AStudyOnDirectedNeutrosophicSocial Networks
YunmingHui,ArindamDey
Abstract—PopularityofsocialnetworkslikeFacebook, LinkedIn,Twitter,Instagram,WhatsAppisincreasinglyincreasingdaybyday.Socialnetworkscholarsareverymuch interestedtocapturetheuncertaintiesofsocialnetwork.The neutrosophicsetisawellknowntooltohandleandrepresentuncertaintiesininformationduetoindeterminateness orincompleteness.Themainapproachofourstudyisto investigatetheneutrosophicapproachtodealtheuncertainties thatmayexistinsocialnetwork.Inthiswork,weintroducea neutrosophicmodeltopresentthesocialnetworkusingdirected neutrosophicgraph.Wecallthisdirectedneutrosophicsocial network(DNSN).Thecentralityofactorsplayaveryimportant theatricalroleinsocialnetworkanalysis.Inthispaper,we introducesomecentralitymeasuresforDNSN,becauseDNSN wherearcsareassociatedwithdirectedneutrosophicrelation wouldconsistmanyinformation.Wedescribesomenewcentralitymeasuressuchasneutrosophicoutdegreecentrality, neutrosophicindegreecentrality,neutrosophicoutcloseness centralityandneutrosophicinclosenesscentralityDFSNs. Wealsoinvestigateaboutdirectedneutrosophicrelationand connectivityforDNSN.Wealsopresenttherobustnessand validnessofourproposedcentralitymeasurementforDNSNby describingthistechniquetosomedirectedneutrosophicgraph anddeterminesatisfactoryresults.
IndexTerms—visual-servoing,tracking,biomimetic,redundancy,degrees-of-freedom.
I.INTRODUCTION
Intwentyfirstcentury,peoplearenowconnectedmore eventhoughonlinesocialnetworkswiththesmartmobile phonesinourdailylife.Itisbasicallyaplatformfor interconnectingwithhugenumberofpeopleineverywhere intheworld.Weexchangeinformationofseveralissues andtopicsinsocialnetworkwhichhelpsfore-commerce ande-business,politicalandsocialcampaigns,influential players(researchers,engineers,employees,organizations, etc.),futureevents,alumni,etc.
Asimplesocialnetworkconsistsofacollectionofsocial units(socialnodesorvertices)describingindividual,groups, companies,etc,whichareinterconnectedbyarcs(linksor edges)describingrelationshipbetweentwosocialunits.The structureofsocialnetworkwithsocialunits(actors)and theirrelationsaregenerallymodeledassimplecrispgraphs. Weusuallyreferthisgraphassociograminsociology. Inasociogram,socialunits(nodes/actors)andrelations respectivelyarerepresentedbynodesandarcsofagraph. Graphsarenaturallyusedtomodelthesocialnetworkas itisefficientandusefulgraphicalmethodtodescribehow objects(itemsorthings)areeitherlogicallyorphysically interconnectedtogether.However,uncertaintiesmayexistin
YunmingHuiiswiththeSchoolofComputerScience,JiangsuUniversityofscienceandtechnology,Jiangsu212003,China:Email.huiyunming@outlook.com
ArindamDeyiswiththeDepartmentofComputerScienceandEngineering,SarojMohanInstituteofTechnology,Hooghly,India;Email. arindam84nit@gmail.com
descriptionofanysocialunitandtheirrelationship.Classical graphisunabletomodeltheuncertaintiesofthesocial networkproperly.Thefuzzysetisanusefultechniqueto thehandletheuncertaintiesininformationofanyreallife problem.
Inmanycases,thesocialnetworkcanberepresentedby fuzzygraphdefinedasfuzzysocialnetwork(FSN)usages type-1fuzzysetasedgeweight.Thedegreeofmembership ofatype-1fuzzyset[1],[2]iscrisp(realnumber),which isdeterminedbyhumanperception.Thereareseveraltypes ofuncertaintiesinthemembershipgrade.Itisveryhardto evaluatethepropermembershipgradeofafuzzyset.Neutrosophicsetisabletohandleandrepresentthisuncertainty.
Smarandache[3]havedescribedtheideaofneutrosophic set(NS).NSisusedtodealandhandletheuncertain informationintherealworldproblemsduetoincompleteness,indeterminacy,imprecisenessandinconsistency.Itis describedbythreememberfunction:truthmembershipgrade (T (B)),anindeterminatemembershipgrade(I (B))and afalsemembershipdegree(F (B)).Thevaluesof T (B), I (B) and F (B) areindependentandwithinthenonstandard unitinterval.TheNSmodelingisawellknownmethod forhandingtheuncertaintiesinreallifescenariosbecause NScanworknotonlyinconsistentinformationanditcan alsohandletheincompletenessandindeterminacyofan information[4],[5],[6],[7],[8],[9],[10],[11],[12],[13], [14],[15],[16],[17].
Theanalysisofsocialnetworkisanimportantstudyfor mappinganddeterminingofrelationsandpositionamong theconnectedsocialunits.Ithelpstodeterminethesignificanceofdifferentsocialunits(actors)intheirnetworkor communicationsanditalsodescribestheiropportunitiesof relationship.Thelessconstraintsofasocialunit(person),the socialnodewillbeinmorefavorablepositiontoexchange theinformation.
Inthestudyofsocialnetwork,aproblemoffindingthe significantsocialunitsofthatsocialnetworkhavebeen researchedforaseveralyears.Forthatpurpose,theideaof thecentralitymeasureofanodeappearedinanetwork.The socialnetworkanalysisassumesverycloselyrelatedtothe ideasofcentralityandabilityaspersoninasocialnetwork. Itdescribesusthatwhoisinthenetwork,whoworksas aleaderinthenetwork,whoworksasamediatorinthe network,whoworksalmostisolatedinthenetwork,orwho appearscentralinthenetwork.
Manysocialscientistshaveintroducedmanytypesofcentralitymeasurementsinsocialnetwork.Degree,betweenness andclosenessofasocialunitarethemostimportantissues intheareaofsocialnetworkanalysis.Thedegreecentrality ofasocialunit/individual/personisusedtomeasurethelevel ofpopularityandcommunicatingactivity,distinguishingthe centralityofasocialunitbasedonthedegree.Thecloseness
centralityofasocialunitisthesumofthelengthsof theshortestpathbetweenasocialunittootherrestofthe socialunits.Thelowervalueofclosenesscentralityis,the higherthecentralityvalue.Closenesscentralitymeasures theindependencyincommunicationandrelationshipor bargainingbetweenthesocialunits.Itisbasicallyusedto measurethepossibilityofcommunicatingwithothersocial unitscalculatingonaminimalnumberofintermediatesocial units.Thebetweennesscentralityofsocialunitiscentrality measurementthatfindsthenumberoftimesthatspecific socialunitliesintheshortestpathfromtherestofthesocial units.Itmeasuresthelevelofcontrolofcommunicationof socialunit.Itisusedtofindthesocialunitwhoinfluences theothersunits.
Centralitymeasurementisappliedmainlyinsocialnetworkanalysisandalsoinbehavioralsciences.However,it isusedincomputerscience,biology,politics,management, chemistry,economicsandsoon.StephensonandZelen[8] haveintroducedthetheconceptofcentralityofanode basedontheshortestpathbetweenthenodes.Theyhave alsointroducedameasurementusingtheideaofdataasit isappliedinthestatisticalestimation.Thishaveusedthe lengthofpathamongtwothesocialunitsintheirdefined measurementandthelengthofeachandeverypathbased onthedataconsistedinit.
Bonacich[18],[19]haveproposedananotherideaof centralitymeasurementofasocialnode.Hehasintroduced thecentralitymeasurementofdifferentsocialnodesby applyingtheeigenvector.Theeigenvectorislinkedwiththe highesteigenvalueintheadjacentmatrix.Brunellietal.[20] proposedanefficientandrobustmeasurementofcentrality. Theirdefinedcentralityiscalculatedbasedonthelevel influenceofothersocialunitsaccordingtotheirvalueof eigenvectorcentrality.SohnandKim[21]haveintroduceda robusttechniquetodeterminethezoneofcentralitymeasure inanurbanizedplace.Kennarrecetal.[22]haveproposed anewtypeofcentralitymeasurement.Theyhaveintroduced thetheideaofthecentralityofsecondorder.Itcanbe calculatedinadistributedway.Qietal.[23]haveintroduced anewtypeofcentralitymeasureforweightednetworks. TheyhavedefinedthiscentralityasLaplaciancentralitymeasure.Itmeasurestheimportancebetweenlocalandglobal characterizationofasocialnode.Themaindisadvantageof Laplaciancentralityisthatitcannotbeusedindirectedsocial networks.Pozoetal.[24]haveproposedananothertypeof centralitymeasuresofadirectedsocialnetworkbasedon thegametheory.Landherreta1.[25]havedescribedthe5 typesofcentralitymeasurements.Thosecentralitymeasures arebasedofthreenecessitiesforthecentralitymeasures.
Sincethesocialnetworksinreallifescenariosareoften uncertain,themodelofreallifesocialnetworkisvery difficultandalwayschallenging.Fuzzysetisanusefultool todealtheuncertaintiesinrealworldproblems.Theideaof afuzzinessinsocialnetworkandthetechniquestoanalysis thosesocialnetworkshaveattractedmanyresearchersinlast fewyears.NairandSarasarnma[26]usedthefuzzyset theorytoanalysisthesocialnetwork.Theyhavemodeled theundirectedsocialnetworkinfuzzyenvironmentusing fuzzygraphwherethesocialunitsarerepresentedasactors ornodesorverticesandtherelationshipbetweenthesocial unitsarerepresentedastheedgesorarcsorlinks.Liaoand
Hu[27]haveintroducedtheideaofundirectedFSNand describedsomepropertiesofit.Theyhavedescribedsome fundamentaldefinitionandrelevanttheoryforfuturestudyof theFSN.Samantaetal.[28],[29],[30],[31],[32]havedone lotsofresearchesinthedomainoffuzzysocialnetwork. Fanetal.[33]havepresentedtheregularandstructural equivalenceinanundirectedfuzzysocialnetworks.Liao eta1.[34]havemodifiedtheideaofcentralitymeasureto thefuzzyenvironment.TheyhaveintroducedtheDefinition ofthreetypesoffuzzycentrality:degree,betweennessand closenessinanundirectedfuzzysocialnetworks.Kunduand Pal[35]haverepresentedthesocialnetworkwithasetof granules.Theyhaveusedthefuzzysetfordescribingthe granular.Theyhavedefinedthismodelasfuzzygranular socialnetwork.Hueta1.[36]havemodifiedtheconception ofcentralizationandcentralitytothefuzzyenvironment. Theyhavedescribedtheclosenesscentralityandgroup closenesscentralityinanundirectedFSNs.Huetal.[37] haveextendedtheideaoffuzzycentralityindirectedfuzzy socialnetwork.TheyhaveintroducedsomenewDefinition offuzzycentralitymeasurementdefinedasindegree,out degreecentrality,inclosenessandoutclosenesscentralityin fuzzyenvironment.
Neutrosophicset[38],[39],[40]isawellknownand populartheorywhichonecandealthenaturalphenomenon ofimprecisionanduncertaintyinrealworldproblem.Neutrosophicnessisextendedversionoffuzzinesstohandlethe uncertainty.Themainobjectiveofthisworkistointroduce amodelofsocialnetworkusingdirectedweightedneutrosophicgraphwhichwillbeverysimpletomodelandanalysis inreallifescenarios.
Tothebestourinformation,thereexistsnostudyonsocial networkusingneutrosophicgraph.Inthisstudy,wepropose amodeltoexpressthesocialnetworkbasedondirected neutrosophicgraph.Wedefinethissocialnetworkasdirected neutrosophicsocialnetwork(DNSN).Thecentralityofsocial unitactsasignificantroletoanalysisinsocialnetwork. SomenewcentralitymeasuresareintroducedforDNSN, becauseDNSNwherelinkbetweensocialunitsarejoined withdirectedneutrosophicrelationwouldconsistseveral information.Wedescribesomenewcentralitymeasuressuch asneutrosophicoutdegreecentrality,neutrosophicindegree centrality,neutrosophicoutclosenesscentralityandneutrosophicinclosenesscentralityinDFSNs.Wealsoinvestigate aboutdirectedneutrosophicrelationandconnectivityfor DNSN.Wealsopresentthevalidnessandrobustnessofour proposedcentralitymeasureforDNSNbydescribingthis techniquetosomedirectedneutrosophicgraphanddetermine satisfactoryresults.
II.PRELIMINARIES
Definition1: Theneutrosophicsetisdescribedbythree membershipfunctions;: TB (m), IB (m) and FB (m).Here, TB (m), IB (m) and FB (m) aretrue,indeterminateand falsemembershipfunctionswhicharealwaysintheinterval ] 0, 1+[ respectively. 0 ≤ sup TB (m))+sup IB (m)+sup FB (m) ≤ 3+ (1) Here, ξ representsanuniversalsetand B representsa neutrosophicset[3]ontheuniversalset ξ
Definition2: Let B representsthesinglevaluedneutrosophicsets(SVNs)[41] B onthe ξ isdescribedasfollowing B = { m : TB (m),IB (x),FB (m)|m ∈ ξ } (2) Thefunctions TB (m) ∈ [0, 1], IB (m) ∈ [0, 1] and FB (m) ∈ [0, 1] aredefinedasdegreeoftruthmembership,degreeof indeterminacymembershipanddegreeoffalsitymembership of x in A,satisfythefollowingcondition:
0 ≤ sup TB (m)+sup IB (m)+sup FB (m) ≤ 3+ (3)
Definition3: Let B = { TB (m),IB (x),FB (m)| } isa SVN,Thescorefunction S [42]ofSVN B iscomputed usingthevalueoftruthmembership(TB (m)),indeterminacy membership(IB (x))andfalsitymembership(FB (m))and itiscalculatedasfollow: S(B)= 1+ p q 2 p =1+(TB (m) 2IB (m) FB (m)) (4) q =(2 TB (m) FB (m))
III.PROBLEMFORMULATION
Insimplesocialnetworkinganalysis,wehaveusedfor binaryrelationsinwhichatwosocialunitsiseitherconnectedordisconnected.However,therelationshipbetween twosocialunitsisgenerallyvagueinnature.Manyresearchershaveusedfuzzysetandfuzzygraphtodescribethis vagueness.However,wecannotmodeltheuncertaintiesdue inconsistentinformationandindeterminateinformationabout anyreallifeproblemusingfuzzygraph.Theneutrosophic settheory,proposedbySmarandache[43]isdesirablefor handlingthistypeofuncertaintiesandimprecision’srelated withinformationrelatingseveralparameters.
Definition4: Anundirectedneutrosophicsocialnetworkisdescribedasanundirectedneutrosophicrelationshipstructure Gun = X, Yun ,where X = {x1, x2,..., xn} isanonemptysetofsocialunits,and Yun =
Severalneutrosophicrelationshipsaredirectional.Aneutrosophicrelationshipisdirectionalifthereexistssometies fromonesocialactortomothernode.
Definition5: Adirectedneutrosophicsocialnetworkis describedasaneutrosophicrelationshipstructure Gdn = X, Ydn ,where X = {x1, x2,..., xn} isanon-emptyset ofsocialunits,and Ydn =
Wehavedescribedtwotypesofneutrosophicsocialnetwork:undirectedanddirected.Themainconceptualdifferencebetweentwotypeofneutrosophicsocialnetworks isconsideredindirectedneutrosophicsocialnetworkand undirectedneutrosophicrelationisconsideredinundirected neutrosophicsocialnetwork.Duetothisreason, exy isand eyx areequalinundirectedneutrosophicsocialnetwork. However, exy isand eyx arenotalwaysequalindirected neutrosophicsocialnetwork.
IV.THENEUTROSOPHICDEGREE CENTRALITY ANALYSISOFDIRECTEDNEUTROSOPHIC SOCIAL NETWORK
Inthissection,wepresentmeasurementofneutrosophic indegreecentrality,neutrosophicout-degreecentralityand neutrosophicdegreecentrality,respectively,inDNSN.
Definition6: Let Gdn = V, Edn beaDNSNandthe singlevaluedneutrosophicset(SVNS)isusedtorepresent thearclengthsof Gdn.Thesumofthelengthsofthearcsthat areadjacenttoasocialnode vx iscalculatedwhichisnoting butaSVNS.Theneutrosophicvalueofnode vx, dI (vx),is calculatedasfollows. dI (vx)= n y=1,y=x
˜ eyx (5) Thesymbol referstoanadditionoperationofSVNS and eyx denotesaSVNSassociatedwiththearc (i,j). dI (vx)= dT I (vx) dI I (vx) ,dF I (vx) isananotherSVNS whichrepresentstheneutrosophicindegreecentrality (NIDC)ofnode vx.Thescorevalueofthecorresponding SVNSisdeterminedandthisscorevalueiscalledasthe NIDCofthenode vx.
ds I (vx)= 1+ p · q 2 p = dT I (vx) 2dI I (vx) dF I (vx) (6) q = 2 dT I (vx) dF I (vx)
Here, ds I (vx) representstheNIDCofnode vx Definition7: Let ˜ Gdn = V, ˜ Edn beaDNSNandthe SVNSisusedtorepresentthearclengthsof Gdn.Thesumof thelengthsofthearcsthatareadjacentfromasocialnode vx iscalculatedwhichisnotingbutaSVNS.Theneutrosophic valueofnode vx, ˜ dO (vx),iscalculatedasfollows. dO (vx)= n j=1,j=i eyx (7)
Thesymbol referstoanadditionoperationofSVNS and eyx denotesaSVNSassociatedwiththearc (i,j) dO (vx)= dT O (vx) dI O (vx) ,dF O (vx) isananotherSVNS whichrepresentstheNIDCofnode vx.Thescorevalueof thecorrespondingSVNSisdeterminedandthisscorevalue iscalledastheNIDCofthenode vx ds O (vx)= 1+ p q 2 p = dT O (vx) 2dI O (vx) dF O (vx) (8) q = 2 dT O (vx) dF O (vx)
Here, ds O (vx) representstheneutrosophicoutdegreecentralityofnode vx Definition8: Let Gdn = V, Edn beaDFSNandthe SVNSisappliedtodescribetheedgeweightsof Gdn.The sumofNIDCandNODCofnode vx iscalledneutrosophic degreecentrality(NDC)of vx.TheNDCofnode vx, d (vx), isdescribedasfollows. d (vx)= ds I (vx)+ ds O (vx) (9)
Here, d (vx) istheNDCofnode vx.
Thesethreetypesofdegreecanbeusedtoreflectthe communicationabilityofanodeinDNSN.TheNIDCofany nodecanbeappliedtodescribethereceptivityorpopularity andtheNODCofanynodecanbeappliedtomeasureof expansiveness.
Definition9: Let Gdn = V, Edn beDNSN.Themean ofNIDCistheaverageofneutrosophicin-degreecentrality ofallthenodesof Gdn anditiscalledasneutrosophic in-degreemeancentralityofneutrosophicsocialnetwork. Theneutrosophicin-degreemeancentralityiscalculatedas follows.
dI =
n x=1 ds I (vx) n (10)
Definition10: Let Gdn = V, Edn beDNSN.Themean ofNODCofgraph Gdn istheaverageofneutrosophicoutdegreecentralityofallthenodesof Gdn anditiscalled asneutrosophicout-degreemeancentralityofthesocial network.Theneutrosophicout-degreemeancentralityis determinedasfollows.
dO =
n x=1 ds O (vx) n (11)
Let Gdn = V, Edn beaDSFN.The ds I and ds O are neutrosophicin-degreemeancentralityandneutrosophicoutdegreemeancentralityof Gdn respectively,then ds I = ds O
Onemightalsobeinterestedinvariabilityofthefuzzyindegreecentralitiesandfuzzyout-degreecentralitics.Unlike themeanfuzzyin-degreecentralitiesandthemeanfuzzy out-degreecentralities,thevarianceofthefuzzyin-degree centralitiesisnotnecessarilythesameasthevarianceofthe fuzzyout-degreecentralities.
σ2 dI and σ2 dO measuresquantifyhowunequaltheactors inaDFSNarewithrespecttoinitiatinggrreceivingfuzzy relations.Thesemeasurementsaresimpleefficientstatistical methodfordescribinghowcentralizedasocialunitisin DNSN.
IntheDNSN,NIDCisusedtomeasureofthepopularity ofasocialunit,andNODCismeasurementofability influenceofasocialunit.BasedonthevalueofNIDCand NODC,thereare4typesofsocialunitsinaDNSN.
1.Thesocialunitisanisolatesocialunit,if dI (vx)= dO (vx)=0
2.Thesocialunitonlyhasneutrosophicrelationstarting from vx,thenitiscalledastransmittersocialunitif dI (vx)=0, dO (vx) > 0
3.Thesocialunitonlyhasneutrosophicrelationterminatingat vx,thenitiscalledasreceiversocialunitif andonlyif dI (vx) > 0, dO (vx)=0.
4.Thesocialunithasfuzzyrelationbothtoandfrom it,thenitiscalledascarrierofordinarysocialunitif dI (vx) > 0, dO (vx) > 0
V.THENEUTROSOPHICCLOSENESSCENTRALITYOF DNSN
Definition13: Let Gdn = V, Edn beaDNSN.The neutrosophicin-closenesscentrality(NICC)ofanode vx is ameasurementoftheclosenessfromallthenode(excluding thenode vx)tothenode vx inaneutrosophicgraph.The NICCofanodeisdeterminedastheinverseofthesumof theneutrosophicweightoftheshortestroutesbetweenthe socialunitandrestofthesocialunitsintheneutrosophic graph.If d(vx,vy ) istheneutrosophiclengthfromthenode vy tonode vx inaneutrosophicgraph,wedeterminethesum (neutrosophicadditionoperation)oftheneutrosophiclengths betweenthenode vx andtherestofallthenodesthatarein samerow(component).Itiscalculatedasfollows.
n i=1 dI (vx) dI 2 n (12)
Definition11: Theaverageofthesquareddifferences fromneutrosophicin-degreemeancentralityiscalledasthe neutrosophicindegreevariancecentrality.Theneutrosophic indegreevariancecentrality, σ2 dI ,isdeterminedasfollows. σ 2 dI =
Thesquarerootoftheneutrosophicindegreevariancecentralityiscalledtheneutrosophicindegreestandarddeviation centrality.Theneutrosophicindegreevariancecentrality providesaroughconceptofvariabilityoftheneutrosophic in-degreecentrality,whilethestandarddeviationcangive moreconcretetoprovidetheexactfromthetheneutrosophic in-degreecentrality.
D (vy , vx)= n j=1,y=x
d (vy , vx) (14)
D (vy , vx) isaSVNS.Thescoreof D (vy , vx) iscomputed byusing().TheNICCistheinverseofthesumoftheneutrosophicconnectedintensityfromallthenodes(excluding vx)to vx.TheNICCof vx iscomputedasfollows.
CCI (vx)= 1 ˜ D (vy , vx) (15)
Definition12: Theaverageofthesquareddifferences fromneutrosophicout-degreemeancentralityiscalledasthe neutrosophicoutdegreevariancecentrality.Theneutrosophic outdegreevariancecentrality, σ2 dO ,isdeterminedasfollows. σ 2 dO =
n i=1 dO (vx) dO 2 n (13)
Thesquarerootoftheneutrosophicoutdegreevariancecentralityiscalledtheneutrosophicindegreestandarddeviation centrality.
Definition14: Let Gdn = V, Edn beaDNSN.The neutrosophicout-closenesscentrality(NOCC)ofanode vx isameasurementoftheclosenessfrom vx toallthenodesin aneutrosophicgraph.TheNOCCofanodeisdeterminedas theinverseofthesumoftheneutrosophiccostoftheshortest pathsbetweenthesocialunitandrestofthesocialunitsinthe neutrosophicgraph.If d(vx,vy ) istheneutrosophicdistance fromthenode vx tonode vy inaneutrosophicgraph,we determinethesum(neutrosophicadditionoperation)ofthe neutrosophiclengthsbetweenthenode vx andtherestofall thenodesthatareinsamerow(component).Itiscalculated asfollows.
˜ DO (vx, vy )= n j=1,y=x
d (vx, vy ) (16)
DO (vy , vx) isaSVNS.Thescoreof DO (vx, vy ) iscomputedbyusing().TheNOCCistheinverseofthesumof theneutrosophicconnectedintensityfromthenode vx tothe restofallothernodes.TheNOCCof vx iscomputedas follows.
CCO (vx)= 1 DO (vy , vx) (17)
Definition15: ThesumofvalueofNOCCandNICCof node vx iscalledneutrosophicclosenesscentralityof vx.The neutrosophicclosenesscentralityof vx canbecalculated
CDC (vx)= CCI (vx)+ CCO (vx) (18)
Definition16: Let Gdn = V, Edn beDNSN.Themean ofNICCistheaverageofneutrosophicin-degreecentrality ofallthenodesof Gdn anditiscalledasneutrosophic in-closenessmeancentralityofneutrosophicsocialnetwork. Theneutrosophicin-closenessmeancentralityiscalculated asfollows.
CCI =
n x=1 CCI (vx) n (19)
Definition17: Let Gdn = V, Edn beDNSN.Themean ofNOCCistheaverageofneutrosophicout-degreecentrality ofallthenodesof Gdn anditiscalledasneutrosophic in-closenessmeancentralityofneutrosophicsocialnetwork. Theneutrosophicout-closenessmeancentralityiscalculated asfollows.
CCO =
n x=1 CCO (vx) n (20)
Insomereallifescenarios,wehavetodeterminetheNICC orNOCCofasocialnetwork.ThemeanvalueofNICCand NOCCarealwayssame,howeverthedeviationoftheNICC andNOCCmaynotbesame.
Definition18: ThestandarddeviationoftheNICC,which werepresentby σCCI isdeterminedas
σCCI =
n i=1 CCI (vx) dCI 2 n (21)
ThesquareofthestandarddeviationoftheNICCiscalled theneutrosophicindegreevariancecentrality.
Definition19: ThestandarddeviationoftheNICC,which werepresentby σCCI isdeterminedas
σCCI =
n i=1 CCI (vx) dCI 2 n (22)
ThesquareofthestandarddeviationoftheNICCiscalled theneutrosophicindegreevariancecentrality.
σCCI and σCCO describethemeasurementhowunequal thenodesinaDNSNarewithrespecttostartingorobtaining indirectanddirectneutrosophicrelationship.TheNICC andNOCCaresimplebutalsoveryefficientstatisticsfor describinghowcentralizedaDNSNis.
ThevalueofNICCdescribesnotonlythepopularityof asocialunitinDNSN,butalsomeasuresthepopularityof thesocialunitbyindirectlyneutrosophicrelation.Thevalue ofNOCCmeasuresnotonlyasocialunitdirectlyinfluence butitcanalsodescribeindirectinfluenceofthesocialnode.
VI.NEUTROSOPHICBETWEENNESSCENTRALITY
Betweennesscentralityisanimportantmeasurementin socialnetworkanalysis,networkdatamodelandcomputer network.Inmanyreallifescenarios,thedistancebetween thesocialunitsnotonlyplaysignificantpropertyinsocial networkanalysis.Itisalsoveryusefultofindthesocialunits withhighernumberoftimesworksasasimplebridgeina shortestpathfromasocialunittoananotherunit.Those typesofsocialunitcancontroltheflowofdatainasocial network.Betweennesscentralityisusedtodeterminethe potentialofasocialunitforcontrolofdatacommunicationinsocialnetwork.Here,weintroducetheconceptof betweennesscentralityofanodeinDNSN.
Definition20: Betweennesscentralityisameasurethe betweennessofasocialunitinaDNSNanditdeterminesof centralityinaDNSNbasedonshortestpaths.Betweenness centralityisthetotalnumberoftimesasocialunitworksas abridgeofaneutrosophicshortestpathbetweentwosocial units.Wedefinethisbetweennesscentralityasneutrosophic betweennesscentrality(NBC).TheNBCofsocialunit vx is computedasfollows.
CNBC (vx)= s=vx=d
σsd(vx) σsd (23) σst(vx) representsthetotalnumberofthoseneutrosophic shortestpathsthatpassnode vx and σst isusedtorepresent thenumberofneutrosophicshortestpathsbetweensource node s anddestinationnode d
VII.NEUTROSOPHICDECAYCENTRALITY
Neutrosophicdecaycentrality(NDC)isameasurementof theclosenessofasocialunittotheotherrestofthesocial unitsinaDNSN.TheNDCofsocialunitiscalculatedbased onthedistanceandanewparametercalledtheneutrosophic decayparameter δ(0 <δ< 1).TheNDCofanode vx for aspecificvalueofthe δ iscomputedasfollows.
NDCδ (vx)= vx=vy
δd(vx,vy ) (24)
VIII.CASE STUDY
Let Gd7 = V, Ed7 isadirectedneutrosophicgraph.Itis usedtomodelaWhatsappgroupoffamilymembers,where V = {v1, v2,..., v7} denotesasetof7familymembers, Ed7 denotesdirectedneutrosophicrelationbetweenthe7 members.Wehavegottheneutrosophiccommunication relationsamong7membersandthissocialnetworkisshown inFig.2.Anundirectedneutrosophicgraphcanbeusedto representthissocialnetwork.ItisshownininFig.1.Inan undirectedneutrosophicsocialnetwork,edgesaregenerally anabsentorpresentinanundirectedneutrosophicrelation withnoanotherdataattached.
WecomputetheNIDC,NODCandNDCofthemember using(6),(8)and(9).WehaveshownthosethreeNDCin TableII.
m1 m2 m3 m4 m5 m6
0.9,0.4,0.1 0.9,0.3,0.3
m7 1.0,0.4,0.1 0.8,0.3,0.1 0.9,0.3,0.2 0.8,0.2,0.3 1.0,0.4,0.1 0.9,0.3,0.1
Fig.1.UndirectedNeutrosophicsocialnetwork m1 m2 m3 m4 m5 m6
0.9,0.4,0.1 0.9,0.3,0.3
m7 1.0,0.4,0.1 0.8,0.3,0.1 0.9,0.3,0.2 0.8,0.2,0.3 1.0,0.4,0.1 0.9,0.3,0.1
Fig.2.DirectedNeutrosophicsocialnetwork
TABLEI
THENEUTROSOPHICDEGREECENTRALITYOFRESEARCHTEAM
Members NIDC NODC NDC m1 0.00 1.100 1.100 m2 0.545 0.555 1.100 m3 0.555 0.535 1.090 m4 0.555 0.500 1.055 m5 0.545 0.500 1.045 m6 0.000 0.600 0.600 m7 0.000 0.500 0.500
TABLEII
THENEUTROSOPHICDEGREECENTRALITYOFRESEARCHTEAM
Members NICC NOCC NCC
m1 0.0000 0.9569 0.9569 m2 0.9569 1.8181 2.7750 m3 0.9569 1.8300 2.7869 m4 0.9478 1.8300 2.7750 m5 1.8300 2.0000 3.8300 m6 0.0000 1.6600 1.6600 m7 0.0000 2.0000 2.000
m1 and m2 arethetoptwosocialunitsofthisDNSN. Thetotalinformationofthesesevenmembersbasedon3 centralitymeasurementsareshowninTableII.Fromthe TableII,wefindthat m3 and m4 obtainthetwomaximum valuesofNIDC.Itindicatesthatseveralothermembers consider m3 and m4 asafriend.While m1, m6 and m7
havegotthethreelowestvalues(=0.00)ofneutrosophicindegreecentrality.Itdescribesthatmembers(m1, m6 and m7)arenotreceivedfriendshipbyothermembers.Basedon ,Themember m1 hasgotthehighestvalueofneutrosophic out-degreecentralityand m1, m6 and m7 havegotthe threelowestvaluesofneutrosophicout-degreecentrality.It indicatesthat m1 appointslotsofothersasmembers,but m1, m6 and m7 havelessinfluenceonothermembers.The m1 and m2 havegotthehighestvalueofNDCwhichindicates that m1 and m2 arejoinedwithothersocialunitswithvery highinfluence.
Usingthe15and17,theNICCandNOCCofthis DNSNarecomputed.Wehavelistedthevaluesbasedon allclosenesscentralitytechniqueinTable2.FromtheTable 2,wefindthat m5 obtainsthehighestvalueoftheNICC. Themember m5 hasverywellsocialrelationandmaximum acceptancepresentsinthefactthatindirectfriendlyrelationshipistakenintocircumstance.Themember m5 hasgotthe maximumvalueofNOCC.Itindicatesthatthemember m5 nominatesmanyothersasmembers.
IX.CONCLUSION
Inthiswork,weintroducetheideaofundirectedand directedneutrosophicgraph.Weproposeamethodtomodel thesocialnetworkusingdirectedneutrosophicgraph.We
definethissocialnetworkasDNSN.Wehavemodifiedthe centralitytheorytoDNSNtoanalysistothissocialnetwork. First,weintroducesomenewcentralitymeasurementfor DNSN:NIDC,NODCandNDC.TheNODCisusedto measureofexpansivenessofasocialunitandtheNIDC isusedtomeasureofpopularityofasocialunit.Inthis work,weassumethesociometricrelationshipbetweenthe members,amemberwithahighNODCisoneunitwho haslinkedwithotherunitsasfriends.Asocialunitwith alowNODCnominatesverylesssocialunitsasfriends. AsocialunitwithahighNIDCisasocialunitwhom manyothersmember(socialunits)nominatethissocialunit asafriendandasocialunitwithalowvalueofNIDC ispreferredbyfewotherssocialunits.Inthiswork,we introducesomeothernewtypesofcentralitymeasurement forDNSN:NICC,NOCCandNCC.Themaindifference betweenNCCandNDCpresentsinthefactthatundirected neutrosophicrelationischosenintoconsideration.Wehave alsodiscussedthemeanofNIDCandmeanNODCforthe DNSN.InDNSN,themeanNIDCandNODCareequal.We haveintroducedtheideaoftheneutrosophicbetweenness centralityofsocialunitofaDNSN.Wehavedefinedthe NDCofasocialunitinaDNSN.Theideaofvariance oftheNIDC,NODC,NICCandNOCCofDNSNare alsoproposedinthisstudy.Thesemeasurementarevery simpleandefficientstatisticalanalysisfordescribinghow centralizedasocialunitisinaDNSN.
Centralityanalysisisawellknownandefficientmethod forsocialnetworkanalysis.Thismeasurementconceptis usedtofindthecentralpositioninthesocialnetwork. WeextendtheideaofcentralityanalysisforDNSN.One simplenumericalexampleofdirectedaswellasundirected weightedneutrosophicgraphtomodelonesmallsocial networks.WecallitasDNSN.Duetothesmallsizeof thesocialnetwork,itisveryeasyandusefultorealize thesignificantoftheDNSN.Therefore,infuturework,we havetorepresentahighdensitysocialnetworkusingthe directedweightedneutrosophicgraph.Weneedtoanalysis thissocialnetwork.Despitetherequirementforfuturework, theintroducedmodeldescribedinthisworkisansignificant initialcontributionstoneutrosophicgraphandsocialnetwork underneutrosophicenvironment.
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