WEIGHTS AND DEGREES IN A RANDOM GRAPH MODEL BASED ON 3-INTERACTIONS

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1 Acta Math. Hungar., 43 04, 3 43 Acta Math. DOI: Hungar., 0.007/s , First published online February 8, DOI: WEIGHTS AND DEGREES IN A RANDOM GRAPH MODEL BASED ON 3-INTERACTIONS Á. BACKHAUSZ,, and T. F. MÓRI, Department of Probability Theory and Statistics, Eötvös Loránd University, Pázmány P. s. /C, H-7 Budapest, Hungary s: agnes@math.elte.hu, mori@math.elte.hu MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary Received April, 03; revised September 9, 03; accepted September 7, 03 Abstract. In a random graph model introduced in ] we give the joint asymptotic distribution of weights and degrees and prove scale-free property for the model. Moreover, we determine the asymptotics of the maximal weight and the maximal degree.. Introduction Many random graph models have been invented recently for modelling large networs lie the internet or social networs,8]. Considering degree distributions, real life networs loo quite different from classical i.e., Erdős Rényi type random graphs. Motivated by this observation, in several models 7,5] the evolution of the graph is driven by the actual degrees. However, in real-world networs larger groups and cliques may also interact and this has a relevant effect on the evolution. Therefore models based on cliques or groups of vertices may be of particular interest 9,,]. Motivated by that, in ] we introduced a random graph model with dynamics based on interactions of three vertices. In our model vertices taing part in an interaction together have larger chance to participate together again, thus it is a ind of preferential attachment structure, while in the models referred to earlier there is no possibility to eep trac of the number of steps where the members of a given group get new edges together. More precisely, at each time step, three randomly chosen vertices interact. In some of the steps a new vertex is added to the graph, and it interacts Corresponding author. The European Union and the European Social Fund have provided financial support to the project under the grant agreement no. TÁMOP 4.../B-09/KMR Key words and phrases: martingale, random graph, preferential attachment, scale free. Mathematics Subject Classification: 05C80, 60G /$ Aadémiai Kiadó, Budapest, Hungary

2 4 Á. BACKHAUSZ and T. F. MÓRI with two randomly chosen old vertices. In the other case three old vertices interact. In both cases the interacting old vertices are chosen randomly, uniformly at random or with probabilities proportional to weights: in this model vertices, edges and triangles have nonnegative weights, increasing randomly in discrete time steps. The weight is the number of interactions that the vertex, pair of vertices or triplets have participated in. In ] we investigated the limit of the ratio of vertices of a given weight and proved that they almost surely exist and decay polynomially. This is the scale-free property of the model. We also determined the asymptotics of the weight of a given vertex. This time we will deal with degrees. This is the number of vertices having interacted with a given one. This is different from the weight of the vertex, which is the total number of interactions. Moreover, there is no deterministic connection between them. If a vertex interacts with another one that it has not been connected to, then both its weight and its degree are increased; on the other hand, if it interacts with one of its neighbours, then only its weight is increased. Therefore we can as about the connection of these two random variables. In this paper we determine the joint asymptotic distribution of weights and degrees; prove scale-free property for degrees; finally, we give the asymptotics of the degree of a given vertex, the maximal weight, and the maximal degree.. The model We start with a single triangle. This has initial weight, and all its three edges have weight. Vertices, edges, and triangles will have nonnegative integer-valued weights, which increase according to the random evolution of the graph. At each step three vertices will interact. Here is the pattern of choosing the interacting vertices. p, q and r are fixed parameters of the model. With probability p, a new vertex is added, and it interacts with two randomly chosen old vertices: With probability r, two old vertices are chosen with probabilities proportional to edge weights. Otherwise two different old vertices are chosen uniformly at random. Otherwise, with probability p, no vertices are added and three old vertices interact: With probability q, three old vertices are chosen with probabilities proportional to edge weights. Otherwise three different old vertices are chosen uniformly at random.

3 WEIGHTS AND DEGREES IN A RANDOM GRAPH MODEL 3 Now we formulate this precisely. At each step there are two possibilities. With probability p, independently of the past, a new vertex is added, which then interacts with two already existing vertices. Otherwise three old vertices interact. We will need 0 <p. Assume that in the nth step a new vertex is added to the graph. With probability r, independently of the past, the choice is done according to the preferential attachment rule, that is, an edge is chosen with probability proportional to its weight, then its endpoints are selected. With probability r two distinct old vertices are chosen uniformly at random. Then the new vertex interacts with the two selected vertices. This means that the triangle they form comes to existence with initial weight, and we increase the weights of all three edges of the 3-interaction by. This is the end of the step where a new vertex is generated. With probability p three of the old vertices will interact. In such a step we have two choices again. With probability q each triangle is selected with probability proportional to its weight. Otherwise, with probability q, three distinct vertices will be chosen at random, uniformly, i.e., each triplet with the same probability. This choice is also independent of the past. In each case, having selected the three vertices to interact, we draw the edges of the triangle that are not present yet. Then the weight of the triangle is increased by, as well as the weights of the three sides of the triangle. Now we define the weights of vertices. The weight of a vertex is the sum of the weights of the triangles that contain it. Note that this is just the half of the sum of weights of edges from it, because whenever a vertex taes part in an interaction, the first sum is increased by, and the latter one is increased by. Our model is parametrized by the triplet of probabilities p, q, r. This construction was introduced in ], where the following properties were proved. The ratio of vertices of weight w converges to x w almost surely as n, where x = hence we have α + β +, x w = β+ Γ+ α x w α Γ+ α β w + α, αw + β αw + β + x w, as w, where α = pq + pr 3, β = p r+3 p q ], Theorem 3.]. We have also studied the rate of growth of the weight of a fixed vertex. It is clear that the weights of the vertices of the starting triangle are interchangeable, therefore it is not necessary to deal with all the three. Let 5

4 6 4 Á. BACKHAUSZ and T. F. MÓRI them be labelled by,, and 0. The further vertices get labels,, etc., in the order they are added to the graph. Let Dn, j] andw n, j] denote the degree and the weight of vertex j after step n, provided it exists. Otherwise let these quantities be equal to zero. Obviously, vertex j cannot exist before step j. According to Theorem 4. of ], for j 0 fixed we have 3 W n, j] ζ j n α almost surely as n, where ζ j is a positive random variable. In the sequel we will denote by F n the σ-field generated by the first n steps, and by V n the number of vertices after the nth step. Thus V 0 = 3. Furthermore, let I be the indicator of the event in the bracet; i.e. it is defined as if the condition within the bracets holds, otherwise let it be Asymptotic joint distribution of degrees and weights We denote the number of vertices of weight w and degree d after n steps by Xn, d, w]. When a vertex is born, its initial weight is one, and its initial degree is two. When it taes part in an interaction, its weight is increased by one, while its degree may not change if it is already connected to the other two interacting vertices, or may increase by one or two. Thus Xn, d, w] > 0 can occur only for pairs of integers d, w with w and d w. The following theorem is about the almost sure convergence of the ratio of vertices of weight w and degree d. Theorem 3.. Given integers w and d w we have Xn, d, w] V n x d,w almost surely as n, where the limits x d,w are positive numbers satisfying the following recurrence equation: x d,w = x, = α + β +, ] α w x d,w + α w x d,w + βx d,w αw + β + for w, where α = pq, α = pr 3, α = α + α, 4 β = p ] p r+3 p q.

5 WEIGHTS AND DEGREES IN A RANDOM GRAPH MODEL 5 Proof. We compute the conditional expectation of Xn, d, w] with respect to the σ-algebra F n. Note that if an old vertex interacts with a new one, its degree must increase. On the other hand, if we choose vertices with probabilities proportional to certain weights, then no new edges are born between old vertices. Having built the graph in n steps we consider a fixed vertex with degree d and weight w. For simplicity we denote by V = V n the number of vertices after n steps. Then d, w can increase w 3n+ new vertex, preferential attach- by, with probability pr ment; by, with probability p r d V, and by, with probability p r V d new vertex, uniform selection; V by 0, with probability pq w n+ old vertices, preferential attachment; by 0, with probability p q d,by, with probability V 3 dv d p q, and by, with probability p q V d V 3 V 3 old vertices, uniform selection. Now it is easy to see that the probability that a vertex of weight w taes part in the interaction at step n is the following see also ]: 5 p r w ] 3n + r + p q w ] V n n + q 3 = αw V n n + βp ; V n this is independent of the degree of the vertex. For d = and w = we also have to tae the new vertex into account: a new vertex is born with probability p, and its degree is surely, while its weight is surely. Summing up, we obtain the conditional expectation of Xn, d, w] in the following form: E Xn, d, w] F n = Xn,d,w] αw n βp 6 V n + Xn,d,w ] p q w ] d + q n Vn 3 w + Xn,d,w ] p r 3n + r d Vn ] 7

6 8 6 Á. BACKHAUSZ and T. F. MÓRI + Xn,d,w ] p q dv n d Vn 3 + Xn,d,w ] p r V n d + p q Vn + pid =,w =. Introduce the normalizing sequence cn, w] = n i= αw i βp, n, w. V i Vn d Vn 3 At each step a new vertex is born with probability p independently of the past. Hence the law of large numbers can be applied to the number of vertices, yielding that 7 V n = pn + o n /+ε ] a.s., for all 0 <ε<. This implies that n log cn, w] = log αw i i= β i + o i /+ε n αw = i i= + β i + o i 3/+ε n =αw + β a.s., where the error term is convergent as n. Therefore 8 cn, w] a w n αw+β i= i + O a.s. as n, where a w is a positive random variable. For n, w, d set Zn, d, w] =cn, w]xn, d, w]. Multiplying both sides of 6 bycn, w] one can see that Zn, d, w], F n is a nonnegative submartingale for all fixed integers w, d. Consider the Doob decomposition Zn, d, w] =Mn, d, w]+an, d, w], where the process Mn, d, w] is a zero mean martingale, and An, d, w] is a predictable increasing process: Mn, d, w] = Zi, d, w] E Zi, d, w] F i, i=

7 WEIGHTS AND DEGREES IN A RANDOM GRAPH MODEL 7 An, d, w] =EZ,d,w]+ E Zi, d, w] F i Zi,d,w]. i= Let us give an upper bound on the conditional variance of the martingale part. Recall that ci, w] isf i -measurable, and since there is only one interaction at each step, the increment of X can not be greater than three. Using 8 we get that 9 Bn, d, w] = = Var Zi, d, w] F i = i= ci, w] Var Xi, d, w] F i i= ci, w] Var Xi, d, w] Xi,d,w] F i i= i= ci, w] E Xi, d, w] Xi,d,w] Fi 9 ci, w] = On αw+β+. i= Let us apply Proposition VII--4 of Neveu 3] with ft = t log t. obtain that 0 Mn, d, w] =o Bn, d, w] / log Bn, d, w] = o n αw+β+. We will later use this estimation to show that Zn, d, w] An, d, w]. Note that Xn, d, w] =0if d w does not hold. Hence x d,w =0in all these cases. We apply induction on w. If the weight of a vertex is equal to, then it could not participate in any interactions except the first one, when it was born. Thus its degree must be equal to two. Therefore Xn, d, ] is zero for d, and it is the number of vertices of weight for d =. In the case w = the proposition follows from. Suppose that the statement holds for all weights less than w, and for all possible degrees d w. Let us compute the asymptotics of An, d, w]. We start from 6: + An, d, w] =EZ,d,w] ci, w]xi,d,w ] p q w ] d + q i Vi 3 i= 9 We

8 30 8 Á. BACKHAUSZ and T. F. MÓRI w + ci, w]xi,d,w ] p r 3i + r d Vi + ci, w]xi,d,w ] p q dv i d + ci, w]xi,d,w ] p r V i d + p q Vi + ci, w]p Id =,w =. Vi 3 Vi d Vi 3 Using the induction hypothesis, the asymptotics of V n in 7 and the regular variation of the normalizing constants cn, w] with exponent αw + β in equation 8, we can compute the asymptotics of An, d, w] by an easy application of the Silverman Toeplitz theorem, leaving out all terms that are of smaller order of magnitude than others: ] ] An, d, w] i= ci, w] pi x d,w pq w i p i= w + ci, w] pi x d,w pr 3i + ci, w] pi x d,w r i + 3 p q pi a w i αw+β w pqw x d,w + pr 3 + r+ ] 3 p q x d,w p ] x d,w p a wn αw+β+ w pqw x d,w + pr x d,w αw + β r+ ] 3 p q x d,w. p

9 WEIGHTS AND DEGREES IN A RANDOM GRAPH MODEL 9 If d and w satisfy d w, then there is at least one term on the righthand side that is positive due to the induction hypothesis. Thus Mn, d, w] = o An, d, w], therefore Zn, d, w] An, d, w] holds almost surely as n. Dividing by the normalizing constants cn, w] we get that Xn, d, w] x d,w pn, from which Xn, d, w] x d,w V n a.s. as n, where x d,w = αw + β + = + r+ pqw x d,w + pr w 3 ] 3 p q x d,w p x d,w αw + β + α w x d,w + α w x d,w + βx d,w ], 3 with α = pq, α = pr 3 p q, β = r+3. p By this the induction step is completed. Moreover, as we noted before, x d,w > 0 holds for d w. Remar 3.. The explicit solution of the recurrence equation in Theorem 3. can be given in the following form. For w set c w =αw + β +αw + β +...α + β +. Let S n 0 =, and for n define S n = Then x d,w = c w w w S w w d = i <i < <i n i i...i. α w d+ α d β, w, d w.

10 3 0 Á. BACKHAUSZ and T. F. MÓRI In other words, x d,w is equal to the coefficient of z d in the expression w iα + α z+βz. c w i= This is not hard to derive, and even easier to chec; however, it does not seem to be very convenient for determining the asymptotics of x d,w as d or w tends to infinity. We rather choose another method for it in the next section. 4. Construction of the two dimensional limit distribution Let W be a positive integer valued random variable with distribution PW = w =x w, w =,,... In addition, let ξ, and let the random variables ξ,ξ 3,... be independent of each other and of W too; moreover, Pξ w =0= α w αw + β, Pξ w == α w αw + β, β Pξ w == αw + β. Define the partial sums S w = ξ + + ξ w. Theorem 4.. We have PS W = d, W = w =x d,w, w, d w. Proof. Clearly, PS W =,W==PW ==x = x,. In addition we have PS W = d, W = w =PS w = d, W = w =PS w = dpw = w = PS w = dpξ w =0+PS w = d Pξ w = + PS w = d Pξ w = ] ] αw + β PW = w αw + β + = PS w = d, W = w α w αw + β + + PS w = d, W= w α w αw + β + β + PS w = d, W= w αw + β +,

11 WEIGHTS AND DEGREES IN A RANDOM GRAPH MODEL thus the probabilities PS W = d, W = w and the limits x d,w satisfy the same recursion. This, combined with Theorem 3., implies that the empirical joint distribution of degree and weight after step n converges almost surely in total variation norm to the distribution of S W,W. As a corollary we obtain that the asymptotic weight distribution x w is just the marginal of the joint distribution x d,w, that is, x w = x,w + + x w,w. Theorem 4.. Suppose both α and α are positive. Then for some positive constant K we have x d,w x w for all d and w. Proof. We have hence ES w = α α α πα α w exp αd α w α α w Kw /, Eξ w = α w +β αw + β = α α + α + α β ααw + β, w + Olog w. Similarly, + O w Varξ w = α α α, VarS w = α α α w + Olog w, as w. The proof can be completed by applying the local limit theorem Theorem VII..5 in 4] to S w. Its conditions are satisfied, namely, lim inf w Hence we have sup d Z w VarS w > 0, lim sup w VarSw PS w = d exp π w w ξ j Eξ j 3 <. j= d ES w VarS w = O. w It easily follows that in ES w can be replaced with a term differing from it by O w log w, and VarS w with a term differing by O w. From Theorem 4. one can derive the asymptotics of the other marginal distribution u d = w d/ x d,w. Clearly, u d is the a.s. limit of the proportion of vertices with degree d. 33

12 34 Á. BACKHAUSZ and T. F. MÓRI Theorem 4.3. We have Proof. Let u d Γ + β+ α α Γ+ β α α + α d, α as d. f = α α d, H = { w : f f /+ε w f + f /+ε}, H = { w : w<f f /+ε}, H + = { w : w>f+ f /+ε}, with some 0 <ε</6. By Hoeffding s well-nown exponential inequality Theorem of 0] for w H we have PS w d P S w ES w d α α w Olog w exp d α α w Olog w w α f w Olog w =exp. α w Here in the numerator f w Olog w f +ε O f /+ε log f,and in the denominator w f. Hence PS w d exp α f ε α + o, thus we have 3 PS W = d, W H +o f exp α f ε α The case of w H + can be treated similarly. PS w d P S w ES w d α α w exp α α w d α w f exp. w α w = o f + α.

13 WEIGHTS AND DEGREES IN A RANDOM GRAPH MODEL 3 This time we use the estimate w f f /+ε + w f f /+ε +w f /+ε w /+ε in the numerator, obtaining Hence PS w d exp 4 PS W = d, W H + w>f Finally, for w H αd α w α α w = α f w α w exp α wε 8α. α wε 8α = o f + α. = α f w +O f /+ε α f 35 = α f w α f + O f /+3ε, consequently Γ + β+ α x d,w α Γ+ α β f + w H α α πα α f exp α w f, α f as d and w H. Since α πα α f exp α w f α f we obtain that + α πα α exp α t dt = α, α α Γ + β+ α 5 PS W = d, W H α Γ+ α β f + α. The proof is completed by 3, 4, and 5 combined.

14 36 4 Á. BACKHAUSZ and T. F. MÓRI 5. Maximal weight, maximal degree In this section our goal is to determine the asymptotics of the maximum of the weights as the number of steps tends to infinity. Let In, j] denote the indicator of the event { W n, j] }. Moreover, we denote by Jn, j] the indicator of the event that vertex j is born at step n, that is, Jn, j] =In, j] In,j]. We fix j, and examine the process W n, j] asn increases. In the first lemma we find martingales that we will use later in the proofs. Then we prove that the maximal weight grows at the same pace as the weight of any fixed vertex does, see 3. Let j,, l be fixed integers, 0 j l,, and let us introduce the sequences bn, ] = n i= + α n, dn,, j] = bi +,] βp i V i i= W i, j]+ with α, β defined in 4. Note that bn, ] is deterministic, while dn,, j] is random, but F n -measurable for all and j. Moreover, we have 6 bn, ] b n α, with b > 0, as n. Lemma 5.. Let W n, j]+ Zn,, j] =bn, ] dn,, j], then Zn,, j]il, j], F n, n l, is a martingale. Proof. Assuming that vertex j already exists after step l, the probability that it participates in an interaction at step n + l is equal to αw n,j] n+ + βp V n. This implies that, for arbitrary positive integers, l, n, we have W n +,j]+ E Il, j] αw n, j] + Il, j] + βp n V n W n, j]+ = Il, j] + Il, j] W n, j]+ F n = Il, j] ] W n, j]+ ] W n, j]+ αw n, j] + βp ] W n, j]+ n V n,

15 WEIGHTS AND DEGREES IN A RANDOM GRAPH MODEL 5 W n, j]+ = Il, j] + α n + Il, j] βp V n W n, j]+ Multiplying both sides by bn +,] we get by definition that W n +,j]+ E bn +,] Il, j] F n = Il, j] bn, ] = Il, j] bn, ] W n, j]+ + βp V n W n, j]+ W n, j]+. bn +,] dn,, j]+dn +,,j] which completes the proof of the lemma, since dn +,,j]isf n -measurable. Then Lemma 5.. For arbitrary fixed integers 0 and m n define Sm, n, ] = E bn, ] Sm, n, ] C W n, j]+ j α with a positive constant C only depending on. In, j] Proof. We prove this by induction on. For = 0 obviously Sm, n, 0] = bn, 0] E In, j] = ]. ], P W n, j] n m +. Suppose that the statement of the lemma holds for. By Lemma 5. we now that Zn,, j]il, j] is a martingale, hence its expectation does not depend on n. The difference of martingales is also a martingale, thus we have the same with Jl, j]. Decomposing In, j] into the sum of terms Jl, j] we obtain that Sm, n, ] = W n, j]+ E bn, ] l=j Jl, j] ] 37

16 38 6 Á. BACKHAUSZ and T. F. MÓRI = = E Zn,, j]+dn,, j] Jl, j] l=j E Zl,, j]+dn,, j] Jl, j] = E l=j bl, ]+dn,, j] dl,, j] Jl, j]. l=j Let us split Sm, n, ] into two parts: Sm, n, ] =S m, n, ]+S m, n, ], where 7 S m, n, ] =E l=j bl, ]Jl, j] bj, ]E Jl, j] l=j bj, ] =b j α +o C For the second part we have S m, n, ] =E = E = E n i=m l=j i=l j α. dn,, j] dl,, j] Jl, j] l=j n bi +,] βp V i bi +,] βp V i = E n i=m i W i, j]+ Jl, j] i W i, j]+ Jl, j] l=j bi +,] βp bi, ] V i i bi, ] ] W i, j]+ W i, j]+ Ii, j]. W i, j]

17 WEIGHTS AND DEGREES IN A RANDOM GRAPH MODEL 7 39 Since W i,j]+ W i,j], we get that S m, n, ] n i=m bi +,] bi, ] E βp V i i ] W i, j]+ bi, ] Ii, j]. For the expectation on the right-hand side we give upper bounds on the events { V i < p/i } and { V i p/i } separately. Remember that I denotes the indicator of the event in bracets. From the induction hypothesis we obtain that E βp I V i pi i bi, ] V i W i, j]+ Ii, j] β i Sm, i, ] βc i i j α. By the Hoeffding bound P V i < p/i e εi with ε>0 only depending on p. Using that V i 3 and the trivial bound on the weights we get that E βp I V i < pi i bi, ] V i uniformly in m. Finally, W i, j]+ βp 3 P V i < pi i i + bi, ] = Oe εi i α i i = o i i bi +,] bi, ] = Oi α. j α Ii, j]

18 40 8 Á. BACKHAUSZ and T. F. MÓRI Putting all these together we obtain that 8 S m, n, ] C i=m i α i j α = C j α i=j i α C j α. We can complete the proof by combining 7 and 8. Next we characterize the growth rate of the maximal weight in the graph. Let W n = max { W n, j] : j n }, the maximal weight after n steps. Theorem 5.. W n μn α almost surely as n, where μ is a finite and positive random variable, namely, μ =sup{ζ j : j }, with ζ j defined in 3. Proof. For m n define Mm, n] = max { W n, j] : j<m }. By 3 it is obvious that lim n n α Mm, n] = max { ζ j : j<m } with probability. All we have to do is to show that 9 lim lim sup n α W n Mm, n] =0. m n Indeed, this implies that lim sup n n α W n Mm, n] is finite for some m. We already now that ζ j is finite for every j, hence the maximum of the first m ones is also finite. This gives us the finiteness of μ. The positivity of μ follows immediately from the positivity of ζ j. From the proof of Lemma 5. it follows that the process W n, j]+ bn, ] Il, j], n l, is a submartingale, hence, the same holds for W n, j]+ W n, j]+ bn, ] = bn, ] In, j], n j. Being the maximum of an increasing number of submartingales, the process Wn Mm, n]+ bn, ], n m,

19 WEIGHTS AND DEGREES IN A RANDOM GRAPH MODEL 9 is also a submartingale. In addition, ] Wn Mm, n]+ 0 E bn, ] Sm, n, ] C by Lemma 5.. Since bn, ] W n Mm, n] bn, ]! bn, ] bn, ] j α Wn Mm, n]+ the nonnegative submartingale bn, ] W n Mm, n] is bounded in L whenever α >. Thus it is convergent with probability, and also in L, for every. Moreover, by 0 and we have E lim n α W n Mm, n] C! j α. n b From this the monotone convergence theorem gives E lim m lim n α W n Mm, n] =0 n if >/α, proving 9. We finally present the asymptotics of the maximal degree as the number of steps tends to infinity. First we will study the growth of the degree of a fixed vertex. Theorem 5.. For j =0,,... we have Dn, j] α α ζ j n α almost surely, as n, where ζ j is a positive random variable, defined in 3. Proof. Starting from the specification of the ways the degree and the weight of a fixed vertex can grow, we can write E W n, j] I, j]dn +,j] F n = I, j] Dn, j]+pr 3n + + p r V n Dn, j] Vn + 3 p q V n Dn, j] Vn, 4

20 4 0 Á. BACKHAUSZ and T. F. MÓRI W n, j] = I, j] Dn, j]+α n + + R n, if n, where 0 R n pβ V n. Introduce ξ n = I, j] Dn, j] Dn,j], then 0 ξ n, hence by Corollary VII--6 of 3] and equation 3 we obtain ξ i Eξ i F i = W i,j] α + R i α i α ζ j n α, i n i n i n a.e. on the event { W,j] }.ThusDn, j] α α ζ j n α on that event. Since we now that lim W, j] =, we have { } P W, j] =, completing the proof. From Theorems 5. and 5. the asymptotic behaviour of the maximal degree immediately follows. Theorem 5.3. Let D n denote the maximal degree in the graph after n steps. Then D n α α μnα almost surely as n, where μ is the finite and positive random variable defined in Theorem 5.. Proof. By the trivial bound Dn, j] W n, j] we obtain max { Dn, j] : j<m } D n max { Dn, j] : j<m } + max { W n, j] : m j n }. Multiplying by n α and letting n we get α α max{ζ j : j<m} lim inf n n α D n lim sup n α D n n α α max { ζ j : j<m } + lim n n α W n Mm, n]. By 9 both sides tend to μα /α as m.

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