美国数学建模论文.docx
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美国数学建模论文
ModelingforCrimeBusting
Abstract:
Inthemathematicalmodeling,wediscussedhowtosolvetheproblemofcrimebusting.First,weestablishedamathematicalmodeltosimulatethenetworkofthestuffinthecompany,bycalculatingtheweightofeachedgetocalculatetheprobabilityofeachnode.Thismodelwithastrongabilitytoadapt,canbeeasilyappliedtoothernetworkstructure.Thenwewillapplythesemanticanalysisandtextanalysisinourmodeltomakethemodelmoreaccurate.Finally,weapplythemodeltothecellnetworkrelationsofinfection
Aimingatthequestionone,weconsiderthatthenetworkofthecompanyissingle-lineandthepropertyofeachedgeisdifferent.So,weclassifyalltheinformationflowaccordingtothetypeofthemessageandthetypeofthetwonodesineachedge.Then,wecalculatetheweightofeachtypeofinformationflow.Andthen,wegettheprobabilityofeachnode,thatistheprobabilityofthestuff.
Aimingatthequestiontwoitisthesameasthemodelinquestionone.Therearetwoconditionschangedinthequestiontwo.Homologous,wechangetheconditionsinquestiononewiththesamemodel,thenwegettheprobabilityofthestuffinquestiontwo.
Aimingatthequestionthree,Weneedtousesemanticanalysisandtextanalysistoamendthemodelestablished.Fortextanalysis,weadoptthemethodofTF-IDF.Wecalculatethefrequencyofallthewordsinthetopics,thenwegetthekeywordabouttheconspiracy.Laterweclassifythetopicsmoremeticulouslytomaketheweightofeachedgeaccurate.
Aimingatthequestionfour,weneedtoapplythemodelintoabroadernetworktype.Inthisproblem,weassumethatasimplecellinfectionofnetworkstructure.Thenwecalculatetheprobabilityofcellinfectionbyusingthemodelandtheresultsprovethatthemodelcanbeusedinothernetworkstructure.
Keywords:
CrimeNetworkAHPTF-IDF
Content
1Introduction
1.1BackgroundandAnalysisoftheProblems
1.2MethodoftheAnalysis
1.3Assumption
2ModelApproach
2.1Howtojudgeone’ssuspicion
2.2TheCrimeNetworkCircle
2.3SetWordNetworkAnalysisModel
2.3.1DefinesomeParameters
2.3.2TheCNCnodepositionsuspiciousdegreemodel
2.4TheWeightVector--AnAHPSolution
2.4.1SetupaStructureModel
2.4.2FigureoutJudgingMatrixandEigenvector
2.5Conclusion
2.5.1TheFinalSuspicionOrderof83Nodes
3ContextAnalysis
3.1BasicAssumptions
3.2TF-IDFModel
3.3Conclusion
4.Themodelincellinfectionnetwork
5.SensitivityAnalysisandImprovements
5.1SensitivityAnalysis
5.2Improvements
6.ModelEvaluation
6.1Strengths
6.2Weaknesses
7.References
1.Introduction
1.1BackgroundandAnalysisoftheProblems
Aconspiracynetworkisembeddedinanetworkofemployeesofacompany,withedgerepresentingamessagesentfromoneemployee（node）toanotherandcategorizedbytopics.Ourorganization,ICM,hasalreadyknownsomeinformation.Thoseinvestigatorsthinkthatinformationwillhelpthemtofindoutthemost-possiblepeopleselectedoftheambiguousconspiratorsandunknownleader.Thegoalofmoldingistofindoutthemost-possibleconspiratorsintheoffice.
●Accordingtoinformation,7peoplehavealreadybeenconfirmedtobeconspirators,15topics（3ofwhichhavebeendeemedtobesuspicious）,and400messagelinks,ourgoalistomakesurewhoareconspiracy,andwhoaretheleader,prioritizethe83nodesbylikelihoodofbeingapartoftheconspiracyanddetermineadiscriminatelineseparatingconspiratorsfromnon-conspirators
●Ifsomealreadyknowninformationchanged,wehavemoreinformation,whatchangeswillhappenwiththeresult
●Whendealingwithmoreinformationormorecomplicatedcircumstance,themodelmustcanbeappliedtoanycondition
1.2MethodoftheAnalysis
Asinvestigators,wehaveknownabout83nodes,400linksover21000wordsofmessagetraffic,15topics（3havebeendeemedtobesuspicious）,7knownconspiratorsand8knownnon-conspirators.Tomakesurewhoaretheconspirators,therearetoomanyfactorstotakeintoconsideration,soweformulateamodeltoaccountfortheimportanceofeveryfactors,andthesefactorswouldaffectthedeterminationofwhoareconspirators.
1.3Assumptions
●Conspiratorstendnottotalkfrequentlywitheachotheraboutirrelevanttopics.
●Theleaderoftheconspiracytriestominimizeriskbyrestrictingdirectcontacts.
●Peoplecanfreelytalktoeachother,andwithoutthelimitofdistance.
●Non-conspiratorsdonothavetheideathatthereareconspiratorsintheircompany,andconspiratorswillnotaddmorepeople.
2.ModelApproach
2.1Howtojudgeone’ssuspicion
Asaconspirator,heorshemusthavesomewordsoractionmakepeoplefeelstrangeandsuspicious.Asagroupofconspirators,theyconnecteachotherthroughwords,soifsomeonereferstomuchinformationaboutsuspicioustopics,thepossibilitythatheisaconspiratorishigh.Inadditiontothiscondition,talkingtoomuchwithconspiratorsisalsomakingsomeonemoresuspicious.Inall,weconcludesomepointsbelowtodiscriminatewhoaremorelikelytobeconspirator:
●Thefrequencyofsendingandreceivingsuspicioustopics
●Thefrequencyofcontactingwithconspirators
●Thefrequencyoftalkingwiththesamepeopleandaboutthesametopic（becausebetweenconspirators,theymaybecontactwithsecretwords）
2.2TheCrimeNetworkCircle
Wecreateamodelcalledthecrimenetworkcircle（CNC）,themodeliscreatedonthebaseofanalysisaboutnetwork,theCNCmodelhasthreeareas:
A（theoutsidering）,B（themiddlering）andC（theinsidering）,Aistheareathatrepresentspeoplewhoareinnocent,Bistheareathatrepresentspeoplewhoaresuspects,andCistheareathatrepresentspeoplewhoareconspirators.
Tobefrank,wedrawa10-peoplesocialnetworkCNCpicture.
2.3SetWordNetworkAnalysisModel
2.3.1DefinesomeParameters
IntheCNCmodel,weuseadottorepresentaperson,wedefineeverydotisanode.Thewordsbetweentwopeopleisalineinthepicture,wedefinethelineisadegree,onepersontalkstoanotherone,thedegreeisanout-degree;likely,ifsomeonereceiveamessagefromothers,thedegreeisanin-degree.What’smore,anotherimportantfactoristhatwhetherthetopicisasuspicioustopic.
Soadegreecanberepresentedintheformofamatrix（x,y,z）.Themeaningofxyandzisinthetablebelow:
Table2
x
Theidentityofspeaker,andthepositionofnodeintheCNCpicture,x=a,b,c
y
Theidentityoflistener,andthepositionofnodeintheCANpicture,y=a,b,c
z
Thetopic.Ifthetopicisasuspiciousone,z=1;otherwise,z=0
Accordingtowhatwehavedefined,wecangivetwoweightvectormatrix:
and
.
isthecasethattopicsarenotsuspicious,and
isthecasethattopicsaresuspicious.
2.3.2TheCNCnodepositionsuspiciousdegreemodel
Accordingtotheassumptions,ateveryjunctionwehave
Where
istheweightvectorofout-degree,
istheweightvectorofin-degree.Wedefine
=0.7,
=0.3.
istheout-degreeofnode
fromareaxtoareayinCNCpicture;familiarly,
meansthein-degreeofnode
fromareaytoareaxinCNCpicture.
Intheequation,
representsproductoftheout-degreeofdoteiandtheweightvectorofout-degreearea,itreflectsthedegreeofcontributionthatnode
’sout-degreegivesuspecttonode
.Atthesameway,
representsproductofthein-degreeofnode
andtheweightvectorofin-degreearea,itreflectsthedegreeofcontributionthatnode
’sin-degreegivesuspecttonode
.Thesetwoindexescomprehensivelyconsiderthesuspectdegreebyin-degreeandout-degreeofnode
whichmakesthemodelmorereasonable.
Ifwecalculatesomeone’s
isveryhigh,thenthispersonhasahigherpossibilitytobeaconspirator.
2.4TheWeightVector--AnAHPSolution
2.4.1SetupaStructureModel
WeuseAHPmainlybecausethisquestiondoesnothaveclearandspecificdatathatwecan’tdosomequantificationanalysis,inordertohavingmoreaccurateanswer,weconvertquantificationjudgmentintocomparingtheimportanceofeveryfactor,avoidinghavingsubjectiveandwrongjudgments.
Thefirststageistosetupastructuremodel.Themodelarebelow:
Structure3
Thejudgmentofaconspirator
A
Targetlayer
Case18
C18
Case3
C3
Case2
C2
Case1
C1
Criterion
Layer….
TheSuspicionofEachIndividual
P
Programlayer
Todemonstratebetterthedifferentcases,wepresentinTable4below:
Table4
Speaker
Listener
Topics
Innocent
Innocent
Suspicious
Innocent
Innocent
NotSuspicious
Innocent
Suspect
Suspicious
Innocent
Suspect
NotSuspicious
Innocent
Conspirator
Suspicious
Innocent
Conspirator
NotSuspicious
Suspect
Innocent
Suspicious
Suspect
Innocent
NotSuspicious
Suspect
Suspect
Suspicious
Suspect
Suspect
NotSuspicious
Suspect
Conspirator
Suspicious
Suspect
Conspirator
NotSuspicious
Conspirator
Innocent
Suspicious
Conspirator
Innocent
NotSuspicious
Conspirator
Suspect
Suspicious
Conspirator
Suspect
NotSuspicious
Conspirator
Conspirator
Suspicious
Conspirator
Conspirator
NotSuspicious
2.4.2FigureoutJudgingMatrixandEigenvector
TogivejudgingmatrixA-C,wefirstdefinethemeaningofthevalueofA
aboutthisquestion:
A
=1:
A
isasimportantasA
（suspicious）;
A
=3:
A
isalittleimportantthanA
;
A
=5:
A
ismuchimportantthanA
;
A
=7:
A
isextremelyimportantthanA
;
A
=2,4,6,8:
theimportanceofA
andA
isbetweenadjacenttwonumberabove.
Weconsiderthatpeoplerefermoresuspicioustopicsandtalkfrequentlywithc
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