard errors were estimated via circular block-bootstrap. While the higher voices in Corelli have higher entropy, Bach's soprano has a lower entropy than all other voices. Bottom: Global O-information (left) and networks of local O-information (middle, right) with red reflecting redundancy () and blue reflecting synergy (). Links characterize the triple interaction between each part of the corresponding dyad and the rest of the system. While Bach's chorales are synergy dominated, the pieces of Corelli are strongly redundant (mainly due to the viola and cello).

The O-information and —introduced in our previous work []—have good agreement.