鲁棒竞争网络

1M.Schmutz,W.Banzhaf.Roustcompetitivenetworks.PhysicalReviewA,1992,Vol45No.6.4132-41552000/4/22[1][2]2[1][3])()1(qVNRq∈)()1(.qVq-∇=42)1(412121)(∑-+-=iiqDDqV2∑=iiqD3.)21(iiiqqDq+-=iqiqiq≥)0(iq0t0)(≥tqi{}NiqRqRiNiN,,1,0L=≥∈=+)(tq)('tq2.4NR+1+mm0mmmNmm1241)1(-⋅-=mmVm341)1(1=V⎩⎨⎧=-==-NmNmqi21)12(00i)(∞→N0)1(0=V81)1(-=∞Vij)0()0(jiqq≥t)()(tqtqji≥)0()0(0iiiqq≠0,100==≠iiiqq0i)0(0=≠iiq)(No)1(V)0(V)1()0(VVV+=4iq)0(VV)0(VnjjiiqqnqV2,)0()(21)(-=∑a,,2,1,0L=≥na)0(V)1()0(VVV+=Vq-∇=.a0=adnad1=n2=nd)(xqdRx∈][][][)1(0(qVqVqV+=ndqdxnqV2)0()(21][∇=∫a42)1(412121][qdxDDqVd∫-+-=2][qdxqDDd∫==5qqVqdd][.-=0)(qxq=0][0][0)(===qVqDxq0)(qxq±=)48(][)12(][)12(1)(2--=-=-=NNqVNNqDnxq∫=ddxN∞→N21][=qD81-=V81][-qVVxV][][qVqV-=)(xq±Vxreqrqsr==-,)(2)(060qsd2][qVd220222)(ddddsqxqdxD--==∫pddddsqxqdx--=∫2)(4024pddnndndsqnxqdx--=∇∫22)]([22022mp)2()2(2)2()2(2dndnnddnΓ+Γ-=-mV),(0sqV),(0sqV0),(0),(000=∂=∂sqVsqVsq1)12(42)2(20=+--nnsqndnam122])1(24[2220=+--+--dddsqdnnndp1=nΙ2=nΠΑ4.94.101=n225202sqa=4.11031382=+-apss4.12pa163=c4.13caa≥7)11(34)(aapacss--==4.144.11⎥⎦⎤⎢⎣⎡--=122132381)(2040qqVa4.1561,34,322520-===cccVsqp4.16acaa112720)8(,)2(--≈≈appasq4.17pa256181--≈V4.18∞→a81-=V∞→aexact4.94.10paaa81),1(420=-=ccq4.19212⎥⎦⎤⎢⎣⎡-=ccsaaaa4.20⎥⎦⎤⎢⎣⎡-=122cVaapa4.21ca4.20a4.20caa81)(-=cVa81)(-aV0)(,0)(0==ccsqaaΙ842≤≤d(4.9)4.101=n]2)2(2)4[(16181202403qdqddVdd----+-+-=4.2242≤≤d81-≥V.Β2=n4.9142220]1)[(2)4(---+=dcddqaa4.23414)4)(2(-⎥⎦⎤⎢⎣⎡++=adds(4.24)⎥⎦⎤⎢⎣⎡-+-=1)(11814dcVaa(4.25)cad124)]4)(2([)]4(2[4-+++=ddddcpa4.2681-≤Vcaa≥IIΙ1=d42≤≤d81-4.321=d4.4dddsq212202--=p4.2700→q0→snd0)(→xq81][-→qVVI[8-10]9)]([xqV)()(rqxq=xr=d220142201214121'2)]([⎥⎦⎤⎢⎣⎡⋅+⎥⎦⎤⎢⎣⎡--⋅=∫∫∞-∞-qrdrqqqnrdrrqVddnddgag5.1'q)(rq)12(/2+Γ=ddddpg5.2d)]([rqV-0)(=rqVdd),('1'')12(')1(2AqUdqdqrdqnqn-=⎥⎦⎤⎢⎣⎡-+--a5.3424121),(qAqAqU+-=5.41)]([2)]([-==rqDrqAA5.5201)]([qrdrrqDdd∫∞-⋅=g5.6),(AqUA5.55.6)(rq-intergo-diffentialA5.35.55.6self-consistency)],([ArqAA=5.7),(Arq81-),(Arq,10∞→V)(rq00∞→rD][qV221D[3.4-3.7]∞→][qVnon-normalizableqr5.31n),(Arq)',(][01qqLrdrqSdd∫∞-⋅=g5.8),('2]',[2AqUqnqqLn-=a(5.9)HamiltonianA),('2122AqUqnnEn+-=a5.10⎩⎨⎧=≤==--=2,01,0'1'2ddqrdEna5.111=d0≠Er[2a2b]0=EA,0)(=rq;0,0=AE0=r210)2(Aq=0=q[2b]1=d81-=V5.100=EnAqUnnq21),()12(2'⎥⎦⎤⎢⎣⎡---=a5.125.72≥dELyapunov[11(5.3)]∞→r),(AqU0≤A∞→r0=q0),(ArqBΙII2,1=n0)(=rq0A),(AqU∞→r0)(=rq0AqU,(03nodeVIIdΙ1=dVI5.1QAq2=6.1nnnAnnanAaa2122212),,(⎥⎦⎤⎢⎣⎡-==-a6.25.12nnQQAaQ2121)1()(1'--=6.3),(AQrr=nQnqqdqAar21211)1()(---⋅-=∫6.4126.3'QQ),(ArQ∫∫∞-=⋅010)('))(',()',(QQQQQFdQQQFdr6.51)(8-=AAdA6.6nnqqdqAaAd2121012)1()()(---⋅=∫(6.7)6.46.76.6n1=n2elementaryΑ6.16.21=n1002cosh)(-⎥⎦⎤⎢⎣⎡=rqqrqa6.80qAAq20=(6.9)0q)11(2)(00aaaaaccqq--==6.10161=ca6.11interst021)8(][qqDa=6.12133021200216)2(4)2(][qqqqVaaa-+-=(6.13)6.10)(00aqq=∫∫∞∞⋅⎥⎦⎤⎢⎣⎡⋅=Δ0022)()(rqdrrqrdrx6.146.142102⎥⎦⎤⎢⎣⎡=Δapqx6.15ca8,220p=Δ=ccxq6.1661,1-==ccVD6.17caapaa4,32120≈Δ≈xq6.1821≈D6.19a768181--≈V6.20Β2=n6.441211)(⎥⎦⎤⎢⎣⎡-==qqtt6.21Α)(rQQ=⎥⎦⎤⎢⎣⎡-Θ+-++++-=)1(212arctan21212ln2)3(2222341tttttttrpa6.22)(Qtt=)(tΘstep140rr≥)(rQpa2)3(410=r6.230rr≤)(rQ20)(341)(rrrQ-≈a6.240r),(AqU0=q6.76.6A1210-=rA6.251)3(2224120-==apAq(6.26)0qcaa4121pa=c6.2720rpa22)3(41≈Δx6.28V6.5216181][qqV--=6.296.136.29ΙII81][-qvVII15scalinglawsscalinglawsa1=aC2=dΙII2,1=d4a-4c1=da0=a∞→aΙIIaaΙ50aΙΙ0a2=dsigulartransition4=d6a-6bIIΙIII78III[8b]VIII16M.SchmutzVolkswagen-StiftungW.BanzhafJ.BellΑHamiltonian5.85.9)',(][01qqLrdrqSdd∫∞-⋅=gA1),('2)',(2AqUqnqqLn-=aA2A11-dr1dA1A1LLrdLdrdqqq∂=∂-+∂''1)(A3HamiltonLpq'∂=(A4))',('),(qpLpqpqH-=(A5)),(''pqqq=HamiltonHqp∂='A6Hprdpq-∂=-+1'A71=dA7))(),((rprqHE=A81d'1)('pqrdrE--=A9A2-17),('1'')12(')12(AqUdqdqrdqnqn-=⎥⎦⎤⎢⎣⎡-+--aA10)12()('-=npqaA11),(1'AqUdqdprdp-=-+A12),('212)(2AqUqnnrEn+-=a(A13)⎩⎨⎧≥≤==--=2,01,0'1)('2ddqrdrEnaA142≥dA11A120)0(=p0=rA12denomiatorr10≤rruln=)()()()(ugrpufrq==A15),()1(')(')12(1AfUdfdegdgagefunu-=-+=-A16-Β-5.31dr0≤Ascalinga1=a1,0=A1=d18EAqUqnnn=+-),('2122B1441)0,(qqU=B241)1,(-qUB3Ε),(Erq0rrr+→rr-→scalingansatzB1E),(Erq1,0-=A⎩⎨⎧-===--)5(1),()4(0),(),(2)1(1214)2(141BArEqEBArEqEErqnnnn)1,()(1==Erqrq1=E2≥dr∞→r0)(rEE=(B4)(B5)EnqrdrE2'1)('--=(B6))(ER)('rE)('rE)()(1'EREerdE⋅--≈(B7)[]∫+=)(2),(')(ERrrnEqdEerr(B8)),(Erq)(Ee)(ER1,0=A)(rEErEk-='(B9)19⎪⎩⎪⎨⎧-=--=--=1132)1(0254)1(AnndAnndkB10B9k-rrE~)(B11B4B5r))(,()(rErqrq=D210)(uqudud∫∞⋅lbgB12⎪⎩⎪⎨⎧-=----=----=1,13)1)(1(10,25)2)(1(1AnndAnndbB13⎪⎩⎪⎨⎧-==--=1,00,)25(2)1(AAnndblB14)(1rq1=ED1-l1-=An2d0=A0≥l1=n42≤≤d2=n2≥dCScalingLaws5.35.4aA5.65.7qra1=a1=a)()(1rqrqml=C1)(1rq1=aansatz1242=-nnmalC22012AAl=C3112]1)1(2[--+-=dDmlC4D12DDd-=mlC52121+-=DKV2DC6∫⎥⎦⎤⎢⎣⎡-∇=4241)(21qqndxKndaC7C214KKd-=mlC811)(xxΔ=Δ-mC9010qql=C101D1K1=a1)(xΔalmC2C4C22=nII41-=amC111142]1)1(2[-+-=DdalC12[1]H.Haken,inComputationalSystem,NaturalandAtificial,ProceedingoftheElmauInternationalSymposiumonsynergetics1987,editedbyH.Haken(Springer,Berlin,1987),p.2.[2]Anotherwaytointroducerobustnessistointerpretthecompetitionastheeffectivedynamicsofentirepopulationsofneurons;seeW.BanzhafandM.Schmutz,Int.J,NeuralSyst.(tobepublished)[3]H.Haken,SynergeticComputerandCognition(Springer,Berlin,1991).21[4]J.A.Feldma