Modeling the IPv6 Internet AS-level Topology

Transcription

1 Modlng h IPv6 Inrn AS-lvl Topology Bo Xao, Lan-dong Lu, Xao-chn Guo, and K Xu * Sa Ky Laboraory of Sofwar Dvlopmn Envronmn, School of Compur Scnc and Engnrng, Bhang Unvrsy, Bng , Chna Absrac To masur h IPv6 nrn AS-lvl opology, a nwork opology dscovry sysm, calld Dolphn, was dvlopd. By comparng h masurmn rsul of Dolphn wh ha of CAIDA s Scampr, was found ha h IPv6 Inrn a AS lvl, smlar o ohr complx nworks, s also scal-fr bu h xponn of s dgr dsrbuon s 1.2, whch s much smallr han ha of h IPv4 Inrn and mos ohr scal-fr nworks. In ordr o xplan hs faur of IPv6 Inrn w argu ha h dgr xponn s a masur of unformy of h dgr dsrbuon. Thn, for h purpos on modlng h nworks, w propos a nw modl basd on h wo maor facors affcng h xponn of h EBA modl. I braks h lowr bound of dgr xponn whch s 2 for mos modls. To vrfy h valdy of hs modl, boh horcal and xprmnal analyss hav bn carrd ou. Fnally, w dmonsra how hs modl can b succssfully usd o rproduc h opology of h IPv6 Inrn. Kywords- IPv6; Inrn opology; dgr xponn; powr-law; modl 1. Inroducon As on of h mos sgnfcan nvnons n h las cnury, h Inrn has provdd human bngs a brand nw nformaon socy. Now h nrn s undrgong a gradual chang. As cor pars of Inrn, TCP/IP proocol sacks prform h asks of packagng and ransmng nformaon. On h IP layr, h Inrn Proocol vrson 4 (IPv4) has h lmaon of addrss spac and som ohr problms such as h QoS, scury and prformanc. Undr hs crcumsanc, h Inrn Proocol vrson 6 (IPv6) has bn pu no praccal us, ladng o h coxsnc of h IPv6 Inrn and h IPv4 Inrn. Currnly, h IPv6 nworks usually connc o ach ohr hrough IPv4 unnls. For h purpos of prdcng how nw chnologs, polcs, or conomc condons wll mpac h Inrn s conncvy srucur a dffrn layrs, h opology of global IPv6 Inrn s ncssary. I mad us of hos unnls and rad hm as h lnks bwn IPv6 domans[1-4]. Th rsarch can b hr a rour lvl or a auonomous sysm (AS) lvl. In fac, mor nrs boomd n AS opologs, bcaus from a macroscopc vw AS opologs ar h sklons of hs complx sysm and mor rprsnav. Th opology modl s h oucom of horcally modlng h ral nworks from h vw of sysmacal voluon or h aspc of rproducng som mporan opology mrcs [5, 6]. Thrfor, opology modlng can xplan h orgns of h xsng proprs of nwork opologs and h modl also conrbus o rsarch on nwork smulaons and srucural analyss. In rcn yars, consdrabl rsarch has bn don on complx nworks whch dscrb a wd vary of sysms n naur and socy ncludng Inrn, World Wd Wb (WWW), socal rlaonshp nworks, conomy nworks, powr nworks, ransporaon nworks and nural nworks [7]. I s also known ha som of h nworks can b rprsnd as scal-fr nworks[8], whos dgr dsrbuon follow h powr-law form pk ( ) k γ whr pk ( ) s h probably ha a randomly slcd nod has xacly k dgs and γ s calld h dgr xponn whch characrzs h dgr dsrbuon of a scal-fr nwork. To undrsand h volvng mchansms of scal-fr nworks, a numbr of volvng opology modls hav bn proposd. A smpl modl of a growng nwork was nroducd by Barabás and Albr (h BA modl [8]) n whch hy found ha h growh of nworks and h prfrnal aachmn wr h orgns of h powr-law dgr dsrbuon and proposd h concp of scal-fr nworks. Th BA modl producs nworks wh h dgr xponnγ = 3. Basd on h BA modl, a lo of ohr volvng modls hav bn nroducd o oban dgr dsrbuons wh varabl dgr xponns [9-18]. Ths rsarch s suppord by Naonal 973 Program of Chna (Gran NO 2005CB321901) and Bng Nova Program (Gran. NO 2005B12). * Corrspondng auhor. Tl.: ; E-mal: kxu@nlsd.buaa.du.cn 1

2 For many obsrvd scal-fr nworks wh 2< γ < 3, hs modls f hr dgr xponn faur vry wll. Howvr, a maor challng arss whn usng hs modls o rproduc h IPv6 Inrn opology wh qu a small dgr xponn bcaus mos of hm hav h lmaon of γ > 2 [7]. In hs papr w propos our modl o brak hs lmaon and rproduc h IPv6 Inrn AS-lvl opology. Th papr s organzd as follows. In scon 2, w frs rvw h volvng modls for nwork opologs. And hn w brfly nroduc h IPv6 Inrn opology dscovry and prsn a nw faur of h AS-lvl opology n scon 3. Th dgr xponn and s mplcaons ar dscussd n scon 4. Thn, basd on h wo maor facors affcng h xponn and h EBA modl, w propos our modl and mak a horcal analyss n scon 5. I s shown n scon 6 ha our modl braks h bound of 2 for h dgr xponn γ, and rproducs h opology of h IPv6 Inrn. Fnally, w conclud our work n scon Evolvng modls for nwork opologs In 1999, svral powr-laws wr obsrvd n h IPv4 Inrn opology [19, 20]. Th dscovry of h powr-law of dgr dsrbuon has sgnfcan mpac on h nwork opology rsarch: h Inrn s nhr as fla as h randommodls (such as h ER modl [21, 22]) dscrb, nor as hrarchcal as h srucural ons (such as h Trs and Trans- Sub modls [23, 24]) dscrb. To modl hs scal-fr propry n h Inrn, many ffors hav bn brough forward o dsgn opology modls ha can produc powr-law dgrs. Thy all consruc random graphs, bu som produc h graphs ncrmnally, whch ar calld volvng modls; and ohrs do no allow h growh of h nwork, whch ar calld nonvolvng modls [25]. Th non-volvng modls can no sasfy our dmands bcaus w wan o prdc h fuur opology bu hs modls only rproduc h sam sz as h gvn on. So w manly dscuss and compar h volvng modls. Barabás and Alr found ha ncras of nworks and prfrnal aachmn wr h orgns of h dgr dsrbuon powr-law. Thy proposd h concpon of scal-fr nworks and a lnar prfrnal aachmn modl (h BA modl) [8] o gnra h scal-fr nworks. Th BA modl conrbud a lo o complx nwork hory by fndng h orgns of h wll-known powr-law characrsc. In h BA modl, h prfrnal aachmn probably s dfnd as: k Π = k. (1) Howvr h BA modl can only consruc nwork opology wh h dgr xponn qual o 3. Thn, Albr and Barabás mprovd hr BA modl and proposd h EBA modl [9]. Th nw modl s sll a lnar prfrnal modl, bu aks no accoun local vns, such as suspnson of addng nods and rwrng of lnks. Th dsrbuon of dgrs n h EBA modl follows h powr-law wh h xponn γ > 2. Th Gnralzd Lnar Prfrnc (GLP) modl [10] was proposd n 2002 by Bu and Towsly who dscovrd ha h nods n Inrn ar mor lkly o aach o h nod wh largr dgr han h probably spcfd n h BA modl. So hy modfd h prfrnal aachmn probably o Π = k β, (2) k β whr β < 1. Th dgr dsrbuon of GLP also follows h powr-law wh h xponnγ > 2. Zhou and Mondragon argud ha an Inrn modl ha dd no rproduc h proprs of h rch-club conncvy would undrsma h acual nwork s roung ffcncy and roung flxbly and would also ovrsma h nwork robusnss undr nod aack [26]. Th Posv Fdback Prfrnc (PFP) modl [11] has succssfully rproducd a wd spcrum of mrcs of h IPv4 Inrn a AS-lvl, ncludng h rch-club conncvy. Ohr modls n hs cagory nclud h hghly opmzd olranc (HOT) Modl [12], h nal aracvnss modl [33], h acclrang growh modl [17] and h gradual agng modl [17]. Howvr, all of hs modls dscussd abov hav h lmaon of γ > 2, as shown lar n scon 4. In hs papr, w wll only choos BA and EBA, h wo mos basc modls, for dscusson and comparson. 2

3 3. Th small dgr xponn: a nw faur of h IPv6 Inrn opology In hs scon w frs analyzd h opology daa from CAIDA s Scampr and Dolphn. W found ha h IPv6 Inrn a AS lvl, smlarly wh ohr complx nworks, s also scal-fr bu s xponn of h dgr dsrbuon s 1.2, whch s much smallr han ha of h IPv4 Inrn and mos ohr scal-fr nworks. 3.1 Topology dscovry on h IPv6 Inrn Now h global IPv6 nworks ar conncd o ach ohr manly hrough IPv4 unnls and can b sn as an ndpndn nwork. Knowldg of hs nw born nwork opology s crucal for undrsandng currn applcaons on IPv6 nworks and dnfyng h problms whn hs applcaons ar dployd. Th IPv6 nrn opology, as h sklon of h communcaon sysm, s of gra mporanc for sng and prdcng h prformanc, robusnss, and scalably of ohr nwork proocols on hs nw IP layr. And h opology can b hlpful for h suds abou roung and sarchng n nworks, robusnss o random nwork falurs and argd aacks, h spd of worms spradng, common srags for raffc ngnrng and nwork managmn [27, 28]. In 1998, h Cooprav Assocaon for Inrn Daa Analyss (CAIDA) [29] bgan h Macroscopc Topology Proc for Inrn opology masurmns. CAIDA has dvlopd a ool namd Skr, whch s dsgnd for h IPv4 Inrn opology dscovry and ohr masurmns. Is counrpar on IPv6 s Scampr [30] whch xnds Skr s approach o h IPv6 proocol. Skr and Scampr ar boh basd on racrou. For IPv6 nworks, racrou obans h IP hops along h pah from h probng pon o a gvn dsnaon by h ICMPv6 proocol. In ordr o masur h global nrn, dsrbuon s rqurd. Unlk h dsrbud proc DIMES[35], whch ralzs dsrbuon by volunrs, Scampr was dployd a 17 dffrn locaons around h world and hn bgan o collc daa. A snapsho of h IPv6 Inrn AS opology on March 4h, 2005 was publshd on h wbs [31].Ths AS map cam from h dscovry of approxmaly 860 globally rouabl IPv6 nworks prfxd by 2,913 IPv6 addrsss and 7,905 IPv6 IP lnks whch ar h lnks bwn adacn addrsss n a racrou pah, as shown n TABLE 1. In arly 2007, CAIDA gav nwork rsarchrs fr accss o h raw daa of Skr and Scampr afr rgsraon. Thr s a ll dffrnc bwn h AS graphs gnrad from h daa w downloadd from CAIDA and h on from h s [31]: h AS numbr of h formr graph s 356 and h lar s 333. In 2004, h Sa Ky Laboraory of Sofwar Dvlopmn Envronmn (SKLSDE) of Bhang Unvrsy (also known as Bng Unvrsy of Aronaucs and Asronaucs or BUAA) bgan a proc namd Global IPv6 Nworks Monorng Proc, on goal of whch was smlar o ha of Scampr,.. o dscovr h accura opology of h IPv6 Inrn. Bu our approach was no smpl racrou. W nocd ha h currn unnl lnkd way of global IPv6 ASs can b usd o dsrbu h probng agns. Exndng hs da, w dsgnd h probng algorhm of h hrd pary dsrbuon and asynchronous racrou. Wh hs wo nw opology dscovry approachs, w pland probng agns by confgurng dffrn unnls nsad of dsrbung h prob program o dffrn gographc locaons. Bcaus of h gra numbr of unnls (up o Oc.2006, hr was 91 usabl unnls n oal), h probng daa wr nrchd dramacally. Th dscovry sysm Dolphn [32] has bn collcng daa of rours, lnks, bandwdhs and h AS sascs of h global IPv6 nworks snc In Mar (h sam m as ha of h daa whch w go from Scampr), Dolphn dscovrd 3022 IPv6 addrsss, 6825 lnks and 419 ASs. Up o Oc. 2006, 7401 IPv6 addrsss, lnks and 508 ASs had bn dscovrd from global backbon IPv6 nworks. Comparng h dscovrd rsuls by h wo sysms n TABLE 1, w can s ha Dolphn s daa ar rchr han Scampr. Bsds, ohr sascs oband by Dolphn ar shown n TABLE 2 and mor nformaon s avalabl on h wbs [32]. As a rsul, w hav wo daa sourcs n hs rsarch: on s Scampr, and h ohr s Dolphn. Th followng scon wll mak a comparson bwn h wo AS opologs gnrad from h daa sourcs whch wr oband a h sam m. TABLE 1 Summary of h IPv6 Inrn opology dscovry by Scampr and Dolphn Scampr(Mar. 2005) Dolphn (Mar. 2005) Dolphn (Oc. 2006) Numbr of probs Numbr of ASs Numbr of IPv6 Addrsss Numbr of IP lnks TABLE 2 3

4 Mor abou Dolphn sysm Sascs Valus Lngh of addrss ls for probng 5256 Numbr of ou-gong packags pr probng 43,046,640 Numbr of dscovrd pv6 rours 6325 Numbr of dscovrd pv6 addrss prfxs 680 Numbr of usd unnls Comparson bwn opologs from h wo daa sourcs From h numbr of dscovrd ASs by Dolphn n TABLE 1, w can s h growh of h IPv6 Inrn. Spcfcally, w dscussd h daa of March 2005 whch s smulanous wh Scampr n h followng. In hs scon w compar h wo opologs basd on a ls of graph mrcs ha hav bn found o b mporan n h nworkng lraur. Th ls of mrcs ncluds dgr dsrbuon, dsanc dsrbuon, local clusrng, normalzd bwnnss and rch-club conncvy as wll as som basc graph proprs such as numbrs of nods and dgs, avrag dgr and maxmum dgr. Ths ls s no compl, bu n hs scon s opology comparson and scon 6 s modl comparson, w blvd ha hos mrcs ar suffcnly dvrs and comprhnsv o b usd as a good ndcaor of graph smlary [6]. Th dgr dsrbuon P(k) spcfs h probably of nods n a graph wh h dgr k. Th dsanc dsrbuon D(d) s h numbr of pars of nods a h dsanc d, dvdd by h oal numbr of pars n 2 (slf-pars ncludd). Th local clusrng C(k) s h rao of h avrag numbr of lnks bwn h nghbors of k-dgr nods o h maxmum numbr of such lnks. Th normalzd bwnnss s a wghd sum of h numbr of shors pahs passng hrough a gvn nod or dg. Th rch-club conncvy φ(r/n) s dfnd as h rao of h numbr of lnks conncng h club mmbrs ovr h maxmum numbr of allowabl lnks r(r-1)/2 n whch r s h rank of a nod dnos s poson n a ls of all nods sord n dcrasng dgr. Th opology from Scampr consss of 356 nods and 1007 dgs. And for h opology from Dolphn, 419 nods and 1812 dgs hav bn found. Th basc graph proprs and avrag valus of h mrcs ar dmonsrad n TABLE 3. Mor dals ar shown n Fg.1. Th wo opologs ar vry smlar n h dgr dsrbuon (Fg.1a, b), h dsanc dsrbuon (Fg.1c), rch-club conncvy (Fg.1d) and normalzd nod bwnnss (Fg.1f) dsp h dffrncs of h avrag dgr and h maxmum dgr. Th local clusrng dsrbuons (Fg.1) ar clos o ach ohr for h small dgrs. Th concluson can b drawn ha h dscovrd rsuls by wo sysms agr wll wh ach ohr n h dsrbuons of dgrs, dsancs, local clusrng, normalzd bwnnss and rch-club conncvy. Th rsuls also ndca ha h IPv6 Inrn opology s a graph wh hgh clusrng, small man-dsanc and obvous rch-club faurs. TABLE 3 Comparson bwn h wo AS opologs a h sam m, March 2005, from h wo sysms CAIDA Scampr Numbr of nods Numbr of dgs Avg dgr Exponn of P(k) Max dgr Avg dsanc BUAA Dolphn Nomalzd avg nod bwnnss 5.435E E-3 Nomalzd avg dg bwnnss E E-3 Avg clusrng Exponn of rch club

5 3.3 Th small dgr xponn of h IPv6 Inrn opology Th dgr dsrbuon s on of h mos mporan mrcs of nworks opology. I s usually dscrbd on a frquncydgr log-log plo. If a nwork s scal-fr, s dgr dsrbuon follows h powr-law. On h plo, s approxmaly lnar and h slop corrsponds o h dgr xponn. Dsp h dffrncs n h dg numbrs and h maxmum dgrs, h opologs oband by Scampr and Dolphn ar qu smlar n h dgr dsrbuon plos. Th log-log plos of dgr dsrbuon of IPv6 opology from boh Dolphn and Scampr ar approxmaly lnar, as shown n Fg.1a and Fg.1b. Th corrlaon coffcn of lnar rgrsson s for Scampr and for Dolphn. Boh ar clos o -1, whch mpls ha hr dgr dsrbuons follow h powr-law. To b clarr, h complmnary cumulav dsrbuon plo s also gvn n Fg.1 g. As a rsul, shows ha h IPv6 Inrn s scal-fr a AS lvl. Bu wha s byond xpcaon s ha h xponn of dgr dsrbuon s only abu 1.2, far smallr han 2.2 of h IPv4 Inrn [19, 20]. Snc h daa sourcs from Dolphn and Scampr ar n good accordanc wh ach ohr, s no a concdnc ha h rsul on h dgr xponn of IPv6 s us 1.2. For h rason ha h powr-law of dgr dsrbuon s a man characrsc of a scal-fr nwork, h bg dffrnc n h powr-law xponn ndcas ha hr s a nw faur n h opology srucur of h IPv6 Inrn dffrng from h IPv4 Inrn. (a) Dgr dsrbuon from dolphn (b) Dgr dsrbuon from scampr (c) Dsanc dsrbuon (d) Rch-club conncvy 5

6 () Local clusrng (f) Normalzd nod bwnnss (g) Complmnary Cumulav dsrbuon of nod dgr Fgur 1. Comparson bwn h wo opologs rspcvly dscovrd by dolphn sysm and scampr sysm frqunc frqunc dgr (a) dgr (b) Fgur 2. Th dgr xponn s a masur for opology unformy of dgr dsrbuon. For nworks of h sam sz, h xponn s bggr n (a), hn s dgr rang s mor concnrav and h opology s mor unform; whl n (b), h xponn s smallr, hrfor h dgr rang s wdr and h opology s lss unform. 6

7 4. Explanaon of small dgr xponns and challngs for nwork volvng modls For dgr dsrbuon whch follows h powr-law, h slop of h rgrsson ln n a log-log plo can rprsn h dgr xponn. For h nworks of h sam sz, s asy o oban ha f h dgr xponn s bggr, hn h dgr rang s mor concnrav, as shown n Fg.2. Espcally whn h dgr xponn s qual o nfny, all nods hav h sam dgr and s a unform opology. On h oppos sd, for h smallr dgr xponn, spcally whn h dgr xponn s 0, all dgrs hav h sam probably o appar n h opology, and hus h maxmum dgr can b much bggr han h mnmum dgr. Thrfor, n gnral, h dgr xponn s a masur of unformy of h dgr dsrbuon. Th bggr h xponn, h mor unform h opology; h smallr, h lss unform. Th IPv6 dgr xponn s much smallr han ha of IPv4, mplyng ha h IPv6 Inrn s lss unform han h IPv4 Inrn. Thr ar svral possbl rasons conrbung o hs. Frs of all, h sz of currn IPv6 Inrn s much smallr han IPv4. Compard wh mor han 10 housand ASs n h IPv4 Inrn, lss han 500 ASs of h IPv6 Inrn consu a rahr small nwork. Thn, IPv6 nworks ar on h sag of ranson from h xprmnal us o praccal global dploymn, and for ha rason, som characrscs of xprmnal nworks ar nvabl. For xampl, mos of h larg IPv6 nrn xprmnal plaforms ar conncd o almos vry AS, whl h smallr ASs ar only lnkd o a fw of hos larg xprmnal plaforms (bcaus no ral applcaon rqur h smallr ASs o drcly connc o ach ohr durng h xprmnal prod). Thn h currn IPv6 Inrn rsuld n h lss unform opology han ha of h IPv4 Inrn. Th dgr xponns of mos scal-fr nworks fall bwn 2 and 3 [7]. Bu bsds h IPv6 Inrn a small numbr of nworks n ohr flds shar h sam faur of small dgr xponns, such as h mal nwork n socal nworks wh h xponn of 1.5 and h sof packag nwork n chnology nworks wh h n-dgr xponn of 1.4 and h oudgr of 1.6 [14]. Th xsnc of scal-fr nworks wh h dgr xponn smallr han 2 s a raly. Bu mos of h xsng volvng modls ar only capabl of consrucng scal-fr nworks wh h xponns bggr han 2, as shown n TABLE 4. Whn w wand o rproduc h small dgr xponn of h IPv6 Inrn, w ncounrd a bg challng. In ordr o solv hs problm, w xnd h EBA modl and propos a nw modl whch has h capably o consruc an volvng scal-fr nwork wh h dgr xponn smallr han 2. TABLE 4 Th volvng modls and hr lms of h xponn γ Evolvng nwork modl Faurs Lm s of γ BA modl [8] Lnar growh, lnar aachmn γ = 3 Nonlnar aachmn modl [13] Π k α only whn α = 1can b scal-fr γ > 2 Inal aracvnss modl [33] Π k+ A γ = 2 f A = 0 A s a consan aracvnss γ f A EBA modl [9] Edg-rwrng γ > 2 Acclrang growh modl [17] γ = 3 f θ 0 θ k = γ =1.5 f θ 1 Drcd nwork I s a drc d nwork modl. Gradual agng mo dl [17] γ 2 f v Π( k ) ( ) v k γ f v 1 PFP modl [11] 1 lgk Π( k ) k δ Th xprmnal rsul shows γ > 2 Ths abl s manly basd on Rf. [7]. 5. Our modl As a masur for unformy of dgr dsrbuon of scal fr nworks, h dgr xponn s affcd by many facors. Basd on currn volvng nwork modls, w mak us of wo maor facors o rproduc h faur of small dgr xponn: h prfrnal aachmn and h dg rwrng n nwork volvng procss. For h prfrnal aachmn, h lnar rlaonshp may b h mos basc on. Alhough raly s far mor complcad han ha, sascs of many ral nworks show ha h aachmn probably s us a ll hghr han h lnar prfrnal aachmn [11]. Ohrws, f w chang h prfrnal aachmn probably as follows: 7

8 k Π =, (3) α α k has bn provn ha h only cas n whch h opology of h nwork s scal fr s ha n whch h prfrnal aachmn s asympocally lnar. Whn α < 1 s a sub-lnar aachmn and h dgr dsrbuon s xponnal; whnα > 1 s supr-lnar aachmn and almos all h lnks ar aracd by on or wo supr nod [13]. Our sarng pon s o carfully choos a lnar aachmn xprsson as follows: Π = k + ε k k + ε k, (4) whr Π s h probably of nod o oban an dg, k s h dgr of nod, currn m. If h valu of s ngav afr calculaon, Π ε s a consan, k s h avrag dgr a h Π wll b s o zro o avod bng mannglss. Th rason for pckng k s ha can brng h faurs of h xsng opology no h nx sag of volvng procss and maks h horc calculaon of h dgr xponn much asr. Th paramrε can adus h nod pckd-up probably hghr or lowr han h pur lnar prfrnal aachmn n h BA modl [8]. Th dg rwrng can b undrsood as adusng rlaonshps bwn nods. Wh, h EBA modl can gnra a scal-fr nwork wh h dgr xponn 2m( 1 q) + 1 p q γ = 1 +. (5) m In ordr o avod gng xponnal opologs, p and q mus b rsrcd nsd h scal-fr rgm. Th EBA modl can g good rsuls of γ > 2 [9] and maks clar ha h dg rwrng mchansm s an mporan facor affcng h dgr xponn. Takng boh h facors no consdraon and basd on h EBA modl, w gv ou our modl as follows: W sar wh fully conncd graph of m 0 nods and a ach m sp w prform on of h followng hr opraons: ()Wh probably p, w add m(m<m 0 ) nw lnks: w randomly slc a nod as h sarng pon of h nw lnk, whl h ohr nd of h lnk s slcd wh probably gvn n (4). ()Wh probably q, w rwr m lnks: w randomly slc a nod and a lnk conncd o. Nx w rmov hs lnk and rplac wh a nw lnk ha conncs wh a nw nod chosn wh probably gvn n (4). ()Wh probably 1-p-q, w add a nw nod: whch has m nw lnks ha wh probably gvn n (4) conncd o nods alrady prsn n h sysm. I s asy o s ha h EBA modl s a spcal cas of our modl wh ε = 1/ k. For our modl, h xprsson of h dgr xponn s: γ = 2(1 q)(1 + ε) + 1. (6) In h followng, w us h Connuum Thory [34] o prov h abov rsul. Frs, w assum ha k changs connuously. Consqunly, h procsss () () all conrbu o k, ach bng ncorporad n h connuum hory as follows. () Addon of m lnks wh probably p: k pm k + εk = + pm () N( ) M ( ), (7) 8

9 whr k () s h dgr of h nod a h m, N() s h numbr of all h nods, a h m N ( ) and M ( ) = ( k ( ) + ε k ) ; = 1 ()Rwrng of m lnks wh probably q: k qm k + εk = + qm ; (8) ( ) N( ) M ( ) ()Addon of a nw nod wh probably 1-p-q: k k + εk = (1 p q) m ( ) M ( ). (9) And a m, w hav: N ( ) = (1 p q), M ( ) = 2(1 q)(1 + ε ) m, (10) N ( ) = 1 k = 2 (1 q) m and N ( ) k = 1 2ε (1 q) m E = ε k = ε =. (11) N( ) 1 p q By addng h conrbuon of h hr procsss, w oban: k A m E B ( ) = ( + + )( ) A E, (12) 1 whr m p q q + A = 2 ( )(1 )(1 ε ), B = 2(1 q)(1 + ε ). 1 p q In h connuum hory, h probably P k ) can b nrprd as h ra a whch k changs: ( B m+ A+ E P[ k() < k] = P > k + A+ E and w oban: B m A E B 1 P + + m + A+ E = k + A+ E = 1 k A E, (13) m0 [ ( ) < k] P k P( k) = k B = B ( m + A + E) B 1 + m 0 ( k + A + E). (14) Thus h dgr dsrbuon P(k) follows a gnral powr-law form and w g: γ = B + 1 = 2(1 q)(1 + ε ) + 1. (15) 6. Modl vrfcaons Whn analyzng sascs on h nwork powr-law dgrs, w adop a usual mhod for opology analyss o capur h powr-law al br, whch s o gnor som (no mor han 5%) nods whos frquncs ar small bu dgrs ar bg[19, 20]. 9

10 In our modl h prfrnal aachmn probably s π ( k) k + εk. To mak sur has physcal manng, π ( k) nds o b abov zro. Howvr n h cas ofε < 0, f h prcnag of nods whos π ( k) < 0 s lss han 15%, h xprmnal rsuls can sll agr wll wh horcal calculaon afr w mak ach nod hav a las on dg. And n h followng, w carry ou h xprmns n hs way. Th modl vrfcaons ar carrd ou hrough a lo of xprmns from hr aspcs: h vrfcaon of h dgr xponn brakng h bound of 2, h vrfcaons for dffrn valus of h xponn, and rproducng h AS opology of h IPv6 Inrn. 6.1 Th vrfcaon for brakng h bound of 2 Our modl can gnra a scal-fr nwork wh h dgr xponn smallr han 2. Th followng paramrs q=0.525, ε =-0.25, p=0.3652, m=1, m 0 =5 wr usd o consruc a larg nwork of 100,000 nods and 434,163 dgs wh h dgr xponn γ =1.7138, shown n Fg.3. And h dgr dsrbuon follows h powr-law wh h corrlaon coffcn R= whch ndcas ha h dgr dsrbuon of h opology s n good accordanc wh h powr-law form. I s asy o s ha h xprmnal valu of h xponn γ = also agrs wll wh h horcal xpcd valu γ = accordng o (6). Fgur 3. Th dgr xponn of h gnrad nwork, whch s γ =1.7138, braks h bound of 2 and s vry clos o h horcal xpcd valu γ = accordng o (6). 6.2 Th vrfcaons for dffrn valus of h xponn Our modl s a gnralzaon of h EBA modl and h BA modl. Wh ε = 1/ k our modl s spcalzd o h EBA and wh ε = 0 and q=0 s xacly h BA modl. No only can our modl consruc a nwork wh h dgr xponn smallr ha 2 bu also has h capably o produc nworks wh dffrn valus of h xponn. Exampls of such fv nworks wh 10,000 nods and 40,000 dgs ar shown n TABLE 5 and Fg.4 (a~). W nocd ha n (6) γ can b adusd by q and ε, so w dd a srs of xprmns and hn pckd ou a group wh h bggs corrlaon coffcns. Ths xprmns show ha: frsly, h powr-law corrlaon coffcn s almos clos o 1, so h opology gnrad by hs modl s n good accordanc wh h powr-law dsrbuon; scondly, xprmnal rsuls agr wll wh h horcal xpcd valu, as shown n TABLE 5. 10

11 TABLE 5 Fv gnrad nworks of h sam sz of 10,000 nods and 40,000 dgs No. q ε γ γ R Fgur Fg.4a Fg.4b Fg.4c Fg.4d Fg.4 In hs abl, q and ε ar h npu paramrs of ach modl; γ s h horcal xpcd valu of h dgr xponn by (6), γ s h xprmnal rsul, and R s h absolu valu of h corrlaon coffcn. (a) γ =1.71, γ =1.71 (b) γ =2.00, γ =1.96 (c) γ =2.50, γ =2.48 (d) γ = 3.00, γ =

12 () γ =3.50, γ =3.44 Fgur 4. Fv nworks of h sam sz of 10,000 nods and 40,000 dgs for dffrn valus of h dgr xponn. In hs fgur, γ s h horcal xpcd valu of h dgr xponn by (6), and γ s h xprmnal rsul. 6.3 Rproducng h IPv6 Inrn AS opology As mnond n scon 4, s dffcul o modl h IPv6 Inrn accuraly bcaus of s small dgr xponn. In h followng, w us our modl o rproduc h IPv6 Inrn AS opology dscovrd by Dolphn and h xprmnal rsuls show ha our modl succssfully capur h nw faur of h small dgr xponn as wll as ohr opology mrcs dscrbd n scon 3. Th xprmns show ha our modl has a convrgnc propry. Tha s whn h numbr of nods grows o h nfny h dgr xponn gradually rachs h horcal valu n (6). Thrfor can gnra a scal-fr nwork of small sz whos dgr xponn of dgr dsrbuon s smallr han h horcal xpcd valu n (6). Bcaus h sz of currn IPv6 opology s rlavly small, w can mak us of hs faur o gnra a nwork. Afr a srs of xprmns, undr hs condon,.. q=0.4, ε =-0.25, p=0.462, m=1, and m 0 =5, w gnra a nwork wh h opology almos h sam as ha of h IPv6 Inrn as shown n TABLE 6 and Fg.5. W also dmonsra h rsuls by h BA and EBA modls n TABLE 6 and Fg.5. In Fg.5a and Fg.5g, s asy o s ha h opology gnrad by our modl closly mach h IPv6 AS graph s dgr dsrbuon. And n Fg.5b, h lnar rgrsson shows our modl succssfully rproduc h small dgr xponn of h IPv6 Inrn AS opology. Fg.5(c~) dmonsra ha h rproducd opology s also n good accordanc wh h IPv6 Inrn AS graph on mrcs of h dsanc dsrbuon, rch-club conncvy, nod bwn-nss and local clusrng. TABLE 6 Rproducng h IPv6 Inrn AS opology AS by Dolphn BA Modl EBA Modl Our Modl Numbr of nods Numbr of dgs Avg dgr Exponn of P(k) Max dgr Avg dsanc Nomalzd avg nod bwnnss 4.2E-3 4.7E-3 5.0E-3 4.2E-3 Nomalzd avg dg bwnnss 1.5E-3 1.8E-3 1.5E-3 1.6E-3 Avg clusrng Exponn of rch club

13 (a) Dgr dsrbuon comparson (b) Dgr dsrbuon lnar rgrsson (c) Dsanc dsrbuon (d) Rch-club conncvy () Local clusrng (f) Normalzd nod bwnnss 13

14 (g) Complmnary Cumulav dsrbuon Fgur 5. Rproducng h IPv6 Inrn AS opology 7. Concluson Basd on h analyss of h AS opologs of h IPv6 Inrn from wo daa sourcs, w fnd ha h IPv6 Inrn s also a scal-fr nwork bu wh a far smallr dgr xponn han ha of IPv4. As a masur of unformy of h dgr dsrbuon, h dgr xponn s affcd by many facors. Basd on h mos mporan wo of hm: h probably of prfrnal aachmn and dg rwrng n nwork volvng procss, w propos our modl whch s a gnralzaon of h EBA modl. I s shown boh horcally and xprmnally ha hs modl braks h bound of 2 for h dgr xponn and can wll consruc nworks wh dffrn valus of h dgr xponn. Fnally usng hs modl w succssfully rproduc h opology of h IPv6 Inrn. Acknowldgmns Th auhors would lk o hank Profssor Guanrong Chn for hs hlpful commns and suggsons and Mahw Luck for hs hlp on Scampr daa analyss and CAIDA for h opology daa of Skr and Scampr. W ar also vry graful o h anonymous rvwrs whos valuabl commns and consrucv crcsm hlpd o mprov h papr sgnfcanly. Rfrncs [1] R. Gllgan, E. Nordmark, RFC(2001) [2] B. Carpnr, K. Moor, RFC(2001) [3] A. Durand, P. Fasano, I. Guardn, D. Lno RFC(2001) [4] D. Kroukov, F. Chung, K. Claffy al. CAIDA Rpor [5] P. Mahadvan, D. Kroukov, M. Fomnkov al. Compu. Commun. Rv. 36 (2006) 6. [6] P. Mahadvan, D. Kroukov, K. Fall, A. Vahda, Procdngs of SIGCOMM [7] R. Albr, A.-L. Barabás, Rv. Mod. Phys. 74(2002) 47. [8] A.-L. Barabás, R. Albr, Scnc 286 (1999) 509. [9] R. Albr, A.-L. Barabás, Phy. Rv. L. 85 (2000) 24. [10] B. Tan, D. Towsly, Procdngs of INFOCOM [11] S. Zhou, R.J. Mondragon Phys. Rv. E 70 (2004) [12] J. M. Carlson, J. Doyl, Phys. Rv. E 60 (1999)

15 [13] P.L. Krapvsky, S. Rdnr, F. Lyvraz, Phy. Rv. L 85 (2000) 4629 [14] M.E.J. Nwman, SIMA Rvw, 45.(2003) 167. [15] Comm on Nwork Scnc for Fuur Army Applcaons, Naonal Rsarch Councl: Nwork Scnc, Th Naonal Acadms Prss, [16] R. Pasor-Saorras, A. Vspgnan: Evoluon and Srucur of h Inrn A Sascal Physcs Approach,Cambrdg Unvrsy Prss, [17] S.N. Dorogovsv, J.F.F. Mnds: Evoluon of Nworks From Bologcal Ns o h Inrn and WWW, Oxford Unvrsy Prss, [18] X.F. Wang, X. L, G. Chn: Complx Nworks Thory and Applcaons (n Chns), Tsnghua Prss, [19] M. Falousos, P. Falousos, C. Falousos, Compu. Commun. Rv. 29 (1999) 251. [20] G. Sganos, M. Falousos, P. Falousos, C. Falousos, IEEE/ACM Trans. on Nworkng 11 (2003) 514. [21] P. Erdös, A. Rény, Publ. Mah. Dbrcn 6 (1959) 21. [22] B.M. Waxman, IEEE Journal on Slcd Aras n Commun, 6 (1988) [23] M.B. Doar, Procdngs. of h GLOBECOM 96. (1996) 86. [24] E.W. Zgura, K.L. Calvr, M.J. Donahoo, IEEE/ACM Trans. on Nworkng, 5 (1997) 770. [25] X.F. Wang, D. Logunov, INFOCOM [26] S. Zhou, R.J. Mondragon, IEEE Commun. L., 8 (2004) 180. [27] G.W. Danl, C. Fangzh, V. Ramsh, Compu. Commun. Rv. 33 (2003) 59. [28] R.Govndam, H.Tangmunarunk, IEEE INFOCOM [29] CAIDA hp:// [30] Scampr hp:// [31] A opology by hp:// [32] hp://pv6.nlsd.buaa.du.cn/ [33] S.N. Dorogovsv, J. F. F. Mnds, A. N. Samukhn, Phys. Rv. L. 85(2000) [34] A.-L. Barabás, R. Albr, H. Jong, Physca A, 272 (1999) 173. [35] hp:// 15