Doktori értekezés Katona Zsolt 2006

Transcription

1 Doktori értekezés Katoa Zsolt 2006

2 Radom Graph Models Doktori értekezés Iformatika Doktori Iskola, Az Iformatika Alapjai program, vezető: Demetrovics Jáos Katoa Zsolt Témavezető: Móri Tamás, doces Eötvös Lorád Tudomáyegyetem Valószíűségelméleti és Statisztika Taszék

3 Cotets 1 Itroductio 2 2 Preferetial attachmet model Results o width ad level-wise degree distributio The shape of the tree Usig martigales Proof of Theorem Degree distributio of levels Geeratig fuctios Proof of Theorem A applicatio: Directory trees 38 4 A modified model: Preferetial attachmet with idepedet selectio The maximum degree Degree distributio A Nash-equilibrium model of the World Wide Web 49 6 Coclusio 51 7 Ackowledgmets 52

4 1 Itroductio The dissertatio studies radom graph models used i describig complex real-world etworks, focusig o radom trees. The theory of radom graphs was itroduced by Erdős ad Réyi i the early 1960 s after Erdős employed radom methods to solve extremal graph theory problems. These methods allow to prove the existece of some structures without costructig them. The first applicatio of this method is by Szele [33], who showed i 1943, that there exists a touramet o vertices, that cotais at least!/2 1 Hamilto cycles. Specifically, if we direct the edges idepedetly with equal probabilities to the two directios, the the expected umber of Hamilto cycles is!/2 1. I 1947, Erdős proved similarly that the Ramsey-umber Rk is at least 2 k/2. The first radom graph models were itroduced idepedetly by differet authors. The Erdős-Réyi model G, m is a radom selectio from all the graphs with vertices ad m edges, with equal probabilities. The model G, p described by [15] is very similar; edges are selected idepedetly with probability p from all the possible edges betwee vertices. If, for example, p = 1/2 the this is equivalet to selectig amog all the graphs with ode with equal probabilities. Of course these models ca be geeralized i several ways. As the Iteret became a importat way of commuicatio, may researchers physicists, computer scietists bega to study its structure. A secodary structure o the Iteret is the World Wide Web WWW, that is, the directed graph formed by the liks coectig the web pages. We ca fid other large etworks with complex structures i the focus of scietific iterest, such as the geetic etwork cosistig of proteis as odes ad the chemical reactios betwee them as liks. The ervous system is a etwork of axos betwee the erve cells, whereas social etworks describe the iteractio betwee people i a orgaizatio or the society. Sice these 2

5 etworks are extremely large it is impossible to capture the exact structure. For example, i 2002, the WWW had aroud pages. The aturally risig idea is to model these etworks with radom graphs. Oe of the first papers that used radom methods to study large realworld etworks is by Watts ad Strogatz [35]. They oticed that real etworks have a small diameter, that is, the average distace betwee odes is small, ad these etworks are highly clustered. The C 1 G clusterig coefficiet is the average of the ode s idividual clusterig coefficiets, that is, the proportio of pairs of the ode s eighbors, that are themselves eighbors. Although the diameter i the origial Erdős-Réyi radom graph model is small, the clusterig coefficiet is also low, icosistetly with the empirical fidigs. The small world model by Watts ad Strogatz [35] solves this problem. I the begiig, the vertices of the graph lie o a lie or a circle ad each vertex is coected to its eighbors ad secod eighbors. I the ext step every edge is reliked idepedetly with probability p, that is, oe ed of the edge is moved to a radomly uiformly selected other vertex. If p = 0, the graph does ot chage, thus, the average path legth is a liear fuctio of ad each degree is four, whereas if p 1, the the average distace is a logarithmic fuctio of. I 1999, Faloutsos, Faloutsos, ad Faloutsos [12] published their paper about the Iteret graph. They focused o the so called degree distributio, that is, the series P k := P d i = k for k = 0, 1, 2,..., where d i deotes the degree of ode i. The results of empirical studies ivestigatig the degree distributios show a iterestig patter. The Iteret graph, for example, has the degree distributio P k c k γ, with γ 2.3. Some papers regardig the World Wide Web, such as the oe studyig the domai d.edu [3], ad [9] yield that this directed graph has a power-law degree distributio for both the i- ad out-degrees with expoets 2.1 ad 2.45 respectively. Figure 1 depicts the i-degree distributio of the Hugaria Web [4]. A much studied etwork is the collaboratio graph of movie actors with almost 500,000 3

6 vertices. The degree distributio agai has the form P k c k γ, where γ = 2, 3 ± 0.1. This graph is also highly clustered, the coefficiet is 100 times higher tha i a radom graph. Aother example is the electric etwork of the wester USA with P k c k γ, where the expoet is aroud 4 ad the coautorship etwork of scietific publicatios with a expoet aroud 3. Similar patters ca observed i the case of cellular etworks, phoe call etworks, citatio etworks ad eural etworks. For a review of empirical results o large etworks see [2]. Iterestigly, power law distributios had bee observed much earlier. I 1926, Lotka [24] claimed that citatios i scietific publicatios follow a power-law distributio. Althoug Lotka did ot talk about degree distributios, this is obviously the out-degree distributio of a citatio etwork. Sice the empirical degree distributios show the same patter, the atural questio is whether this is cosistet with the Erdős-Réyi radom graph model or ot. Cosiderig the model G, p, that is, whe edges are selected idepedetly with probability p from all the possible edges betwee vertices, the degree distributio is asymptotically Poisso with parameter p as [7]. That is, p pk P k e. k! I the G, m model, the degree distributio is also asymptotically Poisso with expectatio 2m I both cases the umber of vertices with degree k is expoetially decreasig ad the expoet chages with the umber of vertices. Therefore these models are ot cosistet with the empirical results. Aother problem is that the etworks uder cosideratio are expadig, whereas the above models cosider the umber of vertices fixed. I order to overcome the problem of power-law degree distributios ad the atural growth of real etworks Barabási ad Albert suggested the preferetial attachmet model [1]. 4

7 1 0.1 lie 1 lie Pk e-05 1e-06 1e-07 1e-08 1e k Figure 1: The i-degree distributio of the Hugaria Web [4]. The horizotal axis represets i-degree k, while the vertical axis measures the umber of odes with a give i-degree P k, both o logarithmic scales. The empirical distributio is P k c k

8 2 Preferetial attachmet model The origial descriptio of Barabási ad Albert [1] is the followig. Startig with a small umber m 0 of vertices, at every time step we add a ew vertex with m m 0 edges that lik the ew vertex to m differet vertices already preset i the system. To icorporate preferetial attachmet, we assume that the probability P that a ew vertex will be coected to vertex i depeds o the degree of that vertex. They poited out that may complex real world etworks caot be adequately described by the classical Erdős-Réyi radom graph model, where the possible edges are icluded idepedetly, with the same probability p. I the case of m = 1, the resultig graph is a tree. These scale-free trees have bee kow sice the 1980s as ouiform radom recursive trees. Two early idetical classes of these trees are radom recursive trees with attractio of vertices proportioal to the degrees ad radom plae-orieted recursive trees see [27] ad [26]. We use the latter model. Startig with a sigle poit, at every step we add a ew vertex ad coect it to oe of the old vertices by a edge. This old vertex is chose radomly with probability proportioal to its degree. This leads to the same model as if we chose a edge radomly, each with equal probability, the oe of the edpoits of that edge. Figure 2 shows a simulated radom tree with 1000 odes. A possible geeralizatio of this model is where the probability of choosig a old vertex is k + β/s, istead of k/2, with a give β > 1, where k is the degree of the vertex ad s = 2 + β + 1 = 2 + β + β is the sum of all weights i the -th step. It was show by Móri i [28] that the proportio of vertices of degree k coverges almost surely to a limit c k, which, as a fuctio of k, decreases at the rate k 3+β. I the origial case β = 0, the formula of the expected proportio of vertices of degree k was determied by Szymański [34] ad strog covergece 6

9 Figure 2: Simulatio of the radom tree process, =

10 of this proportio was show by Lu ad Feg [25]. Rudas, Tóth ad Valkó [32] studied similar trees where geeral weight fuctios were used to build the tree. Similar a.s. results were proved i a paper of Bollobás, Riorda, Specer ad Tusády [6] for geeral graphs, ot oly trees. Bollobás ad Riorda [8] also determied the diameter of these graphs. Cooper ad Frieze [11] obtaied results for the maximum degree ad degree distributio for a aother geeralizatio of the model; they allow ew edges to be iserted betwee existig odes, a variable umber of edges to be added i each step, ad edpoits for ew vertices are chose by a mixture of uiform selectio ad copyig or preferetial attachmet. The recet mathematical results o scale-free graphs are summarized i [5]. The ext sectio summarizes the results obtaied i [19] ad [20] o the Barabási-Albert type tree obtaied with the geeralizatio of Móri, that is, a ew edge is coected to a existig vertex with degree k with probability k + β/s. 2.1 Results o width ad level-wise degree distributio First we study the shape of the tree. Startig from the root 0th level, we divide the tree to levels. The eighbors of the root will be o level 1, the eighbors of these will be o level 2, etc. Let X[, k] deote the size of the k-th level after the -th step the first step is whe we take the first edge. These radom variables determie the shape of the tree. Let W := max{x[, k] : 1 k} be its width ad H := max{k 1 : X[, k] 0} its height. The diameter studied i [8] is i close coectio with H. The results there yield that the height of our origial tree β = 0 is asymptotically 8

11 Olog. O the other had, Pittel proved i [31] that a.s. lim a.s., where y satisfies 1 + βye 1+y = 1. H log = βy Our goal is to determie the width. We use the method of Chauvi, Drmota ad Jabbour-Hattab [10], which they applied to biary search trees for the proof of W followig. Set α = 1+β 2+β. Theorem 1 With probability 1, X[, k] = 4π log. The mai results regardig the shape are the 2απ log exp k α log 2 + O, 2α log log as, where the error term is uiform for all k 0. Corollary 1 As, we have a.s. W = 2απ log 1 + O 1 log. I additio our results also yield that the width is reached at about level α log. The first image i Figure 3 shows the level sizes of a simulated radom tree process with 1000 odes. The curve shows the fuctio exp, which estimates these level sizes. The secod image i k α log 2 2α log 2απ log Figure 3 shows the width of the same tree as icreases from ad the fuctio 2απ log. Kowig the degree distributio ad the shape of the tree, Tamás Móri posed the problem whether the degree distributio is the same o all levels or ot. He oticed that o lower levels it is differet from that of the whole tree, for example o the first level the ratio of vertices with degree j a.s. goes to 1 + β. Hece the degree distributio o the lower level 1 j+β 1 j++β+1

12 Figure 3: Simulated ad theoretical level sizes of a radom tree process, = 1500 ad simulated ad theoretical width of the same realizatio of the process, as icreases from 1 to 1500.

13 is still a power law distributio, but the expoet is 2, idepedetly of the parameter β ad the fixed level k. I this paper we show that the aswer for Móri s questio is yes for the middle levels aroud α log, that cotai almost all vertices, hece determie the degree distributio of the whole tree. Theorem 2 Suppose β = 0. With ay costats 0 < k 1 < k 2, for k 1 log < k 1 2 log < k 2 log the ratio of vertices with degree j coverges a.s. to a limit c j o level k ad c j is equal to the limit of the ratio of j-degree vertices i the whole graph. Remark 1 The theorem holds for ay β > 1, that is, the limit of vertices with degree j o levels aroud α log a.s. coverges to the same limit as i the whole tree. We will oly prove the theorem. Oe ca see that the proof of this geeralizatio goes o the same lies, but eeds loger ad more complicated calculatios. I the ext sectio we will itroduce the way of usig martigales ad prove Theorem 1. The proof of 2 is postpoed to Sectio 2.3. I Sectio 3 we will see a applicatio. V. Batagelj called my attetio to directory trees, which might follow the preferetial attachmet model. We will study some of them ad see how their widths ca be approximated by applyig Theorem The shape of the tree Usig martigales First, itroduce the otatio Y [, k] = X[, k + 1] βx[, k] for k > 1, Y [, 0] = X[, 1] + β, 11

14 for the sum of weights o the level k. Our basic tool is the study of the followig series of complex geeratig fuctios G z = Y [, k]z k. k=0 Let F deote the atural σ-field geerated by the first steps. Lemma 1 For ay fixed z C the sequece M z := G z E z is a martigale with respect to the filtratio F, where E z = 1 j=1 S j βz S j. Proof: ad for k > 0 These yield Easy calculatio gives that EY [ + 1, 0] F = Y [, 0] s + 1 s, EY [ + 1, k] F = Y [, k] s + 1 s EG +1 z F = s + 1 s thus the expectatio G z β s + Y [, k 1] 1 + β s. zg z = s βz G z, s 1 EG z = 1 + β1 + z j=1 S j βz S j = 1 + β1 + ze z, sice G 1 z = 1 + β1 + z. Hece M z is a martigale. The ext lemma is about the asymptotics of the expectatio. 12

15 Lemma 2 For ay compact set of complex umbers C C we have EG z = 1+αz β1 + zγ2α Γ1 + α1 + z uiformly for z C, as. + O αrz 1, E z = 1+αz 1 Γ2α Γ1 + α1 + z + O αrz 1 Proof: As it is i the previous lemma s proof, 1 EG z = 1+β1+z The product is equal to j=1 S j βz S j Γ + α1 + z Γ1 + α1 + z 1 = 1+β1+z Γ2α Γ + 2α 1 Its asymptotitcs ca be determied as i [13] ad [10], provig that Γ + z Γ = z + O Rz 1 uiformly over ay compact set. It yields, that E z = 1+αz 1 Γ2α Γ1 + α1 + z + O αrz 1 uiformly i ay compact set, as. j=1 j + α1 + z j + 2α 1, Next, we are goig to study the covergece of the martigale M z. O this purpose, we determie the covariace fuctio of G z. Lemma 3 For every pair z 1, z 2 C, we have C+1z G 1, z 2 := EG +1 z 1 G +1 z 2 = = b j z 1, z 2 a k z 1, z β z z 2 j=1 k=j+1 a j z 1, z 2, j=1 13

16 with a k z 1, z 2 = βz 1 + z 2 S k, b k z 1, z 2 = = 1 + z 1 + z 1 β1 + z 2 + z 2 β S k EG k z 1 z 2. Proof: We give a liear recursio for C G z 1, z 2. Let k +1 deote the level of the vertex added i the + 1-st step. With this otatio G +1 z G z = z k βz. Thus C G +1z 1, z 2 = E [ E G z 1 + z k z 1 + z 1 βg z 2 + +z k z 2 + z 2 β F ] = = C G z 1, z 2 + E [ E G z 1 z k z 2 + z 2 β + z k z 1 + z 1 βg z 2 + +z k 1 z k z 1 + z 1 β1 + z 2 + z 2 β F ]. The coditioal distributio of k w.r.t. F is the followig { Y [,k] P k = k F = s, if k > 0, Y [,0] s, if k = 0. Hece, the coditioal expectatio is EG z 1 z k z 2 + z 2 β F = 1 + z 2 + z 2 β s G z 1 G z 2. Similarly we have EG z 2 z k z 1 + z 1 β F = 1 + z 1 + z 1 β s G z 1 G z 2, fially, this yields Ez k 1 z k 2 1+z 1 +z 1 β1+z 2 +z 2 β F = 1 + z 1 + z 1 β1 + z 2 + z 2 β s G z 1 z 2. 14

17 Hece C G +1z 1, z 2 = βz 1 + z 2 C G z 1, z 2 + s z 1 + z 1 β1 + z 2 + z 2 β EG z 1 z 2. s This proves the lemma sice C G 1 z 1, z 2 = 1 + β z z 2. Corollary 2 {M z : N} is bouded i L 2 for ay fixed z 1 < 1/α. Thus, there exists a radom variable Mz L 2 such that M z Mz a.s. i L 2 as for z H := {w C : w 1 < 1/α}. Proof: Usig the otatios of Lemma 3, we have 2+αz1 +z a k z 1, z 2 = 1 + O. j j k=j+1 We write A B if there is a costat c > 0 such that A cb for every. By Lemma 2, C G z 1, z 2 = 1 + z 1 + z 1 β1 + z 2 + z 2 β 2 + β Hece, j= β z z 2 j αrz 1z 2 1 j=1 EG j z 1, z 2 j + 2α 1 a k z 1, z 2 j=1 j=k+1 a k z 1, z αRz1 +z αRz 1+z 2 2 j 2+αRz 1+z 2 2 j 2+αRz 1+z 2 z 1 z 2 1. j=1 C M z 1, z 2 := EM z 1 M z 2 = EG z 1 G z 2 EG z 1 EG z 2 j 2+αRz 1+z 2 z 1 z 2 1. j=1 15

18 So, if 2 + αrz + z zz 1 > 1, the the sum is bouded. The iequality above is true exactly i H, hece M z is bouded i L 2 for z H. Also, if z 1, z 2 H the 2 + αrz 1 + z 2 z 1 z 2 1 > 1, hece C M z 1, z 2 coverges to some C M z 1, z 2 uiformly over the compact subsets of H 2 ad C M z 1, z 2 is holomorphic over H 2. To prove the uiform covergece of M z we follow the lies of [10]. The mai idea is the followig result, which ca be proved similarly to Propositio 2 of [10]. Propositio 1 Let I = 1 1/α, 1 + 1/α. The Mt t I has a cotiuous modificatio M such that for ay compact C I, E sup M 2 <. t C Geerally, if γ : R H is cotiuously differetiable, the M γt t R has a modificatio M γ such that for ay compact set C R, E sup M γ t 2 <. t C The uiform covergece of M comes from the followig propositio. The proof beig essetially the same as i [16] will be omitted. Propositio 2 For ay compact set C I, we have M M uiformly over C ad E sup M t Mt 2 0, t C Geerally, let γ : R H be cotiously differetiable ad let M,γ t = M γt ad M γ t = Mγt. The the same result holds for M,γ. 16

19 Corollary 3 M z ad all its derivatives coverge uiformly over the compact subsets of H. Proof: By Propositio 2 M is uiformly coverget over the arc γt = 1 + ρe it for all 0 < ρ < 2, thus for s 1 < ρ we have M s = 1 M z 2πi z s dz, by Cauchy s formula. Thus M ad its derivatives coverge uiformly over the compact subsets of H. I order to prove theorem 1, we will eed two more lemmas o the asymptotics of G z. First, we approximate E G z 2. Lemma 4 For every δ > 0 ad z 1 1/α δ, E G z 2 = O 21+αRz 1. For ay z such that 1/α δ z 1 1/α, E G z 2 = O 21+αRz 1 log, with uiform error terms as. Furthermore, for ay compact C C H, uiformly for z C. E G z 2 = O 1+α z 2 1 log γ Proof: Recall the proof of Corollary 2. It follows that E G z 2 21+αRz 1 17 j 2 α2rz z 2 1. j=1

20 For z 1 1/α δ the expoet of j is at most 1 δ < 1, hece E G z 2 21+αRz 1 j 1 δ 21+αRz 1. O the other had, for 1/α δ z 1 1/α we ca write E G z 2 21+αRz 1 j=1 j 1 δ 21+αRz 1 log. j=1 I the third case, for z 1 > 1/α we have E G z 2 21+αRz 1 j 2 α2rz z 2 1 j=1 21+αRz 1 1 α2rz z α2rz z 2 1. For the uiform equality we eed more. The umerator might ted to 0, so E G z 2 21+αRz α2rz z α2rz z α z e 1 α2rz z 2 1 log+1 1 α2rz z 2 1 This completes the proof. 1 α1+1+β z 2 log Now we approximate G z. Lemma 5 For every 0 < z < 2, we have a.s. G z z 1 log 1 α1+ z + z β. Proof: Obviously, G z G z. By [31] we ow that the height of the tree H log. Hece there exists a 0, for each realizatio of 18

21 the tree, such that X[, k] = 0 a.s. for 0, if k > c log. Hece, for sufficietly large, with probability 1 G z = ky [, k] z k 1 c log Y [, k] z k 1 c log G z. z k=1 k=1 We eed the followig lemma to approximate G z outside H. Sice the proof of this lemma follows from Lemma 4 ad Lemma 5 the same way as the proof of Propositio 3 i [10], it will be omitted. Lemma 6 For ay K > 0 there exists a δ > 0 such that sup G z = O, log K a.s., as. z =1, z 1 1/α δ Remark 2 If β = 0, the same is true for the fuctio Gz 1+z o γδ := {z z = 1, z 1 2 δ, Rz > 0.9} {z Rz = 0.9, z 1}. For ay K > 0 there exists a δ > 0 such that G z 1 + z = O log K a.s., as. sup γδ, Proof: Sice γ evades 1 it is eough to approximate G z o γδ. From here the proof goes exactly the same as that of the previous lemma. 19

22 2.2.2 Proof of Theorem 1 Fially, we ca start to prove the theorem. By defiitio, G z = Y [, k]z k, k=0 G z β βz = X[, k + 1]z k, k=0 if z 1. This exceptio does ot matter if β 0, sice 1 1+β 1+β 1 ad the fuctio ca be expaded to this poit regularly. We ca extract X[, k] from the geeratig fuctio by usig Cauchy s formula. If β 0, the X[, k +1] = 1 2πi z =1 G ξ β 1 π G e it β dξ = βξξk+1 2π π βe it e kit dt. We split the itegral to two parts. Let ϕ = miπ, arccos1 1/2α, ad I 1 := 1 G e it β 2π t ϕ δ βe it e kit dt, I 2 := 1 G e it β 2π βe it e kit dt, π t ϕ δ where δ is the same as i Lemma 6. If β = 0, istead of z = 1 we itegrate o γ = {ξ ξ = 1, Rξ > 0.9} {ξ Rξ = 0.9, ξ 1}. Let I 1 be the same as i the latter case ad I 2 := 1 G ξ dξ, 2πi 1 + ξξk+1 where δ is the same as i Remark 2. γδ By Lemma 6 ad Remark 2, for ay K > 0 we ca approximate the secod itegral i both cases as follows. I 2 1 G ξ β 2π βξ dξ 20 log K, 1

23 where we itegrate o { ξ = 1, ξ 1 1/α δ} i case of β 0 ad o γδ if β = 0. For t ϕ δ, M e it = G e it E e it is a.s. uiformly bouded by Corollary 3. O the other had, Lemma 2 provides us the asymptotics of the deomiator, hece G e it 1 α1+1+βreit = αcos t 1 = e log cos t 1α e c t 2 log for some costat c > 0. By fixig a sufficietly small positive ϑ we have 1 G e it dt 2π log 1 ϑ/2 t φ δ e c t2 log dt e c log ϑ. 2 log 1 ϑ/2 The remaiig part of the itegral is I 0 := 1 G e it 2π t log βe 1 ϑ/2 it e kit dt. Agai we are goig to use M z G z = E zm z = EG z 1 + β1 + z 3 ad Lemma 2, which ca be writte i the form EG z = 1 α1+z+zβ 1 + β1 + zγ2α Γ1 + α1 + z + O Rz 1α = = z 1α 1 + β1 + zγ2α Γ1 + α1 + z + O 1 21

24 uiformly. If t 0 i such a way, that t log 1 ϑ/2, the EG e it βz = elog eit 1α 1 + β1 + e it Γ2α βe it Γ1 + α1 + e it + O 1 = = e αt2 /2 log +itα log 1 it α α2 Γ 1 + 2α αt3 6 i log + Ot2 + t 4 log. 4 O the other had, M 1 = 21 + β, hece M e it 1 + β1 + e it = β itm + Ot β The, by 3, 4 ad 5 we coclude that, with probability 1, G e it e kit βe = /2 log +itα log k it e αt2 1 it α α2 Γ 1 + 2α M β 21 + β αt3 6 i log + Ot2 + t 4 log uiformly with respect to k. For the same reaso as i 2, here we also have e t2 log 1 + t + t 3 log ϑ log e. t log 1 ϑ/2 Hece I 0 = 1 2π e αt2 /2 log +itα log k 1 it α α2 Γ 1 + 2α M β 21 + β 22 αt3 6 i log. dt + Olog 3/2.

25 Itegratio gives + exp 2απ log I 0 = 1 log α k α log 1 + Hece we have X[, k] / 2απ log = exp log α k α log log α k2 2α log log α k 2α log log α k3 6α 2 log 2 + α 1/2 + 2α 2 Γ 1 + 2α M β 21 + β log α k2 2α log log α k 2α log log α k3 6α log 2 + α 1/2 + 2α 2 Γ 1 + 2α M β 21 + β +Olog 3/2. +O a.s., with a error term uiform i k. This completes the proof. 1 log 2.3 Degree distributio of levels Geeratig fuctios The proof goes o similar lies as the previous oe, however, we use geeratig fuctios i a slightly differet way. The mai idea is to cosider the geeratig fuctio G z = G 1 z = X[, k + 1]z k. for ay complex z. The sum is fiite for a fixed, hece G z is holomorphic. Note that G z deotes a geeratig fuctios which is slightly differet from the previous sectio. To study degree distributios, we have to cout 23 k=0

26 the vertices with give degree. Istead of that let X j [, k] be the umber of vertices with degree at least j o level k after step. Let G j z = X j [, k + 1]z k k=0 be the correspodig geeratig fuctio. Obviously, G j 1 is the umber of vertices i the tree with degree at least j excludig the root. As show i [34], the limit of the expected ratio of 4 vertices with degree j is. Oe ca see that summig this quatity jj+1j+2 2 gives the ratio of vertices with degree at least j as. jj+1 To calculate EG j z we use coditioal expectatios. Let F deote the σ-field geerated by the first steps. The umber of vertices with degree at least j o a give level either icreases by oe or does ot chage. For j = 1, the probability of a icrease is P X[ + 1, k] = X[, k] + 1 F = { X[,k]+X[,k 1] X[,1] 2 2, for k > 1,, for k = 1, sice the ew vertex is coected to level k 1 with probability equal to the sum of the degrees o level k 1 over 2. Obviously, the sum of the degrees o level k 1 is X[, k] + X[, k 1] for k > 1 ad X[, 1] for k = 1. For j 2, probability of a icrease is P X j [ + 1, k] = X j [, k] + 1 F = j 1 X j 1 [, k] X j [, k], 2 sice this evet is equivalet to the evet that the ew vertex is coected to a vertex o level k with degree j 1. Thus, for j = k = 1 X[, 1] EX[ + 1, 1] F = X[, 1] = X[, 1] 2 + X[, 1] 1 X[, 1] = 2 + X[, 1] = X[, 1]. 2 24

27 For j = 1, k > 1 we have X[, k] + X[, k 1] EX[ + 1, k] F = X[, k] X[, k] + X[, k 1] + X[, k] 1 = 2 ad fially, for j > 1 = X[, k] + X[, k 1], 2 2 EX j [+1, k] F = X j [, k]+1 j 1X j 1 [, k] X j [, k] X j [, k] 1 j 1X j 1 [, k] X j [, k] = 2 = 2 j + 1 X j [, k] + j X j 1 [, k]. This gives the followig recursive formula for the geeratig fuctios. For j = 1, EG +1 z F = G z + z 2 G z = z G z, 6 2 ad for j 1 +1 z F = 2 j 2 G j+1 z + j 2 G j z. 7 EG j+1 I our calculatios, just as i Sectio 2.2, we will use the fact that i + v Γ1 + w i + w = Rv w Γ 1 + v + O1/, i=1 for ay complex v ad w 1. Sice G 1 z = 1, 1 EG z = EG 1 z = j=1 2j z 2j z = 1/Γ 1+z/2 2 + O1/. 8

28 Remark 3 For ay fixed z C the sequece M z := G z EG z is a martigale with respect to the filtratio F. Proof: It follows from 8 ad 6 that EM +1 z F = EG +1z F EG +1 z = = z G z 2 1 EG +1 z = G z EG +1 z = M z. I geeral, the followig holds for the expectatio of the geeratig fuctios. Lemma 7 For ay fixed j 2 where Proof: EG j z = 1+z 2 c j z = c j z/γ 3 + z + O 2 j 1! z + 2z z + j. 1, We proceed by iductio o j. For j = 1 the equatio is true with c 1 z = 1. Suppose that it is true for j = l 1. By 7 we have EG l+1 +1 z = 2 l 2 EG l+1 z + l Sice G l+1 1 z = 0, this recursive formula gives EG l+1 l 2m l +1 z = 2i EG i z 2m = = c lzl 2Γ 3+z = i=1 i=1 c l zl 2iΓ 3+z 1+z+l l/ z + l/2 2 i1+z m=i+1 2 /i l/2 1 + O O 2 EG l z. 1 = i = c l+1z 1+z Γ 3+z O 1.

29 From ow o, we will study the fuctios U j z := G j z c j zg z. We will show that U j z is a.s. ot far from 0. I order to be able to use martigales, we have to cosider the followig liear combiatio. For j 2 let W j z := j i=2 1 j i j 1 i 1 U i z. 9 The idea is based o a similar combiatio used i [28]. This combiatio will cacel out the differet coefficiets i the proof of Remark 5, origiatig from the recursive formulas for the geeratig fuctios i 7. Easy calculatio shows that where ad for 2 i j Remark 4 For j 2, W j z = j i=1 b j i zg i z, 10 b j 1 z = 1 j 1 j 1 j + z, b j j 1 i z = 1 j i. i 1 U j z = j i=2 j 1 W i z, i 1 Proof: W j z is defied i 9. Pluggig this defiitio ito the right had side of the equatio yields j i=2 j 1 W i z = i 1 j i=2 i k=2 j 1 i 1 1 i k i 1 k 1 27 U k z.

30 By chagig the order of the sums, this is equal to j j k=2 i=k 1 i k j 1 i 1 i 1 k 1 U k z = j 1 = U j z + U k z k=2 j j 1 i 1 1 i k. i 1 k 1 We oly have to show that the last sum is equal to zero for ay 2 k j 1. i=k By maipulatig the biomial formulas, we have j i=k j 1 i 1 1 i k i 1 k 1 j j = k k i=k 1 k i j k j i = 0. Remark 5 For every fixed complex z ad fixed j 2, M j z := W j z is a martigale with respect to F. i=1 2i 2i + 1 j Proof: I order to prove that M j z is a martigale with respect to F, we have to show that EM +1z F j = M j z, or equivaletly, that EW +1z F j = 2+1 j W j 2 z. Usig the recursive formulas 6 ad 7, we have EW j +1z F = j i=1 = b j 1 z z G z+ 2 b j i zeg i +1z F = j i=2 b j 2 j + 1 i z G i z + i G i 1 z 28

31 The coefficiet of G z here is j 1 1 j z j 1 = j + z 2 2 = 1 j 1 j 1 j + z j 2 = b j 1 z j. 2 For j i 2, we have bj i+1 z b j i = j i z i b j 2 i + 1 i z i 2 2 j i i 2, thus the coefficiet of G i z is = b j i z 2 j Thus, EW +1z F j = 2+1 j W j z, completig the proof. The key to the approximatio of U j z is to fid a upper boud for its variace. The followig lemma gives a upper boud for this variace for a arbitrary z, however, it will oly be eeded for z = 1 i the proof of the theorem. Lemma 8 For ay complex z ad fixed k 2, E W k z 2 = O 1+ z 2 2, which yields E U k z 2 = O 1+ z 2 2. Proof: Let C i,j D k z 1, z 2 := EG i z 1 G j z 2, z 1, z 2 := EW k z 1 W k z 2. Usig equatio 10, this ca be rewritte as D k z 1, z 2 = k k l=1 m=1 b k l z 1 b k mz 2 C l,m z 1, z

32 The objective is to give a recursive formula for D k z 1, z 2. Note that G j +1 z = G j z + K j z where the distributio of K j is give by ad for j 2 P K 1 z = z k 1 F = { X[,k]+X[,k 1], for k > 1,, 2 for k = 1,, 2 X[,1] P K j z = z k 1 F = j 1 X j 1 [, k] X j [, k] 2 These yield EK 1 z F = 1 + z 2 G z ad Also we have EK j z F = j 1 2 G j 1 z G j z. EK 1 z 1 K 1 z 2 F = 1 + z 1z 2 G z, 2 EK j z 1 K 1 z 2 F = z 2 j 1 ad for i j 2, 2 G j 1 EK i z 1 K j z 2 F = i 1 It follows for i j 2 that C i,j 2 G i 1 z 1 z 2 G j z 1 z 2 for j 2, z 1 z 2 G i z 1 z 2. +1z 1, z 2 = E [ E ] G i z 1 + K i z 1 G j z 2 + K j z F = [ = E G i z 1 G j z 2 + G i z 1 j 1 G j 1 z 2 G j z i 1 G i 1 z 1 G i z 1 G j z i 1 2 G i 1 z 1 z 2 G i z 1 z 2 ] = = i j EG i z 1 G j z 2 + i j 1 z 1 G j 1 z 2 + i 1 2 EG i 2 EG i 1 2 E G i 1 z 1 G j z 2 + z 1 z 2 G i z 1 z 2. 30

33 This gives the formula C i,j +1z 1, z 2 = i j 2 C i,j z 1, z i 1 2 Ci 1,j z 1, z 2 + j 1 2 Ci,j 1 z 1, z i 1 2 E G i 1 z 1 z 2 G i z 1 z 2. For j 2, similar calculatios lead to ad C j,1 +1 z 1, z 2 = j + z 2 C j,1 z 1, z 2 + j Cj 1,1 z 1, z 2 + j 1 + z 2 2 E G j 1 z 1 z 2 G j z 1 z 2, C 1,1 +1 z 1, z 2 = z 1 + z 2 2 C 1,1 z 1, z z 1z 2 EG z 1 z 2. 2 Notice, that i these recursive formulas, all the EG. z 1 z 2 type expressios are O 1+z 1 z 2 2, accordig to 8 ad Lemma 7. Sice all of them are divided by, all those terms are O z 1 z If we plug the above recursive formulas ito 11, easy but tedious calculatio gives that D +1z k 1, z 2 = k + 1 [ D k z 1, z 2 + O ] z 1 z Sice after the first step there is o vertex with degree at least two, D k 1 z 1, z 2 = 0. Thus, the recursive formula for D k yields D k +1z 1, z 2 = O i=1 i z 1 z = m=i+1 O i=1 m k + 1 = m O i z 1 z i 1 k 2 = O 1+z 1 z

34 Obviously, W k E W k z 2 = W k zw k z 2 = D k z, z = O z = W k 1+zz 2 = O zw k 1+ z 2 2 z, hece. For the secod part, recall that accordig to Remark 4, U k z = k i=2 k 1 W i z, i 1 hece, E U k z 2 c E W k z 2 = O 1+ z 2 2 with some costat c. Now we approximate U j z. Deote by A B if there is a c costat such that A cb. Lemma 9 For every z 0, ad fixed j 2 we have a.s. U j z log z U j z. Proof: Trivially, U j z U j z. I [31] it was show that the height of the tree H c log a.s., where c 4.31 is the greater tha 2 solutio of c log2e/c = 1. Hece, a.s. there exists 0 such that for 0, we have X j [, k] = X[, k] = 0 if k > c + 1 log. Thus, a.s. U j z = k X j [, k] c j z X[, k] z k 1 k=0 log k=0 X j [, k] c j z X[, k] z k 1 = log z U j z. 32

35 2.3.2 Proof of Theorem 2 Before directly eterig the proof we study U j z for z = 1. Lemma 10 For every ε > 0 we have a.s. as. sup z=1 U j z = O 3/4+ε Proof: By Markov s iequality ad Lemma 8, we have P U j z 3/4+ε E U j z 2 3/2+2ε 1/2 2ε Let z, l = expi 2πl for l = 1,..., K, where K = log. These poits K split the circle z = 1 ito K equal arcs. We have Sice P U j z, l 3/4+ε for ay l 1/2 2ε K 1/2 ε 2 1/2 ε <, =1 we ca apply the Borel-Catelli Lemma. Hece for all but fiitely may we have a.s. sup U j z 2, l 2 3/4+ε. 2 l Betwee the poits z 2, l we ca use Lemma 9. Suppose that z = 1 ad 2πl+1 < arg z <. The we have uiformly 2πl K K U j 2 z = U j 2 z 2, l + OU 2z2, l1/k Hece for all but fiitely may we have a.s. lew 2 3/4+ε + O 2 3/4+ε 2 3/4+ε sup U j z 2 3/4+ε. 2 z =1 33

36 Fially, recall that for z = 1 we have G j +1 z G j z = K j z 1, ad G +1 z G z = K 1 z 1. Hece U +1z j U j z G j +1 z G j z c j z G +1 z G z 2. For 1 k 2, we have uiformly U j j 2 +k z = U z + Ok 2 3/4+ε + O 2 + k 3/4+ε. 2 This completes the proof, as it yields for all but fiitely may a.s. sup U j z 3/4+ε. z =1 Now we directly start the proof of the Theorem. Proof: Cauchy s formula. We ca extract X[, k] from the geeratig fuctio by usig X j [, k + 1] c j 1X[, k] = 1 2π = 1 π G j 2π π π π G j e it c j 1G e it e kit dt = e it c j e it G e it dt+ e kit + 1 π c j e it c j 1G e it dt = J + I. 2π π e kit First, we ca approximate J with Lemma 10. We have a.s. J 3/4+ε. The secod itegral ca be approximated just as i Sectio 2.2, we will use two Lemmas to do so. Lemma 11 The martigale M z = Gz EG z ad all its derivatives coverge uiformly over the compact subsets of H := {z C z 1 < 2}. 34

37 Proof: I Remark 3 we have already see that M z is a martigale. Corollary 3 with β = 0 says that 1+zM z ad all its derivatives coverge uiformly over the compact subsets of H := {z C z 1 < 2}. Sice 1 H, this proves the lemma. Lemma 12 Let γδ := {z z = 1, z 1 2 δ, Rz > 0.9} {z Rz = 0.9, z 1}. For ay L > 0 there exists a δ > 0 such that a.s., as. sup G z = O γδ log L, Proof: The fuctio Gz of Remark 2 is equal to the G 1+z z of the preset sectio. Retur to the proof of the Theorem. Sice c j e it c j 1G e it is regular for z < 2, I = 1 π c j e it c j 1G e it dt = 1 2π π e kit 2πi γ where γ = {ξ ξ = 1, Rξ > 0.9} {ξ Rξ = 0.9, ξ 1}. c j ξ c j 1G ξ dξ ξ k+1 We split the itegral I ito two parts. I 1 := 1 c j e it c j 1G e it e kit dt, 2π t π/2 δ I 2 := 1 2πi γδ c j ξ c j 1G ξ dξ, ξ k+1 35

38 with the δ we get from Lemma 12. By the lemma, for ay L > 0 we ca approximate the secod itegral as follows. I 2 sup c j z c j 1 1 G e it dt γδ 2π log. 12 L For t π/2 δ γδ M e it = G e it EG e it is a.s. uiformly bouded by Lemma 11. O the other had, 8 provides us the asymptotics of the deomiator, hece G e it 1+Reit /2 = Reit 1/2 = = cos t 1/2 = e cos t 1log /2 e c t 2 log for some costat c > 0. By fixig a sufficietly small positive ϑ we have 1 2π log 1 ϑ/2 t π/2 δ G e it dt e c t2 log c logϑ dt e log 1 ϑ/2 The remaiig part of the itegral is I 0 := 1 c j e it c j 1G e it e kit dt. 2π t log 1 ϑ/2 Agai, we are goig to use ad 8, which ca be writte i the form log L. 13 G z = EG zm z 14 EG z = 1+z/2 Γ 3+z + O 1 Rz 1/2 = z 1/2 Γ 3+z + O

39 uiformly. If t 0 i such a way that t log 1 ϑ/2, the EG e it = e 1 2 eit 1log 1 1 Γ 3+eit + O = 2 = e t2 /4 log +it/2 log 1 it2 Γ 2 t3 12 i log + Ot4 log. 15 O the other had, M 1 = 1, hece ad trivially M e it = 1 + itm 1 + Ot c j e it c j 1 = ct + Ot with c = c je it t=0. The, by 14, 15 ad 16 we coclude that, with probability 1, G e it e kit = e itlog /2 k t2 /4 log ct + Ot 2 + t 4 log. uiformly with respect to k. Partial itegratio gives e t2 /4 log t 2 + t 4 log dt = 72 πlog 3/2. For the same reaso as i 13, here we also have te t2 log << e t log 1 ϑ/2 Hece Itegratio gives log ϑ I 0 = 1 cte it1/2 log k t2 /4 log dt + Olog 3/2. 2π We ca summarize the results i I 0 1 = O. log X j [, k + 1] c j 1X[, k] = O Comparig this with Theorem 1 completes the proof... log 37

40 3 A applicatio: Directory trees Although there are several examples of etworks that have power-law degree distributios oe of these is a tree. directory trees that should be studied. V. Batagelj called my attetio to The followig examples all have power-law degree distributios P k c k γ with 2 < γ < 3. This allows to compare the width of the tree with the result of Theorem 1. The first example is the directory tree of the mai server of the Departmet of Computer Sciece, Budapest Uiversity of Techology. Figure 4 shows the subdirectory structure ad Figure 5 shows the degree distributio with logarithmic scales. Liear regressio gives that γ Substitutig β = γ 3 = 0.62 ad = to Theorem 1 gives 9162 to the width. We ca compare this with the real width of the tree which is We ca also calculate the β that would give the same theoretical width as the real. From Theorem 1 it is β 0.71 This table shows the results of studyig several directory trees. Directory tree of vertices β real width theor. width β Server of CS Dep., Tech. Uiv Server of CS Dep., Eotvos Uiv Server of Fazekas High School Home Liux Home Widows The servers are all uix systems with may users who create their directories to store their ow files. The ratio of theoretical ad real width is betwee 0.85 ad 1.15 i the examples, hece we ca approximate the width of directory trees with Theorem 1. 38

41 39 Figure 4: Directory Tree, Departmet of Computer Sciece, Budapest Uiversity of Techology.

42 logk logpk Figure 5: Degree distributio, Departmet of Computer Sciece, Budapest Uiversity of Techology. 40

43 4 A modified model: Preferetial attachmet with idepedet selectio Cosider the followig modificatio of the Barabási Albert radom graph [21]. At every step a ew vertex is added to the graph. It is coected to the old vertices radomly, with probabilities proportioal to the degree of the other vertex, ad idepedetly of each other. Sice we are iterested i asymptotic aalysis, the iitial cofiguratio ca be arbitrary, but, for the sake of simplicity, let us start from the very simple graph cosistig of two poits ad the edge betwee them. Let us umber the vertices i the order of their creatio; thus the vertex set of the graph after steps is {0, 1,..., }. Let U[, k] ad V [, k] deote the umber of vertices of degree exactly k ad at least k, resp., after steps. Thus, U[, 0] + U[, 1] + = + 1. Let S = k 1 ku[, k] = k 1 V [, k], the sum of degrees, or equivaletly, the double of the umber of edges. At the th step a old vertex of degree k is coected to the ew oe with probability λk/s 1. This quatity remais below 1, provided the proportioality coefficiet λ is less tha 2, which will therefore be assumed i the sequel. Let F deote the σ-field geerated by the first steps. Let [, k] be the umber of ew edges ito the set of old vertices of degree k at the th step, ad let = k 1 [, k] be the total umber of ew edges. Obviously, the coditioal distributio of [+1, k] with respect to F is biomial with parameters U[, k] ad λ k S, hece E +1 F = λ. The aim of the preset sectio is to study some asymptotic properties of this radom graph as the umber of vertices teds to ifiity. I Sectio 4.1 we prove a strog law of large umbers for the maximum degree, ad i Sectio 4.2 it will be show that the proportio of vertices of degree k coverges a.s. to a costat, which, as a fuctio of k, decreases i the order 41

44 of k 3 as k. 4.1 The maximum degree First we deal with the asymptotics of S. Theorem 3 S = 2λ + o 1/2+ε, ε > 0. Proof: With 1 = 1 defie ζ = j λ = S /2 λ. The ζ, F j=1 is a square itegrable martigale, ad the icreasig process associated with ζ 2 by the Doob decompositio is A = 1 Var j F j 1 = j=2 j=2 k 1 U[j, k] kλ S j It is well kow [30], Propositio VII-2-4 that ζ = o evet A. This completes the proof. 1 kλ λ. S j A 1/2+ε a.e. o the Let us tur to M = max{k : U[, k] > 0}, the maximum degree after steps. Theorem 4 We have lim M / = µ with probability 1, where the limit µ differs from zero with positive probability. Proof: We follow the same lies as i the proof of Theorem 3.1 of [29]. Let W [, j] deote the degree of vertex j after the th step, with the iitial values W [, j] = 0 for < j, W [1, 0] = W [1, 1] = 1, W [j, j] = j. The M = max{w [, j] : j 0}. Let us itroduce the ormalizig terms c[, k] = 1 i=1 S i, 1, k 1. S i +kλ 42

45 For, with probability 1 we have 1 c[, k] = exp kλ i=1 1 + k2 λ 2 S i 2 1 i=1 1 + o1. Si 2 Sice 1 S i = 1 2λi 1 + o i 1/2+ε, we obtai that the expoet i 4.1 differs from k log oly by a term covergig with probability 1. Thus c[, k] 2 γ k k/2, with a appropriate positive radom variable γ k. We clearly have that E W [+1, j] F = W [, j] + λ W [, j] S = W [, j] S +λ S. Hece Z[; j, 1] = c[, 1] W [, j], F, max{j, 1} is either a positive martigale or costat zero, thus it coverges a.s. to some ζ j. To estimate the momets of ζ j, cosider W [, j]+k 1 Z[; j, k] = c[, k]. k Sice W [+1, j] W [, j] is equal to either 1 or 0, we ca write W [+1, j]+k 1 = k W [, j]+k 1 W [, j]+k 1 = + W [+1, j] W [, j] = k k 1 W [, j]+k 1 kw [+1, j] W [, j] = 1 +. k W [, j] By takig coditioal expectatio with respect to F we ca see that W [+1, j]+k 1 W [, j]+k 1 E F = 1 + kλ, k k S 43

46 hece Z[; j, k], F, max{j, 1} is also a coverget martigale. Sice c[, 1] k c[, k], we ca majorize Z[; j, 1] k by k! Z[; j, k]. Now, c[, 1]M = max{z[; j, 1] : 0 j }, beig the maximum of a icreasig umber of oegative martigales, is a submartigale. The proof ca be completed by showig that this submartigale is bouded i L k, for some k 1. Let us start from the estimatio E c[, 1]M k = E max{z[; j, 1] k : 0 j } k! = k!ez[1; 0, k] + k! EZ[j; j, k] = k! + k! j=1 EZ[; j, k] = j=0 j +k 1 Ec[j, k]. k j=1 Here j +k 1 j +k 1 Ec[j, k] = E[E c[j, k] F ] = k k j +k 1 = E[c[j, k]e F ]. k Next we show that, idepedetly of j, j +k 1 π+k 1 E F E, k k where π stads for a Poissoλ radom variable. Rememberig that j = [j, 0] + + [j, j 1], we ca write j +k 1 = [j, 0] [j, j 1] k 1. k l 0 + +l j =k l 0 l j 1 The biomial coefficiets o the right-had side are coditioally idepedet. The coditioal distributio of each [j, i] is biomial. Let ξ be a 44 l j

47 Biomial, p radom variable ad η a Poisso oe, with the same expectatio. The E ξ = l p l pl l l! = E η. l Thus, if all radom variables [j, i] o the right-had side of 4.1 are replaced by coditioally idepedet Poisso variables, the coditioal expectatio caot decrease. Hece 4.1 follows. By 4.1 ad 4.1, for the L k -boudedess of the submartigale c[, 1]M it is sufficiet to check that j=1 Ec[j, k] <. The covergece of j=1 c[j, k] whe k > 2 is clear from the asymptotics obtaied for c[, k], but the itegrability does ot follow immediately. Let k = 8 ad N = max{ : S > 4λ}. The for j > N we have j 1 c[j, 8] = 1 8λ S i +8λ i=1 j 1 i=n = +2 N +1N +2, jj +1 but this is obviously true eve for j N. Thus, for the fiiteess of j=1 Ec[j, 8] it is sufficiet to prove that EN 2 <. By the usual large deviatio argumets we have P N = P S > 4λ = P 2 S/2 > 4 λ E 2 S/2 4 λ. Thus, we have to estimate the momet geeratig fuctio of S. 1 = 1 we ca write S /2 = i=1 i, ad With E 2 S/2 = EE 2 S/2 F \ = E2 S 1 /2 E 2 F \ = 1 = E 2 S 1/2 1 + λj U[ 1,j] E j=1 2 S 1/2 exp { 1 j=1 S 1 λju[ 1, j] S 1 } = e λ E 2 S 1/2. 45

48 Therefore E 2 S/2 e λ, which, combied with 4.1, implies that P N = e/4 λ. Thus EN 2 <, ideed. 4.2 Degree distributio I this sectio we prove that the degree distributio of our graph stabilizes almost surely, as, aroud a power law with expoet 3. First we show that stochastic process of ew vertices approaches a statioary regime. More precisely, the radom variables are asymptotically idepedet ad asymptotically Poissoλ distributed. By LeCam s theorem o Poisso approximatio [23] we have P +1 = k F λk k=0 k! e λ 2 kλ 2 U[, k] 2λ 2 M = o1. Similarly, the coditioal distributio of [+1, k] with respect to F is U[, k] also asymptotically Poisso with parameter λk. S k=1 S S Theorem 5 For every k = 0, 1,... The proportio of vertices of degree k coverges a.s. as : P where lim U[, k] +1 = x k = 1, x 0 = p 0, x k = 2 kk+1k+2 k ii+1p i, i=1 p k = λk k! e λ. Remark 6 x k 2λ2+λ k 3 as k. 46

49 Proof: First we will show by iductio over k that V [, k] P lim +1 = y k = 1, where the costats y k satisfy the followig recursio: y 0 = 1, y k = k 1 k+1 y k 1 + 2q k k+1, q k = p k + p k The sequece U[, 0] obeys the strog law of large umbers, because [11], Corollary VII-2-6 implies that U[, 0] = I i = 0 i=1 P i = 0 F e λ. i=1 Hece we obtai that P lim V [, 1] +1 = 1 e λ = 1. For the iductio step suppose our assertio holds true for k 1. This time we itroduce the ormalizig factors 1 d[, k] = 1 I S i 2k 1 k 1λ 1. S i i=1 Their asymptotic behaviour ca be treated similarly to what we have doe i 4.1. Thus, with probability 1, we ca write 1 d[, k] = exp k 1λ I S i 2k k 12 λ 2 S i 2 i=1 1 i=1 1+o1. Si 2 By Theorem 3 we have 1 = 1 S i 2λi 1 + o i 1/2+ε, thus the expoet differs from k 1 log oly by a a.s. coverget term. Therefore d[, k] 2 δ k k 1/2, with some radom variable δ k > 0. 47

50 Sice V [+1, k] = V [, k] + [+1, k 1] + I +1 k, it is easy to see that E d[+1, k] V [+1, k] F = d[, k] V [, k] + b[, k], where b[, k] = d[+1, k] k 1λ V [, k 1] S + d[+1, k] P +1 k F d[+1, k]λ + 1. I additio, we have Vard[+1, k] V [+1, k] F = = d[+1, k] 2 Var [+1, k 1] + I +1 k 2d[+1, k] 2 Var [+1, k 1] + VarI +1 k 2d[+1, k]b[, k] = O k Let us itroduce a square itegrable martigale by its differeces ξ = d[, k] V [, k] d[ 1, k] V [ 1, k] b[ 1, k]. The icreasig process associated with the square of this martigale is A = E ξi 2 F = i=1 Var d[i, k]v [i, k] F, i=1 which is of order O k by 17. Hece ξ i = d[, k] V [, k] i=1 b[i 1, k] = o k/2+ε. i=1 48

51 From all these we obtai that V [, k] +1 = 1 +1d[, k] b[i 1, k] + o1. Now, by 4.2 ad the iductio hypothesis, k 1 b[i, k] d[i+1, k] y k 1 + q k. 2 By applyig the asymptotics we obtaied for b[i, k] we arrive at the formula which was to be proved. i=1 V [, k] +1 = k 1 k+1 y k 1 + 2q k k+1 + o1, Next we show that the solutio of the recursio 4.2 is y k = 2 kk+1 k iq i. Let r k = q k + q k ad z k = y k + y k , the from the recursio for y k oe ca derive that kz k = k 1z k 1 + 2r k. From that we have i=1 z k = 2 k k r i, i=1 which easily yields the above metioed explicite form of y k, ad fially, the desired expressio for x k. 5 A Nash-equilibrium model of the World Wide Web Although the Barabási ad Albert model is cosistet with empirical data for several differet types of etworks, it has importat shortcomigs i modelig the World Wide Web. First, it treats the etwork members as idetical, 49

52 which is a rather urealistic assumptio i the cotext of Web sites. Furthermore, a model with homogeous etwork members ad thus a fixed umber of out-liks caot say much about the distributio of out-degrees i the graph. The empirical evidece shows that this distributio is similar to the oe for i-degrees but the Barabási ad Albert model does ot predict this. Secod, ad most importatly, this model is ad hoc i the sese that it does ot cosider the etwork members as strategic agets actig deliberately i their ow iterest. Why would odes or agets i the odes select which liks to establish oe after the other without ay iteractio? Why would they cosider solely the degrees to make their decisios? From a ecoomic perspective the iterestig questio is: what icetives drive agets choices of the odes ad how these choices deped o their iheret characteristics e.g. their cotet? The model proposed i [22] explicitly addresses these issues, presetig a approach based o game theory. The basic assumptio is that the odes represet ratioal ecoomic agets e.g. web sites who make simultaeous ad deliberate decisios o the liks they establish betwee themselves. Agets are strategic ad heterogeeous with respect to their edowed cotet, which may be thought of as their value i the eyes of the public/market. The utility of a ode depeds o its cotet ad the structure of the etwork. The objective is to fid the pure strategy Nash-equilibria of the game where players maximize their utilities ad their strategies cosist of establishig liks from oe aother i a simultaeous decisio. A equilibrium of the game represets a graph ad our mai iterest is i describig the structure of this graph. The results show how the i- ad out-degrees deped o the ode s cotets ad that i- ad out-degree distributios are similar. 50

53 6 Coclusio I the past few years, the Iteret ad other large etworks became the foci of etwork research. I order to model these complex researcher i differet disciplies proposed to use radom methods. Sice the origial Erdős-Réyi radom graph model was foud to be icosistet with several empirical patters i these etworks, ew models were proposed. The Watts-Strogatz small world model is a simple mixture of a determiistic etwork ad the Erdős-Réyi model. It is still icosistet with empirical fidigs regardig the degree distributio. Several examples show that real etworks are scale-free with a power law degree distributio. The preferetial attachmet model proposed by Barabási ad Albert solves this problem. Although a variat of the model had bee previously kow i mathematics, its itroductio by Barabási ad Albert geerated a vast amout of research i the field. The review papers [2] ad [5] summarize the empirical fidigs ad the mathematical results i the area. I this dissertatio we studied the radom tree process based o the preferetial attachmet model. We determied the width of the tree ad showed that the degree distributio is similar to that of the whole tree o the middle levels. I the origial preferetial attachmet model the ew vertex is coected to the graph with a give umber of edges radomly with probabilities proportioal to the degrees of old vertices. I Sectio 4, we suggested a modified model where the ew ode is coected to the graph with edges that are established idepedetly with probabilities proportioal to the degrees. We showed that the maximum degree ad the degree distributio is similar to the origial model. The papers [17],[14],[18] summarize the work of the author i the field of extremal combiatorics. 51

54 7 Ackowledgmets I grateful to my dissertatio supervisor, Tamás Móri, for his support ad guidace throughout the PhD program. I also appreciate the suggestios ad commets of the aoymous reviewers of the papers that the dissertatio is based o. 52

55 Refereces [1] A.-L. Barabási ad R. Albert. Emergece of scalig i radom etworks. Sciece, 286: , [2] A.-L. Barabási ad R. Albert. Statistical mechaics of complex etworks. Rev. Mod. Phys., 741:47 96, [3] A.-L. Barabási, R. Albert, ad Hawoog Jeog. Scale-free characteristics of radom etworks: the topology of the world-wide web. Physica A, 281:69 77, [4] A. Beczúr, K. Csalogáy, D. Fogaras, E. Friedma, T. Sarlós, M. Uher, ad E. Widhager. Searchig a small atioal domai prelimiary report. I Proceedigs of WWW, [5] B. Bollobás ad O. Riorda. Hadbook of Graphs ad Networks, chapter Mathematical results o scale-free graphs. Wiley-VCH, Berli, [6] B. Bollobás, O. Riorda, J. Specer, ad G. Tusády. The degree sequece of a scale-free radom graph process. Radom Structures ad Algorithms, 18: , [7] B. Bollobás. Radom Graphs. Academic Press, Lodo, [8] B. Bollobás ad O. Riorda. The diameter of a scale-free radom graph. Combiatorica, 24:5 34, [9] A.Z. Broder, R. Kumar, F. Maghoul, P. Raghava, S. Rajagopala, R. Stata, A. Tomkis, ad J.L. Wieer. Graph structure i the web. Computer Networks, 33: , [10] B. Chauvi, M. Drmota, ad J. Jabbour-Hattab. The profile of biary search trees. The Aals of Applied Probability, 11: ,