{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 1 12 0 0 255 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "Map le Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "Left Justified Maple Ou tput" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 2 0 2 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "#### K is the number of discrete in tervals. You can try large values like 10,000 if you want" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "K:=300;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG\"$+$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f:=i->sum(1/j,j=i..K)/sum(1/j,j=1..K);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"iG6\"6$%)operatorG%&arrowGF(*&-%$sumG6$*& \"\"\"F1%\"jG!\"\"/F2;9$%\"KGF1-F.6$F0/F2;F1F7F3F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot(\{f(i),log(K/i)/log(K),1-(i-1) /K\},i=1..K);" }}{PARA 13 "" 1 "" {GLPLOT2D 605 605 605 {PLOTDATA 2 "6 '-%'CURVESG6$7fn7$$\"\"\"\"\"!F(7$$\"3C$e*[=VL29!#<$\"3!e1-^[G4S*!#=7$ $\"3pm\"zpjoY\"=F.$\"3Ue%*eN'\\_&*)F17$$\"3:](oa&H+AAF.$\"3bGXR28@+')F 17$$\"3RL$eRFP$HEF.$\"3%H>!zpR50$)F17$$\"3I+v$4\"f+WMF.$\"3T2sy7J!>$yF 17$$\"3ymm\"zau'eUF.$\"3aX*z=Aa'fuF17$$\"3g+](=#=,))eF.$\"3=:+x<&z;*oF 17$$\"3bLL$e4\\t^(F.$\"3OCZS9dPjkF17$$\"32]PfR/FN5!#;$\"3RW$p)4XG-fF17 $$\"3zmTgpf!)=8FX$\"33ZM&GK#*yZ&F17$$\"35]i:F8nP;FX$\"3aw19@XB)4&F17$$ 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Fchl7$$\"+:o$)RCFe_m$\"*af)[LFchl7$$\"+]![=]#Fe_m$\"*Wh&[HFchl7$$\"+& \\j_c#Fe_m$\"*#>N\\DFchl7$$\"+FQ:GEFe_m$\"*X'4j@Fchl7$$\"+6N%fo#Fe_m$ \"*YYi\"=Fchl7$$\"+6q<_FFe_m$\"*<'zF9Fchl7$$\"+UqT6GFe_m$\"*A:#)3\"Fch l7$$\"+s#zX(GFe_m$\")xdRtFchl7$$\"+&eL]$HFe_m$\"))p6-%Fchl7$Fd]l$\"(0c I&Fchl-Fh]l6&Fj]l$\")'>!\\&)F]^l$\")#)eqkF]^l$\")?!\\D\"F]^l-%+AXESLAB ELSG6$%\"iGQ!6\"-%%VIEWG6$;F(Fd]l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "#### Precise results -- here you should better keep small value of K" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "K:=4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "#### parameter p " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "p:=0.05;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG$\"\"&!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "#### try some initial conodition" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "#### a) random graph, that is, y[i] is constant" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "# icDE:=seq(y[i](0)=1/K,i=1. .K-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "#### b) short lin ks only" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "# icDE:=y[1](0)= 1,seq(y[i](0)=0,i=2..K-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "#### c) long links only" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "icDE:=seq(y[i](0)=0,i=1..K-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%%icDEG6%/-&%\"yG6#\"\"\"6#\"\"!F-/-&F)6#\"\"#F,F-/-&F)6#\"\"$F,F-" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f:=x->1-sum((K-l)*y[l](x)/ K,l=1..K-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%) operatorG%&arrowGF(,&\"\"\"F--%$sumG6$*(,&%\"KGF-%\"lG!\"\"F--&%\"yG6# F46#9$F-F3F5/F4;F-,&F3F-F-F5F5F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "systemDE:=seq(diff(y[i](x),x)=f(x)*p/K-y[i](x)*i*p/K, i=1..K-1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)systemDEG6%/-%%diffG 6$-&%\"yG6#\"\"\"6#%\"xGF0,*$\"++++]7!#6F.*&$\"+++](=#F4F.F*F.!\"\"*&$ \"++++]i!#7F.-&F,6#\"\"#F/F.F8*&$\"++++DJF " 0 "" {MPLTEXT 1 0 25 "dsol:=dsolv e([systemDE]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%dsolG<%/-&%\"yG6# \"\"#6#%\"xG,.*&%$_C1G\"\"\"-%$expG6#,$*&\"#K!\"\"F-F1F8F1F8*&#\"\"$F+ F1*&%$_C2GF1-F36#*&,&#F1F7F8*&\"$g\"F8\"\"&#F1F+F8F1F-F1F1F1F1*&#F1F+F 1*(F=F1FEFFF>F1F1F8*&F:F1*&%$_C3GF1-F36#*&,&#F1F7F8*&FDF8FEFFF1F1F-F1F 1F1F1*&FFF1*(FLF1FEFFFMF1F1F1#\"\"'\"#DF1/-&F)6#F1F,,.*&#F1F;F1F/F1F8* (F+F1F=F1F>F1F8FIF1*(F+F1FLF1FMF1F8FTF8#\"#7FWF1/-&F)6#F;F,,*#\"\"%FWF 1F/F1F " 0 "" {MPLTEXT 1 0 12 "simplify(% );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<%/-&%\"yG6#\"\"\"6#%\"xG,.*&#F) \"\"$F)*&%$_C1GF)-%$expG6#,$*&\"#K!\"\"F+F)F8F)F)F8*(\"\"#F)%$_C2GF)-F 36#,$*(\"$g\"F8,&\"\"&F)*$FB#F)F:F)F)F+F)F8F)F8*(F;F)F " 0 "" {MPLTEXT 1 0 75 "dsol2:=dsolve([systemDE,icDE ], numeric, range=0..100,output=listprocedure):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 61 "for i from 1 to K-1 do test[i]:=eval(y[i](x),d sol2) ; end do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%testG6#\"\"\"f* 6#%\"xG6'%$resG%%dataG%)solnprocG%)outpointG%(y[1](x)G6#%inCopyright~( c)~2000~by~Waterloo~Maple~Inc.~All~rights~reserved.G6\"C)>%8_EnvDSNume ricSaveDigitsG%'DigitsG>F7\"#9@%/%-_EnvInFsolveG%%trueG>8'-&%&evalfG6# F66#9$>F?-FBFD>8%-%%evalG6#&%8&-_ FI%$GetG6#Q/soln_procedureF3@$4-%%typeG6$F?.%(numericGC$@.-%'memberG6$ FE7+Q&startF3.%&startGQ'methodF3.%'methodGQ%leftF3Q&rightF3Q)leftdataF 3Q*rightdataF3Q+enginedataF3C$@%-F\\o6$FE7$F`oFco>8$-FR6#-%(convertG6$ FE.%'stringG>F`p-FRFD@&-Fen6$F`p.%&arrayGO-FK6$F`pF'0F`pQ)procnameF3OF `p-F\\o6$FE7$Q%lastF3Q(initialF3C$Fhp@$-Fen6$F`p.%%listGO&F`p6#\"\"#-F \\o6$FE7$.%%lastG.%(initialGC$F_p@$F\\rF`r3-Fen6$F?%\"=G-F\\o6$-%$lhsG 6#F?7$FiqFirC$@%-Fen6$-%$rhsGFesF^r>F`p-FR6#/Fiq7$\"\"!-%#opG6#F[t>F`p -FR6#/Fiq7$FcrF[t@$F\\rF`r/F?Q.solnprocedureF3O-FK6#FR/F?Q(sysvarsF3O- FT6#Fbu@%09!%(unknownGO-.FhuFDC$>8(-%(pointtoG6#&-FT6#Q0soln_procedure 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