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M.E.J.NewmanDepartmentofPhysics,UniversityofMichigan,Annarbor,MI48109,USAandSantaFeInstitute,1399HydeParkRoad,SantaFe,NM87501,USA200052ABCABCDABC1.2.DEFGHIAPoissonB1.2.3.4.5.VABCAPriceBBarabasi-AlbertCBarabasi-AlbertDEAB1.SIR2.SISC4123DE567==============================================================81[96]·[244]MarvelComics[10]23α−α4Citeseer()SPIRES()105[3]6768)1(21−nnpnl∑≥+=jiijdnnl)1(2111iijdj)1/()1(+−nnl1−nlnlnml)(mnOllll∑≥−−+=jiijdnnl11)1(211213ijdlnmzα)1(C)2(Crrrllnlognloglnnloglog/log10×=3C3CCC26×=C4CCiiCi=50iCiC∑iiCnC16CC)1(C)2(C11UCInet15C83613013=CiCiCiCiikiCik1−ik∞→n)1(OC=Cn)(1−=nOCnCkkpkpkkpnkp1−k121~−iikC∞→n3616∑∞=′′=kkkkpP7kαα−kpk~1−αα)1(~~−−∞=′−∑′ααkkPkkk8κ/~kkep−κκ//~~kkkkkkkkeepP−∞=′′−∞=′′∑∑=9kkjkpjiijkp17jααmaxk1=knpα−kpk~α/1max~nkmkkmnkmkPpmn−−⎟⎟⎠⎞⎜⎜⎝⎛)1(kPkkhnknkknmmnkmkkPPpPpmnh)1()1()1(1−−−+=−⎟⎟⎠⎞⎜⎜⎝⎛=∑=−10∑=kkkhkmaxkkhkkpdkdP=/0)1()1)((11=−+−+−−−nkknkkkkPpPppdkdp1113)(xfx)()(xbfaxf=18maxk2kknpdkdp−≅12nkhmaxkk≥kp1knp1kPα−kpk~)1/(1max~−αnk13k1=knP14Albert1920iijEjijEijEEEe=14XXiijeji)|(ijPj∑=jijijeeijP/)|(1=∑ijije1)|(=∑jijP1511)|(−−=∑NiiPQi16QQ221eeTrer−−=17rrr21kkk1522n)1(21−nnn16B1723niiRapoportd[346378],ErdösRényi[141,142]PoissonMolloyReed[287,288]Markovp*WattsStrogatz[416]Price[344],BarabásiAlbert[32]24Rapoportd[346378]10ErdösRényi[141]PoissonBernoulliPoissonSec.CFig.6,PoissonPoissonBollobás[63]Jansonetal.[211]Karoński[223]SolomonoffRapoport[378]ErdösRényi[141]n()p1-p18,ErdösRényiGn,p.Gn,pmmpm(1-p)M-mM=1/2n(n-1)ErdösRényiGn,mnm19Gn,pGn,mErdösRényi1960[141142143]nz=p(n-1)Poissonk:nkPoissonpSolomonoffRapoportErdösRényiphasetransitionsizesizeppi.e.O(n)sizesizeukukkEq.(18),sizeuSS=1-u,18(),,n.19,canonicalgrandcanonicalensembles.,,HelmholtaGibbs,,pm.25sizesS,sFig.10.20z1S=0,z1z=1sSsS~(z-1)βs~׀z-1׀-γβ=1γ=1z=1doublejumpsizeO(n2/3)z1O(n)ד=5/23/2sizelzddzlnl=logn/logzA[61][63]pC=psize[416]n-1Fig.6Poissonlocal[238239314318401]PoissonPoissonB.3--B.526pkpkkkini=1…nikistubs”spokes”stubs[287]20211970[4647608889268287288323425]pk[287][288][323][88]Newmanetal.[323]Refs.88,89,287,288.pksizepkkkk1kkpkexcess1excessqkz=∑kkpkn-1O(n-1)O(n-1)22Sec..Bpkqk23Eq.(22)G1(x)=G0(x)/zH1(x)xH120∏iki!,stubs.,,.21ChungLu[88,89].,iki(),(i,j)kikjm=1/2∑ikiPoisson.[78,128,174]22(31),,z,k2n1/2,0.25.23,z,z,x.27qkG1Eq.24Ref.323sizeH0(x)2425sizesizez1=z=k=G0(1)z2=k2-k=G0(1)G1(1)z1=z22327MolloyReed[287]SuEq.(20)u=G1(u)SRef.288uSEq.(26)sizePoissonEq.(28)sizez2/z1PoissonzPoissonβ=γ=1ד=5/2Eq.(3)[136319]PoissonC=z/nzk2/k2Cn-1Fig.6k2/k2C28αζ(α)RiemannζEq.(23)Lin(x)xnEq.27ααc=3.4788…,ααcEq.(29)uα=2u=0S=1size2512ααcEq.(29)Aielloetal.[8]Eq.(31)α3k2~k3-αmaxkmaxEq.(13)kmaxα2αα7/3CPoissonα3C~n-1α=7/3Csizeα7/324α25[319]jkSec..Cpjk24,1..25α7/3,,.Aielloetal.[8],kmax∼n1/α(secIII.C.2),n→∝C→0,α2.,,.29AAAAA[323]excesssize[125323]Sec..ACEOs[167168][104105269][313][416]Senetal[366]µνM,Nµ/M=ν/Nexcessf0(x),f1(x),g0(x)g1(x)f’1(x)g’1(x)=1size[323](projecting)one-mode”one-modeone-modesizeCµnνnn[318,374]30Sec..F[314]Sec..FejkexcessjkEq.(29){uk}detI-m=0mmjk=kejk/qjnon-trivial[58318400]BergLässig[48]Maslovetal[275]InternetInternetMaslovetal[47407]MaslovetalMonteCarloInternetsizeCapoccietal.[83]Sec..BBarabsiAlbert()(transitivity)(Sec.III.B)(Sec.IV.B.4)dual-graph[345][317][194160385][416]Strauss∗p[22,410]Boltzmannniεiβn26,,[48,77].31iβiεZflog−=iβfiβnp−1'G'iεiε32Lkp0=p)24/()33(−−=kkCk4/3LkL4/1=pkLlog/logLkC/2=27,Balletal.[28],,.33pLkp)1(2pLk+k2j34kj2≥kj20=jpkj≥kj0=jpll0→pkLl4/=Lllog~plξp)(xgξ,Lpξ0→pξl)(xgpτξ−p~τ3/2=τlkkp/1=ξ)()(xxfxg=1=τplLLkpk)(xf710=L)(xf35)(xf28,.29,156,166,395,417,418.29,Price,Polyaurn.Price[344].36knkp1=∑kkpmmmkpkk=∑mk0kk+0k10=k1+kkmk)1/()1(++mmpkkkknp1+k1−kknnkp,kpknp1≥k1=kknknkppp==+,1,)12/()1(0++=mmp)/12/(1mkkppkk++=−30,z.z,m,m.m,.37)(/)()(),(bababaB+ΓΓΓ=βbaba−nm/12+=ααmm10=kα0k10≠k0kmmmm1≥mα3=α31KrapivskyRendner[246].38km2mmk2/2/kkkpmmkp=kmknp1+k1−kknnkp,kpknpmkmk=mkknknkppp==+,1,kp)2/(2+=mpm)2/()1(1+−=−kkppkkkkp3~−kpk3=α1=miaana→2/1)/1(−−na391=mjkejkjk∞→nn)(nO)(2nO)1(O)(nOiik0k0kmmjcj+cmmjk+=mck−+mc=k40k0kk+0k0k),(∞−m)2/()2(020kmmkmpm+++=)(/)()(),(bababaB+ΓΓΓ=βkmk/30+=α0k3αkrk1=r1rr1r2rm3241mnanaaBnk+B)1/()1(2BaaB−++=α0=aαiiηiikiikη)(ηρ)(ηρ)(ηρiηi42mmm1m1=mmspSmspsp∑∞==0)(sssxpxH33,,..Web,,,...,,,.431)1(=H)1('Hs=1→xs0122=+−ssm8/1=m8/1=mmmmmmmmkk)1/()2(aa−−=αa44a∈a2/1,032≤≤α45kkkpnkkpq′k′kkkqqkk′−′−⎟⎟⎠⎞⎜⎜⎝⎛′)1(k′kkkkkkkqqkkpp′−′∞′=′−⎟⎟⎠⎞⎜⎜⎝⎛′=∑)1(77α3≤αqcq0≤cqkqkqk∑∞==00)(kkkkxqpxF∑∑−=kkkkkkkpxqkpxF11)(78)(0xH))(()1(1)(1000xHxFFxH+−=79a))(()1(1)(1111xHxFFxH+−=79b1)1(0≠F0F46)1(1)1()1()1(1100FFFFs′−′+=801)1(1=′F)()1(00uFFS−=)()1(111uFFu+−=81kqqk=)1()2()1(−−−−=αζαζαζcq823αcqα1cqcααcα)1(2)2(−=−αζαζcα)(maxkkqk−=θ)(xθmaxkααcqα47iiφ{}iφ)(φf0Φ10Φk/1φ∫=kkdfq/10)(φφ34[229]35SIRB.2SIS36infectiousinfective48βγisrisdtdsβ−=iisdtdiγβ−=idtdrγ=83β)(βiPβγ)(γrP∫∞−−=0/)()(1γβγβγβddePPTri84T)(0xH))(()(100xHxGxH=85a))((1)(111xHTxGTxH+−=85b)(0xG)(1xG)1(/11GTc′=s⎥⎦⎤⎢⎣⎡′−′+=′=)1(1)1(1)1(100