{"id":"https://openalex.org/W2119221712","doi":"https://doi.org/10.1145/1007352.1007441","title":"Sharp thresholds For monotone properties in random geometric graphs","display_name":"Sharp thresholds For monotone properties in random geometric graphs","publication_year":2004,"publication_date":"2004-06-13","ids":{"openalex":"https://openalex.org/W2119221712","doi":"https://doi.org/10.1145/1007352.1007441","mag":"2119221712"},"language":"en","primary_location":{"id":"doi:10.1145/1007352.1007441","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1007352.1007441","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the thirty-sixth annual ACM symposium on Theory of computing","raw_type":"proceedings-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5101748190","display_name":"Ashish Goel","orcid":"https://orcid.org/0000-0003-0938-0401"},"institutions":[{"id":"https://openalex.org/I97018004","display_name":"Stanford University","ror":"https://ror.org/00f54p054","country_code":"US","type":"education","lineage":["https://openalex.org/I97018004"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Ashish Goel","raw_affiliation_strings":["Stanford University, Stanford, CA"],"affiliations":[{"raw_affiliation_string":"Stanford University, Stanford, CA","institution_ids":["https://openalex.org/I97018004"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5075717663","display_name":"Sanatan Rai","orcid":null},"institutions":[{"id":"https://openalex.org/I97018004","display_name":"Stanford University","ror":"https://ror.org/00f54p054","country_code":"US","type":"education","lineage":["https://openalex.org/I97018004"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Sanatan Rai","raw_affiliation_strings":["Stanford University, Stanford, CA"],"affiliations":[{"raw_affiliation_string":"Stanford University, Stanford, CA","institution_ids":["https://openalex.org/I97018004"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5063784062","display_name":"Bhaskar Krishnamachari","orcid":"https://orcid.org/0000-0002-9994-9931"},"institutions":[{"id":"https://openalex.org/I1174212","display_name":"University of Southern California","ror":"https://ror.org/03taz7m60","country_code":"US","type":"education","lineage":["https://openalex.org/I1174212"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Bhaskar Krishnamachari","raw_affiliation_strings":["University of Southern California, Los Angeles, CA","University of Southern California,,,Los Angeles,CA,"],"affiliations":[{"raw_affiliation_string":"University of Southern California, Los Angeles, CA","institution_ids":["https://openalex.org/I1174212"]},{"raw_affiliation_string":"University of Southern California,,,Los Angeles,CA,","institution_ids":["https://openalex.org/I1174212"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5101748190"],"corresponding_institution_ids":["https://openalex.org/I97018004"],"apc_list":null,"apc_paid":null,"fwci":7.4159,"has_fulltext":false,"cited_by_count":88,"citation_normalized_percentile":{"value":0.97336784,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":null,"issue":null,"first_page":"580","last_page":"586"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10246","display_name":"Mobile Ad Hoc Networks","score":0.9995999932289124,"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/T10246","display_name":"Mobile Ad Hoc Networks","score":0.9995999932289124,"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/T10796","display_name":"Cooperative Communication and Network Coding","score":0.9970999956130981,"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/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9937999844551086,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"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/combinatorics","display_name":"Combinatorics","score":0.7900278568267822},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6981134414672852},{"id":"https://openalex.org/keywords/random-graph","display_name":"Random graph","score":0.6412999629974365},{"id":"https://openalex.org/keywords/bernoullis-principle","display_name":"Bernoulli's principle","score":0.6135262250900269},{"id":"https://openalex.org/keywords/random-regular-graph","display_name":"Random regular graph","score":0.6036188006401062},{"id":"https://openalex.org/keywords/random-geometric-graph","display_name":"Random geometric graph","score":0.5835033059120178},{"id":"https://openalex.org/keywords/monotone-polygon","display_name":"Monotone polygon","score":0.5386064052581787},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5036610960960388},{"id":"https://openalex.org/keywords/geometric-graph-theory","display_name":"Geometric graph theory","score":0.4990360736846924},{"id":"https://openalex.org/keywords/chordal-graph","display_name":"Chordal graph","score":0.4940793812274933},{"id":"https://openalex.org/keywords/indifference-graph","display_name":"Indifference graph","score":0.48069077730178833},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.47906550765037537},{"id":"https://openalex.org/keywords/bounding-overwatch","display_name":"Bounding overwatch","score":0.4546625018119812},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.30853012204170227},{"id":"https://openalex.org/keywords/1-planar-graph","display_name":"1-planar graph","score":0.2774791121482849},{"id":"https://openalex.org/keywords/line-graph","display_name":"Line graph","score":0.16496261954307556},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.13901367783546448},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.13740545511245728},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.09384289383888245}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7900278568267822},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6981134414672852},{"id":"https://openalex.org/C47458327","wikidata":"https://www.wikidata.org/wiki/Q910404","display_name":"Random graph","level":3,"score":0.6412999629974365},{"id":"https://openalex.org/C152361515","wikidata":"https://www.wikidata.org/wiki/Q181328","display_name":"Bernoulli's principle","level":2,"score":0.6135262250900269},{"id":"https://openalex.org/C4255713","wikidata":"https://www.wikidata.org/wiki/Q7291988","display_name":"Random regular graph","level":5,"score":0.6036188006401062},{"id":"https://openalex.org/C30609935","wikidata":"https://www.wikidata.org/wiki/Q7291969","display_name":"Random geometric graph","level":5,"score":0.5835033059120178},{"id":"https://openalex.org/C2834757","wikidata":"https://www.wikidata.org/wiki/Q4925424","display_name":"Monotone polygon","level":2,"score":0.5386064052581787},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5036610960960388},{"id":"https://openalex.org/C173547177","wikidata":"https://www.wikidata.org/wiki/Q1503271","display_name":"Geometric graph theory","level":5,"score":0.4990360736846924},{"id":"https://openalex.org/C160446614","wikidata":"https://www.wikidata.org/wiki/Q1322892","display_name":"Chordal graph","level":3,"score":0.4940793812274933},{"id":"https://openalex.org/C74133993","wikidata":"https://www.wikidata.org/wiki/Q3115472","display_name":"Indifference graph","level":3,"score":0.48069077730178833},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.47906550765037537},{"id":"https://openalex.org/C63584917","wikidata":"https://www.wikidata.org/wiki/Q333286","display_name":"Bounding overwatch","level":2,"score":0.4546625018119812},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.30853012204170227},{"id":"https://openalex.org/C102192266","wikidata":"https://www.wikidata.org/wiki/Q4545823","display_name":"1-planar graph","level":4,"score":0.2774791121482849},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.16496261954307556},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.13901367783546448},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.13740545511245728},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.09384289383888245},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C146978453","wikidata":"https://www.wikidata.org/wiki/Q3798668","display_name":"Aerospace engineering","level":1,"score":0.0},{"id":"https://openalex.org/C149530733","wikidata":"https://www.wikidata.org/wiki/Q5597091","display_name":"Graph power","level":4,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C22149727","wikidata":"https://www.wikidata.org/wiki/Q7940747","display_name":"Voltage graph","level":4,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1145/1007352.1007441","is_oa":false,"landing_page_url":"https://doi.org/10.1145/1007352.1007441","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the thirty-sixth annual ACM symposium on Theory of computing","raw_type":"proceedings-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":24,"referenced_works":["https://openalex.org/W390146837","https://openalex.org/W581504706","https://openalex.org/W630423399","https://openalex.org/W1520007244","https://openalex.org/W1529556207","https://openalex.org/W1595613081","https://openalex.org/W1976464463","https://openalex.org/W1976755703","https://openalex.org/W1992260886","https://openalex.org/W1996807175","https://openalex.org/W2007647264","https://openalex.org/W2042337206","https://openalex.org/W2042668891","https://openalex.org/W2084234679","https://openalex.org/W2094776848","https://openalex.org/W2115718483","https://openalex.org/W2128738155","https://openalex.org/W2137775453","https://openalex.org/W2169293240","https://openalex.org/W2610893972","https://openalex.org/W2797135730","https://openalex.org/W2911730943","https://openalex.org/W4241669766","https://openalex.org/W6793127252"],"related_works":["https://openalex.org/W4287117052","https://openalex.org/W2157783000","https://openalex.org/W3176308333","https://openalex.org/W4206705200","https://openalex.org/W2966902162","https://openalex.org/W2028363955","https://openalex.org/W1893557706","https://openalex.org/W2952738468","https://openalex.org/W2963022083","https://openalex.org/W2037549896"],"abstract_inverted_index":{"Random":[0],"geometric":[1,102,112,128],"graphs":[2,40,103],"result":[3],"from":[4],"taking":[5],"n":[6],"uniformly":[7,60],"distributed":[8,59],"points":[9,18,58],"in":[10,61],"the":[11,46,50,68,94],"unit":[12],"cube,":[13],"[0,1]d,":[14],"and":[15,71,82],"connecting":[16],"two":[17,54],"if":[19],"their":[20],"Euclidean":[21],"distance":[22],"is":[23,76,97,114,137],"at":[24,83],"most":[25,84],"r,":[26],"for":[27,36,78,89,100,105,143],"some":[28],"prescribed":[29],"r.":[30],"We":[31,63],"show":[32,72],"that":[33,73],"monotone":[34],"properties":[35],"this":[37,135],"class":[38],"of":[39,56,123],"have":[41,140],"sharp":[42,77],"thresholds":[43],"by":[44],"reducing":[45],"problem":[47],"to":[48,116,139],"bounding":[49],"bottleneck":[51],"matching":[52],"on":[53,67],"sets":[55],"$n$":[57],"[0,1]d.":[62],"present":[64],"upper":[65],"bounds":[66],"threshold":[69,95],"width,":[70],"our":[74],"bound":[75],"d":[79,90],"=":[80],"1":[81],"a":[85,110,118,131],"sublogarithmic":[86],"factor":[87],"away":[88],"\u2265":[91],"2.":[92],"Interestingly,":[93],"width":[96],"much":[98],"sharper":[99],"random":[101,107,111,127,145],"than":[104],"Bernoulli":[106,144],"graphs.":[108,146],"Further,":[109],"graph":[113,129],"shown":[115,138],"be":[117],"subgraph,":[119],"with":[120,130],"high":[121],"probability,":[122],"another":[124],"independently":[125],"drawn":[126],"slightly":[132],"larger":[133],"radius;":[134],"property":[136],"no":[141],"analogue":[142]},"counts_by_year":[{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":5},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":5},{"year":2012,"cited_by_count":2}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
