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1 Ths document s downloaded from the Dgtal Open Access Repostory of VTT Ttle Large clques n a power-law random graph Author(s) Janson, Svante; Luczak, Tomasz; Norros, Ilkka Ctaton Journal of Appled Probablty vol. 47(2010):4, pp Date 2010 URL Rghts Copyrght (2010) Appled Probablty Trust. Ths artcle may be downloaded for personal use only VTT P.O. box 1000 FI VTT Fnland By usng VTT Dgtal Open Access Repostory you are bound by the followng Terms & Condtons. I have read and I understand the followng statement: Ths document s protected by copyrght and other ntellectual property rghts, and duplcaton or sale of all or part of any of ths document s not permtted, except duplcaton for research use or educatonal purposes n electronc or prnt form. You must obtan permsson for any other use. Electronc or prnt copes may not be offered for sale.

2 LARGE CLIQUES IN A POWER-LAW RANDOM GRAPH arxv: v1 [math.co] 5 May 2009 SVANTE JANSON, TOMASZ LUCZAK, AND ILKKA NORROS Abstract. We study the sze of the largest clque ω(g(n, α)) n a random graph G(n, α) on n vertces whch has power-law degree dstrbuton wth exponent α. We show that for flat degree sequences wth α > 2 whp the largest clque n G(n, α) s of a constant sze, whle for the heavy tal dstrbuton, when 0 < α < 2, ω(g(n, α)) grows as a power of n. Moreover, we show that a natural smple algorthm whp fnds n G(n, α) a large clque of sze (1 + o(1))ω(g(n, α)) n polynomal tme. 1. Introducton Random graphs wth fnte densty and power-law degree sequence have attracted much attenton for the last few years (e.g. see Durrett [8] and the references theren). Several models for such graphs has been proposed; n ths paper we concentrate on a Possonan model G(n,α) n whch the number of vertces of degree at least decreases roughly as n α (for a precse defnton of the model see Secton 2 below). We show (Theorem 1) that there s a major dfference n the sze of the largest clque ω(g(n,α)) between the cases α < 2 and α > 2 wth an ntermedate result for α = 2. In the lght tal case, when α > 2 (ths s when the asymptotc degree dstrbuton has a fnte second moment), the sze of the largest clque s ether two or three,.e., t s almost the same as n the standard bnomal model of random graph G(n,p) n whch the expected average degree s a constant. As opposte to that, n the heavy tal case, when 0 < α < 2, ω(g(n,α)) grows roughly as n 1 α/2. In the crtcal case when α = 2 we have ω(g(n,α)) = O p (1), but the probablty that G(n,α) k s bounded away from zero for every k. We also show (Corollary 3) that n each of the above cases there exsts a smple algorthm whch whp fnds n G(n,α) a clque of sze (1 o(1))ω(g(n,α)). Ths s qute dfferent from the bnomal case, where t s wdely beleved that fndng large clque s hard (see for nstance Freze and McDarmd [11]). Smlar but less precse results have been obtaned by Bancon and Marsl [1; 2] for a slghtly dfferent model (see Secton 6.6 below). Date: Aprl 29, Mathematcs Subject Classfcaton. Prmary: 05C80; Secondary: 05C69, 60C05. Key words and phrases. power-law random graph, clque, greedy algorthm. Ths research was done at Insttut Mttag-Leffler, Djursholm, Sweden, durng the program Dscrete Probablty The second author partally supported by the Foundaton for Polsh Scence. 1

3 2 SVANTE JANSON, TOMASZ LUCZAK, AND ILKKA NORROS 2. The model and the results The model we study s a verson of the condtonally Possonan random graph studed by Norros and Rettu [15] (see also Chung and Lu [7] for a related model). For α > 1 t s also an example of the rank 1 case of nhomogeneous random graph studed by Bollobás, Janson, and Rordan [5, Secton 16.4]. In order to defne our model consder a set of n vertces (for convenence labelled 1,...,n). We frst assgn a capacty or weght W to each vertex. For defnteness and smplcty, we assume that these are..d. random varables wth a dstrbuton wth a power-law tal P(W > x) = ax α, x x 0, (2.1) for some constants a > 0 and α > 0, and some x 0 > 0 (here and below we wrte W for any of the W when the ndex does not matter). Thus, for example, W could have a Pareto dstrbuton, when x 0 = a 1/α and P(W > x) = 1 for x x 0, but the dstrbuton could be arbtrarly modfed for small x. We denote the largest weght by W max = max W. Observe that (2.1) mples P(W max > tn 1/α ) n P(W > tn 1/α ) = O(t α ). (2.2) Note also that E W β < f and only f α > β; n partcular, for the heavy tal case when α < 2 we have E W 2 =. Now, condtonally gven the weghts {W } n 1, we jon each par {,j} of vertces by E j parallel edges, where the numbers E j are ndependent Posson dstrbuted random numbers wth means EE j = λ j = b W W j n, (2.3) where b > 0 s another constant. We denote the resultng random (mult)graph by Ĝ(n,α). For our purposes, parallel edges can be merged nto a sngle edge, so we may alternatvely defne G(n,α) as the random smple graph where vertces and j are joned by an edge wth probablty p j = 1 exp( λ j ), (2.4) ndependently for all pars (, j) wth 1 < j n. Then our man result can be stated as follows. Let us recall that an event holds wth hgh probablty (whp), f t holds wth probablty tendng to 1 as n. We also use o p and O p n the standard sense (see, for nstance, Janson, Luczak, Rucńsk [13]). Theorem 1. () If 0 < α < 2, then where ω(g(n,α)) = (c + o p (1))n 1 α/2 (log n) α/2, c = ab α/2 (1 α/2) α/2. (2.5)

4 LARGE CLIQUES IN A POWER-LAW RANDOM GRAPH 3 () If α = 2, then ω(g(n,α)) = O p (1); that s, for every ε > 0 there exsts a constant C ε such that P(ω(G(n,α)) > C ε ) ε for every n. However, there s no fxed fnte bound C such that ω(g(n,α)) C whp. () If α > 2, then ω(g(n, α)) {2, 3} whp. Moreover, the probabltes of each of the events ω(g(n,α)) = 2 and ω(g(n,α)) = 3 tend to postve lmts, gven by (5.10). A queston whch naturally emerges when studyng the sze of the largest clque n a class of graphs s whether one can fnd a large clque n such graph n a polynomal tme. By Theorem 1, whp one can fnd ω(g(n,α)) n a polynomal tme for α > 2, and, wth some extra effort, the same can be accomplshed for α = 2 (see Corollary 3). Thus let us concentrate for the case α < 2, when the large clque s of polynomal sze. Let us suppose that we know the vertex weghts W defned n Secton 2 and, for smplcty, that these are dstnct (otherwse we resolve tes randomly; we omt the detals). Snce vertces wth larger weghts tend to have hgher degrees, they are more lkely to be n a large clque, so t s natural to try to fnd a large clque by lookng at the vertces wth largest weghts. One smple way s the greedy algorthm whch checks the vertces n order of decreasng weghts and selects every vertex that s joned to every other vertex already selected. Ths evdently yelds a clque, whch we call the greedy clque and denote by K gr. Thus K gr = { : j for all j wth W j > W and j K gr }. A smplfed algorthm s to select every vertex that s joned to every vertex wth hgher weght, regardless of whether these already are selected or not. Ths gves the quas top clque studed by Norros [14], whch we denote by K qt. Thus K qt = { : j for all j wth W j > W }. Obvously, K qt K gr. The dfference between the two clques s that f we, whle checkng vertces n order of decreasng weghts, reject a vertex, then that vertex s gnored for future tests when constructng K gr, but not for K qt. A more drastc approach s to stop at the frst falure; we defne the full top clque K ft as the result,.e. K ft = { : j k for all dstnct j,k wth W j,w k W }. Thus K ft s the largest clque consstng of all vertces wth weghts n some nterval [w, ). Clearly, K ft K qt K gr. Fnally, by K max we denote the largest clque (chosen at random, say, f there s a te). Thus K ft K qt K gr K max = ω(g(n,α)). (2.6) The followng theorem shows that the last two nequaltes n (2.6) are asymptotc equaltes, but not the frst one. Here we use for convergence p n probablty, and all unspecfed lmts are as n.

5 4 SVANTE JANSON, TOMASZ LUCZAK, AND ILKKA NORROS Theorem 2. If 0 < α < 2, then G(n,α), K gr and K qt both have sze (1 + o p (1))ω(G(n,α)); n other words On the other hand, K gr / K max p p 1 and K qt / K max 1. K ft / K max p 2 α/2. Thus, whp K gr and K qt almost attan the maxmum sze of a clque, whle K ft falls short by a constant factor. As a smple corollary of the above result one can get the followng. Corollary 3. For every α > 0 there exsts an algorthm whch whp fnds n G(n, α) a clque of sze (1 + o(1))ω(g(n, α)) n a polynomal tme. 3. The proof for the case α < 2 (no second moment) We begn wth a smple lemma gvng an upper bound for the clque number of the Erdős Rény random graph G(n, p) (for much more precse results see, for nstance, Janson, Luczak, Rucńsk [13]). Lemma 4. For any p = p(n), whp ω(g(n,p)) 2log n 1 p. Proof. Denote by X k the number of clques of order k n G(n,p). For the expected number of such clques we have ( ) n ( ne E X k = p (k 2) k k p(k 1)/2) k. If we set k 2log(n)/(1 p), then p (k 1)/2 = ( 1 (1 p) ) (k 1)/2 e (1 p)(k 1)/2 e/n. Consequently, we arrve at ( e 2) k P(ω(G(n,p)) k) = P(X k 1) E X k 0, k snce k 2log n. Proof of Theorems 1() and 2. For s > 0, let us partton the vertex set V = {1,...,n} of G n = G(n,α) nto Vs = { : W s n log n} and V s + = { : W > s n log n}; we may thnk of elements of Vs and V s + as lght and heavy vertces, respectvely. By (2.1), E V s + = n P(W > s n log n) = as α n 1 α/2 log α/2 n. (3.1) Moreover, V s + Bn(n, P(W > s n log n)), and Chebyshev s nequalty (or the sharper Chernoff bounds [13, Secton 2.1]) mples that whp V + s = (1 + o(1)) E V + s = (1 + o(1))as α n 1 α/2 log α/2 n. (3.2)

6 LARGE CLIQUES IN A POWER-LAW RANDOM GRAPH 5 We now condton on the sequence of weghts {W k }. We wll repeatedly use the fact that f and j are vertces wth weghts W = x n log n and W j = y n log n, then by (2.3) (2.4), λ j = bxy log n and In partcular, p j = 1 e λ j = 1 e bxy log n = 1 n bxy. (3.3) p j 1 n bs2, f,j V s, (3.4) p j > 1 n bs2, f,j V + s. (3.5) Consder, stll condtonng on {W k }, for an s that wll be chosen later, the nduced subgraph G n [Vs ] of G(n,α) wth vertex set V s. Ths graph has at most n vertces and, by (3.4), all edge probabltes are at most 1 n bs2, so we may regard G n [Vs ] as a subgraph of G(n,p) wth p = 1 n bs2. Hence, Lemma 4 mples that whp ω(g n [Vs 2log n ]) = 2n bs2 log n. (3.6) n bs2 If K s any clque n G(n,α), then K Vs s a clque n G n [Vs ], and thus K Vs ω(g n[vs ]); further, trvally, K V + s V s +. Hence, K ω(g n [Vs ]) + V + s, and thus ω(g(n,α)) ω(g n [V s + ]) + V. (3.7) We choose, for a gven ε > 0, s = (1 ε)b 1/2 (1 α/2) 1/2 so that the exponents of n n (3.6) and (3.2) almost match; we then obtan from (3.7), (3.2), and (3.6), that whp ω(g(n,α)) (1 + o(1))as α n 1 α/2 log α/2 n = (1 + o(1))(1 ε) α cn 1 α/2 log α/2 n, (3.8) wth c defned as n (2.5). To obtan a matchng lower bound, we consder the quas top clque K qt. Let, agan, s be fxed and condton on the weghts {W k }. If,j V s +, then by (3.5), the probablty that s not joned to j s less than n bs2. Hence, condtoned on the weghts {W k }, the probablty that a gven vertex V s + s not joned to every other j V s + s at most V s + n bs2, whch by (3.2) whp s at most 2as α n 1 α/2 bs2 log α/2 n. We now choose s = (1 + ε)b 1/2 (1 α/2) 1/2 wth ε > 0. Then, for some constant C <, whp P ( / K qt {W k } ) Cn 2ε(1 α/2) and thus E ( V s + \ K qt {W k } ) Cn 2ε(1 α/2) V s +. Hence, by Markov s nequalty, whp V + s \ K qt Cn ε(1 α/2) V + s. s

7 6 SVANTE JANSON, TOMASZ LUCZAK, AND ILKKA NORROS Thus, usng (3.2), whp ω(g(n,α)) K gr K qt V + s V + s \ K qt (1 o(1)) V + s (1 o(1))as α n 1 α/2 log α/2 n = (1 + o(1))(1 + ε) α cn 1 α/2 log α/2 n. (3.9) Snce ε > 0 s arbtrary, (3.8) and (3.9) mply Theorem 1() and the frst part of Theorem 2. In order to complete the proof of Theorem 2, t remans to consder K ft. Defne G n as the complement of G(n,α). Then, usng (3.5) and condtoned on {W }, we nfer that the expected number of edges of G n wth both endponts n V s + s at most n bs2 V s + 2. If we choose s = b 1/2 (2 α) 1/2, then (3.2) mples that ths s whp o(1); hence whp V s + contans no edges of G n,.e., K ft V s +. On the other hand, let 0 < ε < 1/2 and defne, stll wth s = b 1/2 (2 α) 1/2, V = V + (1 2ε)s V (1 ε)s. Then, condtoned on {W }, the probablty of havng no edges of G n n V s, by (3.4), ( ) p j 1 n b(1 ε) 2 s 2 ( V 2 ) ( exp n (1 ε) 2 (2 α) ( V 1) 2 /2 ).,j V By (3.2), whp V 1 = V + (1 2ε)s V + (1 ε)s 1 = (1 + o(1))a ( (1 2ε) α (1 ε) α) s α n 1 α/2 log α/2 n, and t follows from (3.10) that P(K ft V + (1 2ε)s ),j V p j 0. (3.10) Hence, whp K ft V + (1 2ε)s. We have shown that, for any ε (0,1/2), whp V s + K ft V + (1 2ε)s, and t follows by (3.2) and (2.5) (by lettng ε 0) that whp K ft = (1 + o(1)) V + s = (1 + o(1))as α n 1 α/2 log α/2 n = (1 + o(1))2 α/2 cn 1 α/2 log α/2 n = (1 + o(1))2 α/2 ω(g(n,α)), where the last equalty follows from Theorem The case α = 2 (stll no second moment) Proof of Theorem 1() and Corollary 3. Gven the weghts W, the probablty that four vertces,j,k,l form a clque s, by (2.4) and (2.3), p j p k p l p jk p jl p kl λ j λ k λ l λ jk λ jl λ kl = b 6W3 W 3 j W 3 k W 3 l n 6.

8 LARGE CLIQUES IN A POWER-LAW RANDOM GRAPH 7 Thus, f X m s the number of clques of sze m n G(n,α), then the condtonal expectaton of X 4 s E(X 4 {W } n 1 ) b6 n 6 W 3 W j 3 W k 3 W l 3 b 6( n ) 4. 3/2 W 3 (4.1) <j<k<l To show that the number of such quadruples s bounded n probablty, we shall calculate a truncated expectaton of W 3. Usng (2.1), for any constant A > 0, we get ( E W 3 ; W max An 1/2) ( E mn(w,an 1/2 ) 3) = n E mn(w,an 1/2 ) 3 = n An 1/2 0 3x 2 P(W > x)dx = O(nAn 1/2 ), (4.2) and thus, usng (2.2) and Markov s nequalty, for every t > 0 and some constant C ndependent of A, t and n, we arrve at ( P n ) 3/2 W 3 > t ( t 1 E n 3/2 W 3 ; W max An 1/2) + P(W max > An 1/2 ) CAt 1 + CA 2. (4.3) ( Gven t > 0, we choose A = t 1/3 and fnd P n 3/2 ) W 3 > t = O(t 2/3 ). Hence, n 3/2 W 3 = O p (1), and t follows by (4.1) and Markov s nequalty that X 4 = O p (1). Fnally, we observe that, for any m 4, ( P(ω(G(n,α)) m) P X 4 ( ) m ), (4.4) 4 whch thus can be made arbtrarly small (unformly n n) by choosng m large enough. Hence, ω(g(n,α)) = O p (1). To complete the proof of Theorem 1() let us note that for any fxed m n, the probablty that there are at least m vertces wth weghts W > n 1/2 s larger than c 1 > 0 for some absolute constant c 1 > 0, and condtoned on ths event, the probablty that the m frst of these vertces form a clque s larger than c 2 for some absolute constants c 1,c 2 not dependng on n. Fnally, we remark that all clques of sze four can clearly be found n tme O(n 4 ). The number of such clques s whp at most log log n, say, so there exsts an algorthm whch whp fnds the largest clque n a polynomal tme (for example by crudely checkng all sets of clques of sze 4).

9 8 SVANTE JANSON, TOMASZ LUCZAK, AND ILKKA NORROS 5. α > 2 (fnte second moment) Proof of Theorem 1(). Choose ν such that 1/2 > ν > 1/α. Then (2.2) (or (2.1) drectly) mples that whp W max n ν. Furthermore, n analogy to (4.2) and (4.3), ( E and thus ( P n 3/2 W 3 ; W max n ν) E Hence, n 3/2 W 3 ) W 3 > t ( mn(w,n ν ) 3) n ν = n 3x 2 P(W > x)dx = O(nn ν ) = o(n 3/2 ), (5.1) 0 ( t 1 E n 3/2 W 3 ; W max n ν) + P(W max > n ν ) = o(1). (5.2) p 0, and t follows from (4.1) that P(ω(G(n,α)) 4) = P(X 4 1) E ( mn ( 1, E(X 4 {W } n 1 ))) 0. Consequently, whp ω(g(n, α)) 3. Moreover, we can smlarly estmate EX 3 E λ j λ k λ jk = E (b 3 n 3 <j<k <j<k ) W 2 W j 2 W k b3( E W 2) 3 ; (5.3) note that E W 2 < by (2.1) and the assumpton α > 2. Hence, the number of K 3 n G(n,α) s O p (1). To obtan the lmt dstrbuton, t s convenent to truncate the dstrbuton, as we have done t n the prevous secton. We let A be a fxed large constant, and let X3 A be the number of K 3 n G(n,α) such that all three vertces have weghts at most A, and let X3 A be the number of the remanng trangles. Argung as n (5.3), we see easly that EX A b3( E(W 2 ; W A) ) 3 (5.4) E X A 3 b 3( E(W 2 ) ) 2 E(W 2 ; W > A). (5.5) Moreover, f W,W j A, then λ j = O(1/n), and thus p j λ j, and t s easly seen that (5.4) can be sharpened to E X A 3 µ A = 1 6 b3( E(W 2 ; W A) ) 3. (5.6) Furthermore, we may calculate fractonal moments E(X3 A) m by the same method, and t follows easly by a standard argument (see, for nstance, [13, Theorem 3.19] for G(n,p)) that E(X3 A) m µ m A for every m 1, and thus by the method of moments [13, Corollary 6.8] X A 3 d Po(µ A ) (5.7)

10 LARGE CLIQUES IN A POWER-LAW RANDOM GRAPH 9 d as n, for every fxed A, where denotes the convergence n dstrbuton. Fnally, we note that the rght hand sde of (5.5) can be made arbtrarly small by choosng A large enough, and hence lm A lm sup P(X3 A 0) = 0, (5.8) n and that µ A µ = 1 6 (b E(W 2 )) 3 as A. It follows by a standard argument (see Bllngsley [3, Theorem 4.2]) that we can let A n (5.7) and obtan d X 3 Po(µ). (5.9) In partcular, P(X 3 = 0) e µ, whch yelds the followng result: P(ω(G(n,α)) = 2) e µ = e 1 6 (b E(W 2 )) 3, P(ω(G(n,α)) = 3) 1 e µ = 1 e 1 6 (b E(W 2 )) 3. (5.10) Fnally, note that G(n,α) whp contans clques K 2,.e. edges, so clearly ω(g(n,α)) Fnal remarks In ths secton we make some comments on other models of power-law random graphs as well as some remarks on possble varants of our results. We omt detaled proofs. Let us remark frst that, for convenence and to facltate comparsons wth other papers, n the defnton of G(n,α) we used two scale parameters a and b above, besdes the mportant exponent α. By rescalng W tw for some fxed t > 0, we obtan the same G(n,α) for the parameters at α and bt 2 ; hence only the combnaton ab α/2 matters, and we could fx ether a or b as 1 wthout loss of generalty Algorthms based on degrees. As for the algorthmc result Theorem 2, t remans true f we search for large clques examng the vertces one by one n order not by ther weghts but by ther degrees and modfy the defnton of K gr, K qt, and K ft accordngly. (Ths holds both f we take the degrees n the multgraph Ĝ(n,α), or f we consder the correspondng smple graph.) The reason s that, for the vertces of large weght that we are nterested n, the degrees are whp all almost proportnal to the weghts, and thus the two orders do not dffer very much. Ths enables us to fnd an almost maxmal clque n polynomal tme, even wthout knowng the weghts More general weght dstrbutons. Observe that Theorems 1 and 2 reman true (and can be shown by bascally the same argument), provded only that the power law holds asymptotcally for large weghts,.e., (2.1) may be relaxed to P(W > x) ax α as x. (6.1)

11 10 SVANTE JANSON, TOMASZ LUCZAK, AND ILKKA NORROS 6.3. Determnstc weghts. Instead of choosng weghts ndependently accordng to the dstrbuton W we may as well take a sutable determnstc sequence W of weghts (as n Chung and Lu [7]), for example W = a 1/αn1/α, = 1,...,n. (6.2) 1/α All our results reman true also n ths settng; n fact the proofs are slghtly smpler for ths model. A partcularly nterestng specal case for ths model (see Bollobás, Janson, and Rordan [5, Secton 16.2] and Rordan [17]) s when α = 2, where (2.3) and (6.2) combne to yeld λ j = ab j Posson number of vertces. We may also let the number of vertces be random wth a Posson Po(n) dstrbuton (as n, e.g., Norros [14]). Then the set of weghts {W } n 1 can be regarded as a Posson process on [0, ) wth ntensty measure n dµ, where µ s the dstrbuton of the random varable W n (2.1). Note that now n can be any postve real number Dfferent normalzaton. A slghtly dfferent power-law random graph model emerges when nstead of (2.3) we defne the ntenstes λ j by λ j = W W j n k=1 W k (6.3) (see for nstance Chung and Lu [7] and Norros and Rettu [15]). Let us call ths model G(n,α). In the case α > 1, when the mean E W <, the results for G(n,α) and G(n,α) are not much dfferent. In fact, by the law of large numbers, n 1 W k/n E W a.s., so we may for any ε > 0 couple G(n,α) constructed by ths model wth G(n,α) ± constructed as above, usng (2.3) wth b = 1/(E W ε), such that whp G(n,α) G(n,α) G(n,α) +, and t follows that we have the same asymptotc results as n our theorems f we let b = 1/ E W. On the other hand, for α = 1, n 1 W k = (a + o p (1))n log n, and for 0 < α < 1, n 1 W k/n 1/α d Y, where Y s a stable dstrbuton wth exponent α (e.g.see Feller [10, Secton XVII.5]). It follows, argung as n Secton 3, that for α = 2, the largest clque n G(n,α) has (1 + o p (1)) 2an log 1 n vertces, whle for 0 < α < 1 the sze of the largest clque s always close to n; more precsely, ω( G(n,α)) n log α/2 n d Z = a2 α/2 Y α/2, where Z s an absolutely contnuous random varable whose dstrbuton has the entre postve real axs as support. (The square Z 2 has, apart from a scale factor, a Mttag-Leffler dstrbuton wth parameter α, see Bngham,

12 LARGE CLIQUES IN A POWER-LAW RANDOM GRAPH 11 Golde, Teugels [4, Secton 8.0.5].) Thus, for α < 1, ω( G(n,α)) s not sharply concentrated around ts medan; ths s caused by the fact that the normalzng factor W s determned by ts frst terms whch, clearly, are not sharply concentrated around ther medans as well. Interestngly enough, snce n the proof of Theorem 2 we dealt mostly wth the probablty space where we condtoned on W, the analogue of Theorem 2 holds for ths model as well. Thus, for nstance, despte of the fact that nether the largest clque nor the full top clque are sharply concentrated n ths model, one can show the sharp concentraton result for the rato of these two varables The model mn(λ j,1). For small λ j, (2.4) mples p j λ j. In most works on nhomogeneous random graphs, t does not matter whether we use (2.4) or, for example, p j = mn(λ j,1) or p j = λ j /(1 + λ j ) (as n Brtton, Dejfen, and Martn-Löf [6]), see Bollobás, Janson, Rordan [5]. For the clques studed here, however, what matters s manly the probabltes p j that are close to 1, and the precse sze of 1 p j for them s mportant; thus t s mportant that we use (2.4) (cf. Bancon and Marsl [1; 2] where a cutoff s ntroduced). For nstance, a common verson (see e.g. [5]) of G(n,α) replaces (2.4) by p j = mn(λ j,1). (6.4) Ths makes very lttle dfference when λ j s small, whch s the case for most and j, and for many asymptotcal propertes the two versons are equvalent (see agan [5]). In the case W 3 = o p (n 3/2 ), whch n our case wth W governed by (2.1) holds for α > 2 as a consequence of (5.2), a strong general form of asymptotc equvalence s proved n Janson [12]; n the case α = 2, when W 3 = o p (n 3/2 ) by (4.2), a somewhat weaker form of equvalence (known as contguty) holds provded also, say, max j λ j 0.9, see agan [12]. In our case we do not need these general equvalence results; the proofs above for the cases α 2 hold for ths model too, so Theorem 1()() hold wthout changes. If α < 2, however, the results are dfferent. In fact, (6.4) mples that all vertces wth W b 1/2 n 1/2 are joned to each other, and thus form a clque; conversely, f we now defne V = { : W (b + ε) 1/2 n 1/2 }, then p j = λ j b/(b+ε) for,j V, and thus ω(g(n,α)[v ]) = O(log n) whp by Lemma 4. Consequently, argung as n Secton 3, ω(g(n,α)) = (1 + o p (1))n P(W > b 1/2 n 1/2 ) = (1 + o p (1))ab α/2 n 1 α/2, so the logarthmc factor n Theorem 1() dsappears The model λ j /(1+λ j ). Another verson of G(n,α) replaces (2.4) by p j = λ j 1 + λ j. (6.5)

13 12 SVANTE JANSON, TOMASZ LUCZAK, AND ILKKA NORROS Ths verson has the nterestng feature that condtoned on the vertex degrees, the dstrbuton s unform over all graphs wth that degree sequence, see Brtton, Dejfen, and Martn-Löf [6]. In ths verson, for large λ j, 1 p j = 1/(1 + λ j ) s consderably larger than for (2.4) (or (6.4)), and as a consequence, the clque number s smaller. For α 2, stochastc domnaton (or a repetton of the proofs above) shows that Theorem 1()() hold wthout changes. For α < 2, there s a sgnfcant dfference. Argung as n Secton 3, we fnd that, for some constants c and C dependng on a, b and α, whp cn (2 α)/(2+α) ω(g(n,α)) Cn (2 α)/(2+α) (log n) α/(2+α). Although ths only determnes the clque number up to a logarthmc factor, note that the exponent of n s 2 α 2+α, whch s strctly less than the exponent 2 α 2 n Theorem Preferental attachment. Fnally, let us observe that not all powerlaw random graph models contan large clques. Indeed, one of the most popular types of models of such graphs are preferental attachment graphs n whch the graph grows by acqurng new vertces, where each new vertex v s joned to some number k v of old vertces accordng to some random rule (whch usually depends on the structure of the graph we have constructed so far), see, for nstance, Durrett [8]. Clearly, such a graph on n vertces cannot have clques larger than X n = max v n k v + 1, and snce for most of the models X n s bounded from above by an absolute constant or grows very slowly wth n, typcally the sze of the largest clque n preferental attachment random graphs s small. References [1] G. Bancon and M. Marsl, Emergence of large clques n random scalefree networks. Europhys. Lett. 74 (2006), [2] G. Bancon and M. Marsl, Number of clques n random scale-free network ensembles. Preprnt, arxv:cond-mat/ v1. [3] P. Bllngsley, Convergence of Probablty Measures. Wley, New York, [4] N.H. Bngham, C.M. Golde, and J.L. Teugels, Regular varaton. Cambrdge Unv. Press, Cambrdge, [5] B. Bollobás, S. Janson, and O. Rordan, The phase transton n nhomogeneous random graphs. Random Struct. Alg. 31 (2007), [6] T. Brtton, M. Dejfen, and A. Martn-Löf, Generatng smple random graphs wth prescrbed degree dstrbuton. J. Statst. Phys., to appear. [7] F. Chung and L. Lu, Connected components n random graphs wth gven expected degree sequences. Ann. Comb. 6 (2002), [8] R. Durrett, Random Graphs Dynamcs, Cambrdge Unversty Press, Canbrdge, 2007.

14 LARGE CLIQUES IN A POWER-LAW RANDOM GRAPH 13 [9] H. van den Esker, R. van der Hofstad, and D. Znamensk, Dstances n random graphs wth nfnte mean degree sequence. Extremes 8 (2006), [10] W. Feller, An Introducton to Probablty Theory wth ts Applcatons, vol.ii, 2nd edton, Wley, New York, [11] A. Freze, C. McDarmd, Algorthmc theory of random graphs. Random Struct. Alg. 10 (1997), [12] S. Janson, Asymptotc equvalence and contguty of some random graphs. Preprnt, arxv: [13] S. Janson, T. Luczak, and A. Rucńsk, Random Graphs. Wley, New York, [14] I. Norros, A mean-feld approach to some Internet-lke random networks. Preprnt, Insttut Mttag-Leffler no. 04, 2009 sprng. [15] I. Norros and H. Rettu, On a condtonally Possonan graph process. Adv. Appl. Probab. 38 (2006), [16] I. Norros and H. Rettu, Network models wth a soft herarchy : a random graph constructon wth loglog scalablty. IEEE Network 22 (2008), no. 2, [17] O. Rordan, The small gant component n scale-free random graphs. Combn. Probab. Comput. 14 (2005), Department of Mathematcs, Uppsala Unversty, PO Box 480, SE Uppsala, Sweden E-mal address: svante.janson@math.uu.se URL: svante/ Faculty of Mathematcs and Computer Scence, Adam Mckewcz Unversty, ul. Umultowska 87, Poznań, Poland E-mal address: tomasz@amu.edu.pl VTT Techncal Research Centre of Fnland, P.O. Box 1000, VTT, Fnland E-mal address: lkka.norros@vtt.f