The Diameter of Sparse Random Graphs

Transcription

1 Advaces i Applied Mathematics 26, (200 doi:0.006/aama , available olie at o The Diameter of Sparse Radom Graphs Fa Chug ad Liyua Lu Departmet of Mathematics, Uiversity of Califoria, Sa Diego, La Jolla, Califoria Received April 5, 2000; accepted November 20, 2000 this paper is dedicated to the memory of paul erdös. We cosider the diameter of a radom graph G p for various rages of p close to the phase trasitio poit for coectivity. For a discoected graph G, we use the covetio that the diameter of G is the maximum diameter of its coected compoets. We showthat almost surely the diameter of radom graph G p is close to if p. Moreover if = c>8, the the diameter p of G p is cocetrated o two values. I geeral, if = c>c 0, the diameter is cocetrated o at most 2/c 0 +4 values. We also proved that the diameter of G p is almost surely equal to the diameter of its giat compoet if p > Academic Press p. INTRODUCTION As the master of the art of coutig, Erdös has had a far-reachig impact i umerous areas of mathematics ad computer sciece. A recet example, perhaps least expected by Erdös, is the area of Iteret computig. I a atural way, massive graphs that arise i the studies of the Iteret share a umber of similar aspects with radom graphs, although there are sigificat differeces (e.g., there ca be vertices with large degrees i a sparse massive graph. Nevertheless, may of the methods ad ideas [ 3, 4, 6] that are used i modelig ad aalyzig massive graphs have frequetly bee traced to the semial papers of Erdös ad Réyi [2] i 959. Oe topic of cosiderable iterest is determiatio of the diameter of a sparse radom graph. These techiques ad methods ca also be Supported i part by Natioal Sciece Foudatio grat DMS /0 $35.00 Copyright 200 by Academic Press All rights of reproductio i ay form reserved.

2 258 chug ad lu used to examie the coected compoets ad the diameter of Iteret graphs [2, 5]. Let G p deote a radom graph o vertices i which a pair of vertices appears as a edge of G p with probability p. (The reader is referred to [8] for defiitios ad otatio i radom graphs. I this paper, we examie the diameter of G p for all rages of p icludig the rage that G p is ot coected. For a discoected graph G, the diameter of G is defied to be the diameter of its largest coected compoet. We will first briefly survey previous results o the diameter of the radom graph G p. I 98, Klee ad Larma [4] proved that for a fixed iteger d, G p has diameter d with probability approachig as goes to ifiity if p d / 0 ad p d /. This result was later stregtheed by Bollobás [7] ad was proved earlier by Burti [0, ]. Bollobás [9] showed that the diameter of a radom graph G p is almost surely cocetrated o at most four values if p. Furthermore, it was poited out that the diameter of a radom graph is almost p surely cocetrated o at most two values if (see [8, Exercise 2, Chap. 0]. I the other directio, Łuczak [6] examied the diameter of the radom graph for the case of p <. Łuczak determied the limit distributio of the diameter of the radom graph if p /3. The diameter of G p almost surely either is equal to the diameter of its tree compoets or differs by. I this paper, we focus o radom graphs G p for the rage of p > ad p c for some costat c. This rage icludes the emergece of the uique giat compoet. Sice there is a phase trasitio i coectivity at p = /, the problem of determiig the diameter of G p ad its cocetratio seems to be difficult for certai rages of p. Here we ited to clarify the situatio by idetifyig the rages for which results ca be obtaied as well as the rages for which the problems remai ope. For p = c>8, we slightly improve Bollobás result [9] by showig that the diameter of G p is almost surely cocetrated o at most two values p aroud / p.for = c>2, the diameter of G p is almost surely cocetrated o at most three values. For the rage 2 p = c>, the diameter of G p is almost surely cocetrated o at most four values. For the rage p <, the radom graph G p is almost surely discoected. We will prove that almost surely the diameter of G p is + o if p. Moreover, if p = c>c 0 for ay (small costat c ad c 0, the the diameter of G p is almost surely cocetrated o fiitely may values, amely, o more tha 2/c 0 +4 values.

3 diameter of sparse radom graphs 259 TABLE The Diameter of Radom Graphs G p Rage diam(g p Referece p Cocetrated o at most 2 values [8] p = c>8 Cocetrated o at most 2 values Here 8 p = c>2 Cocetrated o at most 3 values Here 2 p = c> Cocetrated o at most 4 values [9] p 0 Cocetrated o at most 2/c 0 +4 Here values >p p c> diamg p= + o Here The ratio diamgp is fiite Here / (betwee ad f c p < diamg p equals the diameter of a tree [6] compoet if p /3 I the rage of, the radom graph G p almost surely has a uique giat compoet. We obtai a tight upper boud of the sizes of its small compoets if p satisfies p c>. We the prove that the diameter of G p almost surely equals the diameter of its giat compoet for the rage p > 353. This problem was previously cosidered by Łuczak [6]. Here we summarize various results i Table. The values of cocetratio for the diameter of G p, whe it occurs, is ear. From the p table, we ca see that umerous questios remai, several of which will be discussed i the last sectio. <p< 2. THE NEIGHBORHOODS IN A RANDOM GRAPH I a graph G, we deote by Ɣ k x the set of vertices i G at distace k from a vertex x: Ɣ k x =y G dx y =k We defie N k x to be the set of vertices withi distace k of x: N k x = k Ɣ i x i=0

4 260 chug ad lu A mai method for estimatig the diameter of a graph is to examie the sizes of eighborhoods N k x ad Ɣ k x. To boud N k x i a radom graph G p, the difficulty varies for differet rages of p. Roughly speakig, the sparser the graph is, the harder the problem is. We will first establish several useful lemmas cocerig the eighborhoods for differet rages of p. Lemma. Suppose p >. With probability at least o, we have Ɣ i x 2i 2 p i for all i N i x 2i 3 p i for all i Lemma 2. Suppose p> c for a costat c 2. The with probability at least o, we have Ɣ i x 9 c pi for all i N i x 0 c pi for all i Lemma 3. Suppose p. For ay ε>0, with probability at least /log 2, we have Ɣ i x + εp i for all i N i x + 2εp i for all i Let X X 2 deote two radom variables. If PrX >a PrX 2 >a for all a, wesayx domiates X 2,orX 2 is domiated by X. We will eed the followig fact. Lemma 4. Let B p deote the biomial distributio with probability p i a space of size.. Suppose X domiates B p. Fora>0, we have PrX <p a e a2 /2p ( 2. Suppose X is domiated by B p. Fora>0, we have PrX >p+ a e a2 /2p+a 3 /p 3 (2 We will repeatedly use Lemma 4 i the followig way. For a vertex x of G p, we cosider Ɣ i x for i = 2. At step i, let X be the radom variable of Ɣ i x give Ɣ i x. We ote that X is ot exactly a biomial distributio. However, it is close to oe if Ɣ i x is small. To be precise, X is domiated by a radom variable with the biomial distributio Bt p where t = Ɣ i x. O the other had, if N i x <m, the X domiates a radom variable Bt p where t = mɣ i x. Thus a upper boud ad lower boud of Ɣ i x ca be obtaied. For differet rages of p, we will derive differet estimates i Lemmas 3.

5 diameter of sparse radom graphs 26 Proof of Lemma. We cosider p satisfyig p >. We wat to show that with probability at least o, we have Ɣ i x 2i 2 p i N i x 2i 3 p i for all i for all i First we will establish the followig: Claim. With probability at least ie λ2 /2+λ 3 / 5, we have Ɣ i x a i p i for all i where a i ( i k satisfies the recurrece formula, a i = a i + λ ai p i/2 for all i with iitial coditio a 0 =. We prove this claim by iductio o i. Clearly, for i = 0, Ɣ 0 x = <, it is true. Suppose that it holds for i. Fori + Ɣ i+ x is domiated by the biomial distributio Bt p where t =Ɣ i x N i x. With probability at least ie λ2 /2+λ 3 / 5, we have Ɣ i x N i x a i p i By Lemma 4, iequality (2, with probability at least i + e λ2 /2+λ 3 / 5 (sice a i p i+ >, we have Ɣ i+ x a i p i p + λ a i p i+ ( p i+ a i + λ ai = a i+ p i+ p i+/2 By choosig λ = 5, we have e λ2 /2+λ 3 / 5 = e = o

6 262 chug ad lu Nowwe showby iductio that a i 2i 2 for all i. Suppose that a j 2j 2, for all j i. The aj a i+ = + λ i j=0 < + ( i 5 + aj + ( 5 + j= p j+/2 i 2j j= i 2 + i/2 < 2i + 2 We have completed the proof of Lemma. For p> c, where c 2 is a costat, the upper boud for Ɣ i x ca be improved. Proof of Lemma 2. Here we focus o the rage p> c for a costat c 2. We wat to show that with probability at least o, we have Ɣ i x 9 c pi for all i N i x 0 c pi for all i We will first prove the followig claim. Claim 2. With probability at least ie λ2 /2+λ 3 /p 5, we have Ɣ i x a i p i for all i where a i ( i satisfies the recurrece formula ai a i = a i + λ for all i p i/2 with iitial coditio a 0 =. Obviously, for i = 0 Ɣ 0 x = = a 0, it holds. Suppose that it holds for i. Fori + Ɣ i+ x is domiated by the biomial distributio Bt p where t =Ɣ i x N i x. With probability at least ie λ2 /2+λ 3 /p 5, we have Ɣ i x N i x a i p i

7 diameter of sparse radom graphs 263 By Lemma 4, iequality (2, with probability at least i + e λ2 /2+λ 3 /p 5 (sice a i p i+ >p, we have Ɣ i+ x a i p i p + λ a i p i+ ( p i+ a i + λ a i p i+/2 = a i+ p i+ We choose λ = 5, ad we have e λ2 /2+λ 3 /p 5 = e 25log+5/c5 = o Nowwe showby iductio that a i 9 c. Suppose that a j 9, for all c j i. The i aj a i+ = + λ j=0 p j+/2 + 5 j=0 9 cp j+/2 = + 45 c p + 45 ( c c + 7 c for c 2.Thus, N i x = 9 c i Ɣ i x j=0 i j=0 9 c pj 0 c pi by the fact that p c. Lemma 2 is proved. If we oly require havig probability o istead, the precedig upper boud ca be stregtheed as follows. Proof of Lemma 3. Suppose p. We wat to show that for ay k< ad ay ε>0, with probability at least /log 2, we have Ɣ i x + εp i for all i k N i x + 2εp i for all i k provided is large eough.

8 264 chug ad lu We will first show the followig: Claim 3. With probability at least ie λ2 /2+λ 3 /p 5, we have Ɣ i x a i p i for all i where a i ( i satisfies the recurrece formula ai a i = a i + λ for all i p i/2 with iitial coditio a 0 =. By choosig λ = 3 log, we have ( ( ke λ2 /2+λ 3 /2p 5 = ko = o log 4 log 3 sice p. By iductio, we will prove a i < + ε for all 0 i k Certaily it holds for i = 0, sice a 0 = < + ε. Suppose that a j < + ε, for all j i. The i aj a i+ = + λ < + λ j=0 p j+/2 i + ε j=0 p j+/2 + λ + ε p + ε by the assumptio λ = 3 log = o p. Therefore, with probability at least o/log 3, we have Ɣ i x + εp i for all i k ad i N i x = + Ɣ j x for large eough. j= + + ε i p j j= = + + ε pi+ p p + 2εp i

9 diameter of sparse radom graphs THE DIAMETER OF THE GIANT COMPONENT Łuczak asked the iterestig questio, is the diameter of the giat compoet the diameter of a radom graph G p? We will aswer this questio for certai rages of p. This result is eeded later i the proof of the mai theorems. Lemma 5. Suppose <c p <, for some costat c. The almost surely the sizes of all small compoets are at most + o p Proof. Whe p> + 22 /2 /3, Bollobás [8] shows that a compoet of size at least 2/3 i G p is almost always uique (so that it is the giat compoet i the sese that all other compoets are at most of size 2/3 /2. Suppose that x is ot i the giat compoet. We compute the probability that x lies i a compoet of size k + < 2/3. Such a coected compoet must cotai a spaig tree. There are ( k ways to select other k vertices. For these k + vertices, there are exactly k + k spaig trees rooted at x. Hece, the probability that a spaig tree exists is at most ( k + k p k p k 2/3 < 2πke k p k e kp /3 k The above probability is o 2 3 if k> p.itiso e if k> +2. Hece, the probability that x lies i a compoet of size p k + +2 is at most p o p o e =o This implies that almost surely all small compoets are of a size that is at most + 2 = + o p p Theorem. Suppose that p > 353; the almost surely the diameter of G p equals the diameter of its giat compoet. Proof. From Lemma 5, the diameter of small compoets is at most + o. O the other had, by Lemma, for ay vertex x, p with probability at least o 2, i N i x = Ɣ j x 2i 3 p i j=0

10 266 chug ad lu This implies the diameter of G p is at least + o p Whe p > 353, we have p >. Hece, the diameter of G p is strictly greater tha the sizes of all small compoets. This completes the proof of the theorem. We ca owprove a lower boud for Ɣ i x. Lemma 6. Suppose p c> with some costat c. For each vertex x i the giat compoet (if G p is ot coected, with probability at least o, we have for i satisfyig i 0 i 3 5 Ɣ i x p 2 pi i 0 where i 0 = 0p/ p 2 + p log2p Proof. First we prove that with probability at least o, there exists a i 0 satisfyig Ɣ i0 x 9 p 2 = d If i 3 5, the by Lemma, with probability at least o, we have Ɣ i x 2/3. Nowwe compute the probability that N i x = k + < 2/3. We wat to show for some k 0 the probability that Ɣ i x <d ad N i x >k 0 is o. We focus o the eighborhood tree formed by breadth-first-search startig at x. There are ( k ways to select other k vertices. For these k + vertices, there are exactly k + k trees rooted at x. Suppose Ɣ i x <d. The probability that such a tree exists is at most ( k + k p k p k d 2/3 <e k p k e k dp /3 k Let k 0 = dp++2 p. The above probability is o 2 if k> dp+3 p. It is o e if k>k 0. Hece, the probability that Ɣ i x <dad N i x = k + >k 0 + is at most o 2 + dp + 3 p o e =o

11 diameter of sparse radom graphs 267 Let i 0 deote the least iteger i satisfyig Ɣ i x d. The above argumets give a crude upper boud for i 0, i 0 k 0 = dp p 0p/ p 2 + p Now, we wat to prove that Ɣ i x grows quickly after i = i 0. Namely, with probability at least o, we have Ɣ i x p 2 pi i 0 for all i satisfyig 3 5 i>i 0. Claim 4. With probability at least o i i 0 e λ2 /2, we have Ɣ i x a i p /3 i i 0 for all i 0 i 3 5. Here a i satisfies the recurrece formula a i = a i λ ai p /3 i i 0/2 for all i 0 i 3 5, with iitial coditio a i0 = λ2 7 ( p /3 We choose λ = 5. Clearly, for i = i 0 Ɣ i0 x d a i0, the statemet of the claim is true. Suppose that it holds for i. Fori + Ɣ i+ x domiates a radom variable with the biomial distributio Bt p where t =Ɣ i x 2/3 with probability at least o ie λ2 /2. By Lemma 4, part, with probability at least i + e λ2 /2, we have Here, 2 Ɣ i i0 +x a i p 2/3 i i 0 p 2/3 λ a i p 2/3 i i0+ p 2/3 i i 0+ ( λ a i a i p 2/3 i i 0+/2 = a i+ p 2/3 i+ o ie λ2 /2 = o e 25 = o

12 268 chug ad lu Sice a i <a i0 for i>i 0, we have Hece, for i i 0, a i = a i0 λ i i j=i 0 aj p 2/3 j i 0+/2 a i0 5 ai0 p 2/3 j+/2 j=i 0 a i0 5a i0 p 2/3 ( 2 2 p 2/3 Ɣ i x a i p /3 i i 0 ( 2 2 p /3 i i 0 p 2/3 p 2 pi i 0 If p>c, the statemet i Lemma 6 ca be further stregtheed. Lemma 7. Suppose p c for some costat c 2. The, for each vertex x i the giat compoet (if G p is ot coected, for each i satisfyig i 0 i 2 3, with probability at least o, we have Ɣ i x 5 c pi i 0 where i 0 satisfies i 0 c +. Proof. We first prove the followig statemet, which is similar to the claim i the proof of the previous lemma. However, we use a differet proof here to obtai a improvemet. With the probability at least o, there exists a i 0 c + satisfyig Ɣ i0 x d where d = 20 c. Let k = c. Sice x is i the giat compoet, Ɣ kx. There exists a path x 0 x x k satisfyig x j Ɣ j x for j k. We write x 0 = x.

13 diameter of sparse radom graphs 269 Let f x j deote the umber of vertices y, where x j y forms a edge but y is ot oe of those vertices x 0 x x k. We compute the probability that f x j d as ( d k Prf x j d = p l p l l l=0 d p l l=0 l! e l k p p d e d k p c d e c d+k+/ e = o c+ɛ for ay small ε>0. Here, f x j s are idepedet radom variables. The probability that f x j d for all 0 j k is at most o c+ɛ k+ =o if ε is small eough. With probability at least o, there is a idex i 0 k + satisfyig f x i0 d. Hece, Ɣ i0 x d. By Lemma, with probability at least o, we have N i x 3/4 for all i 2 3. For i = i 0 +, we have ( Pr Ɣ i0 +x 2 Ɣ i 0 x N i0 xp e Ɣ i 0 x N i0 xp/8 d l=0 l! e dc /4 /8 = o dc/9 = o sice d 0 c. Hece with probability at least o, Ɣ i0 +x 2 Ɣ i 0 x N i0 xp 3 dp For i = i Ɣ i0 +2x domiates a radom variable with the biomial distributio Bt p where t =Ɣ i0 +x N i0 +x. Hece ( Pr Ɣ i0 +2x <Ɣ i0 +x N i0 +xp λ Ɣ i0 +x N i0 +xp <e λ2 /2

14 270 chug ad lu Hece, with probability at least o e λ2 /2, Ɣ i0 +2x Ɣ i0 +x N i0 +xp λ Ɣ i0 +x N i0 +xp Ɣ i0 +x 3/4 p λ Ɣ i0 +xp ( 3 dp2 /4 3λ p 2 By iductio o i i 0 + 2, we ca show that with probability at least o ie λ2 2, ( Ɣ i x d i i 0 3 pi i 0 /3 3λ j=2 p j We choose λ = 3. Sice i<, we have o i i 0 e λ2 /2 = o i 5 = o Therefore, with probability at least o, Ɣ i x d 3 pi i 0 d 3 pi i 0 d 3 pi i 0 ( i i i /4 0 j=2 ( i /4 3λ p ( ( O log 3λ p j p /2 d 4 pi i 0 = 5 c pi i 0 for large eough. Lemma 8. Suppose p c for some costat c>2. For each vertex x belogig to the giat compoet (if G p is ot coected, ad each i satisfyig i 2 3, with probability at least o, we have where c = 2 c ε. Ɣ i x c p i

15 diameter of sparse radom graphs 27 Proof. Let δ be a small positive umber. For i =, we have PrƔ x c + δp=prɣ x p c δp e c δ 2 p/2 e c δ 2 c /2 = c δ 2 c/2 = o where δ is a small value satisfyig c δ 2 c/2 >. (It is always possible to choose such a δ>0, by the assumptio o c. Hece with probability at least o 2, we have Ɣ x c + δp To obtai a better cocetratio result i the rage of c>8, more work is eeded here. However, the argumets are similar to those i Lemmas 6 ad 7. For i = 2, Ɣ 2 x domiates a radom variable with the biomial distributio Bt p where t =Ɣ x /4. We have ( Pr Ɣ 2 x <Ɣ x /4 p λ Ɣ x /4 p <e λ2 /2 Hece, with probability at least o e λ2 /2, we have Ɣ 2 x c + δp ( 2 /4 λ c p 2 By iductio o i 2, it ca be show that with probability at least o 2 ie λ2 /2, ( i Ɣ i x c + δp i /4 λ j=2 c p j By choosig λ = 5, we have o ie λ2 /2 = o i 25 = o sice i<. Therefore, with probability at least o, we have i Ɣ i x c + δp ( i i /4 λ j=2 c p j c + δp ( i i /4 λ c p p /2 ( c + δp ( i O c p i log for large eough.

16 272 chug ad lu 4. THE MAIN THEOREMS We first state the mai theorems that we will prove i this sectio: Theorem 2. If p c for some costat c>8, the diameter of radom graph G p is almost surely cocetrated o at most two values at p. Theorem 3. If p c for some costat c>2, the the diameter of radom graphs G p is almost surely cocetrated o at most three values at p. Theorem 4. If p c for some costat c, the we have logc/ diamg p log33c 2 / c The diameter of radom graph G p is almost surely cocetrated o at most 2 +4 values. c Theorem 5. If >p, the almost surely we have diamg p= + o p Theorem 6. Suppose p c> for some costat c. Almost surely we have + o diamg p p p + 2 0c/ c 2 + c log2c p + Before provig Theorems 2 6, we first state two easy observatios that are useful for establishig upper ad lower bouds for the diameter. Observatio. Suppose there is a iteger k, satisfyig oe of the followig two coditios.. Whe G p is coected, there exists a vertex x satisfyig, almost surely, N k x < ε 2. Whe G p is ot coected, almost surely for all vertices x, N k x < ε

17 diameter of sparse radom graphs 273 (Here ε ca be replaced by ay lower boud of the giat compoet. The we have diamg P >k Observatio 2. Suppose there are itegers k ad k 2, satisfyig Ɣ k xɣ k2 xp >2 + ɛ for all pairs of vertices x y i the giat compoet. If Ɣ k x Ɣ k2 x, the dx y k + k 2.IfƔ k x Ɣ k2 x =, the probability that there is edge betwee them is at least p Ɣ k xɣ k2 x e Ɣ k xɣ k2 xp = o 2 Sice there are at most 2 pairs, almost surely dx y k + k 2 + Thus the diameter of the giat compoet is at most k + k 2 +. Proof of Theorem 2. G p is almost surely coected at this rage. By Lemma 3, almost surely there is a vertex x satisfyig N i x + 2εp i for all i Here, we choose log ε/ + 2ε k = Hece, almost surely, we have log ε/ + 2ε diamg p for ay ε by usig Observatio. O the other had, by Lemma 8, almost surely for all vertices x, Ɣ i x c p i 2 where c = c ε. Nowwe choose log 2 + ε /c k = log2 + ε /c 2 k 2 = ad k

18 274 chug ad lu as i Observatio 2. We ote that k k 2 log2 /c 2 < both satisfy the coditio of Lemma 8. Almost surely we have Ɣ k xɣ k2 yp c p k c p k 2 p 2 + ε Hece, we have log2 + ε /c 2 diamg p k + k 2 + = Therefore, we have proved that almost surely log ε/ + 2ε diamg p log2 + ε/c 2 = for ay ε The differece betwee the upper boud ad the lower boud is at most log2 + ε/c 2 log ε/ + 2ε log2 + ε + 2ε /c 2 ε log2 + ε + 2ε /c 2 ε logc whe ε 0. Therefore, the diameter of G p is cocetrated o at most two values i this rage. Proof of Theorem 3. The proof is quite similar to that of Theorem 2 ad will be omitted. It ca be show that log ε/ + 2ε diamg p log2 + ε/c 2 = for ay ε It is ot difficult to check that i this rage the differece betwee the upper boud ad the lower boud is 2 istead of, for c>2. Therefore, the diameter of G p is cocetrated o at most three values at this rage.

19 diameter of sparse radom graphs 275 Proof of Theorem 4. I this rage, G p may be discoected. However, the diameter of G p is determied by the diameter of its giat compoet by usig Theorem. By Lemma 2, almost surely for all vertices x, we have N i x 0 c pi We choose k = log c/. Note that i this rage, the size of the giat compoet is o. N k x 0 is less tha the giat compoet. Hece, we have logc/ diamg p + O the other directio, by Lemma 7, almost surely for a vertex x i the giat compoet, there exists a i 0 + that satisfies c Ɣ i x 5 c pi i 0 We choose log (33c 2 /400 k = + i 0 ad log33c 2 /400 k 2 = k + i 0 ad k k 2 log33c 2 /400 + i 2 0 < 2 3 The coditio of Lemma 7 is satisfied. Almost surely Ɣ k xɣ k2 y 5 c pk i 0 5 c pk 2 i 0 p Hece, almost surely we have log33c/400 diamg p k + k 2 + = + 2i 0 Therefore, almost surely log c/ log33c 2 /400 diamg p c

20 276 chug ad lu The differece betwee the upper boud ad the lower boud is at most log33c 2 / c log363c/400 log363c/400 logc c c c log c/ Therefore, if c, the diameter of G p is cocetrated o at most +4 values. 2 c Proof of Theorem 5. By Lemma, for almost all x ad i, we have N i x 2i 3 p 3 4 log We owchoose k =. Hece, we have diamg p >k+ = + o O the other had, by Lemma 6, there exists a i 0 satisfyig i 0 0p/ p 2 + p For almost all vertices x, we have We ca the choose Ɣ i x ( = o p 2 pi i 0 k k i 0 Therefore, Ɣ k x Ɣ k x < 2/3. The coditio of Lemma 6 is satisfied. Hece we have diamg p k + k 2 + We obtai + 2i 0 + = + o diamg p= + o

21 diameter of sparse radom graphs 277 Proof of Theorem 6. The proof is very similar to that of Theorem 5, so we will oly sketch the proof here. It ca be show that diamg p + o I the other directio, we choose But ow Hece k k i 0 i 0 0c/ c 2 + log c c log2c p diamg p p + 2 0c/ c 2 + c logc p + 5. PROBLEMS AND REMARKS We have proved that the diameter of G p is almost surely equal to its giat compoet if p > Several questios here remai uaswered: Problem. Is the diameter of G p equal to the diameter of its giat compoet? Of course, this questio oly cocers the rage <p There are umerous questios cocerig the diameter i the evolutio of the radom graph. The classical paper of Erdös ad Réyi [2] stated that all coected compoets are trees or are uicyclic i this rage. What is the the distributio of the diameters of all coected compoets? Is there ay jump or double jump as the coectivity [2] i the evolutio of the radom graphs durig this rage for p? I this paper we proved that almost surely the diameter of G p is + o if p. Whe p = c for some costat c>, we ca p oly showthat the diameter is withi a costat factor of. Ca this p be further improved? Problem 2. Prove or disprove ( ( diam G c = + o log c for costat c>.

22 278 chug ad lu Our method for boudig the diameter by estimatig N i x does ot seem to work for this rage. This difficulty ca perhaps be explaied by the followig observatio. The probability that Ɣ x = is approximately c/e c, a costat. Hece, the probability that Ɣ x=ɣ 2 x= =Ɣ l x = is about c/e c l.forsomelupto ε, this probability is at least c log c ε. So it is quite likely that this may happe for vertex x. I other words, there is a otrivial probability that the radom graph aroud x is just a path startig at x of legth c. The ith eighborhood N i x of x, for i = c, does ot growat all! I Theorems 2 ad 3 we cosider the case of p> c. Do the statemets still hold for p = c? Problem 3. Is it true that the diameter of G p is cocetrated o 2k + 3 values if p = k? It is worth metioig that the case k = is of special iterest. For the rage p = + c, Lemma implies diamg p 3c + o. Ca oe establish a similar upper boud? Problem 4. Is it true that ( diamg p = for p = + c ad c< 3? Łuczak [6] proved that the diameter of G p is equal to the diameter of a tree compoet i the subcritical phase p /3. What ca we say about the diameter of G p whe p /3 c, for some costat c? The diameter problem seems to be hard i this case. A related problem is to examie the average distace of graphs istead of the diameter which is the maximum distace. The problem o the average distace of a radom graph with a give degree sequece has applicatios i so-called small world graphs [3, 7]. Research i this directio ca be foud i [3]. REFERENCES. L. A. Adamic ad B. A. Huberma, Growth dyamics of the World Wide Web, Nature 40 (999, W. Aiello, F. Chug, ad L. Lu, A radom graph model for massive graphs, i Proceedigs of the Thirty-Secod Aual ACM Symposium o Theory of Computig, pp. 7 80, 2000.

23 diameter of sparse radom graphs W. Aiello, F. Chug, ad L. Lu, Radom evolutio i massive graphs, Adv. Appl. Math., to appear. 4. Réka Albert, Hawoog Jeog, ad Albert-László Barabási, Diameter of the World Wide Web, Nature 40 (999, N. Alo ad J. H. Specer, The Probabilistic Method, Wiley, NewYork, Albert-Laszlo Barabasi, ad Reka Albert, Emergece of scalig i radom etworks, Sciece 286 (999, B. Bollobás, The diameter of radom graphs, IEEE Tras. Iform. Theory 36 (990, No. 2, B. Bollobás, Radom Graphs, Academic Press, Sa Diego, B. Bollobás, The evolutio of sparse graphs, i Graph Theory ad Combiatorics, pp , Academic Press, Lodo/NewYork, J. D. Burti, Extremal metric characteristics of a radom graph I, Teor. Veroyatost. i Primee. 9 (974, J. D. Burti, Extremal metric characteristics of a radom graph II, Teor. Veroyatost. i Primee. 20 (975, P. Erdös ad A. Réyi, O radom graphs I, Publ. Math. Debrece 6 (959, F. Chug ad L. Lu, Small world graphs ad radom graphs, preprit. 4. V. Klee ad D. Larma, Diameters of radom graphs, Caad. J. Math. 33 (98, Liyua Lu, The diameter of radom massive graphs, i Proceedigs of the Twelfth ACM-SIAM Symposium o Discrete Algorithms (SODA 200, pp Tomasz Łuczak, Radom trees ad radom graphs, Radom Structures Algorithms 3 (998, D. J. Watts, Small Worlds, Priceto Uiv. Press, Priceto, NJ, 999.