A new computational tool captures hidden group dynamics in complex systems
May 11, 2026
- The study, published in PLOS ONE, introduces THOI, an open-source Python library that dramatically accelerates the analysis of collective patterns in complex systems that are missed by conventional pairwise methods.
- The research is the result of interdisciplinary collaboration between the Instituto de Física Interdisciplinar y Sistemas Complejos (IFISC, UIB-CSIC), Universidad de Buenos Aires, Institut du Cerveau, Imperial College London, University College London, Université Côte d’Azur, INRIA and University of the Balearic Islands.
Understanding how complex systems work, from the human brain to financial markets and ecosystems, often requires more than studying isolated elements or relationships between pairs. Many important behaviors emerge only when several components interact simultaneously. Yet identifying these higher-order interactions –collective interactions beyond pairs– has remained computationally demanding, limiting their practical use in science.
Now, researchers have developed THOI (Torch-based High-Order Interactions), a new software library designed to make this type of analysis faster, more scalable, and widely accessible. Presented in PLOS ONE, THOI enables scientists to examine collective dependencies among many variables at once using standard computers, while also taking advantage of GPUs and other accelerated hardware when available.
Traditional statistical approaches typically focus on pairwise links, such as correlations between two brain regions or two economic indicators. While useful, these methods can miss interactions that only appear when several variables are considered together. Information theory offers tools to quantify such collective effects, including whether systems are dominated by redundancy, where information is repeated across components, or synergy, where the whole contains information not visible in its parts.
“Many real-world systems cannot be understood by looking only at one-to-one relationships”, says Rubén Herzog, researcher at IFISC (UIB-CSIC) and corresponding author of the study. “THOI helps reveal patterns that emerge only when multiple elements are analyzed together”. To tackle this challenge, the team combined mathematical advances in information theory with modern machine-learning infrastructure.
The researchers also demonstrated the software on real neuroscience data. Using functional MRI scans from volunteers in wakefulness and under deep anesthesia, THOI detected marked reductions in complex interaction patterns during anesthesia. In particular, the range of synergistic and redundant interactions was compressed, suggesting that loss of consciousness is associated with simpler large-scale brain dynamics.
Beyond neuroscience, the team applied THOI to more than 900 synthetic and real-world datasets, completing exhaustive analyses in under 30 minutes on a laptop. This broad survey helped identify general dimensions that characterize complex systems, such as overall interdependence and the balance between synergy and redundancy.
“By lowering the computational barrier, we hope more researchers can use higher-order interaction analysis in their own fields”, says Herzog. “These tools can open new questions in areas ranging from cognition to economics and social dynamics”.
The authors note that future versions of THOI could incorporate additional statistical estimators and improved support for large multi-GPU systems. Even so, the current release already offers a practical step forward for studying the hidden collective structure of multivariate data. As increasingly large datasets become common across disciplines, tools such as THOI may help scientists move beyond pairwise thinking and better understand how complex behaviors truly emerge.
Belloli L, Mediano PAM, Cofré R, Slezak DF, Herzog R (2026) THOI: An efficient and accessible library for computing higher-order interactions enhanced by batch-processing. PLoS One 21(5): e0348005. https://doi.org/10.1371/journal.pone.0348005
