A preferential attachment model with random initial degrees

Transcription

1 Mahemaical Saisics Sockholm Universiy A preferenial aachmen model wih random iniial degrees Maria Deijfen Henri van den Esker Remco van der Hofsad Gerard Hooghiemsra Research Repor 2007:2 ISSN

2 Posal address: Mahemaical Saisics Dep. of Mahemaics Sockholm Universiy SE Sockholm Sweden Inerne: hp://

3 Mahemaical Saisics Sockholm Universiy Research Repor 2007:2, hp:// A preferenial aachmen model wih random iniial degrees Maria Deijfen Henri van den Esker Remco van der Hofsad Gerard Hooghiemsra February 2007 Absrac In his paper, a random graph process {G} 1 is sudied and is degree sequence is analyzed. Le {W } 1 be an i.i.d. sequence. The graph process is defined so ha, a each ineger ime, a new verex, wih W edges aached o i, is added o he graph. The new edges added a ime are hen preferenially conneced o older verices, i.e., condiionally on G 1, he probabiliy ha a given edge is conneced o verex i is proporional o d i 1 + δ, where d i 1 is he degree of verex i a ime 1, independenly of he oher edges. The main resul is ha he asympoical degree sequence for his process is a power law wih exponen τ = min{τ W, τ P }, where τ W is he power-law exponen of he iniial degrees {W } 1 and τ P he exponen prediced by pure preferenial aachmen. This resul exends previous work by Cooper and Frieze, which is surveyed. Sockholm Universiy. mia@mah.su.se Delf Universiy of Technology. H.vandenEsker@ewi.udelf.nl Eindhoven Universiy of Technology. rhofsad@win.ue.nl Delf Universiy of Technology. G.Hooghiemsra@ewi.udelf.nl

4 1 Inroducion Empirical sudies on real life neworks, such as he Inerne, he World-Wide Web, social neworks, and various ypes of echnological and biological neworks, show fascinaing similariies. Many of he neworks are small worlds, meaning ha ypical disances in he nework are small, and many of hem have power-law degree sequences, meaning ha he number of verices wih degree k falls off as k τ for some exponen τ > 1. See [18] for an example of hese phenomena in he Inerne, and [24, 25] for an example on he World-Wide Web. Also, Table 3.1 in [26] gives an overview of a large number of neworks and heir properies. Incied by hese empirical findings, random graphs have been proposed o model and/or explain hese phenomena see [14] for an inroducion o random graph models for complex neworks. Two paricular classes of models ha have been sudied from a mahemaical viewpoin are i graphs where he edge probabiliies depend on cerain weighs associaed wih he verices, see e.g. [7, 10, 11, 12, 28], and ii so-called preferenial aachmen models, see e.g. [2, 6, 8, 9, 13]. The firs class can be viewed as generalizaions of he classical Erdős-Rényi graph allowing for power-law degrees. In [10], for insance, a model is considered in which each verex i is assigned a random weigh W i and an edge is drawn beween wo verices i and j wih a probabiliy depending on W i W j. This model, which is referred o as he generalized random graph, leads o a graph where verex i has an asympoic degree disribuion equal o a Poisson random variable wih random parameer W i as he number of verices ends o infiniy, ha is, he asympoic degree of a verex is deermined by is weigh. We refer o [5, 22] for inroducions o classical random graphs. Preferenial aachmen models are differen in spiri in ha hey are dynamic, more precisely, a new verex is added o he graph a each ineger ime. Each new verex comes wih a number of edges aached o i and hese edges are conneced o he old verices in such a way ha verices wih high degree are more likely o be aached o. I can be shown ha his leads o graphs wih power-law degree sequences. Noe ha, in preferenial aachmen models, he degree of a verex increases over ime, implying ha he oldes verices end o have he larges degrees. Consider he degree of a verex as an indicaion of is success, so ha verices wih large degree correspond o successful verices. In preferenial aachmen models, verices wih large degrees are he mos likely verices o obain even larger degrees, ha is, successful verices are likely o become even more successful. In he lieraure his is someimes called he rich-ge-richer effec. In he generalized random graph, on he oher hand, a verex is born wih a cerain weigh and his weigh deermines he degree of he verex, as described above. This will be referred o as he rich-by-birh effec in wha follows. Naurally, in realiy, boh he rich-ge-richer and he rich-by-birh effec may play a role. To see his, consider for insance a social nework, where we idenify verices wih individuals and edges wih social links, ha is, an edge is added beween wo individuals if hey have some kind of social relaion wih each oher. Then, indeed, we would expec o see boh effecs: Firsly, he rich-ge-richer effec should be apparen, since individuals wih a high number of social links will in ime acquire more new social links han individuals wih few social links. Evidenly, more social conacs imply ha he individual is socially more acive, so ha he/she mees more people, and, in urn, each meeing offers a possibiliy o creae a lasing social link. Thus, hese individuals are more likely o ge acquained wih even more individuals. Secondly, he rich-by-birh effec comes in due o he fac ha some individuals are beer in urning a meeing ino a lasing social link han ohers. The social aciviy and skill varies from individual o individual and could be measured for insance by weighs associaed wih he individuals. Hence, in realiy, boh he previous success of a verex and an iniial weigh may play a role in he final success of he verex. Naurally, here are several ways o model how he weigh influences he success of a verex. In he model considered in his paper, individuals arrive in he nework wih a differen iniial number of conacs given o hem a birh and hese iniial numbers form he basis for heir fuure success. Laer on, we shall also discuss oher ways of how his effec can be modeled. 1

5 The aim of he curren paper is o formulae and analyze a model ha combines he rich-gericher and he rich-by-birh effec. The model is a preferenial aachmen model where he number of edges added upon he addiion of a new verex is a random variable associaed o he verex. This indeed gives he desired combinaion of preferenial aachmen and verex-dependency of degrees upon verex-birh. For bounded iniial degrees, he model is included in he very general class of preferenial aachmen models reaed in [13], bu he novely of he model lies in ha he iniial degrees can have an arbirary disribuion. In paricular, we can ake he weigh disribuion o be a power law, which gives a model wih wo compeing power laws: he power law caused by he preferenial aachmen mechanism and he power law of he iniial degrees. In such a siuaion i is indeed no clear which of he power laws will dominae in he resuling degrees of he graph. Our main resul implies ha he mos heavy-ailed power law wins, ha is, he degrees in he resuling graph will follow a power law wih he same exponen as he iniial degrees in case his is smaller han he exponen induced by he preferenial aachmen, and wih an exponen deermined by he preferenial aachmen in case his is smaller. The proof of our main resul requires finie momen of order 1 + ε for he iniial degrees. However, we believe ha he conclusion is rue also in he infinie mean case. More specifically, we conjecure ha, when he disribuion of he iniial degrees is a power law wih infinie mean, he degree sequence in he graph will obey a power law wih he same exponen as he he one of he iniial degrees. Indeed, he power law of he iniial degrees will always be he sronges in his case, since preferenial aachmen mechanisms only seem o be able o produce power laws wih finie mean. In realiy, power laws wih infinie mean are no uncommon, see e.g. Table 3.1 in [26] for some examples, and hence i is desirable o find a model ha can capure his. Unforunaely, we have no been able o give a full proof for he infinie mean case, bu we presen parial resuls in Secion 1.2. We now proceed wih a formal definiion of he model and he formulaion of he main resuls. 1.1 Definiion of he model The model ha we consider is described by a graph process {G} 1. To define i, le {W i } i 1 be an i.i.d. sequence of posiive ineger-valued random variables and le G1 be a graph consising of wo verices v 0 and v 1 wih W 1 edges joining hem. For 2, he graph G is consruced from G 1 in such a way ha a verex v, wih associaed weigh W, is added o he graph G 1, and he edge se is updaed by adding W edges beween he verex v and he verices {v 0, v 1,..., v 1 }. Thus, W is he iniial degree of verex v. Wrie d 0 s,..., d 1 s for he degrees of he verices {v 0, v 1,..., v 1 } a ime s 1. The endpoins of he W edges emanaing from verex v are chosen independenly wih replacemen from {v 0,..., v 1 }, and he probabiliy ha v i is chosen as he endpoin of a fixed edge is equal o d i 1 + δ 1 j=0 d j 1 + δ = d i 1 + δ, 0 i 1, 1.1 2L 1 + δ where L = i=1 W i, and δ is a fixed parameer of he model. Wrie S W for he suppor of he disribuion of he iniial degrees. To ensure ha he above expression defines a probabiliy, we require ha δ + min{x : x S W } > This model will be referred o as he PARID-model Preferenial Aachmen wih Random Iniial Degrees. Noe ha, when W i 1 and δ = 0, we rerieve he original preferenial aachmen model from Barabási-Alber [2]. Remark 1.1 In he PARID-model, we assume ha he differen edges of a verex are aached in an independen way, and we also ake a simple choice for he iniial graph G1. However, he proofs given below are raher insensiive o he precise model definiions, and can be applied o 2

6 slighly differen seings as well. For example, he proof can also be applied o he model used in [9], where W i m for some ineger m 1, and he degrees are updaed during he aachmen of he successive edges i.e., he m edges are no independen. Remark 1.2 We shall give special aenion o he case where PW i = m = 1 for some ineger m 1. This model is closes in spiri o he Barabási-Alber model, and i urns ou ha sharper bounds are possible for he error erms in his case. These resuls will be used in [15], where we sudy he diameer in preferenial aachmen models. 1.2 Heurisics and main resul Our main resul concerns he degree sequence in he resuling graph G. To formulae i, le N k be he number of verices wih degree k in G and define p k = N k / + 1 as he fracion of verices wih degree k. We are ineresed in he limiing disribuion of p k as. This disribuion arises as he soluion of a cerain recurrence relaion, of which we will now give a shor heurisic derivaion. Firs noe ha, obviously, E[N k G 1] = N k 1 + E[N k N k 1 G 1]. 1.3 Asympoically, for large, i is very unlikely ha a verex will be hi by more han one of he W edges added upon he addiion of verex v. Le us hence ignore his possibiliy for he momen. The difference N k N k 1 beween he number of verices wih degree k a ime and ime 1 respecively, is hen obained as follows: a Verices wih degree k in G 1 ha are hi by one of he W edges emanaing from v are subraced from N k 1. The condiional probabiliy ha a fixed edge is aached o a verex wih degree k is k + δn k 1/2L 1 + δ, so ha he expeced number of edges conneced o verices wih degree k is W k + δn k 1/2L 1 + δ. This coincides wih he mean number of verices wih degree k in G 1 hi by edges from v, apar from he case when wo edges are aached o he same verex. b Verices wih degree k 1 in G 1 ha are hi by one of he W edges emanaing from verex v are added o N k 1. By reasoning as above, i follows ha he mean number of such verices is W k 1 + δn k 1 1/2L 1 + δ. c The new verex v should be added o N k 1 if i has degree k, ha is, if W = k. Combining his gives E [N k N k 1 G 1] k 1 + δw 2L 1 + δ N k 1 1 k + δw 2L 1 + δ N k {W =k}, 1.4 where he approximaion sign refers o he fac ha we have ignored he possibiliy of wo or more edges of an arriving verex being conneced o he same end verex. Now assume ha p k converges o some limi p k as, so ha hence N k + 1p k. Also assume ha he disribuion of he iniial degrees has finie mean µ, so ha L 1 / µ. Finally, le {r k } be he probabiliies associaed wih he weigh disribuion, ha is, r k = PW 1 = k, k Subsiuing 1.4 ino 1.3 and replacing L 1 by µ, we arrive, afer aking double expecaions, a E [N k ] E[N k 1] + k 1 + δ E[N k 1 1] θ 3 k + δ E[N k 1] + r k, 1.6 θ

7 where θ = 2 + δ/µ. Sending, and observing ha E[N k ] E[N k 1] p k implies ha 1 E[N k] p k, for all k, hen yields he recursion p k = k 1 + δ θ By ieraion, i can be seen ha his recursion is solved by p k = θ k 1 k + δ + θ r k i i=0 j=1 p k 1 k + δ p k + r k. 1.7 θ i k j + δ, k 1, 1.8 k j + δ + θ where he empy produc, arising when i = 0, is defined o be equal o one. Furhermore, since {p k } saisfies 1.7, we have ha k=1 p k = k=1 r k = 1. Hence, {p k } defines a probabiliy disribuion, and he above reasoning indicaes ha he limiing degree disribuion in he PARIDmodel should be given by {p k }. Our main resul confirms his heurisics: Theorem 1.3 If he iniial degrees {W i } i 1 have finie momen of order 1 + ε for some ε > 0, hen here exiss a consan γ 0, 1 2 such ha lim P max p k p k γ = 0, where {p k } is defined in 1.8. When r m = 1 for some ineger m 1, hen γ can be replaced log by C for some sufficienly large consan C. To analyze he disribuion {p k }, firs consider he case when he iniial degrees are almos surely consan, ha is, when r m = 1 for some posiive ineger m. Then r j = 0 for all j m, and 1.8 reduces o { θγk+δγm+δ+θ p k = Γm+δΓk+1+δ+θ for k m; 0 for k < m, where Γ denoes he gamma-funcion. By Sirling s formula, we have ha Γs + a/γs s a as s, and from his i follows ha p k ck 1+θ for some consan c > 0. Hence, he degree sequence obeys a power law wih exponen 1 + θ = 3 + δ/m. Noe ha, by choosing δ > m appropriaely, any value of he exponen larger han 2 can be obained. For oher choices of {r k }, he behavior of {p k } is less ransparen. The following proposiion assers ha, if {r k } is a power law, hen {p k } is a power law as well. I also gives he aforemenioned characerizaion of he exponen as he minimum of he exponen of he r k s and an exponen induced by he preferenial aachmen mechanism. Proposiion 1.4 Assume ha r k = PW 1 = k = k τ WLk for some τ W > 2 and some funcion k Lk which is slowly varying. Then p k = k τ ˆLk for some slowly varying funcion k ˆLk and wih power-law exponen τ given by τ = min{τ W, τ P }, 1.9 where τ P is he power-law exponen of he pure preferenial aachmen model given by τ P = 3+δ/µ. When r k decays faser han a power law, hen 1.9 remains rue wih he convenion ha τ W =. In deriving he recursion 1.7 we assumed ha he iniial degrees {W i } i 1 have finie mean µ. Assume now ha he mean of he iniial degrees is infinie. More specifically, suppose ha {r k } is a power law wih exponen τ W [1, 2]. Then, we conjecure ha he main resul above remains rue. 4

8 Conjecure 1.5 When {r k } is a power law disribuion wih exponen τ W 1, 2, hen he degree sequence in PARID-model obeys a power law wih he same exponen τ W. Unforunaely, we canno quie prove Conjecure 1.5. However, we shall prove a slighly weaker version of i. To his end, wrie N k for he number of verices wih degree larger han or equal o k a ime, ha is, N k = i=0 1 {d i k}, and le p k = N k / + 1. Since d i W i, obviously E[p k ] = E[N k] + 1 E[ i=1 1 {W i k}] + 1 = PW 1 k + 1 = PW 1 k1 + o1, 1.10 ha is, he expeced degree sequence in he PARID-model is always bounded from below by he weigh disribuion. In order o prove a relaed upper bound, we sar by invesigaing he expecaion of he degrees. Theorem 1.6 Suppose ha k>x r k = PW 1 > x = x 1 τ WLx, where τ W 1, 2 and x Lx is a slowly varying funcion a infiniy. Then, for every s < τ W 1, here exiss a consan C > 0 and a slowly varying funcion x lx such ha, for i {0,..., }, we have ha where x y = max{x, y}. E[d i s s/τw 1 l s, ] C i 1 li Theorem 1.6 gives an upper bound for he expeced degree sequence: Corollary 1.7 If k>x r k = PW 1 > x = x 1 τ WLx, where τ W 1, 2 and x Lx is a slowly varying funcion a infiniy, hen, for every s < τ W 1, here exiss an M independen of such ha E[p k ] Mk s. Proof. For s < τ W 1, i follows from Theorem 1.6 and Markov s inequaliy ha E[p k ] = i=0 Pd i k = Pd i s k s i=0 k s E[d i s ] k s C + 1 i=0 i=0 s/τw 1 l s Mk s, 1.11 i 1 li since, for s < τ 1 and using [17, Theorem 2, p. 283], here exiss a consan c > 0 such ha i 1 s/τw 1 li s = c 1 s τ W 1 l s 1 + o1. i=0 Combining Corollary 1.7 wih 1.10 yields ha, when he weigh disribuion is a power law wih exponen τ W 1, 2, he only possible power law for he degrees has exponen equal o τ W. This saemen is obviously no as srong as Theorem 1.3, bu i does offer convincing evidence for Conjecure 1.5. Theorem 1.6 is proved in Secion 3. 5

9 1.3 Relaed work Before proceeding wih he proofs, we describe some relaed work. In Secion 2.5, he proof of Theorem 1.3 is compared o relaed proofs ha have appeared in he lieraure, and we refer here for he exensive lieraure on power laws for preferenial aachmen models. In his secion, we describe work on relaed models. As menioned in he inroducion, he paper by Cooper and Frieze [13] deals wih a very general class of preferenial aachmen models, including he PARID-model wih bounded iniial degrees. Anoher way of inroducing he rich-by-birh effec in a preferenial aachmen model, is he finess model, formulaed by Barabási and Bianconi [3, 4], and laer generalized by Ergün and Rodgers [16]. We will shorly describe he model of Ergün and Rodgers and some non-rigorous resuls for he degree sequence. The idea wih he model is ha verices have differen abiliy referred o as finess o compee for edges. More precisely, each verex has wo ypes of finess, a muliplicaive and an addiive finess associaed o i. These are given by independen copies of random variables η and ζ, respecively. The dynamics of he model is hen very similar o he dynamics of he PARID-model. However, insead of adding a random number of edges ogeher wih each new verex, new verices come wih a fixed number m of edges. Also, insead of connecing an edge o a given verex wih a probabiliy proporional o he degree plus δ, he probabiliy of connecing o a given verex is proporional o he degree imes he muliplicaive finess plus he addiive finess. Thus, he expression 1.1 for he probabiliy of choosing v i 0 i 1 as he endpoin of a fixed edge is replaced by η i d i 1 + ζ i 1 j=0 η jd j j=0 ζ, 0 i j The original finess model of Barabási and Bianconi is obained when ζ 0. If, in addiion, he muliplicaive finess is he same for all verices, he model reduces furher o he Alber-Barabási model. The rich-by-birh effec is presen since relaively young verices, wih a small degree, can acquire edges a a high rae if he muliplicaive finess or he addiive finess is large. Therefore, his model is someimes called he fi-ge-rich model. Excluding rivial choices for he disribuion of η and ζ, i is no clear before hand if he graph process is driven by he rich-ge-richer effec, by he rich-by-birh effec or by a combinaion of hem. When he addiive finess is zero, Barabási and Bianconi [4] show ha he disribuion of he average degree sequence of G depends on he disribuion of η. For η uniformly disribued on [0, 1], hey show non-rigorously ha he degree sequence {p k } is given by p k c k 1+C log k, where C is he soluion of he equaion exp 2/C = 1 1/C and c > 0 a consan, ha is, he average degree sequence follows a generalized power law. When η is exponenially disribued, numerical simulaions indicae ha he degree sequence also behaves like an exponenial disribuion. For he general model wih non-zero addiive finess here are no explici expressions for he p k s. See however [16] for some special cases. We menion also ha [3] provides a coupling of he finess model o a so-called Bose gas. This coupling gives a way of predicing non-rigorously wheher he rich-ge-richer or he rich-by-birh effec will be dominan. 2 Proof of Theorem 1.3 and Proposiion 1.4 In his secion, we prove Theorem 1.3 and Proposiion 1.4. We sar by proving Proposiion 1.4, since he proof of Theorem 1.3 makes use of i. 6

10 2.1 Proof of Proposiion 1.4 Recall he definiion 1.8 of p k. Assume ha {r k } is a power law disribuion wih exponen τ W > 2, ha is, assume ha r k = Lkk τ W, for some slowly varying funcion k Lk. We wan o show ha hen p k is a power law disribuion as well, more precisely, we wan o show ha p k = ˆLkk τ, where τ = min{τ W, 1 + θ} and k ˆLk is again a slowly varying funcion. To his end, firs noe ha he expression for p k can be rewrien in erms of gamma-funcions as p k = By Sirling s formula, we have ha and θ Γk + δ Γk + δ θ k m=1 Γm + δ + θ r m. 2.1 Γm + δ Γk + δ Γk + δ θ = k 1+θ 1 + O k 1, k, 2.2 Γm + δ + θ Γm + δ Furhermore, by he assumpion, r m = Lmm τ W. I follows ha = m θ 1 + O m 1, m. 2.3 k m=1 Γm + δ + θ r m 2.4 Γm + δ is convergen as k if θ τ W < 1, ha is, if τ W > 1+θ. For such values of τ W, he disribuion p k is hence a power law wih exponen τ P = 1 + θ. When θ τ W 1, ha is, when τ W τ P, he series in 2.4 diverges and, by [17, Lemma, p. 280], i can be seen ha k k m=1 Γm + δ + θ r m Γm + δ varies regularly wih exponen θ τ W + 1. Combining his wih 2.2 yields ha p k compare 2.1 varies regularly wih exponen τ W, as desired. 2.2 Proof of Theorem 1.3 The proof of Theorem 1.3 consiss of wo pars: in he firs par, we prove ha he degree sequence is concenraed around is mean, and in he second par, he mean degree sequence is idenified. We formulae hese resuls in wo separae proposiions Proposiion 2.1 and 2.2 which are proved in Secion 2.3 and 2.4 respecively. The resul on he concenraion of he degree sequence is as follows: Proposiion 2.1 If he iniial degrees {W i } i 1 in he PARID-model have finie momen of order 1 + ε, for some ε > 0, hen here exiss a consan α 1 2, 1 such ha lim P N k E[N k ] α = 0. max When r m = 1 for some m 1, hen α can be replaced by C log for some sufficienly large C. Idenical concenraion esimaes hold for N k. As for he idenificaion of he mean degree sequence, he following proposiion says ha he expeced number of verices wih degree k is close o + 1p k for large. More precisely, i assers ha he difference beween E[N k ] and + 1p k is bounded, uniformly in k, by a consan imes β, for some β [0, 1. 7

11 Proposiion 2.2 Assume ha he iniial degrees {W i } i 1 in he PARID-model have finie momen of order 1 + ε for some ε > 0, and le {p k } be defined as in 1.8. Then here exis consans c > 0 and β [0, 1 such ha max E[N k] + 1p k c β. 2.5 When r m = 1 for some m 1, hen he above esimae holds wih β = 0. Wih Proposiions 2.1 and 2.2 a hand i is no hard o prove Theorem 1.3: Proof of Theorem 1.3: Combining 2.5 wih he riangle inequaliy, i follows ha P Nk + 1p k c β + α P Nk E[N k ] α. max By Proposiion 2.1, he righ side ends o 0 as and hence, since p k = N k / + 1, we have ha lim P max p k p k cβ + α = The heorem follows from his by picking 0 < γ < 1 max{α, β}. Noe ha, since 0 β < 1 and 1 2 < α < 1, we have 0 < γ < 1 2. The proof for r m = 1 is analogous. 2.3 Proof of Proposiion 2.1 This proof is an adapion of a maringale argumen, which firs appeared in [9], and has been used for all proofs of power-law degree sequences since. The idea is o express he difference N k E[N k ] in erms of a Doob maringale. Afer bounding he maringale differences, which are bounded in erms of he random number of edges {W i } i 1, he Azuma-Hoeffding inequaliy can be applied o conclude ha he probabiliy of observing large deviaions is suiably small, a leas when he iniial number of edges has bounded suppor. When he iniial degrees {W i } i 1 are unbounded, and exra coupling sep is required. The argumen for N k is idenical, so we focus on N k. We sar by giving an argumen when W i a for all i and some a 0, 1 2. Firs noe ha max N k 1 k ln l 1 k l=k l=1 ln l = L k. 2.6 Thus, E[N k ] µ/k. For α 1 2, 1, le η > 0 be such ha η + α > 1 he choice of α will be specified in more deail below. Then, for any k > η, he even N k E[N k ] α implies ha N k α, and hence ha L kn k > η+α. I follows from Boole s inequaliy ha P max N k E[N k ] α η k=1 P N k E[N k ] α + PL > η+α. Since η + α > 1 and L / µ, he even L > η+α has small probabiliy. To esimae he probabiliy P N k E[N k ] α, inroduce M n = E[N k Gn], n = 0,...,, where G0 is defined as he empy graph. Since E[M n ] <, he process is a Doob maringale wih respec o {Gn} n=0. Furhermore, we have ha M = N k and M 0 = E[N k ], so ha N k E[N k ] = M M 0. 8

12 Also, condiionally on he iniial degrees {W i } i=1, he incremens saisfy M n M n 1 2W n. To see his, noe ha he addiional informaion conained in Gn compared o Gn 1 consiss in how he W n edges emanaing from v n are aached. This can affec he degrees of a mos 2W n verices. By he assumpion ha W i a for all i = 1,...,, we obain ha M n M n 1 2 a. Combining all his, i follows from he Azuma-Hoeffding inequaliy see e.g. [19, Secion 12.2] ha P N k E[N k ] α { 2 exp 2α } 8 = 2 exp { 2α 1 2a /8 }, i=1 2a so ha we end up wih he esimae P max N k E[N k ] α 2 η exp { 2α 1 2a /8 } + PL > η+α. 2.7 Since a < 1/2, he above exponenial ends o 0 for α < 1 saisfying ha α > a + 1/2. When he iniial degrees are bounded, he above argumen can be adaped o yield ha he probabiliy ha max N k E[N k ] exceeds C log is o1 for some C > 0 sufficienly large. We omi he deails of his argumen. We conclude ha we have proved he saemen for graphs saisfying ha W i a for some a 0, 1/2 and all i = 1,...,. Naurally, his assumpion may no be rue. When he iniial degrees are bounded, he assumpion is rue, even wih a replaced by m, bu we are ineresed in graphs having iniial degrees wih finie 1 + ε-momens. We nex exend he proof o his seing by a coupling argumen. Wrie W i = W i a, L s = s i=1 W i, 2.8 where x y = min{x, y}. Then, he above argumen shows ha he PARID-model wih iniial degrees {W i } i=1 saisfies he claim in Proposiion 2.1. Denoe he graph process wih iniial degrees {W i } i=1 by {G i} i=1 and, for i s, he degree of verex i in G s by d i s. We now presen a coupling beween {Gi} i=1 and {G i} i=1 which is such ha, wih high probabiliy, he number of edges ha differ is bounded by b for some b 0, 1. Define he aachmen probabiliies in {Gi} i=1 and {G i} i=1 by p i s = d is 1 + δ 2L s 1 + δs, p is = d i s 1 + δ δs 2L s 1 Now, we couple he edge aachmens such ha he l h edge of verex s in boh graphs is aached o i wih probabiliy p i s p i s. Oherwise, he edge is miscoupled. We shall give a bound on he expeced number of miscouplings. The number of miscouplings in Gs and G s is denoed by U s, and is defined in more deail as follows. We define U 0 = 0 and explain recursively how o consruc U s from U s 1. The number of miscouplings is adjused afer each edge which is conneced. We consider he edges o be direced, and call a direced edge poining owards i an in-edge for i, and a direced edge poining away from i an ou-edge for i. For convenience laer on, we regard an edge from s o i as boh an in-edge for i as well as an ou-edge for s. By he above definiions, he number of in-edges of i a ime s is he in-degree of i a ime s, and he number of ou-edges of i a ime s is he ou-degree of i a ime s. If we denoe he in- and ou-degrees of verex i in Gs by d i,in s and d i,ou s, hen clearly d i s = d i,in s + d i,ou s. The same holds for he in- and ou-degrees d i,in s and d i,ou s of verex i in G s. The edges which are aached from verex s are numbered 1,..., W s. When W s > a, hen an edge wih a number beween a and W s adds 2 o U s 1, and we call boh he in-edge of i and he ou-edge of s as belonging o he miscoupled se. The size of he miscoupled se a any ime s = 0,..., will be equal o U s. 9

13 When he edge number is in beween 1 and W s, hen we add 1 o U s 1 precisely when he edge is conneced differenly for Gs and G s. In his case, we say ha he in-edge of i belongs o he miscoupled se, bu he ou-edge of s does no. The miscoupled se remains unchanged when an edge is aached in he same way in Gs and in G s. We nex define he weigh of every in- and ou-edge of verex i a ime s o be equal o 1. The oal weigh of a verex i in Gs a ime s is he sum of weighs of he in- and ou-edges of i in Gs plus δ. The oal weigh of a verex i in G s is defined in a similar manner. The probabiliies in 2.9 are precisely proporional o he oal weigh of he verex i a ime s 1. As a resul, for an ou-edge of verex s wih number in beween 1 and W s, a miscoupling occurs wih probabiliy equal o U s 1 /TW s 1, where TW s is he oal weigh of all verices a ime s i.e., all weighs in Gs and G s combined plus δs. To see his, we can choose an edge wih probabiliy equal o he oal weigh of he end verex of he edge divided by TW s. If his edge is no in he miscoupled se, hen we are done, and he wo direced edges in Gs and G s are equal o an in-edge in he verex which is conneced o he chosen direced edge, and an ou-edge from he verex s. If he chosen direced edge is in he miscoupled se, hen i is an edge eiher for Gs 1 or for G s 1, bu no for boh, and i is chosen wih he correc condiional probabiliy. The above rule hen consrucs he in- and ou-edges corresponding o he edge we wish o aach. Say he chosen edge is an edge for Gs 1, hen we choose he edge for G s 1 from he edges of G s 1 wih probabiliy equal o p i s 1. As a resul, we do no creae any miscoupling when he iniial edge drawn was no in he miscoupled se. The probabiliy of a miscoupling a ime s is herefore equal o U s 1 /TW s 1. Noe ha TW s 2L s + δs + 1. The following lemma bounds he expeced value of U : Lemma 2.3 There exiss a consans K > 0 and b 0, 1 such ha E[U ] K b Proof of Lemma 2.3: We prove Lemma 2.3 by inducion. The inducion hypohesis is ha, for all 0 s, s E[U s ] K b The bound in 2.10 follows from he one in 2.11 by subsiuing s =. We now prove For s = 0, we have U 0 = 0, which iniializes he inducion hypohesis. To advance he inducion hypohesis, we noe ha U s is equal o U s 1 + 2W s W s + R s, where R s is he number of ou-edges for s wih number in beween 1 and W s ha are miscoupled. As a resul, we have E[U s ] = E[U s 1 ] + 2E[W s W s] + E[R s ] By he fac ha for each ou-edge for s wih number in beween 1 and W s, a miscoupling occurs wih probabiliy equal o U s 1 /TW s 1, we have ha [ ] [ E[R s ] = E E[R s W s ] = E W s U ] [ s 1 = E[W Us 1 ] TW s]e, 2.13 s 1 TW s 1 where he las equaliy follows from he independence of W s and U s 1, TW s 1. Now, we use ha TW s 1 2L s 1 + δs 1, ogeher wih he fac ha L s is concenraed around is mean, o conclude ha L s 1 µ εs 1 wih high probabiliy. Thus, [ ] E[R s ] = E E[R s W s ] Using he inducion hypohesis, we arrive a E[R s ] µ s 12µ + δ 2ε E[U s 1] + µp L s 1 µ εs µ s 12µ + δ 2ε K s 1 b + µpl s 1 µ εs 1 K b 1 µ 2µ + δ 2ε + µp L s 1 µ εs

14 We furher bound E[W s W s] = E[W s a 1 {Ws > a }] aε E[W 1+ε s ] C aε Therefore, by aking b 1 = aε, we ge ha E[U s ] E[U s 1 ] + 2E[W s W s] + E[R s ] Ks 1 b 1 + 2C b 1 + K b 1 µ = K b 1 {s 1 + 2C/K + Ks b 1, 2µ + δ 2ε + µpl s 1 µ εs 1 } µ + µpl s 1 µ εs 1 2µ + δ 2ε 2.17 by noing ha P L s µ εs is exponenially small in s, for s, and using ha, since 2µ + δ > µ, we can ake ε > 0 so small ha µ/2µ + δ 2ε < 1, and, afer his, we can ake K so large ha µ 2µ + δ 2ε + 2C K < 1. Wih hese choices, we have advanced he inducion hypohesis. We now complee he proof of Proposiion 2.1. The Azuma-Hoefding argumen proves ha N k, he number of verices wih degree k in G, saisfies he bound in Proposiion 2.1, i.e., ha recall 2.7 P max N k E[ N k ] α 2 η exp { 2α 1 2a /8 } + PL > η+α, 2.18 for α 1 2, 1 and η > 0 such ha α + η > 1 and a 0, 1 2. Moreover, we have for every k 1, ha N k N k U, 2.19 since every miscoupling can change he degree of a mos one verex. By 2.19 and 2.10, here is a b 0, 1 such ha E[Nk ] E[N k ] E[U ] K b Also, by he Markov inequaliy, 2.19 and 2.10, for every α b, 1, we have ha P max N k N k > α P U > α α E[U ] = o Now fix α b a + 1 2, 1, where x y = max{x, y}, and decompose max Nk E[N k ] max N k E[N k ] + max E[Nk ] E[N k ] + max Nk N k The firs erm on he righ hand side is bounded by α wih high probabiliy by 2.18, he second erm is, for sufficienly large and wih probabiliy one, bounded by α by 2.20 while he hird erm is bounded by α wih high probabiliy by This complees he proof. 2.4 Proof of Proposiion 2.2 For k 1, le N k = E[N k {W i } i=1]

15 denoe he expeced number of verices wih degree k a ime given he iniial degrees W 1,..., W, and define ε k = N k + 1p k, k Also, for Q = {Q k } a sequence of real numbers, define he supremum norm of Q as Q = sup Q k. Using his noaion, since E[ N k ] = E[N k ], we have o show ha here are consans c > 0 and β [0, 1 such ha E[ε] = sup E[ N k ] + 1p k c β, for = 0, 1,..., 2.25 where ε = {ε k } k=1. The plan o do his is o formulae a recursion for ε, and hen use inducion in o esablish The recursion for ε is obained by combining a recursion for N = { N k }, ha will be derived below, and he recursion for p k in 1.7. The hard work hen is o bound he error erms in his recursion; see Lemma 2.4 below. Le us sar by deriving a recursion for N. To his end, for a real-valued sequence Q = {Q k } k 0, wih Q 0 = 0, inroduce he operaor T, defined as compare o 1.6 T Q k = 1 k + δ Q k + k 1 + δ 2L 1 + δ 2L 1 + δ Q k 1, k When applied o N 1, he operaor T describes he effec of he addiion of a single edge emanaing from he verex v, he verex v iself being excluded from he degree sequence. Indeed, here are on he average N k 1 1 verices wih degree k 1 a ime 1 and a new edge is conneced o such a verex wih probabiliy k 1 + δ/2l 1 + δ. Afer his connecion is made, he verex will have degree k. Similarly, here are on he average N k 1 verices wih degree k a ime 1. Such a verex is hi by a new edge wih probabiliy k + δ/2l 1 + δ, and will hen have degree k + 1. The expeced number of verices wih degree k afer he addiion of one edge is hence given by he operaor in 2.26 applied o N. Wrie T n for he n-fold applicaion of T, and define T = T W. Then T describes he change in he expeced degree sequence N when all he W edges emanaing from verex v have been conneced, ignoring verex v iself. Hence, N saisfies N k = T N 1 k + 1 {W=k}, k Inroduce a second operaor S on sequences of real numbers Q = {Q k } k 0, wih Q 0 = 0, by compare o 1.7 SQ k = k 1 + δ Q k 1 k + δ Q k, θ θ k 1, 2.28 where θ = 2 + δ/µ and µ is he expecaion of W 1. The recursion 1.7 is given by p k = Sp k + r k, wih iniial condiion p 0 = 0. I is solved by p = {p k }, as defined in 1.8. Observe ha + 1p k = p k + Sp k + r k = T p k + r k κ k, k 1, 2.29 where κ k = T p k Sp k p k Combining 2.24, 2.27 and 2.29, and using he lineariy of T, i follows ha ε = {ε k } saisfies he recursion ε k = T ε 1 k + 1 {W =k} r k + κ k, 2.31 indeed, ε k = N k + 1p k = T N 1 k + 1 {W=k} T p k r k + κ k = T ε 1 k + 1 {W =k} r k + κ k. 12

16 Now we define k = η, where η µ, 2µ + δ. Since, by 1.2, δ > min{x : x S W } µ, he inerval µ, 2µ+δ. Also, by he law of large numbers, L k, as, wih high probabiliy. Furher, we define ε k = ε k 1 {k k } and noe ha, for k k, he sequence { ε k } saisfies ε k = 1 {k k }T ε 1 k + 1 {W =k} r k + κ k, 2.32 where κ k = κ k 1 {k k }. I follows from E [ 1 {W =k}] = rk and he riangle inequaliy ha E[ε] E[ε ε] + E[ ε] E[ε ε] + E [ 1,k ] T ε 1 ] + E[ κ], 2.33 where 1,k ]k = 1 {k k }. Inequaliy 2.33 is he key ingredien in he proof of Proposiion 2.2. We will derive he following bounds for he erms in Lemma 2.4 There are consans C ε, C ε 1, C ε 2 and C κ, independen of, such ha for sufficienly large and some β [0, 1, a E[ε ε] C ε 1 β, b E [ 1,k ] T ε 1 ] c E[ κ] C κ 1 β. 1 C1 ε E[ε 1] + C2 ε, 1 β When r m = 1 for some ineger m 1, hen he above bounds hold wih β = 0. Given hese bounds, Proposiion 2.2 is easily esablished. Proof of Proposiion 2.2: Recall ha we wan o esablish We shall prove his by inducion on. Fix 0 N. We sar by verifying he inducion hypohesis for 0, hus iniializing he inducion hypohesis. For any 0, we have E[ε] sup E [ Nk ] sup p k , 2.34 since here are precisely verices a ime 0 and p k 1. This iniializes he inducion hypohesis, when c is so large ha c β 0. Nex, we advance he inducion hypohesis. Assume ha 2.25 holds a ime 1 and apply Lemma 2.4 o 2.33 o ge ha E[ε] E[ε ε] + E [ 1,k ] T ε 1 ] + E[ κ] 1 C ε 1 β + C ε 1 c 1 β + C ε 2 C κ + 1 β 1 β c β c C 1 ε C 2 ε + C ε + C κ 1 β, as long as 1 C1 ε 0, which is equivalen o C ε 1. If we hen choose c large so ha c C ε 1 C ε 2 + C ε + C κ and c β 0 recall 2.34 and 0 C ε 1, hen we have ha E[ε] c β, and 2.25 follows by inducion in. I remains o prove Lemma 2.4. We shall prove Lemma 2.4 a-c one by one, saring wih a. 13

17 Proof of Lemma 2.4a: of ε, we ge ha We have E[ε ε] E[ ε ε ], and, using he definiion ε ε = sup k>k Nk + 1p k sup k>k Nk sup k>k p k. The maximal possible degree of a verex a ime is L, implying ha sup k>k Nk = 0, when L k. The laer is rue almos surely when r m = 1 for some ineger m, when is sufficienly large, since for large L = m η = k, where η m, 2m + δ, by he fac ha µ = m and δ > m. On he oher hand, by 2.6, wih N k replaced by N k we find N k L k for k k, and we obain ha E[sup Nk ] k 1 E[L 1 {L >k }] k>k Wih k = η for some η µ, 2µ + δ, we have ha E[L 1 {L >k }] k ε E[L 1+ε 1 {L >k }] k ε E[ L µ 1+ε ] + µ 1+ε k ε PL > k, 2.36 and, by he Markov inequaliy PL > k P L µ 1+ε > k µ 1+ε k µ 1+ε E[ L µ 1+ε ]. Combining he wo laer resuls, we obain µ 1+ε E[L 1 {L >k }] k ε 1 + E[ L µ 1+ε ] η µ To bound he las expecaion, we will use a consequence of he Marcinkiewicz-Zygmund inequaliy, see e.g [20, Corollary 8.2 in 3], which runs as follows. Le q [1, 2], and suppose ha {X i } i 1 is an i.i.d. sequence wih E[X 1 ] = 0 and E[ X 1 q ] <. Then here exiss a consan c q depending only on q, such ha [ q] E X i c q E[ X q 1 ] i=1 Applying 2.38 wih q = 1 + ε, we obain E[sup k>k Nk ] k 1+ε µ 1+ε 1 + E[ L µ 1+ε ] c 1+ε ε η µ Furhermore, since by Proposiion 1.4, we have p k ck γ, for some γ > 2 see also 1.9, we have ha sup k>k p k c γ for some consan c. I follows ha + 1 sup k>k p k C p γ 1, and, since γ > 2, par a is esablished wih C ε = c 1+ε + C p, and 1 β = ε γ 1. Proof of Lemma 2.4b: Moving on o b, we will sar by showing ha for sufficienly large, E [ 1,k] T ε 1 ] 1 C 1 ε E [ 1,k] ε 1 ] + C 3 ε, β which is b when we condiion on W = 1. We shall exend he proof o he case where W 1 a a laer sage. To prove 2.40, we shall prove a relaed bound, which also proves useful in he 14

18 exension o W 1. Indeed, we shall prove, for any real-valued sequence Q = {Q k } k 0, saisfying i Q 0 = 0 and ii sup k + δ Q k C Q L 1, 2.41 ha here exiss a β 0, 1 independen of Q and a consan c > 0 such ha for sufficienly large, E [ 1,k] T Q ] 1 C ε 1 E [ 1 ],k] Q + cc Q β Here we sress ha Q can be random, for example, we shall apply 2.42 o ε 1 in order o derive In order o prove 2.42, we recall ha E[T Q k ] = E [ 1 k + δ 2L 1 + δ Q k + k 1 + δ ] 2L 1 + δ Q k 1, k In bounding his expecaion we will encouner a problem in ha Q k, which is allowed o be random, and L 1 are no independen for example when Q = ε 1. To ge around his, we add and subrac he expression on he righ hand side bu wih he random quaniies replaced by heir expecaions, ha is, for k 1, we wrie k + δ E[T Q k ] = 1 E[Q k ] + k 1 + δ 2µ 1 + δ 2µ 1 + δ E[Q k 1] 2.44 [ ] 2L 1 2µ 1 + k + δe Q k L 1 + δ2µ 1 + δ [ ] 2µ 1 2L 1 + k + δ 1E Q k L 1 + δ2µ 1 + δ Noe ha, when r m = 1 for some ineger m 1, hen L = µ = m. Hence he erms in 2.45, 2.46 are boh equal o zero, and only 2.44 conribues. We firs deal wih Observe ha k k = η, wih η µ, 2µ + δ, implies ha k 2µ + δ 1 for sufficienly large, and hence I follows ha, for sufficienly large, sup k + δ 1 k k 2µ 1 + δ 1 1 2µ 1 + δ 1 k + δ 2µ 1 + δ E[Q k ] + k 1 + δ 2µ 1 + δ E[Q k 1] 2.48 E [ 1 ],k] Q 1 C ε 1 [ ] E 1,k] Q, for some consan C ε 1. This proves 2.42 wih C Q = 0 when he number of edges is a.s. consan since 2.45, 2.46 are zero. I remains o bound he erms 2.45, 2.46 in he case where he number of edges is no a.s. consan. We will prove ha he supremum over k of he absolue values of boh hese erms are bounded by consans divided by 1 β for some β [0, 1. Saring wih 2.45, by using he assumpion ii in 2.41, as well as 2L 1 + δ L 1 for sufficienly large, i follows ha [ ] sup k + δe 2L 1 2µ 1 Q k cc Q E[ L 1 µ 1 ]. 2L 1 + δ2µ 1 + δ To bound he laer expecaion, we combine 2.38 for q = 1+ε, wih Hölders inequaliy, o obain E[ L µ ] E [ L µ 1+ε] 1/1+ε [ c 1+ε E W 1 µ 1+ε] 1/1+ε c 1/1+ε,

19 since W i have finie momen of order 1 + ε by assumpion, where, wihou loss of generaliy, we can assume ha ε 1. Hence, we have shown ha he supremum over k of he absolue value of 2.45 is bounded from above by a consan divided by 1 β, where β = 1/1 + ε. Tha he same is rue for he erm 2.46 can be seen analogously. This complees he proof of To prove 2.40, we noe ha, by convenion, ε 0 1 = 0, so ha we only need o prove ha sup k + δ ε k 1 cl 1. For his, noe from 2.6, he bound p k ck γ, γ > 2, and from he lower bound L ha sup k + δ ε k 1 + δ ε k 1 k k + δ N k δ p k k L 1 + δ 1 + k + δ p k cl 1, 2.50 for some consan c. This complees he proof of To complee he proof of Lemma 2.4b, we firs show ha 2.42 implies, for every 1 n, and all k 1, E [ 1 {k k } T n ε 1 k ] 1 C ε 1 E [ 1,k ] ε 1 ] + 3 nc ε β To see 2.51, we use inducion on n. We noe ha 2.51 for n = 1 is precisely equal o 2.40, and his iniializes he inducion hypohesis. To advance he inducion hypohesis, we noe ha 1 {k k } T n ε 1 k = 1 {k k }T Qn 1 k, 2.52 where Q k n 1 = 1 {k k } T n 1 ε 1. We wish o use 2.42, and we firs check he assumpions i-ii. By definiion, Q 0 n 1 = 0, which esablishes i. For assumpion ii, we need o do k some more work. According o 2.26, and using ha 2L 1 + δ > L 1 1, for sufficienly large, and hence, by inducion, k + δ T Q k k=1 k + δ T n 1 Q k k= k + δ Q k, k= n 1 k + δ Q k. k=1 Subsiuing Q k = ε k 1 and using ε k 1 N k 1 + p k, yields k + δ T n 1 N 1 k + k + δ T n 1 p k k k k k n 1 k + δ N k 1 + k= n 1 k + δ p k n 1 cl 1, 2.53 according o Using he inequaliy 1 +x e x, x 0, ogeher wih n, his in urn yields, sup k + δ Q k n 1 exp1cl 1, 2.54 k=1 16

20 which implies assumpion ii. By he inducion hypohesis, we have ha, for k k, E[Q k n 1] 1 C ε 1 E [ 1,k] ε 1 ] + so ha we obain, from 2.42, wih Q = 1,k ] T ε 1, E [ 1 {k k} T n ε 1 k ] 1 C ε 1 E [ 1,k] ε 1 ] + which advances he inducion hypohesis when C 3 ε > cc Q. By 2.56, we obain ha, for W, n 1C 3 ε 1 β, 2.55 n 1C 3 ε + cc Q 1 β, 2.56 E [ 1 {k k} T ε 1 ] W k = 1 C ε 1 1 C 1 ε E[ε 1 W ] + W C 3 ε 1 β E[ε 1] + W C 3 ε 1 β, where we use ha ε 1 is independen of W. In he case ha W >, we bound, similarly as in 2.50, sup T ε 1 k cl, 2.57 k k so ha E [ 1 {k k} T ε 1 ] W k 1 C ε 1 E[ε 1] + W C ε 3 1 β The bound in b follows from his by aking expecaions on boh sides, using E[L 1 {W >}] = µ 1PW > + E[W 1 {W >}] + ce[l 1 {W>} W ] µ ε + 1 ε E[W 1+ε ], 2.59 afer which we use ha β = 1/1 + ε 1 ε and choose he consans appropriaely. complees he proof of Lemma 2.4b. This Proof of Lemma 2.4c: For par c of he lemma, recall ha κ k = κ k 1 {k k} wih κ k = T Ip k Sp k, 2.60 where T is defined in 2.26, T = T W, S is defined in 2.28, and where I denoes he ideniy operaor. In wha follows, we will assume ha k k, so ha κ k = κ k. We sar by proving a rivial bound on κ k. By 2.31, we have ha κ k = ε k T ε 1 k 1 {W =k} + r k, 2.61 where sup ε k cl by 2.50 and sup 1 k k T ε 1 k cl by 2.57, so ha hence sup κ k C η L k k recall ha k = η where η µ, 2µ + δ. For x [0, 1] and w N, we denoe f k x; w = I + xt I w p k. 17

21 Then κ k = κ k ; W, where κ k ; w = [f k 1; w f k 0; w] Sp k, 2.63 and x f k x; w is a polynomial in x of degree w. By a Taylor expansion around x = 0, f k 1; w = p k + w T Ip k f k x k; w, 2.64 for some x k 0, 1, and, since I + xt I and T I commue, f k I x; w = ww 1 + xt I w 2 T I 2 p We nex claim ha, on he even {k 2L 1 + 1δ}, sup I + xt IQ k sup Q k. k k k k Indeed, I + xt I = 1 xi + xt, so he claim follows when sup k k T Q k sup k k Q k. The laer is he case, since, on he even ha k + δ 2L 1 + δ, and arguing as in 2.48, we have sup T Q k sup k k k k [ 1 1 k + δ 2L 1 + δ 1 2L 1 + δ k. Q k + k 1 + δ ] 2L 1 + δ Q k 1 sup k k Q k. Since k k, he inequaliy k + δ 2L 1 + δ follows when k 2L 1 + 1δ. As a resul, on he even {k 2L 1 + 1δ}, we have ha max sup f k x; w ww 1 sup T I 2 p x [0,1] k k k k k Now recall he definiion 2.28 of he operaor S, and noe ha, for any sequence Q = {Q k } k=1, we can wrie θ T IQ k = 2L 1 + δ SQ k = 1 µ SQ k + R Q k, 2.66 where he remainder operaor R is defined as k + δ R Q k = 2µ + δ k + δ 2L 1 + δ Q k + k 1 + δ 2L 1 + δ k 1 + δ Q k µ + δ Combining 2.63, 2.64, 2.65 and 2.66, on he even {k 2L 1 + 1δ} and uniformly for k k, we obain ha w κ k ; w µ 1 Sp k + w sup R p k + 1 ww 1 sup T I 2 p, 2.68 k k 2 k k k ogeher wih a similar lower bound wih minus signs in fron of he las wo erms. Indeed, κ k ; w = [f k 1; w f k 0; w] Sp k and 2.68 follows from his ideniy and = w T Ip k + 1 f k 2 x k; w Sp k = w µ Sp k + wrp k Sp k + 1 f k 2 x k; w, 18

22 Wih 2.68 a hand, we are now ready o complee he proof of c. We sar by reaing he case where r m = 1 for some ineger m 1. In his case, wih w = W = m = µ, we have ha w µ 1Sp k 0. Furhermore, he inequaliy k 2L 1 + 1δ is rue almos surely when is sufficienly large. Hence, we are done if we can bound he las wo erms in 2.68 wih w = W. To do his, noe ha, by he definiion 2.26 of T and he fac ha 2L 1 + δ k = η, wih η > µ, sup T IQ k 2 η sup Applying 2.69 wice yields ha T I 2 p k 4 η 2 2 sup k + δ 2 p k, k + δ Q k and hence, since by Proposiion 1.4, p k ck γ for some γ > 2, here is a consan C p such ha Finally, since L = m, we have ha R p k 2 m 1 sup supk + δ 2 p k C p k + δ p k 2 C p m 1. Summarizing, we arrive a he saemen ha here exiss c m,δ such ha sup κ k ; m c m,δ, k k which proves he claim in c when r m = 1, and wih β = 0. We now move o random iniial degrees. For any a 0, 1, we can spli κ k = κ k 1 {W a } + κ k 1 {W > a } On he even {k 2L 1 + 1δ}, he firs erm of 2.71 can be bounded by he righ side of 2.68, i.e., κ k 1 {W a } W /µ 1Sp k + W sup R p k + W W 1 sup T I 2 p 1 k k 2 k {W a }, k k wih a similar lower bound where he las wo erms have a minus sign. From 2.62, we obain he upper bound κ k 1 {W> a } C η L 1 {W> a }. Combining hese wo upper bounds wih 2.71, and adding he erm W /µ 1Sp k 1 {W> a } o he righ side, yields ha on he even ha {k 2L 1 + 1δ}, W κ k µ 1 Sp k + W 1 {W a } sup R p k 2.72 k k + W 2 1 {W a } sup T I 2 p + 1 k {W > a }C η L, k k and similarly we ge as a lower bound, W κ k µ 1 Sp k W 1 {W a } sup R p k 2.73 k k W 2 1 {W a } sup T I 2 p W k 1{W> a } C s k k µ 1 + Cη L, 19

23 where we used ha sup Sp k C s. We use 2.72 and 2.73 on {k 2L 1 + 1δ}, and 2.62 on he even {k > 2L 1 + 1δ} o arrive a W κ k µ 1 Sp k + W 1 {W a } sup R p k 2.74 k k + W 2 1 {W a } sup T I 2 p k + 1{W > a } + 1 {k >2L 1 + 1δ} W C s k k µ 1 + Cη L, wih a similar lower bound where he las hree erms have a minus sign. We now ake expecaions on boh sides of 2.74 and ake advanage of he equaliy E[W /µ] = 1 and he propery ha Sp k is deerminisic, so ha he firs erm on he righ side drops ou. Moreover, using ha W and L 1 are independen, as well as ha k > 2L 1 + 1δ implies ha L 1 k, we arrive a [ W ] E[κ k ] E 1 {W > a } C s µ 1 + Cη C η k + E[W ] + C s E [ W µ 1 ] Pk > 2L 1 + 1δ 2.76 [ +E sup R p k k k [ +E[W 2 1 {W a }]E ] [ ] E W 1 {W> a } sup k k T I 2 p k 2.77 ] We now bound each of hese four erms one by one. To bound 2.75, we use ha W has finie 1 + ε-momen, o obain ha E [ ] [ 1 {W> a }W = E 1{W> a }W ε ] aε E [ W 1+ε ] = O aε, and, which bounds 2.75 as W 1+ε E [ ] 1 {W > a } = P W 1+ε > a1+ε 1 a1+ε E [ W 1+ε ] = O 1 a1+ε, [ W ] E 1 {W > a } C s µ 1 + Cη = O b, 2.79 wih b = max{ aε, 1 a1 + ε}. To bound 2.76, we use ha when k > 2L 1 + 1δ, hen L 1 < 1 2 η δ 1 = 1 2 η δ η. Now, since η µ, 2µ + δ, we have ha 1 2 η δ < µ. Sandard Large Deviaion heory and he fac ha he iniial degrees W i are non-negaive give ha he probabiliy ha L 1 < σ 1, wih σ < µ, is exponenially small in. As a resul, we obain ha C η k + E[W ] + C s E [ W µ 1 ] P k > 2L 1 + 1δ = O To bound 2.77, we use ha 2L 1 + δ L 1 1 /2, and also use 2.70, o obain ha [ ] E sup R p k c k k 2 E L 1 µ supk + δ p k c 2 E L 1 µ. Thus, [ ] [ ] E sup R p k E W 1 {W > a } c k k E L 1 µ aε O aε ε/1+ε, 2.81 where he final bound follows from Finally, o bound 2.78, noe ha E[W 2 1 {W a }] = E[W 1 ε W 1+ε 1 {W a }] a1 ε E[W 1+ε 20 ] = O a1 ε,

24 and, by 2.26 and he fac ha 2L 1 + δ η for some η > 0, we have [ E sup T I 2 p ] c k k k 2 sup k + δ 2 p k This leads o he bound ha [ E[W 2 1 {W a }]E sup T I 2 p ] k k k O a1 ε Combining he bounds in 2.79, 2.80, 2.81 and 2.83 complees he proof of par c of Lemma 2.4, for any a such ha 1/ε + 1 < a < Discussion and relaed resuls In his secion, we discuss he similariies and he differences of our proof of he asympoic degree sequence in Theorem 1.3 as compared o oher proofs ha have appeared in he lieraure. Virually all proofs of asympoic power laws in preferenial aachmen models use he wo seps presened here in Proposiions 2.1 and 2.2. For bounded suppor of W i, he concenraion resul in Proposiion 2.1 and is proof are idenical in all proofs. For unbounded W i, an addiional coupling argumen is required. The differences arise in he saemen and proof of Proposiion 2.2. Our Proposiion 2.2 proves a sronger resul han ha for δ = 0 appearing in [9] for he fixed number of edges case, and in [13] in he random number of edges case, in ha he resul is valid for a wider range of k values and he error erm is smaller. Indeed, in [13], he equivalen of Proposiion 2.2 is proved for k 1/21, and in [9] for k 1/15. In [13], also a random number of edges {W i } i 1 is allowed. However, i is assumed ha he suppor of W i is bounded, in which case he Azuma-Hoeffding argumen presened in Secion 2.3 simplifies considerably. The nice feaure of allowing an unbounded number of edges is he compeiion of he exponens in 1.9. The model in [13] is much more general han he model discussed here, and a every ime allows for he creaion of a new verex wih a random number of edges or he addiion of a random number of edges o an old verex. Under reasonable assumpions on he parameers of he model, a power law is proved for he degree sequence of he graph, indicaing ha he occurrence of power laws is raher robus in he model definiion in conras o he value of he power-law exponen. Due o he complexiy of he model in [13], he asympoic degree sequence saisfies a more involved recurrence relaion see [13, Eq. 2] han he one in 1.7. We close his discussion by reviewing some relaed resuls. Similar resuls for various random graph processes where a fixed number of edges is added can be found in [21], where also similar error bounds are proved for models where a fixed number of edges is added. In [6], a direced preferenial aachmen model is invesigaed, and i is proved ha he degrees obey a power law similar o he one in [9]. Finally, in [1], he error bound in Proposiion 2.1 is proved for m = 1 for several models. The resul for fixed {W } 1 and m > 1 is, however, no conained here. We inend o make use of his resul in order o sudy disances in preferenial aachmen models in [15]. For relaed references, see [21] and [29]. We finally menion he resuls in [23]. There, a scale-free graph process is sudied where, condiionally on G, edges are added independenly wih a probabiliy which is proporional o he degree of he verex. In his case, as in [9], he power-law exponen can only ake he value τ = 3, bu i can be expeced ha by incorporaing an addiive δ-erm as in 1.1, he model can be generalized o τ 3. Since δ < 0 is no allowed in his model by he independence of he edges, a degree is zero wih posiive probabiliy, we expec ha only τ 3 is possible. 21