The accumulative law and its probability model - Journals

The observable phenomena presented by the Pareto distribution are commonly referred to as the Pareto principle, or '80-20 rule'. The rule says ... Logintoyouraccount Email Password Forgotpassword? Keepmeloggedin NewUser InstitutionalLogin ChangePassword OldPassword NewPassword TooShort Weak Medium Strong VeryStrong TooLong Congrats! Yourpasswordhasbeenchanged Createanewaccount Email Returninguser Can'tsignin?Forgotyourpassword? Enteryouremailaddressbelowandwewillsendyoutheresetinstructions Email Cancel Iftheaddressmatchesanexistingaccountyouwillreceiveanemailwithinstructionstoresetyourpassword. Close RequestUsername Can'tsignin?Forgotyourusername? Enteryouremailaddressbelowandwewillsendyouyourusername Email Close Iftheaddressmatchesanexistingaccountyouwillreceiveanemailwithinstructionstoretrieveyourusername YouhaveaccessMoreSectionsViewPDF ToolsAddtofavoritesDownloadCitationsTrackCitations ShareShareonFacebookTwitterLinkedInRedditEmail Citethisarticle FengMinyu, DengLiang-Jian, ChenFeng, PercMatjažand KurthsJürgen 2020Theaccumulativelawanditsprobabilitymodel:anextensionoftheParetodistributionandthelog-normaldistributionProc.R.Soc.A.4762020001920200019http://doi.org/10.1098/rspa.2020.0019SectionYouhaveaccessResearcharticlesTheaccumulativelawanditsprobabilitymodel:anextensionoftheParetodistributionandthelog-normaldistributionMinyuFengMinyuFenghttp://orcid.org/0000-0001-6772-3017CollegeofArtificialIntelligence,SouthwestUniversity,Chongqing400715,People’sRepublicofChinaGoogleScholarFindthisauthoronPubMedSearchformorepapersbythisauthor,Liang-JianDengLiang-JianDengSchoolofMathematicalSciences,UniversityofElectronicScienceandTechnologyofChina,Chengdu611731,People’sRepublicofChinaGoogleScholarFindthisauthoronPubMedSearchformorepapersbythisauthor,FengChenFengChenCollegeofArtificialIntelligence,SouthwestUniversity,Chongqing400715,People’sRepublicofChinaGoogleScholarFindthisauthoronPubMedSearchformorepapersbythisauthor,MatjažPercMatjažPerchttp://orcid.org/0000-0002-3087-541XFacultyofNaturalSciencesandMathematics,UniversityofMaribor,Koroškacesta160,2000Maribor,SloveniaDepartmentofMedicalResearch,ChinaMedicalUniversityHospital,ChinaMedicalUniversity,Taichung404,Taiwan[email protected]GoogleScholarFindthisauthoronPubMedSearchformorepapersbythisauthorandJürgenKurthsJürgenKurthsPotsdamInstituteforClimateImpactResearch,14473Potsdam,GermanyDepartmentofPhysics,HumboldtUniversity,12489Berlin,GermanyGoogleScholarFindthisauthoronPubMedSearchformorepapersbythisauthorMinyuFengMinyuFenghttp://orcid.org/0000-0001-6772-3017CollegeofArtificialIntelligence,SouthwestUniversity,Chongqing400715,People’sRepublicofChinaGoogleScholarFindthisauthoronPubMedSearchformorepapersbythisauthor,Liang-JianDengLiang-JianDengSchoolofMathematicalSciences,UniversityofElectronicScienceandTechnologyofChina,Chengdu611731,People’sRepublicofChinaGoogleScholarFindthisauthoronPubMedSearchformorepapersbythisauthor,FengChenFengChenCollegeofArtificialIntelligence,SouthwestUniversity,Chongqing400715,People’sRepublicofChinaGoogleScholarFindthisauthoronPubMedSearchformorepapersbythisauthor,MatjažPercMatjažPerchttp://orcid.org/0000-0002-3087-541XFacultyofNaturalSciencesandMathematics,UniversityofMaribor,Koroškacesta160,2000Maribor,SloveniaDepartmentofMedicalResearch,ChinaMedicalUniversityHospital,ChinaMedicalUniversity,Taichung404,Taiwan[email protected]GoogleScholarFindthisauthoronPubMedSearchformorepapersbythisauthorandJürgenKurthsJürgenKurthsPotsdamInstituteforClimateImpactResearch,14473Potsdam,GermanyDepartmentofPhysics,HumboldtUniversity,12489Berlin,GermanyGoogleScholarFindthisauthoronPubMedSearchformorepapersbythisauthorPublished:06May2020https://doi.org/10.1098/rspa.2020.0019AbstractThedivergencebetweentheParetodistributionandthelog-normaldistributionhasbeenobservedpersistentlyoverthepastcoupleofdecadesincomplexnetworkresearch,economics,andsocialsciences.Toaddressthis,wehereproposeanapproachtermedastheaccumulativelawanditsrelatedprobabilitymodel.WeshowthattheresultingaccumulativedistributionhaspropertiesthatareakintoboththeParetodistributionandthelog-normaldistribution,whichleadstoabroadrangeofapplicationsinmodellingandfittingrealdata.Wepresentallthedetailsoftheaccumulativelaw,describethepropertiesofthedistribution,aswellastheallocationandtheaccumulationofvariables.Wealsoshowhowtheproposedaccumulativelawcanbeappliedtogeneratecomplexnetworks,todescribetheaccumulationofpersonalwealth,andtoexplainthescalingofinternettrafficacrossdifferentdomains.1.IntroductionDuringthedevelopmentoftheprobabilitytheory,ParetodistributionnamedaftertheItalianeconomistandsociologistVilfredoPareto,whichisalsoknownasthepower-lawdistributionforaspecificcase,hasbecomeanindispensablecomponentinresearchfields.Itisacontinuousprobabilitydistributionofarandomvariablewhosecurveislong-tailed,andspecifically,thezetadistributionisitsdiscretecase.TheobservablephenomenapresentedbytheParetodistributionarecommonlyreferredtoastheParetoprinciple,or‘80-20rule’.Therulesaysthat,e.g.fornetworks,80%ofthedegreeofacomplexnetworkisheldby20%ofitsvertices[1].Thisdistributionhasalreadybeenfoundempiricallytofitawiderangeofsituationsincludingthefrequencyofoccurrenceofuniquewordsinanovel[2],thesizedistributionofgenefamilies[3],theasymptoticdecayofthetotalconductanceofsubcriticaltrees[4],thesizesofhumansettlements[5],theproteinsequencealignments[6],eventhedistributionofartistsbytheaveragepriceoftheirartworks[7],etc.Anotherfrequentlyusedcontinuousprobabilitydistributionisthelog-normaldistributionconsistingofarandomvariablewhoselogarithmisnormallydistributed.Alog-normaldistributiondescribesnumerousgrowingprocessesbasedontheaccumulationofsmallpercentagechangesthatisalogscale.Sincealog-normallydistributedvariableonlytakesrealpositivevaluesandisnon-symmetric,itbettersimulatesanon-negativeandnon-uniformdistributionthanthenormaldistribution,especiallyinvariousphenomenaofeconomicsandsocialsciences[8].Someoftheusuallog-normallydistributedcasesincludefiresizes[9],thesizeofcities[10],thefatiguelifetimeofamaintainablesystems[11],thetripdurationfortakingataxi[12]andindeedmanymore.BoththeParetodistributionandthelog-normaldistributionplayasignificantroleintheprobabilitytheoryandstatisticalapplications.However,someimportantunaddressedissuespersist,leadingtoaconsiderabledivergenceinmanycases.Takingthenetworkscienceasanexample,Barabásihighlightedthedegreedistribution,theprobabilitydistributionofdegreesoverthewholenetwork,asafundamentalconcept[13].Heusedthemean-fieldandcontinuoustheorytocalculatethedegreedistributionofscale-freenetworks,andtheoutcomeistheveryfamouspower-lawdistributionindependentoftime,aparticularcaseoftheParetodistribution[14].However,duetothemethodBarabásiusedtoobtainthepower-lawdistribution,heignoredthediscreteandcontinuousproblemsandidealizedthevariationsofthenetworks,andsomequeriescamealong.Thecentralquestionistheconceptandderivationofapower-lawdistribution.Soonafterthescale-freenetworkswereproposed,Bollobásetal.suggestedthatthemodellingprocessofscale-freenetworksbyBarabásiisnotprecise,andtheypresentedamoremathematical-basedmethodtoconstructnetworks[15].Later,Lietal.claimedthattherewasnoconsistent,precisedefinitionofscale-freenetworksandonlyafewrigorousproofsofmanyoftheirclaimedproperties[16].Then,Krioukovetal.usedageometricapproachtoobtainthedistributionconsideringhyperbolicspaces[17],andothersappliedittoreal-worldcases[18,19].Mayetal.indicatedthatthesubnetworksofscale-freenetworksarenotscale-free,i.e.notfollowingapower-lawdistribution[20].Insteadofthepower-lawdistribution,Fangetal.proposedadoubleParetolog-normaldistributionforcomplexnetworks[21].Besidesthedegreedistribution,therearealsooutstandingchallengesineconomics.Forexample,RobertGibratproposedaprincipletodescribethattheproportionalrateofgrowthofafirmisindependentofitsabsolutesize[22].Generally,processescharacterizedbytheGibrat’sLawconvergetoalimitingdistribution,oftenproposedtobethelog-normal,orapower-law,dependingonmorespecificassumptionsaboutthestochasticgrowthprocess.Nevertheless,thetailofthelog-normalsometimesdropsfastanditsprobabilitydensityfunctionisnotmonotonic,butratherhasaY-interceptofzeroprobabilityatthebeginning.TheParetodistribution’stailcannotsimulatethedeclineinthetailwhenthesizeislarge,andwhichdoesnotextenddownwardstozero,shouldbetruncatedatsomepositiveminimumvalueinstead.Thus,manyresearchersproposedotherdistributionstheyclaimedasabetterresultfortheGibrat’sprocess,suchastheWeibulldistribution[23].ConsideringtheinsufficiencyofrigorousproofontheGibrat’sLaw,Stanleyetal.evendisagreedwiththetheoryandresultofGibrat’sLawandproposedthattheoutcomeonlyworksempiricallyinthestudyoffirms[24].Rozenfeldetal.alsopointedoutthatthecitysizeispartiallyagainsttheGibrat’sLaw[25].Aswecansee,boththeParetoandlog-normaldistributionsalsohavesomeproblemsinfittingpracticaldistributions.Inthisarticle,weaimtoaddresstheissueontheinaccuracyoffittingcertaindistributionsinrealsituations.Forthatpurpose,wefirstinvestigatethosepracticalsituationsthattheParetodistributionandthelog-normaldistributioncannotperfectlyfit,e.g.thedegreedistribution,firm’ssize,personalwealthdistribution,etc.Allofthemhaveacommoncharacteristic:theyhaveacontinuousgrowingprocess.Takingthescale-freenetworksasanexample,theconnectionofavertexkeepsgrowingsincethenewverticesgenerateandbringconnectionsallocatingtothenetwork,thenthedegreewillbeanaccumulationofconnections.Anothercommonfeaturedisplaysastheunequalallocationoftheincrementsorincome.Forpeopleinacompany,thetotalincomeappealstoinequalityinallocation,thusdifferentpeopleobtaindifferentwealth,usuallythericharegettingricher,andthepooraregettingpoorer.Takingbothprocessesintoconsideration,oursolutionisnotsimplytousetheParetodistributionorlog-normaldistributiontoempiricallyfitthepracticaldistribution,instead,weproposeamathematicalmodelbasedonthegrowingprocessandallocationprocess.Thealgorithmofthismodelconcludesastheaccumulativelawsincethemodelshowsanaccumulativepropertytodescribetheunequallyaccumulativephenomenon.Throughthislaw,wederiveanovelprobabilitydistributionanditsvariantformtowidelyfitunequallyallocateddistribution.Thestatisticalpropertiesarestudiedtobetterunderstandthebehaviouroftheaccumulativelaw.Thegoodnessoffitoftheaccumulativedistributionisanapplicationaswellasanevidencetoshowthatourdistributionshouldhaveabetterresultindescribingrealsystems,e.g.thedegreedistribution,personalwealthdistribution,websitevisitdistribution,etc.Theorganizationofthispaperisasfollows:Aconcretetheoryoftheaccumulativelawanditsprobabilitymodelisdiscussedin§2.Accumulativefunctionandsomestatisticalpropertiesarecalculatedin§3.Therealsystemsbasedontheaccumulativelawaredescribedin§4.Besides,wedisplaysomesimulationstudiesin§5.Finally,someconclusionsaredrawnin§6.2.AccumulativelawanditsprobabilitymodelAsdescribedabove,thecontroversyoverthelog-normalandpower-lawdistributionhasneverendedineconomicsandvariousotherfields.However,theybothdonotstrictlydescribetheallocatedandaccumulativeprocess.Inthissection,weproposealinearlygrowingmodelbythetimewhoseallocationisdecidedbythe‘Mattheweffect’whichisprovenusefulinempiricaldata[26]andcalculateitsaccumulativedistribution.Itisworthnotingthattheaccumulativedistributioninthispaperisdifferentfromthecumulativedistributionfunction,whichisanaccumulationofincrementsinsteadofprobabilities.(a)AccumulativelawInessence,ourprobabilitymodelundertheaccumulativelawdescribesagrowingsystembytimethatcontainslargeindividualsandkeepsreceivingnewindividuals.Eachnewindividualarrivesatthesystemataspecificrateandsomeincrementscomealong,andtheseincrementsareallocatedamongtheexistingindividualsbytheruleofthe‘Mattheweffect’,i.e.thosepossessingmoreaccumulativeincrementsaremorelikelytobedistributedthanthefewerones.Theseincrementsgraduallyconstitutetheaccumulationofdistributedindividuals.Astimepasses,thestationaryaccumulativedistributionofthesystemisourobjectfunction.Indetail,theaccumulativelawconsistsoftwofeaturedprocessesofgrowthandallocation: (1)Growingprocess:thenumberofindividualsinthissystemkeepsgrowingataspecificrateandmakesthesizeofthesystemalsogrow;theincrementscarriedbynewindividualsalsokeepgrowingalongwiththenumberofnewindividualsandturnintotheaccumulationoftheexistingindividuals.(2)Allocatingprocess:theincrementsofeachnewindividualaredividedintoportions,eachportionismorepossiblyallocatedtoanexistingindividualwithahigheraccumulation.Basedonthislaw,wecalculatetheaccumulativeprobabilitydistribution.Tosimplifyandclarifythecalculationprocess,somefundamentalconceptionsarerequiredanddescribedasfollows:Astheobjectfunction,weprimarilyproposethedefinitionoftheaccumulativedistributioninthispaper.Definition2.1.LetZ(t)denotetheaccumulationinthesystemattimetanddefinetheaccumulativedistributionf(z),z > 0,by f(z)=limt→∞Pz(t)=limt→∞P{Z(t)λt(y/z)αf(x)f(y) dx dy=∫0z∫λt(y/z)αλt12πσyλt e−(ln⁡y−μ)2/2σ2 dx dy=∫0z1−(y/z)α2πσy e−(ln⁡y−μ)2/2σ2 dy.2.17Theprobabilitydensityfunctionf(z)isthederivationofequation(2.17)leadingto f(z)=F′(z)=∫0z∂∂z1−(y/z)α2πσy e−(ln⁡y−μ)2/2σ2 dy=αzα+1∫0zyα−12πσ e−(ln⁡y−μ)2/2σ2 dy.2.18Consideringbothequations(2.17)and(2.18),thelimitationofthisprobabilityisirrelevanttothetimeifthetimeislargeenough.Therefore,referringtodefinition2.1,theaccumulativefunctionisdisplayedasequation(2.18).Fromthederivation,wesuccessfullyconfirmtherelationshipamongtheaccumulation,individualsandincrementsinthisgrowingsystem.Theirfunctiondefinesthattheaccumulativedistributionshouldbeaprobabilitydistributiondisplayedasafunctionoftwovariables.Thederivationalsoindicatesthattheinequalityofthe‘Mattheweffect’willfinallyhaveanindirectinfluenceontheboundaryoftheprobabilitydensityfunction.Bytheprincipleofaccumulativebehaviour,wecanbetterunderstandandsimulaterealaccumulativesystemsanddiscovertheconsequenceoftheactionoftheaccumulationofincrementsandtheirunbalancedallocationstoindividuals.(c)ThevariationsoftheaccumulativelawBasedonthepreviousdescription,theaccumulativedistributiondependsonthefactthattheincrementvariablesfollowalog-normaldistributionwhichisdescribedastheassumption(3)in§2a.However,theincrementsmaybeotherpotentialprobabilitymodels,evennotavariable.Therefore,weherebydiscussanalternativeformoftheaccumulativedistribution.(i)ConstantincrementThefirstextensionoftheaccumulativefunctionisveryspecial,sinceitsincrementremainsunchangedinsteadofbeingavariable,i.e.yisaconstant.Inthatcase,basedontheresultsin§2b,wehavethecumulativedistribution F(z)=∫zx(t)λt(y/z)αf(x) dx=1−(yz)α.2.19Then,theprobabilitydensityfunctionis f(z)=F′(z)=αyαzα+1.2.20Obviously,thisfunctionistheprobabilitydensityofaParetodistribution,alsoknownasapower-lawdistribution.Thuswesay,aParetodistributionisaspecialcaseofanaccumulativedistributionfollowingtheaccumulativelaw.(ii)ExponentialincrementInsomesituations,theincrementmaybememoryless,whichmeansthatwewillnotgetanyinformationaboutthemovertime.Therefore,weconsiderthattheincrementdistributesexponentiallyasanotherextensionoftheaccumulativefunctionandassumethattheincrementYisavariablefollowinganexponentialdistributionwitharateparameterβ,i.e.Y ∼ exp(β).Inthatcase,thecumulativedistributionisderivedas F(z)=∫0z∫λt(y/z)αλtβλt dx dy=∫0z[1−(yz)α]β e−βy dy=1−αzα∫0zyα−1 e−βy dy.2.21Then,wehavetheprobabilitydensityfunction f(z)=F′(z)=αβzα+1(1−e−βz−βz e−βz).2.22Thisaccumulativedistributionfunctionisdifferentfromtheoriginalone,sincetheexponentialdistributionhasadistinctpropertyfromthelog-normaldistribution.Asaresult,thisexponential-basedaccumulativedistributionisdecreasing,whichisavariationofthepower-lawdistributionbutmorecomplicated.Inpractical,theaccumulativedistributionanditsextensiondistributionscandescribemanyrealphenomena,wewilldiscussitin§4.3.AccumulativefunctionandsomestatisticalpropertiesThederivedprocessofaccumulativedistributionisfuzzy,andinthepracticalsituations,itsdefinitionandstatisticalpropertiesaremoreserviceable.Inthissection,foraclearinvestigationoftheaccumulativedistributionandapotentialapplicationonthestatisticalstudyontherealsystems,wedisplayitsstrictdefinition,variantexpressionandsomeusefulstatisticalproperties.Particularly,weemploytheoriginalaccumulativelaw,i.e.theincrementsarelog-normallydistributed.Onthebasisofthepreviousresearch,weshowthefundamentaldescriptionoftheaccumulativeprobabilityasfollow:Definition3.1.Thefunctionf(x),x ≥ 0iscalledtobeanaccumulativedistributionofacontinuousrandomvariableXhavingarateparameterα,α > 0,if (1)f(0) = 0.(2)Thevariablehasindependentincrementsfollowingalog-normaldistributionwithparametersμandσ.(3)TheprobabilityoftheaccumulationofXistheprobabilitydensityfunction,thatis,forallx ≥ 0 f(x)=∫0xαyα−12πσxα+1 e−(ln⁡y−μ)2/2σ2 dy.3.1Thedefinitionincludesallconditionswhichanaccumulativedistributionhastofollowandthepreciseprobabilitydensityfunction.Condition(i)simplyshowsthattheaccumulationofthesystemstartsatt = 0,andthecondition(ii)isanothersignificantdescriptionthattheincrementsintheprocessofaccumulationshouldbeindependentandlog-normallydistributed.Condition(iii)isthefinalmathematicalexpressionofthisprobabilitydistribution.Toensurethecorrectnessofthisexpression,wenextprovethatitisaprobabilitydensityfunctioninstatistics.Theorem3.2.Thefunctionf(x)describedindefinition3.1isaprobabilitydensityfunction.Proof.Inordertoprovethatf(x)isanaturalprobabilitydensityfunction,weneedtoproofthatthisfunctionisnon-negativeeverywhere,anditsintegralovertheentirespaceisequaltoone.Basedoncondition(i),forx ≤ 0,f(x) = 0.Otherwise,byequation(3.1),f(x) > 0.Therefore,thenon-negativefeatureholds.Forally,having0  1.Proof.Consideringthecontinuityoftheaccumulativevariableandtheintegrationorder,alsointroducingt=(ln⁡y−μ)/2σ,wehavetheexpectedvalue E[x]=∫0+∞xf(x) dx=∫0+∞∫y+∞αyα−12πσxα e−(ln⁡y−μ)2/2σ2 dx dy=αα−1∫0+∞12πσ e−(ln⁡y−μ)2/2σ2 dy=αα−1∫−∞+∞12πσ e−t2/2+tσ+μ dt=αα−1 eμ+σ2/2.3.7Theresultfollows. ▪Apparently,ifweletα ≤ 1,thenthevalueofE[x]isequalto+∞.Wesaythattheexpectedvaluenolongerconvergesinthiscase.Inotherwords,theexpectedvalueofanaccumulativevariableexistsonlyifα > 1.Anotherfactwecanlearnfromtheexpressionoftheexpectedvalueisthatitispositivelycorrelatedwiththeparametersμandσ,whilenegativelycorrelatedwithα.Practicallyspeaking,fromthederivation,weknowthatthevalueofμandσisrelatedtothevalueoftheincrement,andαindicatestheproportionrateofthetotalaccumulationtotheincrement.Wecantheninferthatthemoreincrementsanditslessproportionrateleadtoalargeraccumulationofanindividual,andviceversa.Theresultperfectlyfollowsequation(2.16)in§2,whichdisplaystherelationshipoftheaccumulationvariable,theincrementvariableandrate.Basedonthis,wemayeasilyestimatethepossiblevalueofdifferentaccumulationdistributions.Basedontheexpectedvaluewhichisalsocalledthefirstmoment,wecanextendittoanothercentrestatisticalproperties,thenthmoment.Thenthmomentisknownasapowerfultooltostudytheshapeofourfunction,itsexpressionisasfollows:Theorem3.6.Thenthmomentofanaccumulativevariableis(α/(α−n)) enμ+(1/2)n2σ2,ifα > n.Proof.Analogouslytotheorem3.5,andintroducingt=(ln⁡y−μ)/2σ,wehavethentheexpressionofthenthmomentas μn=∫0+∞xnf(x) dx=∫0+∞∫y+∞αyα−12πσxα+1−n e−(ln⁡y−μ)2/2σ2 dx dy=αα−n∫0+∞yn−12πσ e−(ln⁡y−μ)2/2σ2 dy=αα−n∫−∞+∞12πσ e−t2/2+ntσ+nμ dt=αα−n enμ+n2σ2/2.3.8Theresultfollows. ▪Fromthisderivation,welearnthatthenthmomentexistsonlyifα > n.Obviously,themomentisalsopositivelycorrelatedwiththeparametersμandσoftheincrementvariable,whilenegativelyrelatedtotheproportionrateofα.Themomentshowsamoregeneralwaytostudytheaccumulativedistribution,sincewecaneasilygettheskewness,kurtosis,andevenhigher-ordermomentfromit.Apartfromthecentreproperties,wepresentadispersionproperty,standarddeviation,whichmeasuresthedispersionofasetoftheaccumulativedatafromtheexpectedvalue.Theorem3.7.Thestandarddeviationofanaccumulativevariableisα e2μ+σ2[(1/(α−2))eσ2−α/(α−1)2],ifα > 2.Proof.Byusingtheorems3.5(n = 2)and3.6,thestandarddeviationiscalculatedas SX=E[(x−E[x])]2=μ2−E[x]2=α e2μ+σ2[1α−2eσ2−α(α−1)2]3.9Theresultfollows. ▪Corollary3.8.Thestandarddeviationofanaccumulativevariableforα > 2exitsonlyifσ > ln(α(α − 2)/(α − 1)2).Proof.Theexistenceofthestandarddeviationisrelatedtotheparameterσ.Thereasonisthatthevalueunderthesquarerootshouldbelargerthanzero,thus α e2μ+σ2[1α−2 eσ2−α(α−1)2]>0.3.10Obviously,itequalsto 1α−2 eσ2−α(α−1)2>0.3.11Thenwehave σ>ln⁡α(α−2)(α−1)23.12 whichisanotherconditionthatthestandarddeviationwillexit.Theresultfollows. ▪Sincethestandarddeviation,whichmeasureshowfarasetofrandomnumbersarespreadoutfromtheiraveragevalue,usesthesecondmomentandthevalueunderthesquarerootshouldbepositive,itexistsonlyifα > 2andσ > ln(α(α − 2)/(α − 1)2).Differentfromtheabovestatisticalproperties,thestandarddeviationispositivelycorrelatedwithallparametersα,μandσ,i.e.thelargerincrementanditsproportionratewillletthevalueoftheaccumulativedatamoredispersed.Basedonthisresult,thevarianceisαe2μ+σ2[(1/(α−2)) eσ2−α/(α−1)2],whichisthesquareofthestandarddeviation.Furthermore,weshowtheskewnessoftheaccumulativedistributiontomeasuretheasymmetryofthedistributionaboutitsexpectedvalue.Theorem3.9.Theskewnessofanaccumulativevariableisexpressedas1α−3 e3σ2−3α(α−1)(α−2) eσ2+2α2(α−1)3α1/2[1α−2eσ−α(α−1)2]3/2,if α>3.Proof.BythedefinitionofPearson’smomentcoefficientofskewness,theorems3.6(n=2and3)and(3.7),wehave γ=E[(x−E[x]s)3]=E[x3]−3E[x]E[x2]+2E[x]2s3=μ3−3E[x]μ2+2E[x]2s3=1α−3 e3σ2−3α(α−1)(α−2) eσ2+2α2(α−1)3α1/2[1α−2eσ−α(α−1)2]3/2.3.13Theresultfollows. ▪Obviously,theskewnessinvolvesthethirdmoment,wethusrequireα > 3toletitexist.Whatwecareaboutourdistributionisthatwhetherithasapositiveoranegativeskew,whichappearsasaleft-leaningcurveoraright-leaningcurve.Toaddressthisissue,wehavethefollowingcorollary.Corollary3.10.Theaccumulativedistributionforα > 3,ifitsstandarddeviationexists,hasapositiveskew.Proof.Toprovetheskewthatanaccumulativedistributionhasispositive,weonlyneedtoprovethatitsexpressionofskewnessisgreaterthan0.First,weconsiderthenon-negativenessofitsnumeratoras 1α−3 e3σ2−3α(α−1)(α−2) eσ2+2α2(α−1)3>1α−2 e3σ2−1(α−2)2 eσ2>1α−2 e3σ2−1(α−2)2 e3σ2=(α−3)(α−2)2 e3σ2>0.3.14Then,forthedenominator,referringtoremark2,weknowthatthestandarddeviationexists α1/2[1α−2 eσ−α(α−1)2]3/2>0.3.15Overall,weapparentlyconcludethatγ > 0,i.e.thedistributionhasapositiveskew.Theresultfollows. ▪Fromthiscorollary,wesaythattheaccumulationisright-skewedandright-tailed,rightreferstotherighttailbeingdrawnoutandthemeanbeingskewedtotherightofatypicalcentreofthedata.Then,weeasilyinferthatthoseindividualsholdingalargevalueoftheaccumulationistheminority,andmostoftheindividualsonlypossessalowvalue,whichisaconsequenceofMattheweffectandalsoasignificantpropertyofourdistribution.Sofar,weshowthemoststatisticalpropertiesoftheaccumulativedistributionfrequentlyused.Ourdistributionhasmathematicalexpressionsforthemanddisplaysitsinimitablecharacteristic.4.TheaccumulativelawappliedtorealsystemsTheaccumulativelawanditsprobabilitymodeldisplaytheircharacteristicsinverymanyfields,lettingaphenomenonoftheaccumulativedistributionverycommon.Fromthepracticalperspective,wespeciallypresentthemodellingofcomplexnetworksstudiedinnaturalsciences,theaccumulatedwealthinvestigatedbyeconomistsandtheInternettrafficininformationscienceinthissection.Allofthemfollowtheaccumulativelawtocreatetheirsystemsandaccumulativelydistributedincertainways.(a)ComplexnetworkanddegreedistributionComplexnetworks,asatypicalrepresentativeofcomplexsystems,showvarioustopologicalcharacteristics.Oneofthemostsignificantonesisthenon-uniformdistributionofdegreesofvertices.Here,wemainlyproposeacomplexnetworkmodelbasedontheextendedscale-freenetworkswhichhavethepropertyofthevertexgrowth,variableconnectionsandpreferentialattachmentresultinginaspecificdegreedistribution.Wewilldemonstratehowexactlythenetworkfollowstheaccumulativelawandthedegreedistributionfollowsanaccumulativedistribution.(i)ModellingprocessFirst,wedemonstratethemodellingprocessofimprovedscale-freenetworks.Initialization:Assumingthatthenumberoftotalverticesoftheinitialnetworkisn.Eachofthemlinkstokneighboursandhastheprobabilityptolinktoothers,whichisasmall-worldnetwork.Growthisasignificantstepforanimprovedscale-freenetwork,whichconsistsofthevertexgrowingrateanditsconnectiontoexistingvertices.Ateachtimet,weaddλvertices.Intheinterval[t,t+△t],theprobabilityofthenumberofnewverticesisthen P{N(t+δt)−N(t)=k}=(λδt)kk! eδt,k=0,1,…4.1 Besides,foreachvertex,weconnectmedgestothemdifferentverticesalreadyexistinginthenetwork,wheremfollowsalog-normaldistributionwithparametersμandσ.Connectionissimplylinearlydependentonthedegreeofthetargetbasedontheknowledgeofscale-freenetworks,ϕ(i),theprobabilityofaconnectiontoavertexi,isdenotedas ϕ(i)=ki∑jkj,4.2 wherekiisthedegreeofvertexi.Terminationiscontrolledbytimet,whichdirectlyaffectsthescaleofnetworks.Obviously,thedegreeofverticesistheaccumulativevariableofanetwork,whichisdenotedbyp(k).Fromthemodellingprocessofimprovedscale-freenetworks,wecanapparentlyseethatitfollowstheaccumulationlaw.TheGrowthstepandConnectionstep,respectively,interpretthegrowingprocessandallocatingprocessofthelaw.Thuswesay,theverticesareconsideredasindependentindividualsofanetwork,andtheconnectionstoexistingverticesareincrements.Forabetterunderstanding,wepresentthemappingtermsbetweenthecomplexnetworksandaccumulationlaw(describedin§2a)intable1.Fromthistable,weillustratethepracticaldescriptionofanetworkcorrespondingtotheabstracttermsintheaccumulationlaw.Thusthenetworkturnsintoanaccumulationmodelanditsaccumulativedistributionofdegreecanbecalculatedbyourmethod. Table 1.Themappingbetweentheaccumulationlawandimprovedscale-freenetworks. Collapse thetermsinaccumulationlawthetermsinimprovedscale-freenetworksindividualxvertexiincrementyconnectionmproportionrateαequalto2probabilityPprobabilityϕaccumulationzdegreekaccumulativedistributionf(z)degreedistributionp(k)(ii)DegreedistributionWeapplyourmethodfrom§2btocalculatethedegreedistribution,anditisworthnotingthattheproportionrateofcomplexnetworksareequalto2sinceeachedgehastwovertices.Then,thedifferentialequationofdegreeis ∂ki(t)∂t=ki2tandki(iλ)=m.}4.3 Itssolutionis ki=m(λti)4.4 whichshowsthefunctionalrelationshipamongthedegree,connectionandvertexnumber.Furthermore,thecumulativefunctionisexpressedas P{ki(t)