A Random Graph Model for Power Law Graphs

Transcription

1 A Random Gaph Modl fo Pow Law Gaphs Wllam Allo, Fan Chung, and Lnyuan Lu CONTNTS. Intoducton. A Random Gaph Modl 3. Th Connctd Componnts 4. Th Szs of Connctd Componnts n Ctan Rangs fo 5. On th Sz of th Scond Lagst Componnt 6. Compasons wth Ralstc Massv Gaphs 7. Opn Qustons Acnowldgmnts Rfncs W popos a andom gaph modl whch s a spcal cas of spas andom gaphs wth gvn dg squncs whch satsfy a pow law. Ths modl nvolvs only a small numb of paamts, calld logsz and log-log gowth at. Ths paamts captu som unvsal chaactstcs of massv gaphs. Fom ths paamts, vaous popts of th gaph can b dvd. Fo ampl, fo ctan angs of th paamts, w wll comput th pctd dstbuton of th szs of th connctd componnts whch almost suly occu wth hgh pobablty. W llustat th consstncy of ou modl wth th bhavo of som massv gaphs dvd fom data n tlcommuncatons. W also dscuss th thshold functon, th gant componnt, and th voluton of andom gaphs n ths modl. Rsach suppotd n pat by NSF Gant No. DMS INTRODUCTION Is th Wold Wd Wb compltly connctd? If not, how bg s th lagst componnt, th scond lagst componnt, tc.? Anyon who has sufd th Wb fo any lngth of tm wll undoubtdly com away flng that f th a dsconnctd componnts at all, thy must b small and fw n numb. Is th Wb too lag, dynamc and stuctulss to answ ths qustons? Pobably ys, f th szs of th lagst componnts a qud to b act. Rcntly, howv, som of th stuctu of th Wb has com to lght whch may nabl us to dscb gaph popts of th Wb qualtatvly. Kuma t al. [999a; 999b] and Klnbg t al. [999] hav masud th dg squncs of th Wb and shown that t s wll appomatd by a pow law dstbuton. That s, th numb of nods, y, of a gvn dg s popotonal to fo som constant >. Ths was potd ndpndntly by Albt, Baabas and Jong [Albt t al. 999; Baabas and Albt 999; Baabas t al. ]. Th pow law dstbuton of th dg squnc appas to b a vy obust popty of th Wb dspt ts dynamc natu. In c A K Pts, Ltd / $.5 p pag pmntal Mathmatcs, pag 53

2 54 pmntal Mathmatcs, Vol. (), No. fact, th pow law dstbuton of th dg squnc may b a ubqutous chaactstc, applyng to many massv al wold gaphs. Indd, Abllo t al. [998] hav shown that th dg squnc of call gaphs s ncly appomatd by a pow law dstbuton. Call gaphs a gaphs of calls handld by som subst of tlphony cas fo a spcc tm pod. Faloutsos t al. [999] hav shown that th dg squnc of th Intnt out gaph also follows a pow law. Just as many oth al wold pocsss hav bn ctvly modld by appopat andom modls, n ths pap w popos a pasmonous andom gaph modl fo gaphs wth a pow law dg squnc. W thn dv connctvty sults that hold wth hgh pobablty n vaous gms of ou paamts. Fnally, w compa th sults fom th modl wth th act connctvty stuctu fo som call gaphs computd by Abllo t al. [998]. An tndd abstact of ths pap has appad n th Pocdngs of th Thtyscond Annual ACM Symposum on Thoy of Computng [Allo t al. ]. In ths pap, w hav ncludd th complt poofs fo th man thoms and addtonal thoms focusd on th scond lagst componnts of pow gaphs n vaous angs. In addton, w gv som cnt fncs; s also [Hays ]. Pow Law Random Gaphs Th study of andom gaphs dats bac to th smnal paps of d}os and Rny [96; 96], whch lad th foundaton fo th thoy. Th a th standad modls fo what w wll call n ths pap unfom andom gaphs [Alon and Spnc 99]. ach has two paamts, on contollng th numb of nods n th gaph and th oth th dnsty o numb of dgs. Fo ampl, th andom gaph modl G(n; ) assgns unfom pobablty to all gaphs wth n nods and dgs, whl n th andom gaph modl G(n; p) ach dg s chosn wth pobablty p. Ou pow law andom gaph modl P (; ) also has two paamts. Thy only oughly dlnat th sz and dnsty, but thy a natual and convnnt fo dscbng a pow law dg squnc. Th modl s dscbd as follows. Lt y b th numb of nods wth dg. P (; ) assgns unfom pobablty to all gaphs wth y (wh slf loops a allowd). Not that s th ntcpt and s th (ngatv) slop whn th dg squnc s plottd on a log-log scal. Th s also an altnatv pow law andom gaph modl analogous to th unfom gaph modl G(n; p). Instad of havng a d dg squnc, th andom gaph has an pctd dg squnc dstbuton. Th two modls a bascally asymptotcally quvalnt, subjct to a boundng of o stmats fo th vaancs; ths wll b futh dscussd n [Allo t al. ]. Ou Rsults Just as fo th unfom andom gaph modl wh gaph popts a studd fo ctan gms of th dnsty paamt and shown to hold wth hgh pobablty asymptotcally n th sz paamt, n ths pap w study th connctvty popts of P (; ) as a functon of th pow whch hold almost suly fo sucntly lag gaphs. By, w show that whn <, th gaph s almost suly connctd. Fo < < th s a gant componnt, that s, a componnt of sz (n). Moov, all small componnts a of sz O(). Fo < < th s a gant componnt and all small componnts a of sz O(log n). Fo th small componnts a of sz O(log nlog log n). Fo > th gaph almost suly has no gant componnt. In addton w dv sval sults on th szs of th scond lagst componnt. Fo ampl, w show that fo > 4 th numb of componnts of gvn szs can b appomatd by a pow law as wll. Pvous Wo Stctly spang ou modl s a spcal cas of andom gaphs wth a gvn dg squnc, fo whch th s a lag ltatu. Fo ampl, Womald [98] studd th connctvty of gaphs whos dgs a n an ntval [; R], wh 3. Lucza [99] consdd th asymptotc bhavo of th lagst componnt of a andom gaph wth gvn dg squnc as a functon of th numb of vtcs of dg. Hs sult was futh mpovd by Molloy and Rd [995; 998], who consdd a andom gaph on n vtcs wth th followng dg dstbuton. Th numb of vtcs of dg ; ; ;

3 Allo, Chung, and Lu A Random Gaph Modl fo Pow Law Gaphs 55 a about n, n,..., spctvly, wh th 's sum to. In [Molloy and Rd 995] t s shown that f Q P ( ) > and th mamum dg s not too lag, thn such andom gaphs hav a gant componnt wth pobablty tndng to as n gos to nnty, whl f Q < thn all componnts a small wth pobablty tndng to as n!. Th pap also amns th thshold bhavo of such gaphs. In ths pap, w wll apply ths tchnqus to dal wth th spcal cas that concns ou modl. Sval oth paps hav tan an appoach to modlng pow law gaphs dnt fom th on tan h [Allo t al. ; Baabas and Albt 999; Baabas t al. ; Klnbg t al. 999; Kuma t al. 999b]. Th ssntal da of ths paps s to dn a andom pocss fo gowng a gaph by addng nods and dgs. Th ntnt s to show that th dnd pocsss asymptotcally yld gaphs wth a pow law dg squnc wth vy hgh pobablty. Whl ntstng and mpotant, ths appoach has sval dcults. Fst, th modls a dcult to analyz goously, snc th tanston pobablts a thmslvs dpndnt on th th cunt stat. Fo ampl, [Baabas and Albt 999; Baabas t al. ] mplctly assum that th pobablty that a nod has a gvn dg s a contnuous functon. Kuma t al. [] o a patal analyss of th stuaton. Scond, whl th modls may gnat gaphs wth pow law dg squncs, t mans to b sn f thy gnat gaphs that duplcat oth stuctual popts of th Wb, th Intnt, and call gaphs. Fo ampl, th modl n [Baabas and Albt 999; Baabas t al. ] cannot gnat gaphs wth a pow law oth than c 3. Moov, all th gaphs can b dcomposd nto m dsjont ts, wh m s a paamt of th modl. Th (; ) modl n [Kuma t al. 999b] s abl to gnat gaphs fo whch th pow law fo th ndg s dnt than th pow law fo th outdg as s th cas fo th Wb. Howv, to do so, th modl qus that th b nods that hav only ndg and no outdg and vc vsa. Whl ths may b appopat fo call gaphs (.g., custom svc numbs) t may not b ght fo modlng th Wb. Thus, whl th andom gaph gnaton appoach holds th poms of accuatly pdctng a wd vaty of stuctual popts of many al wold massv gaphs, much wo mans to b don. In ths pap w ta a dnt appoach. W do not attmpt to answ how a gaph coms to hav a pow law dg squnc. Rath, w ta that as a gvn. In ou modl, all gaphs wth a gvn pow law dg squnc a qupobabl. Th goal s to dv stuctual popts that hold wth pobablty asymptotcally appoachng. Such an appoach, whl potntally lss accuat than th dtald modlng appoach abov, has th advantag of bng obust th stuctual popts dvd n ths modl wll b tu fo th vast majoty of gaphs wth th gvn dg squnc. Thus, w blv that ths modl wll b an mpotant complmnt to andom gaph gnaton modls. Th pow law andom gaph modl wll b dscbd n dtal n th nt scton. In Sctons 3 and 4, ou sults on connctvty wll b dvd. Scton 5 dscusss th szs of th scond lagst componnts. Scton 6 compas th sults of ou modl to act connctvty data fo call gaphs. A shot lst of opn qustons concluds th atcl. A subsqunt pap [Allo t al. ] amns futh sval aspcts of pow law gaphs, ncludng th voluton, th \scal nvaanc", and th asymmty of n-dgs and out-dgs.. A RANDOM GRAPH MODL W consd a andom gaph wth th followng dg dstbuton dpndng on two gvn valus and. Th a y vtcs of dg, wh and y satsfy log y log In oth wods, fv dg v g y Bascally, s th logathm of th sz of th gaph and s th log-log gowth at of th gaph. Th numb of dgs should b an ntg. To b pcs, th psson abov fo y should b oundd down to b c. If w us al numbs nstad of oundng down to ntgs, ths may caus som o tms n futh computaton, but w wll s that th o tms can b asly boundd. Fo smplcty and convnnc, w wll us al numbs

4 56 pmntal Mathmatcs, Vol. (), No. wth th undstandng that th actual numbs a th ntg pats. Anoth constant s that th sum of th dgs should b vn. Ths can b assud by addng a vt of dg f th sum s odd. Futhmo, fo smplcty, w assum h that th a no solatd vtcs. W can dduc th followng facts fo ou gaph () Th mamum dg of th gaph s. Not that log y log. () Th numb n of vtcs can b computd as follows By summng y() fo fom to, w hav n 8 >< > () f >, f, ( ) f < <, P wh (t) n n t s th Rmann zta functon. (3) Th numb of dgs s gvn by 8 >< > ( ) f >, 4 f, ( ) f < <. Th css of th al numbs n (){(3) ov th ntg pats can b stmatd as follows Fo th numb n of vtcs, th o tm s at most. Fo, t s o(n), whch s a low od tm. Fo < <, th o tm fo n s latvly lag. In ths cas, w hav n Thfo, n has sam magntud as ( ). Th numb of dgs can b tatd smlaly. Fo, th o tm of s o(), a low od tm. Fo < <, has th sam magntud as n th fomula of tm (3). In ths pap w dal manly wth th cas >. Th cas < < s consdd only n th nt scton, wh w f to as a constant. By usng al numbs nstad of oundng down to th ntg pats, w smplfy th agumnts wthout actng th conclusons. To study th andom gaph modl, w must consd lag n. W say a popty holds almost suly (a. s.) f th pobablty that t holds tnds to as th numb n of th vtcs gos to nnty. Thus w consd to b lag but s d. W us th followng andom gaph modl fo a gvn dg squnc Th modl. Fom a st L contanng dg v dstnct cops of ach vt v.. Choos a andom matchng of th lmnts of L. 3. Fo two vtcs u and v, th numb of dgs jonng u and v s qual to th numb of dgs n th matchng of L jonng cops of u to cops of v. W ma that th gaphs that w a consdng a n fact mult-gaphs, possbly wth loops. Ths modl s a natual tnson of th modl fo -gula gaphs, fomd by combnng andom matchngs. Fo fncs and undnd tmnology, s [Alon and Spnc 99; Womald 999]. Ths andom gaph modl s slghtly dnt fom th unfom slcton modl P (; ) dscbd n Scton.. Howv, by usng th tchnqus of [Molloy and Rd 998, Lmma ], t can b shown that f a andom gaph wth a gvn dg squnc a. s. has popty P und on of ths two modls, thn t a. s. has popty P und th oth modl, povdd som gnal condtons a satsd. 3. TH CONNCTD COMPONNTS Molloy and Rd [995] showd that fo a andom gaph wth ( + o())n vtcs of dg, wh th a nonngatv valus that sum to, th gant componnt mgs whn Q X ( ) > ; so long as th mamum dg s lss than n 4 ". Thy also show that almost suly th s no gant componnt whn Q < and th mamum dg s lss than n 8 ". H w comput Q fo ou (; )-gaphs j X Q ( ) ( ) ( ) f > 3 W a thus ld to consd th valu , whch s a soluton to ( ) ( )

5 Allo, Chung, and Lu A Random Gaph Modl fo Pow Law Gaphs 57 If >, w hav j ( ) < W summaz ou sults h. Whn > , th andom gaph almost suly has no gant componnt. Whn < , th s almost suly a unqu gant componnt.. Whn < < , almost suly th scond lagst componnts hav sz (log n). Fo any < (log n), th s almost suly a componnt of sz. 3. Whn, almost suly th scond lagst componnts a of sz (log nlog log n). Fo any < (log nlog log n), th s almost suly a componnt of sz. 4. Whn < <, th scond lagst componnts a almost suly of sz (). Th gaph s almost suly not connctd. 5. Whn < <, th gaph s almost suly connctd. 6. Fo , th cas s complcatd. It cosponds to th doubl jump of a andom gaph G(n; p) wth p n. 7. Fo, th s a nontval pobablty fo th cas that th gaph s connctd o dsconnctd. W ma that fo > 8, Molloy and Rd's sult mmdatly mpls that almost suly th s no gant componnt. Whn 8, addtonal analyss s ndd to dal wth th dg constants. W wll pov n Thom 4. that almost suly th s no gant componnt whn >. In Scton 5, w wll dal wth th ang <. W wll show n Thom 5. that almost suly th s a unqu gant componnt whn <. Futhmo, w wll dtmn th sz of th scond lagst componnt wthn a constant facto. 4. TH SIZS OF CONNCTD COMPONNTS IN CRTAIN RANGS FOR Fo > , almost suly th s no gant componnt. Ths ang s of spcal ntst snc t s qut usful lat fo dscbng th dstbuton of small componnts. Thom 4.. Fo (; )-gaphs wth > 4, th dstbuton of th numb of connctd componnts s as follows. Fo ach vt v of dg d (), lt b th sz of th connctd componnt contanng v. Thn wh P d c > c dc c ; c ( ) ( ) and c ( 3) ( ) ( ) ( ) a constants and wh d " wth " an abtay small postv numb and d a (slowly) ncasng functon of n. In oth wods, th vt v almost suly blongs to a connctd componnt of sz d c + O(d +" ). Th numb of connctd componnts of sz s almost suly at last and at most wh ( + o()) c c 3 log n + ; c 3 4+ c ( )c + s a constant dpndng only on. 3. A connctd componnt of th (; )-gaph almost suly has sz at most (+) (n (+) log n) In ou poof of ths sult w us th scond momnt, whos convgnc dpnds on > 4. In fact fo 4 th scond momnt dvgs as th sz of th gaph gos to nnty, so ou mthod no long appls. Thom 4. stngthns th followng sult whch can b dvd fom [Molloy and Rd 995, Lmma 3] fo th ang of > 4. Thom 4.. Fo > , a connctd componnt of th (; )-gaph almost suly has sz at most C (n log n), wh C 6c s a constant dpndng only on.

6 58 pmntal Mathmatcs, Vol. (), No. Th poof of Thom 4., whch w by dscb h bcaus t s ndd n povng Thom 4., uss th banchng pocss mthod. Pc any vt v n th gaph, pos ts nghbos, thn th nghbos of ts nghbos, patng untl th nt componnt s posd. W pos only on vt at ach stag. At stag, lt L th st of vtcs posd and X b th andom vaabl that counts th numb of vtcs n L. W ma all vtcs n L as th lv o dad. A vt n L whos nghbos hav not all bn posd yt s mad lv. On whos nghbos hav all bn posd s mad dad. Lt O b th st of lv vtcs and Y th andom vaabl that s th numb of vtcs n O. At ach stp w ma act on dad vt, so th total numb of dad vtcs at th -th stp s. W hav X Y +. Intally w assgn L O fvg. Thn at stag, w do th followng. If Y, stop and output X.. Othws, andomly choos a lv vt u fom O and pos ts nghbos n N u. Thn ma u dad and ma ach vt lv f t s n N u but not n L. W hav L L [ N u ; O (O n fug) [ (N u n L ) Suppos that v has dg d. Thn X d +, and Y d. vntually Y wll ht f s lag nough. Lt dnot th stoppng tm of Y, namly, Y. Thn X Y + masus th sz of th connctd componnt. W st comput th pctd valu of Y and thn us Azuma's Inqualty [Molloy and Rd 995] to pov Thom 4.. Suppos that vt u s posd at stag. Thn N u \ L contans at last on vt v, whch was posd to ach u. Howv, N u \L may contan mo than on vt. W call an dg fom u to a vt n L oth than v a bacdg. Bacdgs caus th ploaton to stop soon, spcally whn th componnt s lag. Howv n ou cas > , th contbuton of bacdgs s qut small. W st Z #fn u g and W #fn u \ L g, so Z masus th dg of th vt posd at stag, whl W masus th numb of bacdgs. By dnton, Y Y Z W W hav (Z ) ( ) ( ) + O(n3 ) ( ) + O(n ) ( ) ( ) + O(n3 ) Now w bound W. Suppos th a m dgs posd at stag. Thn th pobablty that a nw nghbo s n L s at most m. W hav (W ) X m + O m m m ( m) ; (4 ) povdd that m o(). Whn C, m s at most C 3. Hnc, m O(n3 log n) o() W hav (Y ) Y + d + X j X (Y j Y j ) (Z j W j ) j ( ) d + ( ) ( ) O(n 3 log n) d c ( ) + o() Poof of Thom 4.. Snc jy j Y j j, by Azuma's matngal nqualty, w hav P jy (Y )j > t t ( ) ; wh (6c ) log n and t c. Snc w hav (Y ) + t d c ( ) + o() + c c + d + c + o() < ; P > (6c ) log n P > P(Y ) P Y > (Y ) + t p t n

7 Allo, Chung, and Lu A Random Gaph Modl fo Pow Law Gaphs 59 Hnc, th pobablty that th sts a vt v such that v ls n a componnt of sz gat than s at most 6 log n c n n n o() Th poof of Thom 4. uss th mthodology abov as a statng pont whl ntoducng th calculaton of th vaanc of th abov andom vaabls. Poof of Thom 4.. W follow th notaton and pvous sults of Scton 4. Und th assumpton > 4, w consd Va(Z ) ( ) (Z ) 3 (Z ) ( 3) + O(n4 ) ( ) + O(n ) ( ) ( ) + O(n 3 ) ( 3) ( ) ( ) + O(n 4 ) ( ) c + o(); snc > 4. W nd to comput th covaanc. Th a modls fo andom gaphs n whch th dgs a n dpndntly chosn. Thn, Z and Z j a ndpndnt. Howv, n th modl basd on andom matchngs, th s a small colaton. Fo ampl, Z slghtly cts th pobablty of Z j y. Namly, Z j has slghtly lss chanc, whl Z j y 6 has slghtly mo chanc. Both dncs can b boundd by Hnc Cova(Z ; Z j ) (Z ) ( ) O n f 6 j Now w wll bound W. Suppos that th a m dgs posd at stag. Thn th pobablty that a nw nghbo s n L s at most m. W hav Va(W ) Cova(W ; W j ) Cova(Z ; W j ) X m 3 (W ) m(m + ) O m ( m) 3 ; m m + O q Va(W ) Va(W j ) m + O m ; q Va(Z ) Va(W j ) O m Whn O( ), m O( ), w hav ( ) (Y ) d + ( ) ( ) + O(n 3 ) + m and d ( )c + O(n 4 ) d ( )c + o() Va(Y ) Va Va X j d + X j (Z j W j ) (Y j Y j ) X Va(Zj ) + Va(W j ) j X + Cova(Zj ; Z ) Cova(Z j ; W ) j6 + Cova(W j ; W ) c + o() + O(n) + O( p ( ) ) + O( ( ) ) c + o() + O( ( ) ) + O( (3 ) ) c + o() Chbyshv's nqualty gvs P jy (Y )j > < ; wh s th standad dvaton of Y, and p c + o( p ). St d dc ; c c c d + dc c c c

8 6 pmntal Mathmatcs, Vol. (), No. Thn (Y ) d ( )c +o() p c +o( p ) dc d c o( p d) c c Hnc, dc c o( p d) > P( < ) P(Y ) Smlaly, P Y < (Y ) (Y )+ d ( )c +o()+ p c +o( p ) dc d + c +o( p d) c c Hnc, dc c +o( p d) < P( > ) P(Y > ) Thfo P P Y > (Y ) + d dc > c c c Fo a d v and a functon slowly ncasng to nnty, th pcdng nqualty mpls that almost suly w hav dc + O( p d). Almost all componnts gnatd by vtcs of dg hav sz about dc. On such componnt can hav at most about c vtcs of dg d. Hnc, th numb of componnts of sz dc s at last c d. Lt d c. Thn th numb of componnts of sz s at last c + o() Th agumnt abov actually gvs th followng sult. Th sz of vy componnt whos vtcs hav dg at most d s almost suly Cd log n, wh C 6c s th sam constant as n Thom 4.. St Cd log n and consd th numb of componnts of sz. A componnt of sz almost suly contans at last on vt of dg gat than d. Fo ach vt v wth dg d d, by pat n th statmnt of Thom 4., w hav d P d dc > c c c d Lttng w hav d c Cd log n c 4 c d ; P( Cd log n) P > d + d dc c d C 3 d 4 log n ; wh C 3 3c (c 3 C ) c c 8 s a constant dpndng only on. Snc th a only d vtcs of dg d, th numb of componnts of sz at last s at most wh dd d C d 3 d 4 log n C 3 d 4 log n c c X dd d C 3 d 4 log n C 3 ( )d + log n c 3 log n + ; c 3 C 3 ( ) C + 4+ c ( )c + d Fo (+), th pcdng nqualty mpls that th numb of componnts of sz at last s at most o(). In oth wods, almost suly no componnt has sz gat than (+) Ths complts th poof of Thom ON TH SIZ OF TH SCOND LARGST COMPONNT Fo < , w consd th gant componnt as wll as th sz of th scond lagst componnt. Thom 5.. Consd an (; )-gaph wth < Th s a unqu gant componnt of sz (n).

9 Allo, Chung, and Lu A Random Gaph Modl fo Pow Law Gaphs 6. Whn < <, almost suly th sz of th scond lagst componnt s (log n). 3. Whn, almost suly th sz of th scond lagst componnt s (log nlog log n). 4. Whn <, almost suly th sz of th scond lagst componnt s (). 5. Whn < <, almost suly th gaph s connctd. Poof. Whn <, th banchng pocss mthod s no long fasbl whn vtcs of lag dgs a nvolvd. Thus, w cannot apply Azuma's matngal nqualty fo boundng Y as w dd n al poofs. W wll modfy th banchng pocss mthod as follows. (a) Choos a numb (to b spcd lat dpndng on ). (b) Stat wth Y lv vtcs and Y C log n. All oth vtcs a unmad. (c) At th -th stp, choos on lv vt u and posd ts nghbos. If th dg of u s lss than o qual to, pocd as n Scton 4, by mang u dad and all vtcs v N u lv (povdd v s not mad bfo). If th dg of u s gat than, ma actly on vt v N u lv and oths dad, povdd v s unmad. In both cass u s mad dad. Th man da s to show that Y, a tuncatd vson of Y, s wll-concntatd aound (Y ). Although t s dcult to dctly dv such a sult fo Y bcaus of vtcs of lag dgs, w wll b abl to bound th dstbuton Y. Indd, w wll show that th st of mad vtcs (lv o dad) gows to a gant componnt f Y cds a ctan bound. W consd th angs fo. Cas < <. W consd th postv constant Q j ( ) Th s a constant ntg satsfyng X j ( ) > Q W choos satsfyng ( ) Q 4 If th componnt has mo than dgs, t must hav (n) vtcs snc >. So t s a gant componnt and w a don. W may assum that th componnt has no mo than dgs. W now choos and apply th modd banchng pocss. Thn, Y satss Y dc log n, wh C 3 Q s a constant dpndng only on. Y Y. Lt W b th numb of bacdgs as dnd n Scton 4. By nqualty (4{) and th assumpton that th numb of dgs m n th componnt s at most n, w hav Hnc, (Y (W ) Y ) ( ) Q 4 X Q Q 4 Q 4 By Azuma's matngal nqualty, P Y Q 8 P Y j ( ) (W ) (Y ) Q 8 < p (Q8) o(n ) povdd that > C log n. Th pcdng nqualty mpls that wth pobablty at last o(n ), w hav Y > Q8 > whn > dc log n. Snc Y dcass by at most at ach stp, Y cannot b zo f dc log n. So Y > fo all. In oth wods, a. s. th banchng pocss wll not stop. Howv, t s mpossbl to hav Yn > a contadcton. Thus w conclud that th componnt must hav at last n dgs. So t s a gant componnt. W not that f a componnt has mo than dc log n dgs posd, thn almost suly t s a gant componnt. In patcula, any vt wth dg mo than dc log n s almost suly n th gant componnt. Hnc, th scond componnts hav sz of at most (log n). Nt w show that th scond lagst has sz at last (log n). W consd th vtcs v of dg c, wh c s som constant. Th s a postv pobablty that all nghbong vtcs of v hav dg. In ths cas, w gt a connctd componnt

10 6 pmntal Mathmatcs, Vol. (), No. of sz + (log n). Th pobablty of ths s about c ( ) Th a (c) vtcs of dg. Thus th pobablty that non of thm has th pcdng popty s about (c) p ( ) c ( ) c (c) wh c 8 < p o(); f 3, log( ) ( ) c (c) f 3 > >. In oth wods, a. s. th s a componnt of sz c + (log n). Thfo, th scond lagst componnt has sz (log n). Moov, th agumnt stll holds f w plac c by any small numb. Hnc, small componnts hbt a contnuous bhavo. Cas. W choos. W not that a componnt wth mo than 3 dgs must b unqu. W wll pov that almost suly th unqu componnt contans all vtcs wth dg gat than. So t contans ( o()) dgs and t s th gant componnt. W futh modfy th banchng pocss by statng fom Y d vtcs. If th componnt has mo than dgs, w a don. Othws, 3 th pctd numb of bacgs s small. (W ) 3 ( 3) 6 fom nqualty (4{). Hnc, Y (Y Y d ; Y Y ; Y ) X satss j ( ) (W ) > 6 By Azuma's matngal nqualty, P(Y ) P(Y (Y ) < p o(n ) () povdd that. Ths nqualty mpls that wth poablty at last o(n ), w hav Y > whn > d. Snc Y dcass at most by at ach stp, Y cannot b zo f d. So Y > fo all. In oth wods, a. s. th banchng pocss wll not stop. Howv, t s mpossbl to hav Yn > a contadcton. Thus w conclud that th componnt must hav at last dgs. W 3 not that a. s. all vtcs wth dg mo than d a n th unqu componnt wth at last dgs, hnc th gant componnt. 3 Th pobablty that a andom vt s n th gant componnt s at most X log Th pobablty that th a log vtcs not n th gant componnt s at most log log (+o()) o(n ) Snc th s at most n connctd componnts, w conclud that a. s. a connctd componnt of sz gat that log n log log log n must b th gant componnt. Now w nd a vt v of dg wth 9log. Th pobablty that all ts nghbos a of dg s (). Th pobablty that no such vt sts s at most p p o() Hnc, almost suly th s a vt of dg 9log that, whch foms a connctd componnt of sz +. Ths povs that a. s. th scond lagst componnt has sz (log nlog log n).

11 Allo, Chung, and Lu A Random Gaph Modl fo Pow Law Gaphs 63 Cas 3 < <. W us th modd banchng pocss by choosng p (5 ) (6 ) If a componnt has mo than 3 dgs, t s th unqu gant componnt and w a don. Othws, 3 (W ) ( 3) 6 Hnc, Y (Y satss Y 5 ( ) p C (3 ) ; Y Y ) Y p (5 ) (6 ) ; p (5 ) (6 ) X C () j ( ) (W ) H C s a constant dpndng only on. By Azuma's matngal nqualty, P Y C() < P Y (Y povdd that ) C() < p ( C() ) p (5 ) (6 ) o(n ) 5 ( ) p C (3 )! Ths nqualty shows that wth pobablty at last o(n ), w hav Y > C() > povdd that > 5 ( ) p C (3 ) Snc Y dcass at most by at ach stp, Y cannot b zo f 5 ( ) p C (3 ) So Y > fo all. In oth wods, a. s. th banchng pocssng wll not stop. Howv, t s mpossbl to hav Yn > a contadcton. So, a. s. all vtcs wth dg mo than 5 ( ) p C (3 ) a n th gant componnt. Th pobablty that a andom vt s n th gant componnt s at most 5 ( ) p C X (3 ) Th pobablty that all j 3 + p ( ) (3 ) vtcs a not n th gant vt s at most p ( ) 3 + o(n ) (3 ) Snc th a at most n connctd componnts, w conclud that a. s. a connctd componnt of sz gat that j 3 () must b th gant componnt. Fo < <, w a vt v of dg. Th pobablty that th oth vt that conncts to v s also of dg s Thfo th pobablty that no componnt has sz of s at most ( ) o() In oth wods, th gaph a. s. has at last on componnt of sz. Fo < <, w want to show that th andom gaph s a. s. connctd. Snc th sz of th possbl scond lagst componnt s boundd by a constant M, all vtcs of dg M a almost suly n th gant componnt. W only nd to show th pobablty that th s an dg connctng two small dg vtcs s small. Th a only MX j C vtcs wth dg lss than M. Fo any andom pa of vtcs (u; v), th pobablty that th s an dgs connctng thm s about ( )

12 64 pmntal Mathmatcs, Vol. (), No. Hnc th pobablty that th s dg connctng two small dg vtcs s at most X u;v (C ) ( ) o() Thus vy vt s a. s. connctd to a vt wth dg M, whch a. s. blongs to th gant ponnt. Hnc, th andom gaph s a. s. connctd. 6. COMPARISONS WITH RALISTIC MASSIV GRAPHS Ou (; )-andom gaph modl was ognally dvd fom massv gaphs gnatd by long dstanc tlphon calls. Ths so-calld call gaphs a tan ov dnt tm ntvals. Fo th sa of smplcty, w consd all th calls mad n on day. vy compltd phon call s an dg n th gaph. vy phon numb that th ognats o cvs a call s a nod n th gaph. Whn a nod ognats a call, th dg s dctd out of th nod and contbuts to that nod's outdg. Lws, whn a nod cvs a call, th dg s dctd nto th nod and contbuts to that nod's ndg. y() 7 Th patcula call gaph w usd fo th statstcs n ths scton cospond to th dat August, 998, a typcal day. Th data w compld by J. Abllo and A. Buchsbaum of AT&T Labs fom aw phon call cods usng, n pat, th tnal mmoy algothm of [Abllo t al. 998] fo computng connctd componnts of massv gaphs. In Fgu, w plot th numb of vtcs vsus th ndg and th outdg fo th call gaph. Lt y() b th numb of vtcs wth ndg. Fo ach such that y() >, a dot on th lft plot s placd at ; y(). Th plot on th ght s bult n th sam way. Plots of th numb of vtcs vsus th ndg o outdg fo th call gaphs of oth days a vy smla. Fgu plots fo th sam call gaph th numb of connctd componnts fo ach possbl sz. Th dg squnc of th call gaph dos not oby pfctly th (; )-gaph modl. Th numb of vtcs of a gvn dg dos not vn dcas monotoncally wth ncasng dg. Moov, th call gaph s dctd fo ach dg th s a nod that ognats th call and a nod that cvs th call. Th ndg and outdg of a nod nd not b th sam. Claly th (; )-andom y(o) FIGUR. Lft numb of vtcs y() vsus ndg, plottd on a log-log scal, fo a psntatv al-lf gaph. Rght numb of vtcs vsus outdg o fo th sam gaph. o

13 y(s) Allo, Chung, and Lu A Random Gaph Modl fo Pow Law Gaphs 65 lna. Ths suggsts that aft movng th gant componnt, on s lft appomatly wth an (; )-gaph wth > 4. (Thom 4. ylds a loglog lna laton btwn numb of componnts and componnt sz fo > 4.) Ths sms ntutvly asonabl, snc th gat th dg, th fw nods of that dg w pct to man aft dltng th gant componnt. Ths wll ncas th valu of fo th sultng gaph FIGUR. Lft numb of connctd componnts fo ach possbl componnt sz s fo ou ampl gaph. Not th gant componnt on th low ght. gaph modl dos not captu all of th andom bhavo of th al wold call gaph. Nonthlss, ou modl dos captu som of th bhavo of th call gaph. To s ths w st stmat and n Fgu. Rcall that fo an (; )- gaph, th numb of vtcs as a functon of dg s gvn by log y log. By appomatng Fgu by a staght ln, can b stmatd usng th slop of th ln to b appomatly. Th valu of fo Fgu s appomatly 3 6. Th total numb of nods n th call gaph can b stmatd by () Fo btwn and, th (; )-gaph wll hav a gant componnt of sz (n). In addton, a. s. all oth componnts a of sz O(log n). Moov, fo any O(log n), a componnt of sz sts. Ths s qualtatvly tu of th dstbuton of componnt szs of th call gaph n Fgu. Th on gant componnt contans naly all of th nods. Th mamum sz of th nt lagst componnt s ndd ponntally small than th sz of th gant componnt. Also, a componnt of naly vy sz blow ths mamum sts. Intstngly, th dstbuton of th numb of componnts of sz small than th gant componnt s naly log-log s 7. OPN QUSTIONS Numous qustons man to b studd. Fo ampl, what s th ct of tm scalng? How dos t cospond wth th voluton of? What a th stuctual bhavos of th call gaphs? What a th colatons btwn th dctd and undctd gaphs? It s of ntst to undstand th phas tanston of th gant componnt n th alstc gaph. In th oth dcton, th numb of tny componnts of sz s ladng to many ntstng qustons as wll. Claly, th s much wo to b don n ou undstandng of massv gaphs. ACKNOWLDGMNTS W a gatful to J. Fgnbaum, J. Abllo, A. Buchsbaum, J. Rds, and J. Wstboo fo th assstanc n ppang th gus and fo many ntstng dscussons on call gaphs. W a vy thanful to th anonymous f fo nvaluabl commnts. RFRNCS [Abllo t al. 998] J. Abllo, A. L. Buchsbaum, and J. R. Wstboo, \A functonal appoach to tnal gaph algothms", pp. 33{343 n Algothms SA '98 (Vnc, 998), dtd by G. Blad t al., Lctu Nots n Comp. Sc. 46, Spng, Bln, 998. [Allo t al. ] W. Allo, F. Chung, and L. Lu, \A andom gaph modl fo massv gaphs", pp. 7{8 n Pocdngs of th 3nd Annual ACM Symposum on Thoy of Computng (Potland, OR, ), ACM Pss, Nw Yo,. [Allo t al. ] W. Allo, F. Chung, and L. Lu, \Random voluton of pow law gaphs", n Handboo of massv data sts, vol., dtd by J. Abllo t al. To appa.

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