数学建模论文汇总.docx

数学建模论文汇总.docx

《数学建模论文汇总.docx》由会员分享，可在线阅读，更多相关《数学建模论文汇总.docx（26页珍藏版）》请在冰豆网上搜索。

数学建模论文汇总

Contents

1.Introduction

2.RestatementoftheProblem

3.Convention

3.1Terminology

3.2Variables

3.3Assumptions

4.TheModel

4.1theco-authornetworkoftheErdos1authors

4.1.1Completeco-authornetwork

4.1.2Streamlinedco-authornetwork

4.2NMIMModelToFindtheMostInfluentialAuthor

4.2.1WhyisNMIM（Network’smulti-attributeindicatormethod）

4.2.2DefinitionofMeasureIndicators

4.2.3ModelEstablishment

4.2.4SummaryoftheNMIM

4.3TheJournalCitationNetworksModel

4.3.1Modelestablishment

4.3.2Modelsolutionandanalysis

4.3.3Theinfluencemeasureofanindividualnetworkresearcher

4.3.4Theinfluencemeasureofajournal

5.ApplicationoftheModel:

theInfluentialActorsinFeng’s

6.FurtherDiscussion

6.1DiscussionabouttheScience,UnderstandingandUtility

6.2FurtherDiscussion:

HelpSocietyinPreventingspreadofrumor

7.Conclusion

8.StrengthsandWeaknesses

9.Reference

1.Introduction

Asanordinaryperson,wewouldneverthinkofthatthereisanintersectionpointwiththeworld-famousmathematician.Ifwewanttoknowtheintersectionpoint,soeasy,justvisitthewebsitesaboutErdösNumber.Asweallknow,PaulErdösisaverytalentedmathgenius,hehaspublished1457papersandhascollaboratedwith511peopleinhislife.Becausetheco-authorsaretoomany,soErdösnumberisborn.Erdös’sErdösNumberiszero.TheErdösNumberofpeoplewhohaswritedpapersdirectlywithErdösisone,ifsomeonehaswritedwithanErdösNumberoneperson,hisErdösNumberwillbetwo,andsoon.Soifwegototest,itispossibletofindtheErdösNumberthatbelongstoourselves.WhatismarvelousisErdösisnotonlythecenterofErdösNumbernetwork,butalsofocusonthestudyofnetworkscience.

Networkscienceisaninterdisciplinaryacademicfieldwhichstudiesthequalitativeandquantitativelawsofcomplexnetworkssuchassocialnetworks,computer,telecommunicationnetworks,biologicalnetworksandsoon.TheTheUnitedStatesNationalResearchCouncildefinesnetworkscienceas"thestudyofnetworkrepresentationsofphysical,biological,andsocialphenomenaleadingtopredictivemodelsofthesephenomena.

2.RestatementoftheProblem

2.1Problemsweareconfronting

Inaword,ourtaskistoanalyzeinfluenceandimpactinresearchnetworksandotherareasofsociety.Andspecificproblemsareasbelow:

●First,weareallowedtobuildtheco-authornetworkoftheErdos1authors.Thenanalyzethepropertiesofthenetwork.

●CreateamethodtomeasuretheinfluencesoastodeterminewhointhisErdos1networkhassignificantinfluencewithinthenetwork.

●Theproblemtalksaboutanothertypeofinfluencemeasuretocomparethesignificanceofaresearchpaper.Usethepapersintheattachedlisttobuildamodeltodeterminewhichpaperisinthecenterofthecitationnetwork.Besides,discussthesimilarmethodtomeasuretheinfluenceofnetworkresearcher,university,departmentorjournalinthenetwork.

●Applythemodelintodifferenttypesofnetworkandanalyzetheimportanceofnodesinthenet.

●Finally,discusstheinfluencemethodologytosolvetheactualproblemsinthelife.

2.2Solutionstosolvetheseproblems

●AccordingtotheappendixErdos1.htm,wegetamassofdataabout511researcherswhocoauthoredapaperwithErdösandtheirlinksandweuseMicrosoftExcelandJavaCompilertodisposal data.Forthefirstquestion,wecandrawacomplexnetworktobuildtheco-authornetworkoftheErdos1authors.Beforewedothis,weshouldgetridofpeopleoutsidetheErdos1network.Then,regardeverypersonasanode.Weuse

toanalyzeandvisualizelargenetworks.Andweshouldtakesomemeasures,justlikereducingthenumberofedges,deletingsomenodesthathaveminoreffectonthewholenetwork.Finally,wecananalyzethepropertiesofthisnetworkwiththehelpofnetworkfigures.

●Afterreferencingsomepapersassociatedwithco-authornetwork,wehavelearnedseveralcentralityindicatorsdecidingtheimportanceofnodes.Weknowthatsingleindexmaycausetheassessingbecomeonesidedinevitably.SoweadopttheNMIMmodelalsoas”keynodesincomplexnetworksidentifiedbymulti-attributeindicatormethod”tofindthemostinfluentialnodesinthenetwork.Therearefourkindsofindictorsareintroducedtoobtainacomprehensiveresult.Use

tocalculatethevalueofthesecentralityindicators.ThenweadoptAnalyticHierarchyProcesstogettheweightofeachindicator.AccordingtotheintegratedanalysiswecandrawtheconclusionthatthemostinfluentialauthorinErdosnetworkisALON,NOGAM.

●Thethirdproblemissimilartothesecondone.Thedifferenceisthattheconnectionsbetweenpapershavedirections.Wecanalsocalculatethevalueofcentralityindicatorsusing

.Someofthemaredividedintooutdegreeindicatorandindegreeindicator.Besides,weshouldconsidertheimportanceofotherpapersconnectedwiththepaperwestudy.Finally,listTop5papersineachindicatorinatable.Analyzethedatainthetabletogetresults.Thelefttwosmallproblemsaretheapplicationandimprovementofthecitationnetworkmodel.Calculateresultsandmodifysomepartsofmodelcansolvetheproblemeasily.

●Thefourthproblemrequiresustoimplementouralgorithmonacompletelydifferentnetwork.Weshouldchoosetherepresentativenetwork,anditmustbeeasytoachive.ourchoosepositionedintheactorswhohaveperformedinFeng’smovie（directedbyXiaogangFeng）.ThenusetheNMIMmodeltoselecttheinfluentialactors.

●First,weshoulddiscussthetheScience,UnderstandingandUtilityofourmodelbasingontheprocessofmodelestablishment.Thenanalyzethepossibilitythattohelpsocietyinpreventingspreadofrumorbyourmodels..

3.Convention

3.1Variables

3.2Assumptions

●Wethinktheinfluenceandimpactmentionedintheproblemareequivalenttosignificanceandthedestructivenessofthenetworkafterdeletingthespecificnode.Inourpaper,theinfluenceandimpactofsomeoneorsomethinginnetwork

●Weassumethatintheco-authornetworktheinfluenceisonlyrelatedtothefour

4.TheModel

4.1theco-authornetworkoftheErdos1authors

4.1.1Completeco-authornetwork

Wepickoutthe511authorinsidetheErdos1networkbyutilizingMicrosoftExcelandJavaprogram.Sortthem alphabetically.For convenience sake,weuse1to511tostandforthe511authorrespectively（astheirID）.Getan adjacencymatrixwith511rowsand511columnstoshowtherelationshipbetweenthecoauthors.Itisobviousthattherelationshipbetweenthecoauthorsismutual（ifAcollaborateswithB,Bcollaborates withA）.Sotheadjacencymatrixissymmetric.Wecanjustdrawanundirected graph.ByadoptingPajek,acomplexnetworkispresentedinFigure1.

Figure1.Completeco-authornetwork

InFigure1,wecandistinguishthediffirentdegreesofnodesaccordingtothecolorsofnodes.ThecorrespondencesbetweenthedegreesandthecolorsbaseontheDefaultVertexColorinwherewesetupthePartitionColorinPajek.Tabel1showspartofthecorrespondences.

Table1Thecorrespondencesbetweenthedegreesandthecolors

partition

0

1

2

3

4

5

6

7

Color

Cyan

Yellow

LimeGreen

Red

Blue

Pink

White

Orange

partition

8

9

10

11

12

13

14

15

Color

Purple

CadetBlue

TealBlue

OliveGreen

Gray

Black

Maroom

LightGreen

4.1.2Streamlinedco-authornetwork

However,thenumberofedgesissolargethatthelinescovereachother.Wehardlylookintothecharacteristicandlawsamongnodes,edges,andcolors.Sowelimitthesizeofnetwork.Foranode,wesuspectthatcloserelationstoothernodesmeansmajorinfluence.Wehavedegreesofnodesembodythoserelationsforthetimebeing.So,weselectthenodeswhosedegreeis greater than210.Thereare21nodesthatmeetrequirements.Wedrawthenetworkdrawingconsistingofthesenodesandedgesassociatedwiththenodes.Now,wecanobserveclearinFigure2.

Figure2Streamlinedco-authornetwork

InFigure2,weisinvirtueofnotonlycolor,butalsodiameterdistinguishthediffirentdegreesofnodes.Wejusthaveunderstoodhowcolorrealizes thispurpose.ThecorrespondencesbetweenthedegreesandthecolorsinFigure2aresamewiththeminFigure1.Usingdiametertoattainlikecoloriseasytoo.Thatis:

largerthenodesare,largerthediameterofnodesare.

WereachthefollowingpropertiesofthisnetworkforthepresentfromFigure2:

●Thenumber10（Liu,AndyC.F.）hasmostdegree,andhehasthemostdirectconnectionwithothers.Fromthepointofviewofthis,thenumber10willbethemostinfluential.

●Tofacilitatetheexplanation,wecallthesenodesthathavemajordirectlyconnectededgesBigNote.AndwefoundthatmostofdirectlyconnectededgesofBigNotearedirectlyconnectededgesofotherBigNote.Thatistosay,thedirectrelationbetweenBigNoteismuchmorethanothercombinations.

Theseconclusionbyperceivingsubjectivelyandanalyzingqualitativelywillgetmodificationandperfectioninthenextmodeling.

4.2NMIMModelToFindtheMostInfluentialAuthor

4.2.1WhyisNMIM（Network’smulti-attributeindicatormethod）

The“Network’smulti-attributeindicatormethod”（NMIM）,knownalsoas”keynodesincomplexnetworksidentifiedbymulti-attributeindicatormethod”.TheErdos1network（donotincludeErdős）isalsoinvolvesasetofitems（authors）,whichwewillcallnodesorsometimesvertices,withconnectionsbetweenthem,callededges.

Ineffect,theresearcherwhohassignificantinfluencewithinthenetworkisthekeynode.

4.2.2DefinitionofMeasureIndicators

ForagivengraphG=（V,E）isanon-directionalnetworkwithoutself-rings,

issetofallthenodesand

isthesetofedgesbetweenthe