{"id":"https://openalex.org/W2729208231","doi":"https://doi.org/10.3390/e19090451","title":"Invariant Components of Synergy, Redundancy, and Unique Information among Three Variables","display_name":"Invariant Components of Synergy, Redundancy, and Unique Information among Three Variables","publication_year":2017,"publication_date":"2017-08-28","ids":{"openalex":"https://openalex.org/W2729208231","doi":"https://doi.org/10.3390/e19090451","mag":"2729208231"},"language":"en","primary_location":{"id":"doi:10.3390/e19090451","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e19090451","pdf_url":"https://www.mdpi.com/1099-4300/19/9/451/pdf?version=1504000656","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/19/9/451/pdf?version=1504000656","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5070301037","display_name":"Giuseppe Pica","orcid":"https://orcid.org/0000-0002-4506-0600"},"institutions":[{"id":"https://openalex.org/I30771326","display_name":"Italian Institute of Technology","ror":"https://ror.org/042t93s57","country_code":"IT","type":"facility","lineage":["https://openalex.org/I30771326"]},{"id":"https://openalex.org/I4210127676","display_name":"Center for Neuroscience and Cognitive Systems","ror":"https://ror.org/045yf0774","country_code":"IT","type":"facility","lineage":["https://openalex.org/I30771326","https://openalex.org/I4210127676"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"Giuseppe Pica","raw_affiliation_strings":["Neural Computation Laboratory, Center for Neuroscience and Cognitive Systems @UniTn, Istituto Italiano di Tecnologia, 38068 Rovereto (TN), Italy"],"raw_orcid":"https://orcid.org/0000-0002-4506-0600","affiliations":[{"raw_affiliation_string":"Neural Computation Laboratory, Center for Neuroscience and Cognitive Systems @UniTn, Istituto Italiano di Tecnologia, 38068 Rovereto (TN), Italy","institution_ids":["https://openalex.org/I4210127676","https://openalex.org/I30771326"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5030475896","display_name":"Eugenio Piasini","orcid":"https://orcid.org/0000-0003-0384-7699"},"institutions":[{"id":"https://openalex.org/I30771326","display_name":"Italian Institute of Technology","ror":"https://ror.org/042t93s57","country_code":"IT","type":"facility","lineage":["https://openalex.org/I30771326"]},{"id":"https://openalex.org/I4210127676","display_name":"Center for Neuroscience and Cognitive Systems","ror":"https://ror.org/045yf0774","country_code":"IT","type":"facility","lineage":["https://openalex.org/I30771326","https://openalex.org/I4210127676"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Eugenio Piasini","raw_affiliation_strings":["Neural Computation Laboratory, Center for Neuroscience and Cognitive Systems @UniTn, Istituto Italiano di Tecnologia, 38068 Rovereto (TN), Italy"],"raw_orcid":"https://orcid.org/0000-0003-0384-7699","affiliations":[{"raw_affiliation_string":"Neural Computation Laboratory, Center for Neuroscience and Cognitive Systems @UniTn, Istituto Italiano di Tecnologia, 38068 Rovereto (TN), Italy","institution_ids":["https://openalex.org/I4210127676","https://openalex.org/I30771326"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5018721964","display_name":"Daniel Chicharro","orcid":"https://orcid.org/0000-0002-4038-258X"},"institutions":[{"id":"https://openalex.org/I136199984","display_name":"Harvard University","ror":"https://ror.org/03vek6s52","country_code":"US","type":"education","lineage":["https://openalex.org/I136199984"]},{"id":"https://openalex.org/I30771326","display_name":"Italian Institute of Technology","ror":"https://ror.org/042t93s57","country_code":"IT","type":"facility","lineage":["https://openalex.org/I30771326"]},{"id":"https://openalex.org/I4210127676","display_name":"Center for Neuroscience and Cognitive Systems","ror":"https://ror.org/045yf0774","country_code":"IT","type":"facility","lineage":["https://openalex.org/I30771326","https://openalex.org/I4210127676"]}],"countries":["IT","US"],"is_corresponding":false,"raw_author_name":"Daniel Chicharro","raw_affiliation_strings":["Department of Neurobiology, Harvard Medical School, Boston, MA 02115, USA","Neural Computation Laboratory, Center for Neuroscience and Cognitive Systems @UniTn, Istituto Italiano di Tecnologia, 38068 Rovereto (TN), Italy"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Neurobiology, Harvard Medical School, Boston, MA 02115, USA","institution_ids":["https://openalex.org/I136199984"]},{"raw_affiliation_string":"Neural Computation Laboratory, Center for Neuroscience and Cognitive Systems @UniTn, Istituto Italiano di Tecnologia, 38068 Rovereto (TN), Italy","institution_ids":["https://openalex.org/I4210127676","https://openalex.org/I30771326"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5014847103","display_name":"Stefano Panzeri","orcid":"https://orcid.org/0000-0003-1700-8909"},"institutions":[{"id":"https://openalex.org/I30771326","display_name":"Italian Institute of Technology","ror":"https://ror.org/042t93s57","country_code":"IT","type":"facility","lineage":["https://openalex.org/I30771326"]},{"id":"https://openalex.org/I4210127676","display_name":"Center for Neuroscience and Cognitive Systems","ror":"https://ror.org/045yf0774","country_code":"IT","type":"facility","lineage":["https://openalex.org/I30771326","https://openalex.org/I4210127676"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"Stefano Panzeri","raw_affiliation_strings":["Neural Computation Laboratory, Center for Neuroscience and Cognitive Systems @UniTn, Istituto Italiano di Tecnologia, 38068 Rovereto (TN), Italy"],"raw_orcid":"https://orcid.org/0000-0003-1700-8909","affiliations":[{"raw_affiliation_string":"Neural Computation Laboratory, Center for Neuroscience and Cognitive Systems @UniTn, Istituto Italiano di Tecnologia, 38068 Rovereto (TN), Italy","institution_ids":["https://openalex.org/I4210127676","https://openalex.org/I30771326"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5014847103","https://openalex.org/A5070301037"],"corresponding_institution_ids":["https://openalex.org/I30771326","https://openalex.org/I4210127676"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":4.7832,"has_fulltext":true,"cited_by_count":55,"citation_normalized_percentile":{"value":0.9572546,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":91,"max":100},"biblio":{"volume":"19","issue":"9","first_page":"451","last_page":"451"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10211","display_name":"Computational Drug Discovery Methods","score":0.9646999835968018,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10211","display_name":"Computational Drug Discovery Methods","score":0.9646999835968018,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12805","display_name":"Cognitive Science and Mapping","score":0.9133999943733215,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/pairwise-comparison","display_name":"Pairwise comparison","score":0.69984370470047},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5337415933609009},{"id":"https://openalex.org/keywords/invariant","display_name":"Invariant (physics)","score":0.5148319602012634},{"id":"https://openalex.org/keywords/entropy","display_name":"Entropy (arrow of time)","score":0.5119979977607727},{"id":"https://openalex.org/keywords/generality","display_name":"Generality","score":0.47354525327682495},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4658791422843933},{"id":"https://openalex.org/keywords/axiom","display_name":"Axiom","score":0.4321785271167755},{"id":"https://openalex.org/keywords/redundancy","display_name":"Redundancy (engineering)","score":0.426744282245636},{"id":"https://openalex.org/keywords/theoretical-computer-science","display_name":"Theoretical computer science","score":0.3568556606769562},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.16401243209838867}],"concepts":[{"id":"https://openalex.org/C184898388","wikidata":"https://www.wikidata.org/wiki/Q1435712","display_name":"Pairwise comparison","level":2,"score":0.69984370470047},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5337415933609009},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.5148319602012634},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.5119979977607727},{"id":"https://openalex.org/C2780767217","wikidata":"https://www.wikidata.org/wiki/Q5532421","display_name":"Generality","level":2,"score":0.47354525327682495},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4658791422843933},{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.4321785271167755},{"id":"https://openalex.org/C152124472","wikidata":"https://www.wikidata.org/wiki/Q1204361","display_name":"Redundancy (engineering)","level":2,"score":0.426744282245636},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.3568556606769562},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.16401243209838867},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C15744967","wikidata":"https://www.wikidata.org/wiki/Q9418","display_name":"Psychology","level":0,"score":0.0},{"id":"https://openalex.org/C542102704","wikidata":"https://www.wikidata.org/wiki/Q183257","display_name":"Psychotherapist","level":1,"score":0.0},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.3390/e19090451","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e19090451","pdf_url":"https://www.mdpi.com/1099-4300/19/9/451/pdf?version=1504000656","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},{"id":"pmh:oai:openaccess.city.ac.uk:27163","is_oa":false,"landing_page_url":"https://openaccess.city.ac.uk/view/creators_id/daniel=2Echicharro.html>","pdf_url":null,"source":{"id":"https://openalex.org/S4306401940","display_name":"City Research Online (City University London)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I180825142","host_organization_name":"City, University of London","host_organization_lineage":["https://openalex.org/I180825142"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":null,"raw_type":"PeerReviewed"},{"id":"pmh:oai:arXiv.org:1706.08921","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1706.08921","pdf_url":"https://arxiv.org/pdf/1706.08921","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"pmh:oai:doaj.org/article:f02f49bb593b4d369310e839b3cb47bf","is_oa":true,"landing_page_url":"https://doaj.org/article/f02f49bb593b4d369310e839b3cb47bf","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy, Vol 19, Iss 9, p 451 (2017)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/e19090451","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e19090451","pdf_url":"https://www.mdpi.com/1099-4300/19/9/451/pdf?version=1504000656","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320311063","display_name":"Fondation Bertarelli","ror":"https://ror.org/04h7fev25"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2729208231.pdf","grobid_xml":"https://content.openalex.org/works/W2729208231.grobid-xml"},"referenced_works_count":55,"referenced_works":["https://openalex.org/W1592663088","https://openalex.org/W1812423768","https://openalex.org/W1866114900","https://openalex.org/W1890016494","https://openalex.org/W1899852503","https://openalex.org/W1933178270","https://openalex.org/W1969116803","https://openalex.org/W1995875735","https://openalex.org/W1998050452","https://openalex.org/W2025670747","https://openalex.org/W2027665171","https://openalex.org/W2036381840","https://openalex.org/W2037667080","https://openalex.org/W2038276856","https://openalex.org/W2056594786","https://openalex.org/W2081988641","https://openalex.org/W2083250197","https://openalex.org/W2095439994","https://openalex.org/W2100850281","https://openalex.org/W2102456970","https://openalex.org/W2116811583","https://openalex.org/W2139708568","https://openalex.org/W2141771360","https://openalex.org/W2143891888","https://openalex.org/W2151910747","https://openalex.org/W2152587044","https://openalex.org/W2163296115","https://openalex.org/W2165559465","https://openalex.org/W2248909638","https://openalex.org/W2266454428","https://openalex.org/W2278576239","https://openalex.org/W2286271878","https://openalex.org/W2346453965","https://openalex.org/W2400942168","https://openalex.org/W2518518020","https://openalex.org/W2585967819","https://openalex.org/W2587049423","https://openalex.org/W2588151410","https://openalex.org/W2594429360","https://openalex.org/W2621961142","https://openalex.org/W2623462877","https://openalex.org/W2624014641","https://openalex.org/W2737989178","https://openalex.org/W2887002214","https://openalex.org/W2993383518","https://openalex.org/W3099289621","https://openalex.org/W3099837250","https://openalex.org/W3100370408","https://openalex.org/W3100917360","https://openalex.org/W3102073019","https://openalex.org/W3105841952","https://openalex.org/W3145915828","https://openalex.org/W4293776717","https://openalex.org/W4297825563","https://openalex.org/W4300972832"],"related_works":["https://openalex.org/W2045049461","https://openalex.org/W1978893398","https://openalex.org/W2201908702","https://openalex.org/W4381094582","https://openalex.org/W2369625323","https://openalex.org/W2364579609","https://openalex.org/W3141784573","https://openalex.org/W3122935400","https://openalex.org/W2598398909","https://openalex.org/W1967353875"],"abstract_inverted_index":{"In":[0],"a":[1,25,114],"system":[2],"of":[3,12,40,48,83,100,108,119],"three":[4,50],"stochastic":[5],"variables,":[6,166],"the":[7,17,43,46,49,59,77,97,101,109,137,154,160,164,168,177,180,184,194,202,205],"Partial":[8],"Information":[9],"Decomposition":[10],"(PID)":[11],"Williams":[13],"and":[14,37,55,67,131,167,174],"Beer":[15],"dissects":[16],"information":[18,31,122,149,179],"that":[19,33,62,124,132],"two":[20,53,151],"variables":[21,51],"(sources)":[22],"carry":[23,182],"about":[24,96,183],"third":[26],"variable":[27],"(target)":[28],"into":[29,52,150],"nonnegative":[30,121,152],"atoms":[32],"describe":[34,80,136],"redundant,":[35],"unique,":[36],"synergistic":[38,178],"modes":[39,61,82],"dependencies":[41,84],"among":[42,139],"variables.":[44,102],"However,":[45],"classification":[47],"sources":[54,181],"one":[56],"target":[57],"limits":[58],"dependency":[60],"can":[63],"be":[64],"quantitatively":[65],"resolved,":[66],"does":[68,172],"not":[69],"naturally":[70,146],"suit":[71],"all":[72,140,201],"systems.":[73],"Here,":[74],"we":[75,112],"extend":[76],"PID":[78,106,116,141],"to":[79,127,135,176,192],"trivariate":[81],"in":[85,204],"full":[86],"generality,":[87],"without":[88],"introducing":[89],"additional":[90],"decomposition":[91],"axioms":[92],"or":[93],"making":[94],"assumptions":[95],"target/source":[98,129],"nature":[99],"By":[103],"comparing":[104],"different":[105,128],"lattices":[107],"same":[110],"system,":[111],"unveil":[113],"finer":[115,144],"structure":[117,145,188],"made":[118],"seven":[120],"subatoms":[123],"are":[125,133],"invariant":[126,187],"classifications":[130],"sufficient":[134,191],"relationships":[138],"lattices.":[142],"This":[143],"splits":[147],"redundant":[148],"components:":[153],"source":[155,165],"redundancy,":[156,170],"which":[157,171],"arises":[158],"from":[159],"pairwise":[161],"correlations":[162],"between":[163],"non-source":[169],"not,":[173],"relates":[175],"target.":[185],"The":[186],"is":[189],"also":[190],"construct":[193],"system\u2019s":[195],"entropy,":[196],"hence":[197],"it":[198],"characterizes":[199],"completely":[200],"interdependencies":[203],"system.":[206]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":5},{"year":2023,"cited_by_count":19},{"year":2022,"cited_by_count":6},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":4},{"year":2018,"cited_by_count":7},{"year":2017,"cited_by_count":5}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
