{"id":"https://openalex.org/W2749661016","doi":"https://doi.org/10.3390/e19080417","title":"Power-Law Distributions from Sigma-Pi Structure of Sums of Random Multiplicative Processes","display_name":"Power-Law Distributions from Sigma-Pi Structure of Sums of Random Multiplicative Processes","publication_year":2017,"publication_date":"2017-08-17","ids":{"openalex":"https://openalex.org/W2749661016","doi":"https://doi.org/10.3390/e19080417","mag":"2749661016"},"language":"en","primary_location":{"id":"doi:10.3390/e19080417","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e19080417","pdf_url":"https://www.mdpi.com/1099-4300/19/8/417/pdf?version=1503277692","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":["crossref","datacite","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/19/8/417/pdf?version=1503277692","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5008983572","display_name":"Arthur Matsuo Yamashita Rios de Sousa","orcid":"https://orcid.org/0000-0003-3813-1295"},"institutions":[{"id":"https://openalex.org/I114531698","display_name":"Tokyo Institute of Technology","ror":"https://ror.org/0112mx960","country_code":"JP","type":"education","lineage":["https://openalex.org/I114531698"]}],"countries":["JP"],"is_corresponding":true,"raw_author_name":"Arthur Sousa","raw_affiliation_strings":["Department of Mathematical and Computing Science, School of Computing, Tokyo Institute of Technology, Yokohama 226-8502, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical and Computing Science, School of Computing, Tokyo Institute of Technology, Yokohama 226-8502, Japan","institution_ids":["https://openalex.org/I114531698"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5002007070","display_name":"Hideki Takayasu","orcid":"https://orcid.org/0000-0002-4452-5517"},"institutions":[{"id":"https://openalex.org/I114531698","display_name":"Tokyo Institute of Technology","ror":"https://ror.org/0112mx960","country_code":"JP","type":"education","lineage":["https://openalex.org/I114531698"]},{"id":"https://openalex.org/I4210122684","display_name":"Sony Computer Science Laboratories","ror":"https://ror.org/02nc46417","country_code":"JP","type":"facility","lineage":["https://openalex.org/I4210122684"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Hideki Takayasu","raw_affiliation_strings":["Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 226-8502, Japan","Sony Computer Science Laboratories, Tokyo 141-0022, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 226-8502, Japan","institution_ids":["https://openalex.org/I114531698"]},{"raw_affiliation_string":"Sony Computer Science Laboratories, Tokyo 141-0022, Japan","institution_ids":["https://openalex.org/I4210122684"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5057002364","display_name":"Didier Sornette","orcid":"https://orcid.org/0000-0001-7874-7962"},"institutions":[{"id":"https://openalex.org/I114457229","display_name":"University of Geneva","ror":"https://ror.org/01swzsf04","country_code":"CH","type":"education","lineage":["https://openalex.org/I114457229"]},{"id":"https://openalex.org/I114531698","display_name":"Tokyo Institute of Technology","ror":"https://ror.org/0112mx960","country_code":"JP","type":"education","lineage":["https://openalex.org/I114531698"]},{"id":"https://openalex.org/I35440088","display_name":"ETH Zurich","ror":"https://ror.org/05a28rw58","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I35440088"]},{"id":"https://openalex.org/I54157616","display_name":"Swiss Finance Institute","ror":"https://ror.org/02cgn2a44","country_code":"CH","type":"education","lineage":["https://openalex.org/I54157616"]}],"countries":["CH","JP"],"is_corresponding":false,"raw_author_name":"Didier Sornette","raw_affiliation_strings":["Department of Management, Technology and Economics, ETH Z\u00fcrich, Z\u00fcrich 8092, Switzerland","Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 226-8502, Japan","Swiss Finance Institute, University of Geneva, Geneva 1211, Switzerland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Management, Technology and Economics, ETH Z\u00fcrich, Z\u00fcrich 8092, Switzerland","institution_ids":["https://openalex.org/I35440088"]},{"raw_affiliation_string":"Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 226-8502, Japan","institution_ids":["https://openalex.org/I114531698"]},{"raw_affiliation_string":"Swiss Finance Institute, University of Geneva, Geneva 1211, Switzerland","institution_ids":["https://openalex.org/I114457229","https://openalex.org/I54157616"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5079617036","display_name":"Misako Takayasu","orcid":"https://orcid.org/0000-0002-2467-614X"},"institutions":[{"id":"https://openalex.org/I114531698","display_name":"Tokyo Institute of Technology","ror":"https://ror.org/0112mx960","country_code":"JP","type":"education","lineage":["https://openalex.org/I114531698"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Misako Takayasu","raw_affiliation_strings":["Department of Mathematical and Computing Science, School of Computing, Tokyo Institute of Technology, Yokohama 226-8502, Japan","Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 226-8502, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical and Computing Science, School of Computing, Tokyo Institute of Technology, Yokohama 226-8502, Japan","institution_ids":["https://openalex.org/I114531698"]},{"raw_affiliation_string":"Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 226-8502, Japan","institution_ids":["https://openalex.org/I114531698"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5008983572"],"corresponding_institution_ids":["https://openalex.org/I114531698"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2227},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2227},"fwci":0.1499,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.4621593,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":95},"biblio":{"volume":"19","issue":"8","first_page":"417","last_page":"417"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.9988999962806702,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12261","display_name":"Statistical Mechanics and Entropy","score":0.9988999962806702,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11270","display_name":"Complex Systems and Time Series Analysis","score":0.9983000159263611,"subfield":{"id":"https://openalex.org/subfields/2002","display_name":"Economics and Econometrics"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T10591","display_name":"Theoretical and Computational Physics","score":0.9934999942779541,"subfield":{"id":"https://openalex.org/subfields/3104","display_name":"Condensed Matter Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/multiplicative-function","display_name":"Multiplicative function","score":0.8663437366485596},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6785774230957031},{"id":"https://openalex.org/keywords/independence","display_name":"Independence (probability theory)","score":0.6695157289505005},{"id":"https://openalex.org/keywords/sigma","display_name":"Sigma","score":0.6578102111816406},{"id":"https://openalex.org/keywords/exponent","display_name":"Exponent","score":0.6485915184020996},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.6020462512969971},{"id":"https://openalex.org/keywords/power-law","display_name":"Power law","score":0.5682337880134583},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.4144824147224426},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.35957232117652893},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.26348447799682617},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.25595688819885254},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.2033182978630066},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.10691246390342712}],"concepts":[{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.8663437366485596},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6785774230957031},{"id":"https://openalex.org/C35651441","wikidata":"https://www.wikidata.org/wiki/Q625303","display_name":"Independence (probability theory)","level":2,"score":0.6695157289505005},{"id":"https://openalex.org/C2778049214","wikidata":"https://www.wikidata.org/wiki/Q7512234","display_name":"Sigma","level":2,"score":0.6578102111816406},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.6485915184020996},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.6020462512969971},{"id":"https://openalex.org/C87040749","wikidata":"https://www.wikidata.org/wiki/Q428971","display_name":"Power law","level":2,"score":0.5682337880134583},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.4144824147224426},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.35957232117652893},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.26348447799682617},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.25595688819885254},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.2033182978630066},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.10691246390342712},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.3390/e19080417","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e19080417","pdf_url":"https://www.mdpi.com/1099-4300/19/8/417/pdf?version=1503277692","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:doaj.org/article:df30e2472ecb4b85bbf96fe53afdbcc6","is_oa":true,"landing_page_url":"https://doaj.org/article/df30e2472ecb4b85bbf96fe53afdbcc6","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 8, p 417 (2017)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/1099-4300/19/8/417/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/e19080417","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy; Volume 19; Issue 8; Pages: 417","raw_type":"Text"},{"id":"pmh:oai:www.research-collection.ethz.ch:20.500.11850/214303","is_oa":true,"landing_page_url":"http://hdl.handle.net/20.500.11850/214303","pdf_url":null,"source":{"id":"https://openalex.org/S4306402302","display_name":"Repository for Publications and Research Data (ETH Zurich)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I35440088","host_organization_name":"ETH Zurich","host_organization_lineage":["https://openalex.org/I35440088"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy, 19 (8)","raw_type":"info:eu-repo/semantics/article"},{"id":"doi:10.3929/ethz-b-000191179","is_oa":true,"landing_page_url":"https://doi.org/10.3929/ethz-b-000191179","pdf_url":null,"source":{"id":"https://openalex.org/S7407051236","display_name":"ETH Z\u00fcrich Research Collection","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","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"doi:10.3390/e19080417","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e19080417","pdf_url":"https://www.mdpi.com/1099-4300/19/8/417/pdf?version=1503277692","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":[],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2749661016.pdf"},"referenced_works_count":47,"referenced_works":["https://openalex.org/W1643412971","https://openalex.org/W1665885516","https://openalex.org/W1872698747","https://openalex.org/W1972501654","https://openalex.org/W1973149840","https://openalex.org/W1983874169","https://openalex.org/W1993803315","https://openalex.org/W1995511076","https://openalex.org/W2000042664","https://openalex.org/W2000356308","https://openalex.org/W2005924742","https://openalex.org/W2017891730","https://openalex.org/W2019364974","https://openalex.org/W2020959454","https://openalex.org/W2036671379","https://openalex.org/W2055951984","https://openalex.org/W2060830753","https://openalex.org/W2075286485","https://openalex.org/W2075514521","https://openalex.org/W2084862934","https://openalex.org/W2095402301","https://openalex.org/W2110617809","https://openalex.org/W2113924996","https://openalex.org/W2121821841","https://openalex.org/W2135869163","https://openalex.org/W2144038585","https://openalex.org/W2147569252","https://openalex.org/W2149614433","https://openalex.org/W2161613033","https://openalex.org/W2478227765","https://openalex.org/W2490074880","https://openalex.org/W2801982793","https://openalex.org/W2950627632","https://openalex.org/W3022050436","https://openalex.org/W3102497017","https://openalex.org/W3103362336","https://openalex.org/W3122938238","https://openalex.org/W3125736847","https://openalex.org/W4212853637","https://openalex.org/W4231687540","https://openalex.org/W4292334561","https://openalex.org/W4388331996","https://openalex.org/W6636106449","https://openalex.org/W6654704927","https://openalex.org/W6674428577","https://openalex.org/W6684996598","https://openalex.org/W6948047203"],"related_works":["https://openalex.org/W1999290596","https://openalex.org/W2742063177","https://openalex.org/W1994753018","https://openalex.org/W2386091227","https://openalex.org/W2008499310","https://openalex.org/W2026675405","https://openalex.org/W4378214560","https://openalex.org/W1978408548","https://openalex.org/W2017291714","https://openalex.org/W3010644611"],"abstract_inverted_index":{"We":[0,54,92],"introduce":[1],"a":[2,77,88,157],"simple":[3],"growth":[4,41,134],"model":[5],"in":[6],"which":[7,72],"the":[8,28,44,56,60,63,66,70,74,96,102,108,111,115,118,122,130,133,138,143,149,162,167,174],"sizes":[9,61],"of":[10,38,46,59,62,69,76,79,110,117,121,152,169],"entities":[11,64],"evolve":[12],"as":[13,65],"multiplicative":[14],"random":[15],"processes":[16],"that":[17,95,148],"start":[18],"at":[19],"different":[20],"times.":[21],"A":[22],"novel":[23],"aspect":[24],"we":[25,34,146],"examine":[26],"is":[27,125,166],"dependence":[29,39,50,97,131],"among":[30],"entities.":[31],"For":[32],"this,":[33],"consider":[35],"three":[36],"classes":[37],"between":[40,132],"factors":[42,135],"governing":[43],"evolution":[45],"sizes:":[47],"independence,":[48],"Kesten":[49],"and":[51,185],"mixed":[52],"dependence.":[53],"take":[55],"sum":[57,78,123],"X":[58,124,153],"representative":[67],"quantity":[68],"system,":[71],"has":[73,87],"structure":[75,116],"product":[80],"terms":[81],"(Sigma-Pi),":[82],"whose":[83],"asymptotic":[84,103,163],"distribution":[85],"function":[86],"power-law":[89,104,164],"tail":[90,105,112,165],"behavior.":[91],"present":[93],"evidence":[94],"type":[98],"does":[99],"not":[100],"alter":[101],"behavior,":[106],"nor":[107],"value":[109],"exponent.":[113],"However,":[114],"large":[119,150],"values":[120,151],"found":[126],"to":[127,173],"vary":[128],"with":[129,176,183,186],"(and":[136],"thus":[137],"entities).":[139],"In":[140],"particular,":[141],"for":[142],"independence":[144],"case,":[145],"find":[147],"are":[154],"contributed":[155],"by":[156],"single":[158,171],"maximum":[159,178],"size":[160],"entity:":[161],"result":[168],"such":[170],"contribution":[172],"sum,":[175],"this":[177],"contributing":[179],"entity":[180],"changing":[181],"stochastically":[182],"time":[184],"realizations.":[187]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2018,"cited_by_count":1}],"updated_date":"2026-07-30T17:31:21.811387","created_date":"2025-10-10T00:00:00"}
