Our ability to understand and control complex systems of many interacting parts remains limited. A key challenge is that we still do not know how best to describe—and quantify—the many-to-many dynamical interactions that characterize their complexity. To address this limitation, we introduce the mathematical framework of Integrated Information Decomposition, or Φ ID. Φ ID provides a comprehensive framework to disentangle and characterize the information dynamics of complex multivariate systems. On the theoretical side, Φ ID reveals the existence of previously unreported modes of collective information flow, providing tools to express well-known measures of information transfer, information storage, and dynamical complexity as aggregates of these modes, thereby overcoming some of their known theoretical shortcomings. On the empirical side, we validate our theoretical results with computational models and examples from over 1,000 biological, social, physical, and synthetic dynamical systems. Altogether, Φ ID improves our understanding of the behavior of widely used measures for characterizing complex systems across disciplines and leads to new more refined analyses of dynamical complexity.
Mediano et al. (2025) studied this question.