美国数学建模英语论文规范格式.docx

美国数学建模英语论文规范格式.docx

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美国数学建模英语论文规范格式

InterpretingGraph-likeArtifacts

Summary

Inordertointerpret“signatures”containedinthegiantgraphanddeterminethemostlikelygraph-creation-modes,theneuralnetworkpatternrecognitionmodelbasedonthefuzzyclusteringisestablishedandcomputersimulationsareestablished.Properresultsareobtainedthroughthemodelandsimulations.

Intask1，firstlyusingtheknowledgeofthegraphtheory,thedegree,theedge,thesimilarityandtheclusteringcoefficientofnodescanbecalculatedaccordingtothegivennetwork.Itcanbefoundoutthatnodesinthenetworkfollowthepowerexponentialdistribution.Inaddition,thereare3489isolatednodesandahugeconnectedcomponentof5929nodes.Secondly,acomprehensiveanalysisofthedegree,theedge,thesimilarityandtheclusteringcoefficientistaken.Thirdly,sixnodesarepickedoutasthecentersofthesub-graphs.Finally,thenodesaredividedbasedonthesimilarity.

Intask2，inordertodeterminethemostlikelygraph-creation-modes,thefuzzyclusteringmodelissetupandtheneuralnetworkpatternrecognitionbasedonfuzzyclusteringisdesigned.Firstly,thenodesinthenetworkcanbedividedbyusingfuzzyclustering.Bycomparison,wecanfindoutthattheresultisagreewiththeresultintask1.Then,thedefinitionsofgraph-creation-modesmentionedinthisproblemaregiven.Wetakethepatternrecognitiontothecategorythataredivided,usingtheneuralnetworkpatternrecognitionmodelbasedonfuzzyclustering.

Intask3，inordertotestandverifythegeneralityofthemodel,1000nodesareselectedtoformthesub-graphinitially,andafterwardsothernodesareaddedtothesub-graphrandomly.Throughtheneuralnetworkpatternrecognitionmodelbasedonfuzzyclusteringandcomputersimulations,themostlikelygraph-creation-modesaredetermined.

Finally,thestrengthsandtheweaknessesofourmodelaregivenandconclusionsrelatedtothemodelareobtained.

Keywords:

theneuralnetworkpatternrecognitionmodel;thecomputersimulations;thegraphtheory;theclusteringcoefficient

1.Introduction

Inmanyfieldsofphysical,biologicalandsocialscienceswherethesizeanddynamicnatureofthedatasetssimilarlydonotallowforexactanswers,aniterativeinterplaybetweenexperimentaldataandmodelingisnecessary.However,boththedataandthemodelingoftenhavearandomorstatisticalbasis.Suchinterplayisinitsearlystagesforthestudyofseveralmassive,dynamicgraphssuchastheWorldWideWeb.Thedegreedistributionsofseveraldifferentmassivegraphsfollowapowerlaw.Inthispaperwesimulateandanalyzemanyrandomsub-graphevolutionwithpowerlawdegreedistributionstofindthemostlikelyruleofthelargegraph’creation-mode.

ResearchrelatedtoRandomGraphisdiscussed.Extensiveandprofoundprogresshasbeenmadeinthisareabymanywellknownexperts.Afteranalyzingtheirpapers,weproposeseveralnewideastoimprovetheirmodelssoastomakethesemodelssuitabletothisproblem.Somebasicassumptionsaremadebasedontheneedsofthisproblem.Goalsandapproachesofmodelsinthispaperarelisted.Severalgraph-creation-modesproposedinthisproblemaredefinedandillustratedintheformofpropertiesofgraph.Itcanbehelpfultojudgethecreation-modeofacertaingraph.

Firstly,weshoulddevelopmodelsanddeterminethe"signatures"（i.e.,thecollectionsofgraph-traits）correspondingtoasmanyoftheabove"graph-creation-modes".

Secondly,weshoulddesignalgorithm（s）todeterminethelikelycreation-modeofagivenGraph.

Thirdly,weshouldtestouralgorithmsonsimulatedgraphs（illustratingdifferentcreation-modes）andonsomelargegraphavailableontheWeb.

2.Backgrounds

Intherealworld,therearelotsoflargeentitysystemsintheformofNetworkstructure,suchasthesocialrelationshipsofinteraction,theliteraturecitedineachresearchandthecollaborationnetworkofauthors.Sincenetworksareratherlargeandtheinteractionamongentitiesiscomplicatedintherealworld,theirspecificformsareoftenhidden.

Earlierscientiststhoughtthattherelationshipamongtherealsystemelementscanbedescribedbysomerulegeometricfigures.Therefore,thestudyonnetworkcanbestartedbasedontherulegeometricfigures.Thisiscalledtherulenetworkstage.Attheendofthe1950s,complexlarge-scalenetworkwithnospecificdesignprincipleismainlyusedfortherandomgraphswhicharesimpleandeasytobeacceptedbymostpeople.Thisiscalledtherandomnetworkstage.Inrecentyears,withtherapiddevelopmentofthetechnologyincommunicationandelectronicinformationfields,scientistsfoundthatlotsofrealnetworksneitherbelongtotherulenetwork,northerandomnetwork.However,theyhavestatisticalpropertiesincommonwhicharesignificantlydifferentfromtheabovetwo.Theyhaverathercomplexityinscale,structureandfactors.Thus,thisiscalledthecomplexnetwork.

3.Assumptions

（1）Therearenomorethantwoedgesbetweentwonodes.

（2）Theconnectivityofsomenodesexistsbecausethesenodescontainthesameorsimilarinformation.

（3）Thegraph-creation-modescanbereflectedthroughthegrowthofnodesandedges.

（4）Thegraph-creation-modeofanysub-graphinalargegraphisthesame,andalsothesametothelargegraph.

（5）Astoasub-graph,thegreateritstotaldegreeis,thelargerthisgraphis.

（6）Allconnectedsub-graphsareregardedasconnectedcomponents.

4.Symbols

（1）

:

the

node;

（2）

:

theedgebetweenthe

nodeandthe

node;

（3）

:

theclusteringcoefficientofthe

node;

（4）

:

thedegreeofthe

node;

（5）

:

thesimilarityofthe

nodewithrespecttothe

node.

5.Modelestablishmentsandsolutions

5.1Task1

5.1.1Thedegreeofnodes

Thefollowingformulaisusedtodescribeagraph

.

Amongthem,eachelementof

istheundirectedpairofverticesin

knownastheedgesof

.Twoverticesineachpairmustbedifferent.Thus,thedescriptionofthegraphis

.

Inthegraph

thedegreeofthenode

istheadjacentnodenumberofthenode

.Ifthedegreeofanodeis0,thenthisnodeiscalledtheisolatednode.

Inordertoworkoutthedegreeofthenode,theadjacencymatrix

mustbefound.Itis

.

Thus,theadjacencymatrix

canbeexpressedasfollows

.

Furthermore,theexpressionofthenodedegreeisobtainedby

.　　　　　　　　

（1）

Basedonformula

（1）,thedegreesofallnodescanbecalculated.Thenstatisticalanalysistothedegreesofallnodes（seeAppendix）isworkedoutandexpressedbythefollowingfigure.

Note:

Y-coordinateistakenaslog10.

Figure1Therelationshipbetweenthedegreeandthenumberofnodes

FromFigure1,weknowthatifthedegreeofnodesbecomesgreater,thenumberofnodesbecomessmallerandthenumberofnodesdecreasesexponentiallyapproximatelyasthedegreeincreases.Inaddition,thetotalsumofallnodedegreesis31938,andtheaveragedegreeis3.3.Ofallnodesinthegivengraph,about3500nodesareisolated.Mostofthenodeshaveadegreewhichisbelow50.

Thenodeswhosedegreeisrankingfrom1to20areselected（SeeTable1）.

Table1Thenodeswhosedegreeisrankingfrom1to20

Ranking

Degree

Num

Ranking

Degree

Num

1

199

1806

11

87

7755

2

172

1079

12

85

82

3

168

235

13

85

4823

4

155

9

14

82

8671

5

139

0

15

80

8652

6

134

2078

16

78

1812

7

131

14

17

78

3020

8

109

6427

18

77

70

9

100

31

19

76

257

10

95

8687

20

76

1617

5.1.2Theclusteringcoefficientofnodes

Ingeneral,weassumethatnode

connectswiththeothersby

edges.Thus,thecorresponding

nodesaretheneighboursofthenode

.Furthermore,thereare

edgesatmostbetweenthe

nodes.Theratiooftheedgesbetweenthe

nodestothetotal

iscalledtheclusteringcoefficientofthenodes

.Namely,itis

.

（2）

Theclusteringcoefficientoftheentiregraph

isthemeanofallthenodes

.Thus,therangeoftheclusteringcoefficientis

.

representsthatallthenodesareisolated.

representsthatthecorrespondingnetworkisglobalcouplingwhichmeansthatanytwopointsaredirectlyconnectedinthegraph.

Themeanclusteringcoefficient

isobtainedbyrunningMatlab.Thus,themeancouplingofallthenodesis0.0504.

5.1.3Theedgeofnodesandthescoreoftheedges

Itcanbefoundinthedatathatfewnodesareconnectedwiththesamenodebytwoedges.Thus,wethinkthatitmaybemoresuitablethanthedegreediscussedabovebyintroducingtheconceptoftheedge.

Anodemayconnectwithseveralnodesdirectlyandscores1timeasthenumberoftheedgesbetweenthem;thesenodesmayalsoconnectwithmanyothernodesdirectlyandscores1/2timeasthenumberoftheedgesbetweenthem;thesenodeswhichweobtainrecentlymayalsoconnectwithmanyothernodesdirectlyandscores1/3timeasthenumberoftheedgesbetweenthem…theyarerespectivelycalledthefirstprocesscorrespondingtothefirstedgescore,thesecondprocesscorrespondingtothesecondedgescoreandthethirdprocesscorrespondingtothethirdedgescore…（SeeTables2,3and4）.

Table2Thenodeswhosefirstedgescoreisrankingfrom1to20

Ranking

Score1

Num

Ranking

Score1

Num

1

199

1806

11

90

4823

2

172

1079

12

89

7755

3

168

235

13

85

82

4

155

9

14

82

8671

5

143

0

15

80

8652

6

134

2078

16

79

257

7

132

14

17

78

1812

8

109

6427

18

78

3020

9

100

31

19

77

70

10

95

8687

20

76

1617

Table3Thenodeswhosesecondedgescoreisrankingfrom1to20

Ranking

Score2

Num

Ranking

Score2

Num

1

1217

1080

11

705.5

2125

2

1212

236

12

704.5

834

3

903.