A Stochastic Model for Topology Discovery of Tree Networks
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Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. A Stochastic Model for Topology Discovery of Tree Networks Youssef Azzana — Fabrice Guillemin — Philippe Robert. N° 5424 Deembre 2004. N 0249-6399. ISRN INRIA/RR--5424--FR+ENG. Thème COM. apport de recherche.
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(28) . /102.3 4 /1*()/5 /5637084 #. ^*_c){ywtmrAb wAqcxbc]tAn/r£©n_ac)n{wAqc]{prAsmnpcq]ww=ac_arAlonl'wn'np_achc:wÀ¡=c:l¬r£©n_ac)n{pc]c)tacxn}¦'rA{p§qxw=t 1cloc]cxt wAl w mkCtw=bu q]w l}nrCqy_w=lonpuvq!a{rmqc:lplhw=l)£¯r= r¦l )u taunpuvw kZn_ac p 1r=u tSnl w{cewnn_ac{prCr=n r=£Gnp_cen{pc]c=8n_acxt¢np_ac]k¢w{c n_a{pr¦t {w=tmrAb kZr=t¢np_ac lpr=tl9!wn n_acetc®Cn l}ncxNn_acloslocxnhr=£ 1r=u tSnl*wn* cx¡=c] 2 w{cn_a{r¦t[{w=tmrAb kwbr=t np_c d dlo1rAtl]Sw=tlprerAtN¨x¨]¨osatSnu8n_acxk{c]wAqy_np_ac 2 c]wÀ¡Ac]l/r£np_ac n{pc]c=¨ -°unp_ np_u lmc:lpqx{pu mnurAtX¦_ac]tTn_acZrAutSnylw{c@wn cx¡Acx X cn mcxtrnpc:lnp_ac ¡Ac]qnpr={@mc]lq{u aut np_ac-tCsab!1cx{@r£ 1r=u tSnl@wn@c]w=qy_ r£ np_c jl tarmm(X c:l 1jw,n . .cx.¡A, cxX ljj¨ ) ^*_auvl¡Ac]qnpr={ j. j. Ãà ¾AZÞ[é[.
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(35) Gw{u r=slhl}nsmu c]l_wÀ¡=cw {pc:w=mkZlp_ar¦tOnp_wnhnp_acsatmc]{p kCut A{w=a_l4r£np_ac¤¥tSncx{tacnGwt!r=£np_c¦'c]ecx®C_uaun¬np_c/1r¦/cx{ wÀ¦ a{r=1cx{pn}k £¯r={ np_cmcxA{pc]c'r=£tarmmc:lx¨ jmcxc >w=rAsmnlprAl "B 6 & :î¨ . ( *4 "! -* x
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(53) %. E(D(N, p)) =. 1 HSB > r={. n X. p 1 lj 1 − 1 − lj j=1 −. n X 1 p−1 . l j=2 j. *Q,. a wtarmmc A wn/n_ac j np_Z cx¡=c]©_w=l*trn1cxcxtOmuvlqr¡=c]{pc:u£ c]u n_acx{ i ¦u n_Onp_c!mkCtwbuvq auvqnsa{pcA¦_ac]tZnp_acerAutSnyl)w{c wnc]¡=cx j 1tarAtac r£ np_cxb u lwn)trCac A ¨ ^*_ac a{r=w=au un}kr£4np_auvlc]¡=cxtSnuvl*=u ¡=cxt@Ck 1≤j≤n. . 1−. i w n) cx¡Acx j < n 1w n_ac p 1r=u tSnlw{c wn)tarmmc c]r¦ A ¨Xjmcxc>4uAsa{pc & ¨. 1 lj. p. A. wtOw=Ër£¬np_cxb qy_arAlpc np_aclpw=bc losaanp{cxc. p points p points. >4uAsa{pc &. jCu tqc n_acx{c w{c. lj −1. A. j. ^*_c ^˦/r[jCu nswnurAtl*¦_ac]{pc;rmmc. dj. vu l*tar=n))uvlpqxr¡=c]{pc: losqy_ np{cxc:lxCn_ac a{r=w=au un}kr£4np_auvlc]¡=cxtSnuvl*=u ¡=cxt@Ck dj. Ãà ¾AZÞ[é[. 1 1 1 = dj p ljp dpj lj+1. A.
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(55) + . ^*_ac wÀ¡Acx{ywAc)tCsab!1cx{r=£tarmmc:lmuvlpqxr¡=c]{pc:u lnp_Csl*=u ¡=c]t Ck n X j=1. lj. ". . 1 1− 1− lj. p. − dj. 1 p lj+1. {j<n}. # =. p X n 1 1 − lj 1 − 1 − , p−1 lj l j=2 j j=1. n X. w=t 07GSswnurAt *L ,'£¯r= r¦lx¨ -°_ac]t uvl w={pAc=aunuvltar=n)mu ª[qxsa nnpr[lpcxc n_wnn_ac!lpc]qxr=t npc]{pb r=t@np_c {pu =_Sn_w=tOlpu mc r=£ 0TGAswnpu r=t *Lp ,Gu l'tacxAu =u a c *Òlpu tqc ,¨ ¤în/u l'br={cxr¡=c]{GqrAtS¡Acxtau cxtSn¬nr!¦{u ncnp_ac {yl}nXnpcx{b lj ≥ 2j−1 wAl n X. . 1 lj P m p ≤ l j j=1. *& ,. . ¦_cx{c wt (U ) w{ceu tmc]c]tmc]tAn {ywtmr=b&¡w={puvwa c]lhsatu £¯rA{pb k¢muvl L i , 1 ≤ i ≤ p) n{pu asmnc]mprAt =[0,inf(U @ ¨ * ^ a _ v u l x c C ® p { : c p l p l u rAt uvl u ti £Òw=qnewtcGAsu¡w cxtSn!r=£ E(D(N, p)) ¦_acxt p u l vw{=cA¨ 1] «)tmc]{*np_auvlqrAtmunpu r=tCn_ac ¡Àw={puvwc qrAtS¡Acx{=c:lGu t@muvlonp{uasanpu r=t nprnp_ac {ywtar=b ¡w{u w=a c X ¦_u qy_@u l)wt@c®mrAtacxtSnu w= kmuvl}n{pu asmnpm c]@p{ywtmrAb ¡w{u w=ac ¦unp_Zw={w=bcxnpc]{ 1 P (pmp ≥ x) =. . x 1− p. p. ∼ e−x .. ¦_cxt npc]talhnr u t taun}k=¨d'r=tqxcx{taut[np_clpw=tatau ta[n{pc]cer=£/muvlpqxr¡=c]{pc:Z{prAsmnpc]{l]8¦'c_wÀ¡=c!np_ac £¯rA r¦u ptae{c]lpsan]¨ TJ = S J M#
(56) S/J ! W # · · Òm ´ A 0≤j<n p>1 S/A A $$ @# A. K 1! $! A . . %. P(H(N, p) ≤ n − j) =. H S " A. j. 1. p−1 , lj+1. A A +# A %. p 1 1 1− 1− − dj p lj lj+1. {j<n} .. 1 HSB ^*_cXloc:qrAt u acxtSnpun}kuvlËqxc:w{©£¯{r=b np_c'w1r¡=c a{rCr£}¨4^Ër{pr¡Acn_ac {lonËr=tc=:u n4u llosmª[qxuc]tAn nr[{cxbw{§[n_wn:u tNr={ymc]{npr _wÀ¡=c tartarmmc!r=£ cx¡Acx j au lqr¡Acx{c]©mn_acxt w np_c!1r=u tAnylbesl}nhc u tOlorAbc lpsamn{pc]c ¦_arAlpc {prCrn*uvlwn*np_c (j + 1)np_Zc]¡=c]Q¨. ̯Í4ÃÌλ.
(57)
(58) ! "$# %. . 88. .
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(61) :Ë£¯r={mcxnwu vl$,¨ ^*_ac@SG sw=tSnpun}k. uvl4np_aca{r=wau un}k np_wnG¦unp_ n{wAqc]{prAsmnpc*qxw=wa c/_arAlonl]=w n_ac/tarmmc]l wn¬ cx¡Acx P(D wj {jc'(N, muvlqp)r¡==c]{pc:lj©)¨4d'rAtS¡Acx{yloc]kA]rAtac/bwÀk qxr=tlpu apcx{4np_ac/a{r=a cxb r=£mcxnpcx{bututa n_ac tCsab!1cx{*r£np{yw=qxcx{r=smnc_arSl}nyl/r=tachtacxc:al/nprmuvlqr¡=c]{'w=8n_ac tarmmc]lwn/ cx¡Acx j ¨ Qcn ν mcxtrnpc j n_auvl{w=tmrAb ¡Àw={puvwcA¨ >{prAb n_acqr=srAt¢qrA c]qnpr={a{r=a cxbZ1np_aca{r=w=au un}k uvlAu¡AcxtOCkF>arA{pbesa w<*Q,¨ ^*_chbc:wtZ¡w sac r£4np_ac {ywtar=b ¡w{u w=a c ν uvl=u ¡=c]t Ck P(νj = p) j. E(νj ) = 1 +. Ãà ¾AZÞ[é[. lj lj lj + + · · · + + lj lj − 1 lj − 2 2.
(62) 8À. "
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(65)
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(67) 8]. "
(68) + .
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(70)
(71) ^rsatacx{yl}nywthbr={cGq rAlpcx kn_acGwAlouvq a{r=1cx{pnpu c]l8r£Cnr=1r= r==kmu lqr¡Acx{kCkbc]w=tl©r=£An{wAqcx{r=sanpc= u nGuvlGw=llosbc:utnp_u l'loc:qnpu r=tenp_wnGn_ac{wnpu r r£8np_actCsabec]{ p r£1np{yw=qxcx{r=smncqxwwa c*_arAlonl nr np_ac nrnyw1tSsb!1cx{ N r£Ë{r=smncx{yl¬u t[np_c)tacxn}¦'rA{p§u l a®mc]wtcGSswnprw!qrAtlonwtSn λ w=t¦/c)losarSloc n_wn N ArSc:l4npr u t taun}k=¨ --cw{c*lp1c]quq]w keutSnpc]{pc:l}nc]eutn_acc]_wÀ¡CurA{¬r£ E(D(N, p))/N ¦_acxt npc]tal nr ut tau n}kA¨¬^*_au l GSsw=tSnpun}k =u ¡=c:lu tm£¯rA{pbwnurAtw=rAsmn np_ac{ywnur r=£np_ctSsb!1cx{¬r£1{prAsmnpc]{l N au lqr¡Acx{c]nrnp_ac nrnw=©tCsab!1cx{r=£4tarmmc:l*ut@n_ac tacn}¦/r={§8¨ . . . / 6
(72) * . . / /5. ¤¥tZw {lonl}ncxm¦/cu tC¡=c:l}nuSwnpc)np_ac q]w=lpc¦_ac]tn_ac mcxA{pc]c)r=£tarmmc:l'uvlqrAtlonwtSn:ScGSsw1npr. . ·X· Ò· . # " A . E(D(N, p)) ∼ T d (λ) N. . J # . d≥2. . +∞
(73) X d−1 =. 1 HSB «lou ta n_wn. N → +∞. j=1. dj. wtn_ac)cGSsau¡w=c]tAnI* & G, w=l N ∼ dn /(d − 1). N. . d≥2. p/N → λ. λdj 1 − exp − d−1 . . . .. * ,. =rCc]l¬npr u t taun}k=Sr=tc=cxnl¬n_wn. 1 E(D(N, p)) N. w=tnp_ac]t. ∼ (d−1). n X dj−1 j=1. dn. n. P pmp ≤. d bλ d−1 c. dj−1. !. n X 1 λ 1 j E(D(N, p)) ∼ (d − 1) P X ≤ d N dj (d − 1) j=1. ¦_cx{c u l wtec®m1r=tac]tAnu w= k muvl}n{pu asmnc]e{w=tmrAb ¡Àw={puvwc'¦u n_esatu n¬bc]w=t=lputqxc/wAl4bc]tSnpu r=tac: u tNjCc]qnpXu r=t & Cn_ac {w=tmrAb±¡w{u w=ac pm qxr=tC¡=c]{pAc]l'utOmuvl}n{pu asmnurAtnr X ¨ p --c'tr¦nw§AcX1cxtac an4np_ac*q rAlpc] £¯rA{pb r=£anp_ac/{ywnpu rr=£muvlpqxr¡=cx{c] trCac]l=u ¡=cxt!Sk cGAswnpu r=t * , u t¢rA{acx{npr@u sl}n{wnpcenp_aclo1cxc:¢r=£Gnp_cecx®CrA{wnpu r=tNa{rmqc:lpl]¨ >ar={hn_au lhas{p1rAlpc=©¦/celonpsakOnp_ac acx1cxtmc]tqc r=£Ën_ac tCsab!1cx{r=£4muvlqr¡=c]{pc:[tarmmc:l*r=t@np_c {wnpu r λ ∼ p/N ¦_cxt N uvlvw{=ch¦_au c uvl lpbw=L¨¤în!mc]lq{u1c]lhu t-lpr=bclpcxtlpc!n_acu tau nu w=Glo1cxc:¢r=£Xn_acc®ma r={ywnpu r=t a{rCqxc]llnp_a{r=saA_ λ n_actacn}¦/r={§8¨¤¥tmc]c]©8n_acew=tw kmlouvlAu¡Ac]l)np_ace{wnpu rr£Xau lqr¡Acx{c]@tarmmc]l¦unp_ w lobw=4tCsab!1cx{ r=£Ën{wAqcx{r=sanpc qxwwa c _arAlonl]¨. ̯Í4ÃÌλ. ¨.
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(79) J N " d ≥ # 2 A A9 $λ. . . * 8:,. T d (λ) ∼ −λ logd (λ).. 1 HSB lu nqxw=tTc]wAlou k-1cZloc]cxt¬n_acOw=lpkCbanprnu qc]_wÀ¡CurA{er=+£ 07GSswnpu r=t *S8] ,!¦_ac]t npc]tal nr 0 uvlIGAsu ncmc]uvqxwnpclou tqc!np_acloc]{pu c]l)u lhmu ¡=c]{pAcxtSn]¨)^*_c npc:qy_atau GSsac!slpc]OnprAcn wtNcx®CλwtlpurAt {cx u c]lrAtNwqxr=tC¡=c]tauc]tSn{pc]¦{punpu ta[r=£¬np_acecHGSswnurAtZnprA=cn_acx{¦u n_Nn_ac!slpc!r=£N>asaau tau l ^*_ac]r={cxb £¯rA{tar=t LLtac]Awnpu ¡=c£¯satqnurAtl]¨@*QjCc]c ~r=1cx{pn6"8À:£¯r={wAcxtac]{w=8ac]lq{umnurAt r=£Ën_ac bcnp_rC¨ , --c _wÀ¡=c +∞ X 1 λdj 1 − exp − dj d−1 j=1 =. Z +∞ X 1 +∞ dj 0 j=1. =. Z. 0. +∞ +∞ X j=1. 1 dj. n. n. o e−u. du. o e−u. du,. j. λd u≤ d−1. j. λd u≤ d−1. ¦_cx{c ¦/c _wÀ¡=c@sloc:Y>saau tau ln_acxrA{pc]b npr-c®aqy_wt=c np_acOlpsab wt np_c@u tSnpc]={ywlou =tlx¨T^*_u l c]tSnwu vl/np_wn Z λa +∞ X 1 λdj d 1 − exp − = e−u du j d d − 1 d − 1 0 j=1 +. 1 d−1. Z. +∞ λa. 1 e−u du, dblogd (u(d−1)/λ)c. *S88,. ¦ unp_ a = d/(d − 1) ¨ ^*_ac{lonnpc]{pb r=t@np_c {pu =_Sn_w=tZlouvmc r=£ 0TGAswnpu r=t *S88,/¡w=tau lp_ac:l¦_acxt npc]tal*nra¨ >ar={ n_ac!lpc]qxr=t npc]{pbZ¦'c tar=npc n_wn¦unp_¢{pc]Aw{ynrnp_c!w=lpkCbanprnu q 1cx_wÀ¡Su r={*£¯rA{ λλ q rAlpcnranp_ac u tSnpc]={ywnpu r=tutSnpc]{p¡w= [λa, ∞] qxw=t1c{cx wAqc]Ckw=tCkutSnpc]{p¡w= [λa, b] ¦u n_ b > λa ¨G^Ërlou bau£¯k= ¦/chnw=§=c b = a lprnp_wn lpu tqch£¯r={. u≥a. Ãà ¾AZÞ[é[. . Z. +∞ a. 1 dblogd (u(d−1)/λ)c. e−u du ≤ λ(1 − e−a ),. blogd (u(d − 1)/λ)c ≥ − logd λ..
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