From Statistical to Structural Synergy: A Predictability Framework to Quantify the Effects due to High-Order Mechanisms

High-order interactions are increasingly recognized as a hallmark of collective dynamics in complex systems.

The relationship between high-order behaviours (HOBs), observed as synergistic or redundant statistical dependencies, and high-order mechanisms (HOMs), related to the structural or dynamical rules of the data-generating process, remains difficult to establish from data.

We introduce a predictability-based framework to disentangle these two levels of description in complex network systems.

Structural synergy is defined as the excess predictive power gained when two sources are considered jointly beyond the best additive description and is estimated through polynomial regression by comparing a model with interaction terms against an additive model.

Simulations show that dependencies among sources reflecting HOBs can arise even in the absence of HOMs, while synergy due to a non-additive mechanism may remain hidden when the observed synergy-redundancy balance is dominated by redundancy and become detectable only through structural synergy.

Applications to climate and source-reconstructed cortical EEG dynamics reveal significant non-additive predictive components despite predominantly redundancy-dominated HOBs.

These findings emphasize that HOBs and HOMs can dissociate: a system may display redundancy-dominated HOBs while still containing a significant synergistic mechanism.

The proposed framework supports a mechanistically informed interpretation of complex-system dynamics and may help to identify when mechanism-based models are needed to predict the response of a system to perturbations or interventions, while also recognizing that structural synergy should be interpreted as evidence of non-additive predictive structure rather than as a direct identification of the underlying generative mechanism.