{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,5]],"date-time":"2026-01-05T11:11:15Z","timestamp":1767611475468,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,12,21]],"date-time":"2021-12-21T00:00:00Z","timestamp":1640044800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["41975064"],"award-info":[{"award-number":["41975064"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Information flow provides a natural measure for the causal interaction between dynamical events. This study extends our previous rigorous formalism of componentwise information flow to the bulk information flow between two complex subsystems of a large-dimensional parental system. Analytical formulas have been obtained in a closed form. Under a Gaussian assumption, their maximum likelihood estimators have also been obtained. These formulas have been validated using different subsystems with preset relations, and they yield causalities just as expected. On the contrary, the commonly used proxies for the characterization of subsystems, such as averages and principal components, generally do not work correctly. This study can help diagnose the emergence of patterns in complex systems and is expected to have applications in many real world problems in different disciplines such as climate science, fluid dynamics, neuroscience, financial economics, etc.<\/jats:p>","DOI":"10.3390\/e24010003","type":"journal-article","created":{"date-parts":[[2021,12,21]],"date-time":"2021-12-21T09:50:43Z","timestamp":1640080243000},"page":"3","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["The Causal Interaction between Complex Subsystems"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8472-3211","authenticated-orcid":false,"given":"X. San","family":"Liang","sequence":"first","affiliation":[{"name":"Department of Atmospheric & Oceanic Sciences, Institute of Atmospheric Sciences, Fudan University, Shanghai 200438, China"},{"name":"IRDR ICoE on Risk Interconnectivity and Governance on Weather\/Climate Extremes Impact and Public Health, Fudan University, Shanghai 200438, China"},{"name":"Shanghai Qi Zhi Institute (Andrew C. Yao Institute for Artificial Intelligence), Shanghai 200232, China"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,21]]},"reference":[{"key":"ref_1","unstructured":"Intergovenmental Panel on Climate Change (IPCC) (2021, November 15). The Sixth Assessment Report, Climate Change 2021: The Physical Science Basis. 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