{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:22:12Z","timestamp":1772252532768,"version":"3.50.1"},"reference-count":43,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2017,5,18]],"date-time":"2017-05-18T00:00:00Z","timestamp":1495065600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"European Union Horizon 2020 research and innovation programme","award":["H2020-641321"],"award-info":[{"award-number":["H2020-641321"]}]},{"name":"EON Long Term Visitor Project"},{"name":"Japan Society for the Promotion of Science Doctoral Fellowship (Short-Term)"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We present a first formal analysis of specific and complete local integration. Complete local integration was previously proposed as a criterion for detecting entities or wholes in distributed dynamical systems. Such entities in turn were conceived to form the basis of a theory of emergence of agents within dynamical systems. Here, we give a more thorough account of the underlying formal measures. The main contribution is the disintegration theorem which reveals a special role of completely locally integrated patterns (what we call \u03b9-entities) within the trajectories they occur in. Apart from proving this theorem we introduce the disintegration hierarchy and its refinement-free version as a way to structure the patterns in a trajectory. Furthermore, we construct the least upper bound and provide a candidate for the greatest lower bound of specific local integration. Finally, we calculate the    \u03b9   -entities in small example systems as a first sanity check and find that    \u03b9   -entities largely fulfil simple expectations.<\/jats:p>","DOI":"10.3390\/e19050230","type":"journal-article","created":{"date-parts":[[2017,5,23]],"date-time":"2017-05-23T01:47:33Z","timestamp":1495504053000},"page":"230","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Specific and Complete Local Integration of Patterns in Bayesian Networks"],"prefix":"10.3390","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1670-6855","authenticated-orcid":false,"given":"Martin","family":"Biehl","sequence":"first","affiliation":[{"name":"Araya Incorporation, 2F Mori 15 Building, 2-8-10 Toranomon, Minato-ku, Tokyo 105-0001, Japan"},{"name":"School of Computer Science, University of Hertfordshire, Hatfield AL10 9AB, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Takashi","family":"Ikegami","sequence":"additional","affiliation":[{"name":"Department of General Systems Studies, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Daniel","family":"Polani","sequence":"additional","affiliation":[{"name":"School of Computer Science, University of Hertfordshire, Hatfield AL10 9AB, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,5,18]]},"reference":[{"key":"ref_1","unstructured":"Zalta, E.N. 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