{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,11]],"date-time":"2025-11-11T15:48:49Z","timestamp":1762876129617,"version":"build-2065373602"},"reference-count":90,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2019,8,8]],"date-time":"2019-08-08T00:00:00Z","timestamp":1565222400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["DP140104402"],"award-info":[{"award-number":["DP140104402"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The concept of a \u201cflow network\u201d\u2014a set of nodes and links which carries one or more flows\u2014unites many different disciplines, including pipe flow, fluid flow, electrical, chemical reaction, ecological, epidemiological, neurological, communications, transportation, financial, economic and human social networks. This Feature Paper presents a generalized maximum entropy framework to infer the state of a flow network, including its flow rates and other properties, in probabilistic form. In this method, the network uncertainty is represented by a joint probability function over its unknowns, subject to all that is known. This gives a relative entropy function which is maximized, subject to the constraints, to determine the most probable or most representative state of the network. The constraints can include \u201cobservable\u201d constraints on various parameters, \u201cphysical\u201d constraints such as conservation laws and frictional properties, and \u201cgraphical\u201d constraints arising from uncertainty in the network structure itself. Since the method is probabilistic, it enables the prediction of network properties when there is insufficient information to obtain a deterministic solution. The derived framework can incorporate nonlinear constraints or nonlinear interdependencies between variables, at the cost of requiring numerical solution. The theoretical foundations of the method are first presented, followed by its application to a variety of flow networks.<\/jats:p>","DOI":"10.3390\/e21080776","type":"journal-article","created":{"date-parts":[[2019,8,8]],"date-time":"2019-08-08T11:05:32Z","timestamp":1565262332000},"page":"776","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Maximum Entropy Analysis of Flow Networks: Theoretical Foundation and Applications"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5781-1259","authenticated-orcid":false,"given":"Robert K.","family":"Niven","sequence":"first","affiliation":[{"name":"School of Engineering and Information Technology, The University of New South Wales, Northcott Drive, Canberra, ACT 2600, Australia"}]},{"given":"Markus","family":"Abel","sequence":"additional","affiliation":[{"name":"Ambrosys GmbH, 14469 Potsdam, Germany"},{"name":"Institute for Physics and Astrophysics, University of Potsdam, 14469 Potsdam, Germany"}]},{"given":"Michael","family":"Schlegel","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Str\u00f6mungsmechanik und Technische Akustik, Technische Universit\u00e4t Berlin, 10623 Berlin, Germany"}]},{"given":"Steven H.","family":"Waldrip","sequence":"additional","affiliation":[{"name":"School of Engineering and Information Technology, The University of New South Wales, Northcott Drive, Canberra, ACT 2600, Australia"}]}],"member":"1968","published-online":{"date-parts":[[2019,8,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1126\/science.286.5439.509","article-title":"Emergence of scaling in random networks","volume":"286","author":"Barabasi","year":"1999","journal-title":"Science"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"268","DOI":"10.1038\/35065725","article-title":"Exploring complex networks","volume":"410","author":"Strogatz","year":"2001","journal-title":"Nature"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1079","DOI":"10.1080\/00018730110112519","article-title":"Evolution of networks","volume":"51","author":"Dorogovtsev","year":"2002","journal-title":"Adv. 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