{"id":"https://openalex.org/W2403365012","doi":"https://doi.org/10.4086/toc.2014.v010a016","title":"","display_name":"","publication_year":2014,"publication_date":"2014-01-01","ids":{"openalex":"https://openalex.org/W2403365012","doi":"https://doi.org/10.4086/toc.2014.v010a016","mag":"2403365012"},"language":"en","primary_location":{"id":"doi:10.4086/toc.2014.v010a016","is_oa":true,"landing_page_url":"https://doi.org/10.4086/toc.2014.v010a016","pdf_url":"http://www.theoryofcomputing.org/articles/v010a016/v010a016.pdf","source":{"id":"https://openalex.org/S142665794","display_name":"Theory of Computing","issn_l":"1557-2862","issn":["1557-2862"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310315690","host_organization_name":"University of Chicago","host_organization_lineage":["https://openalex.org/P4310315690"],"host_organization_lineage_names":["University of Chicago"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Theory of Computing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"http://www.theoryofcomputing.org/articles/v010a016/v010a016.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5044565966","display_name":"Bernard Chazelle","orcid":"https://orcid.org/0000-0001-8542-0247"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Bernard Chazelle","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5044565966"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.30538655,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"10","issue":"1","first_page":"421","last_page":"451"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10249","display_name":"Distributed Control Multi-Agent Systems","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/T10249","display_name":"Distributed Control Multi-Agent Systems","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/T11252","display_name":"Evolutionary Game Theory and Cooperation","score":0.979200005531311,"subfield":{"id":"https://openalex.org/subfields/3312","display_name":"Sociology and Political Science"},"field":{"id":"https://openalex.org/fields/33","display_name":"Social Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T12784","display_name":"Modular Robots and Swarm Intelligence","score":0.972599983215332,"subfield":{"id":"https://openalex.org/subfields/2210","display_name":"Mechanical Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/logarithm","display_name":"Logarithm","score":0.6500113606452942},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.5959369540214539},{"id":"https://openalex.org/keywords/flock","display_name":"Flock","score":0.5771497488021851},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5764774680137634},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4919850528240204},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.40370506048202515},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.34981703758239746}],"concepts":[{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.6500113606452942},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5959369540214539},{"id":"https://openalex.org/C134215735","wikidata":"https://www.wikidata.org/wiki/Q120997","display_name":"Flock","level":2,"score":0.5771497488021851},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5764774680137634},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4919850528240204},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.40370506048202515},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.34981703758239746},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.4086/toc.2014.v010a016","is_oa":true,"landing_page_url":"https://doi.org/10.4086/toc.2014.v010a016","pdf_url":"http://www.theoryofcomputing.org/articles/v010a016/v010a016.pdf","source":{"id":"https://openalex.org/S142665794","display_name":"Theory of Computing","issn_l":"1557-2862","issn":["1557-2862"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310315690","host_organization_name":"University of Chicago","host_organization_lineage":["https://openalex.org/P4310315690"],"host_organization_lineage_names":["University of Chicago"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Theory of Computing","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.4086/toc.2014.v010a016","is_oa":true,"landing_page_url":"https://doi.org/10.4086/toc.2014.v010a016","pdf_url":"http://www.theoryofcomputing.org/articles/v010a016/v010a016.pdf","source":{"id":"https://openalex.org/S142665794","display_name":"Theory of Computing","issn_l":"1557-2862","issn":["1557-2862"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310315690","host_organization_name":"University of Chicago","host_organization_lineage":["https://openalex.org/P4310315690"],"host_organization_lineage_names":["University of Chicago"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Theory of Computing","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1844007987","display_name":"Data-Powered Algorithms","funder_award_id":"0634958","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G4117300866","display_name":null,"funder_award_id":"832797","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G4289525447","display_name":null,"funder_award_id":"CCF-0832797","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2403365012.pdf","grobid_xml":"https://content.openalex.org/works/W2403365012.grobid-xml"},"referenced_works_count":16,"referenced_works":["https://openalex.org/W1615818243","https://openalex.org/W1963737727","https://openalex.org/W1964126948","https://openalex.org/W2015410655","https://openalex.org/W2044247492","https://openalex.org/W2070842753","https://openalex.org/W2103714988","https://openalex.org/W2105850748","https://openalex.org/W2125668987","https://openalex.org/W2128928412","https://openalex.org/W2144579655","https://openalex.org/W2150312211","https://openalex.org/W2165744313","https://openalex.org/W2796575992","https://openalex.org/W4226347946","https://openalex.org/W4234406933"],"related_works":["https://openalex.org/W1964655178","https://openalex.org/W2032854653","https://openalex.org/W3027407481","https://openalex.org/W1976296680","https://openalex.org/W2158769936","https://openalex.org/W2889385621","https://openalex.org/W2327661222","https://openalex.org/W1971588791","https://openalex.org/W1970514258","https://openalex.org/W1993390207"],"abstract_inverted_index":{"We":[0,63,76],"derive":[1],"a":[2,11,19,35,46,53,85,91],"tight":[3,71],"bound":[4,68],"on":[5,94],"the":[6,42,59,73,81,95,104],"time":[7],"it":[8],"takes":[9],"for":[10],"flock":[12,83],"of":[13,31,48,55,61,97,106],"birds":[14],"to":[15],"reach":[16],"equilibrium":[17],"in":[18,58,72,109],"standard":[20],"model.":[21],"Birds":[22],"navigate":[23],"by":[24,79],"constantly":[25],"averaging":[26],"their":[27,32],"velocities":[28],"with":[29],"those":[30],"neighbors":[33],"within":[34],"fixed":[36],"distance.":[37],"It":[38],"is":[39,69],"known":[40],"that":[41,65],"system":[43],"converges":[44],"after":[45],"number":[47,60],"steps":[49],"no":[50],"greater":[51],"than":[52],"tower-of-twos":[54],"height":[56],"logarithmic":[57],"birds.":[62],"show":[64],"this":[66],"astronomical":[67],"actually":[70],"worst":[74],"case.":[75],"do":[77],"so":[78],"viewing":[80],"bird":[82],"as":[84],"distributed":[86],"computing":[87],"device":[88],"and":[89],"deriving":[90],"sharp":[92],"estimate":[93],"growth":[96],"its":[98],"busy-beaver":[99],"function.":[100],"The":[101],"proof":[102],"highlights":[103],"use":[105],"spectral":[107],"techniques":[108],"natural":[110],"algorithms.":[111]},"counts_by_year":[{"year":2019,"cited_by_count":1}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2016-06-24T00:00:00"}