Scale-Free Property for Degrees and Weights in a Preferential Attachment Random Graph Model

Abstract

A random graph evolution mechanism is defined. The evolution studied is a combination of the preferential attachment model and the interaction of four vertices. The asymptotic behaviour of the graph is described. It is proved that the graph exhibits a power law degree distribution; in other words, it is scale-free. It turns out that any exponent in (2,∞) can be achieved. The proofs are based on martingale methods.

1. Introduction

During the last 15 years the behaviour of many types of real-world networks was investigated. Such networks are the WWW, the Internet, and social and biological networks (see [1] for an overview). The main common characteristic of such networks is their scale-free nature, in other words the power law degree distribution, that is, 𝑝_𝑘∼𝐶𝑘^−𝛾, as 𝑘→∞. Using real-life data, the exponents were determined for several cases. For the WWW the in-degree and the out-degree of web pages follow power law with 𝛾_in=2.1 and 𝛾_out=2.7, for the Internet 𝛾=2.3, for the movie actor network 𝛾=2.3, and for the collaboration graph of mathematicians 𝛾=2.4 (see [1] for details). To describe the phenomenon, in [2] the preferential attachment model was introduced. In that model the growing procedure of the random graph is the following. At every time 𝑡=2,3,… a new vertex with 𝑚 edges is added so that the edges link the new vertex to 𝑚 existing vertices. The probability 𝜋_𝑖 that the new vertex will be connected to the old vertex 𝑖 depends on the degree 𝑘_𝑖 of vertex 𝑖, so that 𝜋_𝑖=𝑘_𝑖/∑_𝑗𝑘_𝑗. Then several papers were devoted to the proof of the power law in the preferential attachment model (see, e.g., [3]).

There are versions of the preferential attachment model (see [4, 5]). It turned out that besides the degrees of the vertices other characteristics of the graph can be important (see [5]). In [6] a model based on the interaction of three vertices was introduced. Then the power law degree distribution in that model was obtained in [7].

In this paper, we extend the model and the results of [6, 7] to interactions of four vertices. Our model is the following. For short, a complete graph with four vertices we will call tetragon. The starting point at time 𝑛=0 is one tetragon. The initial weight of this graph is one. This graph contains 4 vertices, 6 edges, and 4 triangles. Each of these objects has initial weight 1. After the initial step we start to increase the size of the graph. The main feature of the procedure is that at each step 𝑛=1,2,… we consider four vertices and draw all nonexisting edges between these vertices. So we obtain a tetragon. The weight of this tetragon and the weights of all the objects in the tetragon are increased by 1. (That is, we increase the weights of 4 vertices, 6 edges, 4 triangles, and 1 tetragon.) The choice of the four vertices is the following.

There are two possibilities at each step. With probability 𝑝 we add a new vertex that interacts with three old vertices, on the other hand, with probability (1−𝑝) four old vertices interact. Here 0<𝑝≤1 is fixed.

When we add a new vertex, then we choose 3 old vertices and they together will form a tetragon. However, to choose the three old vertices we have two possibilities. With probability 𝑟 we choose a triangle from the existing triangles according to the weights of the triangles. It means that a triangle of weight 𝑤_𝑡 is chosen with probability 𝑤_𝑡/∑_ℎ𝑤_ℎ (preferential attachment rule). On the other hand, with probability 1−𝑟, we choose among the existing vertices uniformly; that is, all three vertices have the same chance.

When four old vertices interact, we have again two possibilities. With probability 𝑞, we choose among the existing tetragons according to their weights. It means that a tetragon of weight 𝑤_𝑡 is chosen with probability 𝑤_𝑡/∑_ℎ𝑤_ℎ (preferential attachment rule). On the other hand, with probability 1−𝑞, we choose among the existing vertices uniformly; that is, all four vertices have the same chance.

Our aim is to prove that the above mechanism produces a scale-free graph. We follow the lines of [6, 7]. Let 𝑋(𝑛,𝑑,𝑤) denote the number of vertices of weight 𝑤 and degree 𝑑 after the 𝑛th step. Let 𝑉_𝑛 denote the number of vertices after the 𝑛th step. Let ℱ_𝑛−1 denote the 𝜎-algebra of observable events after the (𝑛−1)th step. First we calculate the conditional expectation 𝔼{𝑋(𝑛,𝑑,𝑤)∣ℱ_𝑛−1}, see Lemma 2. Then we prove (Theorem 3) that 𝑋(𝑛,𝑑,𝑤)/𝑉_𝑛→𝑥_𝑑,𝑤 almost surely (a.s.) as 𝑛→∞, where 𝑥_𝑑,𝑤 are fixed nonnegative numbers. The main tool of the proof is the Doob-Meyer decomposition of submartingales. We remark that in the 3-interaction model of [6] the limit 𝑥_𝑑,𝑤 is always positive (see [7]). As in our case the limit can be zero, we should modify the proof presented in [7].

We show that 𝑥_𝑑,𝑤, 𝑑=3,4,…,3𝑤, 𝑤=1,2,…, is a proper two-dimensional discrete probability distribution (Lemma 4). Then we turn to the scale-free property for the weights. Let 𝑋(𝑛,𝑤) denote the number of vertices of weight 𝑤 after the 𝑛th step. Then for all 𝑤=1,2,… we have 𝑋(𝑛,𝑤)𝑉_𝑛⟶𝑥_𝑤=𝑥_3,𝑤+𝑥_4,𝑤+⋯+𝑥_3𝑤,𝑤 (1) almost surely and 𝑥_𝑤∼𝐶𝑤^{−(1+1/𝛼)}, as 𝑤→∞ (Theorem 5). To derive the above results from Theorem 3, we need only some known facts about the Γ-function, see Lemma 4 and Theorem 5. Finally, we obtain the scale-free property for the degrees. Let us denote by 𝑈(𝑛,𝑑) the number of vertices of degree 𝑑 after the 𝑛th step. For any 𝑑≥3 we have 𝑈(𝑛,𝑑)𝑉_𝑛⟶𝑢_𝑑 (2) almost surely as 𝑛→∞, where 𝑢_𝑑 are positive numbers. Furthermore, 𝑢_𝑑∼Γ1+𝛽+1/𝛼𝛼₂Γ1+𝛽/𝛼𝛼𝑑𝛼₂^{−(1+1/𝛼)}, (3) as 𝑑→∞, where 𝛼, 𝛽, and 𝛼₂ are appropriate constants (see Theorem 8). In both cases the exponent is 1+1/𝛼. We can see that its value can be any number in (2,∞).

If we compare the results and methods of the present paper with the ones in [6, 7], we can see that the calculations for the 4-vertice model are much longer than those of the 3-vertice model (see the proofs of Lemma 2 and Theorem 3). One can think that when we want to extend our model to the interaction of 𝑁 vertices, then formulae will be burdening. But it is not the case. Analysing the proof of Theorem 3, it turns out that several terms are asymptotically negligible and this phenomenon remains true for the 𝑁-vertices model, too. Therefore the asymptotic results of the 3- and 4-vertice models can be extended to the 𝑁-vertices model. Moreover, several versions of the 𝑁-vertices model can be constructed and their fit to real-life data can be studied.

2. The Evolution of the Graph

Throughout the paper 0<𝑝≤1, 0≤𝑟≤1, 0≤𝑞≤1 are fixed numbers. Let 𝑋(𝑛,𝑑,𝑤) denote the number of vertices of weight 𝑤 and degree 𝑑 after the 𝑛th step. Let 𝑉_𝑛 denote the number of vertices after the 𝑛th step.

Remark 1. Each vertex has initial weight 1 and initial degree 3. When a vertex takes part in an interaction, then its weight is increased by 1 and its degree may be increased by 0, 1, 2, or 3. Therefore 𝑋(𝑛,𝑑,𝑤) can be positive only for 1≤𝑤 and 3≤𝑑≤3𝑤.
Let ℱ_𝑛−1 denote the 𝜎-algebra of observable events after the (𝑛−1)th step. We compute the conditional expectation of 𝑋(𝑛,𝑑,𝑤) with respect to ℱ_𝑛−1 for 𝑤≥1.
Let 𝛼₁=1−𝑝𝑞,𝛼₂=34𝑝𝑟,𝛼=𝛼₁+𝛼₂,()+4𝛽=31−𝑟1−𝑝1−𝑞𝑝. (4) The following lemma contains the basic equation of the paper.

Lemma 2. We have 𝔼𝑋(𝑛,𝑑,𝑤)∣ℱ_𝑛−1(𝑤=𝑋𝑛−1,𝑑,𝑤)1−𝑛𝑝𝛼+𝑉_𝑛−1𝛽()×𝑞+𝑋𝑛−1,𝑑,𝑤−11−𝑝𝑤−1𝑛+1−𝑞^𝑑3_{𝑉𝑛−14}()×𝑝𝑟3()+𝑋𝑛−1,𝑑−1,𝑤−1𝑤−1+()4𝑛1−𝑟^2𝑑−1_{𝑉𝑛−13}+1−𝑝1−𝑞^2𝑑−1𝑉_𝑛−1−𝑑_{𝑉𝑛−14}()×𝑝()()𝑉+𝑋𝑛−1,𝑑−2,𝑤−11−𝑟𝑑−2_𝑛−1−𝑑+1_{𝑉𝑛−13}+()1−𝑝1−𝑞𝑑−2^𝑉𝑛−1₂^−𝑑+1_{𝑉𝑛−14}()×𝑝()+𝑋𝑛−1,𝑑−3,𝑤−11−𝑟^{𝑉𝑛−12−𝑑+2}_{𝑉𝑛−13}+1−𝑝1−𝑞^{𝑉𝑛−13−𝑑+2}_{𝑉𝑛−14}+𝑝𝛿_𝑑,3𝛿_𝑤,1 (5) for 𝑤≥1 and 3≤𝑑≤3𝑤. Here 𝛿_𝑘,𝑙 denotes the Dirac-delta.

Proof. The total weight of tetragons after (𝑛−1) steps is 𝑛. The total weight of triangles after (𝑛−1) steps is 4𝑛. The total weight of the triangles having a fixed common vertex of weight 𝑤 is 3𝑤. Moreover, after (𝑛−1) steps, we have the following. When we choose three vertices randomly, then the probability that a given vertex is chosen is ^{𝑉𝑛−12−1}_{𝑉𝑛−13}=3𝑉_𝑛−1. (6) When we choose four vertices randomly, then the probability that a given vertex is chosen is ^{𝑉𝑛−13−1}_{𝑉𝑛−14}=4𝑉_𝑛−1. (7) Therefore the probability that an old vertex of weight 𝑤 takes part in the interaction at step 𝑛 is 𝑝𝑟3𝑤+()34𝑛1−𝑟𝑉_𝑛−1+𝑞𝑤1−𝑝𝑛+41−𝑞𝑉_𝑛−1=𝑤𝑛𝑝𝛼+𝑉_𝑛−1𝛽, (8) where 𝛼 and 𝛽 are defined by (4). A new vertex always takes part in the interaction. At each step with probability 𝑝 a new vertex with weight 1 and with degree 3 is born. This explains term 𝑝𝛿_𝑑,3𝛿_𝑤,1 in (5).
Consider a fixed vertex with weight 𝑤 and degree 𝑑. The probability of the event that in the 𝑛th step (i) neither its degree 𝑑 nor its weight 𝑤 change is 𝑤1−𝑛𝑝𝛼+𝑉_𝑛−1𝛽; (9)(ii) its degree does not change but its weight is increased by 1 is 1−𝑝𝑞𝑤−1𝑛+1−𝑞^𝑑3_{𝑉𝑛−14}; (10)(iii) both its degree and its weight are increased by 1 is 𝑝𝑟3(𝑤−1)+()4𝑛1−𝑟^2𝑑−1_{𝑉𝑛−13}+1−𝑝1−𝑞^2𝑑−1𝑉_𝑛−1−𝑑_{𝑉𝑛−14}; (11)(iv) its degree is increased by 2 and its weight is increased by 1 is 𝑝()()𝑉1−𝑟𝑑−2_𝑛−1−𝑑+1_{𝑉𝑛−13}+(1−𝑝1−𝑞𝑑−2)^𝑉𝑛−1₂^−𝑑+1_{𝑉𝑛−14}; (12) (v) its degree is increased by 3 and its weight is increased by 1 is 𝑝(1−𝑟)^{𝑉𝑛−12−𝑑+2}_{𝑉𝑛−13}+1−𝑝1−𝑞^{𝑉𝑛−13−𝑑+2}_{𝑉𝑛−14}. (13)Using the above formulae, we obtain (5).

The following theorem is an extension of Theorem  3.1 in [7].

Theorem 3. Let 0<𝑝<1, 𝑞>0, 𝑟>0 and (1−𝑟)(1−𝑞)>0. For any fixed 𝑤 and 𝑑 with 1≤𝑤 and 3≤𝑑≤3𝑤 one has 𝑋(𝑛,𝑑,𝑤)𝑉_𝑛⟶𝑥_𝑑,𝑤 (14) almost surely as 𝑛→∞, where 𝑥_𝑑,𝑤 are fixed nonnegative numbers. Furthermore, the numbers 𝑥_𝑑,𝑤 satisfy the following recurrence: 𝑥_3,1=1𝛼+𝛽+1,𝑥_𝑑,1𝑥=0,𝑓𝑜𝑟𝑑≠3,_𝑑,𝑤=1𝛼𝛼𝑤+𝛽+1¹(𝑤−1)𝑥_{𝑑,𝑤−1}+𝛼₂(𝑤−1)𝑥_{𝑑−1,𝑤−1}+𝛽𝑥_{𝑑−3,𝑤−1}, (15) for 𝑤≥2, 3≤𝑑≤3𝑤, where 𝛼₁, 𝛼₂, 𝛼, and 𝛽 are given by (4). One has 𝑥_𝑑,𝑤>0 if 𝑤≥1, 3≤𝑑≤3𝑤 unless 𝑑=3𝑤−1. If 𝑑=3𝑤−1, then one has 𝑥_𝑑,𝑤=0; moreover, in this case 𝑋(𝑛,𝑑,𝑤)𝑉_𝑛=𝑜𝑛^−𝑎, (16) where 𝑎 is a positive number which may depend on 𝑤 and 𝑑. If 3≤𝑑≤3𝑤 does not satisfied, then 𝑥_𝑑,𝑤=0.

Proof. During the proof we take care of the case 𝑥_𝑑,𝑤=0 which does not appear in [7]. Introduce notation 𝑐(𝑛,𝑤)=^𝑛_{𝑖=𝑤−1}1−𝛼𝑤𝑖−𝛽𝑝𝑉_𝑖−1⁻¹,𝑛≥𝑤−1,𝑤≥1. (17)𝑐(𝑛,𝑤) is an ℱ_𝑛−1-measurable random variable. Applying the Marcinkiewicz strong law of large numbers to the number of vertices, we have 𝑉_𝑛=𝑝𝑛+𝑜𝑛^1/2+𝜀 (18) almost surely, for any 𝜀>0.
Using (18) and the Taylor expansion for log(1+𝑥), we obtain log𝑐(𝑛,𝑤)=−^𝑛_{𝑖=𝑤−1}log1−𝛼𝑤𝑖−𝛽𝑖+𝑜𝑖^1/2+𝜀=𝛼𝑤+𝛽^𝑛_{𝑖=𝑤−1}1𝑖+𝑂(1), (19) where the error term is convergent as 𝑛→∞. Therefore 𝑐(𝑛,𝑤)∼𝑎_𝑤𝑛^{𝛼𝑤+𝛽} (20) almost surely, as 𝑛→∞, where 𝑎_𝑤 is a positive random variable.
Let 𝑍(𝑛,𝑑,𝑤)=𝑐(𝑛,𝑤)𝑋(𝑛,𝑑,𝑤)for1≤𝑤,3≤𝑑≤3𝑤. (21) Using (5), we can see that {𝑍(𝑛,𝑑,𝑤),ℱ_𝑛,𝑛=𝑤−1,𝑤,𝑤+1,…} is a nonnegative submartingale for any fixed 1≤𝑤, 3≤𝑑≤3𝑤. Define 𝑍(𝑛,𝑑,𝑤)=0 for 𝑛=1,2,…,𝑤−2. Applying the Doob-Meyer decomposition to 𝑍(𝑛,𝑑,𝑤), we can write 𝑍(𝑛,𝑑,𝑤)=𝑀(𝑛,𝑑,𝑤)+𝐴(𝑛,𝑑,𝑤), (22) where 𝑀(𝑛,𝑑,𝑤) is a martingale and 𝐴(𝑛,𝑑,𝑤) is a predictable increasing process. The general form of 𝑀(𝑛,𝑑,𝑤) and 𝐴(𝑛,𝑑,𝑤) is the following: 𝑀(𝑛,𝑑,𝑤)=^𝑛_𝑖=1𝑍(𝑖,𝑑,𝑤)−𝔼𝑍(𝑖,𝑑,𝑤)∣ℱ_𝑖−1,𝐴()()+𝑛,𝑑,𝑤=𝔼𝑍1,𝑑,𝑤_𝑛𝑖=2𝔼𝑍(𝑖,𝑑,𝑤)∣ℱ_𝑖−1−𝑍(𝑖−1,𝑑,𝑤), (23)(24) where ℱ₀ is the trivial 𝜎-algebra. Using (5) and (24), we obtain 𝐴()()+𝑛,𝑑,𝑤=𝔼𝑍1,𝑑,𝑤_𝑛𝑖=2𝑐()𝑋()×𝑞𝑖,𝑤𝑖−1,𝑑,𝑤−11−𝑝𝑤−1𝑖+1−𝑞^𝑑3_{𝑉𝑖−14}()𝑋()×𝑝𝑟3()+𝑐𝑖,𝑤𝑖−1,𝑑−1,𝑤−1𝑤−1+()4𝑖1−𝑟^2𝑑−1_{𝑉𝑖−13}+1−𝑝1−𝑞^2𝑑−1𝑉_𝑖−1−𝑑_{𝑉𝑖−14}()𝑋()×𝑝()()𝑉+𝑐𝑖,𝑤𝑖−1,𝑑−2,𝑤−11−𝑟𝑑−2_𝑖−1−𝑑+1_{𝑉𝑖−13}+()1−𝑝1−𝑞𝑑−2^𝑉𝑖−1₂^−𝑑+1_{𝑉𝑖−14}()𝑋()×𝑝()+𝑐𝑖,𝑤𝑖−1,𝑑−3,𝑤−11−𝑟^{𝑉𝑖−12−𝑑+2}_{𝑉𝑖−13}+1−𝑝1−𝑞^{𝑉𝑖−13−𝑑+2}_{𝑉𝑖−14}+𝑐(𝑖,𝑤)𝑝𝛿_𝑑,3𝛿_𝑤,1. (25) Let 𝐵(𝑛,𝑑,𝑤) denote the sum of the conditional variances of the process 𝑍(𝑛,𝑑,𝑤). Now we will give an upper bound of 𝐵(𝑛,𝑑,𝑤) as follows: 𝐵()=𝑛,𝑑,𝑤_𝑛𝑖=2𝔻²𝑍(𝑖,𝑑,𝑤)∣ℱ_𝑖−1=^𝑛_𝑖=2𝔼𝑍(𝑖,𝑑,𝑤)−𝔼𝑍(𝑖,𝑑,𝑤)∣ℱ_𝑖−1²∣ℱ_𝑖−1=^𝑛_𝑖=2𝑐(𝑖,𝑤)²×𝔼𝑋(𝑖,𝑑,𝑤)−𝔼𝑋(𝑖,𝑑,𝑤)∣ℱ_𝑖−1²∣ℱ_𝑖−1≤^𝑛_𝑖=2𝑐(𝑖,𝑤)²×𝔼(𝑋(𝑖,𝑑,𝑤)−𝑋(𝑖−1,𝑑,𝑤))²∣ℱ_𝑖−1≤16^𝑛_𝑖=2𝑐(𝑖,𝑤)²=𝑂𝑛^{2(𝛼𝑤+𝛽)+1}. (26) Above we used that 𝑐(𝑖,𝑤) is ℱ_𝑖−1-measurable and at each step four vertices can interact. Jensen’s inequality implies that 𝑀²(𝑛,𝑑,𝑤) is a (nonnegative) submartingale if 𝑀(𝑛,𝑑,𝑤) is a martingale. Now we can apply the Doob-Meyer decomposition to 𝑀²(𝑛,𝑑,𝑤). It is known that 𝐵(𝑛,𝑑,𝑤), that is the sum of the conditional variances of terms 𝑍(𝑛,𝑑,𝑤) from formula (26), is the same (up to an additive constant) as the increasing predictable process in the Doob-Meyer decomposition of the nonnegative submartingale 𝑀²(𝑛,𝑑,𝑤). Therefore the Doob-Meyer decomposition is 𝑀²(𝑛,𝑑,𝑤)=𝑌(𝑛,𝑑,𝑤)+𝐵(𝑛,𝑑,𝑤), (27) where 𝑌(𝑛,𝑑,𝑤) is a martingale and the predictable increasing process 𝐵(𝑛,𝑑,𝑤) is given by (26).
We use induction on 𝑤. Let 𝑤=1. We can see that a vertex of weight 1 could take part in an interaction when it was born. Therefore its degree must be equal to 3. By (25) and (20), 𝐴(𝑛,3,1)∼𝑝^𝑛_𝑖=2𝑐(𝑖,1)∼𝑝^𝑛_𝑖=2𝑎₁𝑖^𝛼+𝛽∼𝑝𝑎₁𝑛^𝛼+𝛽+1𝛼+𝛽+1 (28) almost surely as 𝑛→∞. By (26), 𝐵(𝑛,3,1)=𝑂(𝑛^{2(𝛼+𝛽)+1}) and therefore (𝐵(𝑛,3,1))^1/2log𝐵(𝑛,3,1)=𝑂(𝐴(𝑛,3,1)). It follows from Proposition VII-2-4 of [8] (see Proposition 10 in the Appendix) that 𝑍()(){𝐴()}𝑛,3,1∼𝐴𝑛,3,1a.s.ontheevent𝑛,3,1⟶∞as𝑛⟶∞. (29) As, by (28), 𝐴(𝑛,3,1)→∞ almost surely, therefore using (18) and (20), relation (29) implies 𝑋(𝑛,3,1)𝑉_𝑛=𝑍(𝑛,3,1)𝑐(𝑛,1)𝑉_𝑛∼𝐴(𝑛,3,1)𝑐(𝑛,1)𝑉_𝑛∼𝑝𝑎₁𝑛^𝛼+𝛽+1/𝛼+𝛽+1𝑎₁𝑛^𝛼+𝛽=1𝑝𝑛𝛼+𝛽+1=𝑥_3,1>0 (30) almost surely. So (14) is valid for 𝑤=1.
Now let 𝑤=2. In this case the degree of the vertex must be 3≤𝑑≤6. If 𝑤=2 and 𝑑=3,4,or6, then we will show 𝑥_𝑑,𝑤>0. By (18), (20), and (25), we can find the asymptotic behaviour of 𝐴(𝑛,𝑑,2) as 𝐴(𝑛,3,2)∼𝑝𝑎₂𝑛^2𝛼+𝛽+12𝛼+𝛽+11−𝑝𝑞𝑥_3,1𝐴()⟶∞,𝑛,4,2∼𝑝𝑎₂𝑛^2𝛼+𝛽+132𝛼+𝛽+14𝑝𝑟𝑥_3,1𝐴()⟶∞,𝑛,6,2∼𝑝𝑎₂𝑛^2𝛼+𝛽+1×3()+42𝛼+𝛽+11−𝑟1−𝑝1−𝑞𝑝𝑥_3,1⟶∞. (31) Moreover, by (26), 𝐵(𝑛,𝑑,2)=𝑂(𝑛^{2(2𝛼+𝛽)+1}); thus, (𝐵(𝑛,𝑑,2))^1/2log𝐵(𝑛,𝑑,2)=𝑂(𝐴(𝑛,𝑑,2)) for 𝑑=3, 4, and 6. Therefore in these cases 𝑍(𝑛,𝑑,2)∼𝐴(𝑛,𝑑,2) almost surely on {𝐴(𝑛,𝑑,2)→∞} as 𝑛→∞. It implies that 𝑋(𝑛,𝑑,2)𝑉_𝑛=𝑍(𝑛,𝑑,2)𝑐(𝑛,2)𝑉_𝑛∼𝐴(𝑛,𝑑,2)𝑐(𝑛,2)𝑉_𝑛∼𝑝𝑎₂𝑛^2𝛼+𝛽+1/2𝛼+𝛽+1𝑎₂𝑛^2𝛼+𝛽𝑇𝑝𝑛^𝑑,2=𝑇_𝑑,22𝛼+𝛽+1 (32) with appropriate 𝑇_𝑑,2. In particular, we have 𝑋(𝑛,3,2)𝑉_𝑛⟶1−𝑝𝑞𝑥_3,12𝛼+𝛽+1=𝑥_3,2𝑋()>0,𝑛,4,2𝑉_𝑛⟶(3/4)𝑝𝑟𝑥_3,12𝛼+𝛽+1=𝑥_4,2𝑋()>0,𝑛,6,2𝑉_𝑛⟶3()+41−𝑟1−𝑝1−𝑞𝑝𝑥_3,12𝛼+𝛽+1=𝑥_6,2>0, (33) as 𝑛→∞. So (14) is valid for 𝑤=2 and 𝑑=3,4,6.
However, the case 𝑑=5, 𝑤=2 is different from the previous cases. As 𝑥_5,2=0, the relation 𝑋(𝑛,5,2)/𝑉_𝑛→𝑥_5,2 would not be enough to proceed the induction. By (25) and using Remark 1, we have 𝐴()()+𝑛,5,2=𝔼𝑍1,5,2_𝑛𝑖=2𝑐()𝑋()×𝑝()3𝑉𝑖,2𝑖−1,3,11−𝑟^𝑖−1−4_{𝑉𝑖−13}+31−𝑝1−𝑞^{𝑉𝑖−12−4}_{𝑉𝑖−14}. (34) Using (18), (34), and the limit of 𝑋(𝑖−1,3,1)/𝑉_𝑖−1, we obtain 𝐴(𝑛,5,2)∼^𝑛_𝑖=2𝑎₂𝑖^2𝛼+𝛽𝑥_3,1×()𝑖𝑝18𝑝1−𝑟𝑝𝑖²+361−𝑝1−𝑞𝑝𝑖²∼𝑎₂𝑥_3,118𝑝^𝑛_𝑖=2𝑖^{2𝛼+𝛽−1}𝑝(1−𝑟)+21−𝑝1−𝑞∼𝑎₂𝑥_3,118𝑝𝑝()×𝑛1−𝑟+21−𝑝1−𝑞_2𝛼+𝛽2𝛼+𝛽⟶∞,as𝑛⟶∞. (35) Now we have 𝑋(𝑛,5,2)𝑉_𝑛=𝑍(𝑛,5,2)𝑐(𝑛,2)𝑉_𝑛=𝑀(𝑛,5,2)+𝐴(𝑛,5,2)𝑐(𝑛,2)𝑉_𝑛. (36) We denoted by 𝐵(𝑛,𝑑,𝑤) the increasing predictable process in the Doob-Meyer decomposition of 𝑀²(𝑛,𝑑,𝑤). We know, by (26), that 𝐵(𝑛,5,2)=𝑂(𝑛^{4𝛼+2𝛽+1}) and so (𝐵(𝑛,5,2))^1/2log𝐵(𝑛,5,2)=𝑂(𝑛^{2𝛼+𝛽+1/2+𝜀}) with arbitrary small positive 𝜀.
Applying Propositions VII-2-3 and VII-2-4 of [8] (see Proposition 9 in the Appendix), we have 𝑀(𝑛,5,2)=𝑜(𝐵(𝑛,5,2))^1/2()𝑛log𝐵𝑛,5,2=𝑜^{2𝛼+𝛽+1/2+𝜀}a.e.on{𝐵(𝑛,5,2)⟶∞}. (37) Moreover, on the set {𝐵(∞,5,2)<∞}, the sequence 𝑀(𝑛,5,2) is almost surely. convergent. So 𝑀(𝑛,5,2)=𝑜(𝑛^{2𝛼+𝛽+1/2+𝜀}) almost surely. Therefore, using (18), (20), and (35), we obtain 𝑋(𝑛,5,2)𝑉_𝑛=𝑀(𝑛,5,2)+𝐴(𝑛,5,2)𝑐(𝑛,2)𝑉_𝑛𝑛≤𝐶^{2𝛼+𝛽+1/2+𝜀}𝑛^2𝛼+𝛽𝑛1=𝐶𝑛^𝑎⟶0, (38) if 𝑛→∞ where 1/4<𝑎<1/2. So the proposition is valid for 𝑤=1 and 𝑤=2.
Now, suppose that the statement is true for all weights less than 𝑤 and for all possible degrees. First we study the positive limits. Consider 𝐴(𝑛,𝑑,𝑤) in (25) and assume that at least one of the coefficients 𝑥_{𝑑,𝑤−1}, 𝑥_{𝑑−1,𝑤−1}, 𝑥_{𝑑−3,𝑤−1} is positive. Then by (18), (20), and using the induction hypothesis, we see that 𝐴()∼𝑛,𝑑,𝑤_𝑛𝑖=2𝑐(𝑖,𝑤)𝑥_{𝑑,𝑤−1}𝑝𝑖1−𝑝𝑞𝑤−1𝑖+𝑐(𝑖,𝑤)𝑥_{𝑑−1,𝑤−1}3()𝑝𝑖𝑝𝑟𝑤−1()𝑥4𝑖+𝑐𝑖,𝑤_{𝑑−3,𝑤−1}×()𝑝𝑖3𝑝1−𝑟+4𝑝𝑖1−𝑝1−𝑞∼𝑝𝑖_𝑛𝑖=2𝑎_𝑤𝑖^{𝛼𝑤+𝛽}×𝑥_{𝑑,𝑤−1}𝑝1−𝑝𝑞(𝑤−1)+𝑥_{𝑑−1,𝑤−1}34𝑝²𝑟(𝑤−1)+𝑥_{𝑑−3,𝑤−1}3𝑝(1−𝑟)+41−𝑝1−𝑞∼𝑝𝑎_𝑤𝑛^{𝛼𝑤+𝛽+1}×𝑞()𝑥𝛼𝑤+𝛽+11−𝑝𝑤−1_{𝑑,𝑤−1}+34𝑝𝑟(𝑤−1)𝑥_{𝑑−1,𝑤−1}+3()+41−𝑟1−𝑝1−𝑞𝑝𝑥_{𝑑−3,𝑤−1}. (39) In the above computation we deleted all terms having asymptotically smaller degree than the largest one.
Consider the case when 3≤𝑑≤3𝑤 and 𝑑≠3𝑤−1. In this case, by the induction hypothesis, there is at least one positive term in (39). Therefore (39) implies 𝐴(𝑛,𝑑,𝑤)∼𝑝𝑎_𝑤𝑛^{𝛼𝑤+𝛽+1}𝑥_𝑑,𝑤→∞ (because 𝑥_𝑑,𝑤>0). In this case (𝐵(𝑛,𝑑,𝑤))^1/2log𝐵(𝑛,𝑑,𝑤)=𝑂(𝐴(𝑛,𝑑,𝑤)). So, using Proposition VII-2-4 of [8] (see Proposition 10 in the Appendix), we have 𝑍(𝑛,𝑑,𝑤)∼𝐴(𝑛,𝑑,𝑤). Therefore 𝑋(𝑛,𝑑,𝑤)𝑉_𝑛=𝑍(𝑛,𝑑,𝑤)𝑐(𝑛,𝑤)𝑉_𝑛∼𝐴(𝑛,𝑑,𝑤)𝑐(𝑛,𝑤)𝑉_𝑛∼𝑝𝑎_𝑤𝑛^{𝛼𝑤+𝛽+1}𝑥_𝑑,𝑤𝑎_𝑤𝑛^{𝛼𝑤+𝛽}𝑝𝑛=𝑥_𝑑,𝑤a.s.as𝑛⟶∞, (40) where, by (39), 𝑥_𝑑,𝑤=1×𝛼𝛼𝑤+𝛽+1₁(𝑤−1)𝑥_{𝑑,𝑤−1}+𝛼₂(𝑤−1)𝑥_{𝑑−1,𝑤−1}+𝛽𝑥_{𝑑−3,𝑤−1}, (41) with 𝛼₁, 𝛼₂, 𝛼, and 𝛽 defined by (4).
To handle the case when the limit is 0, we argue as follows. Consider the case when 𝑑=3𝑤−1. By (25) and using the induction hypothesis, we have 𝐴()()+𝑛,3𝑤−1,𝑤=𝔼𝑍1,3𝑤−1,𝑤_𝑛𝑖=2𝑐()𝑋()×𝑝()(𝑉𝑖,𝑤𝑖−1,3𝑤−3,𝑤−11−𝑟3𝑤−3)_𝑖−1−3𝑤+2_{𝑉𝑖−13}+()1−𝑝1−𝑞3𝑤−3^𝑉𝑖−1₂^−3𝑤+2_{𝑉𝑖−14}()𝑋()×𝑝()+𝑐𝑖,𝑤𝑖−1,3𝑤−4,𝑤−11−𝑟^{𝑉𝑖−12−3𝑤+3}_{𝑉𝑖−13}+1−𝑝1−𝑞^{𝑉𝑖−13−3𝑤+3}_{𝑉𝑖−14}∼^𝑛_𝑖=2𝑎_𝑤𝑖^{𝛼𝑤+𝛽}𝑥_{3𝑤−3,𝑤−1}𝐶₁11𝑝𝑖+𝑂𝑖^𝑎≤𝐶₁𝑛^{𝛼𝑤+𝛽}𝛼𝑤+𝛽+𝐶₂𝑛^{𝛼𝑤+𝛽+1−𝑎}𝑛𝛼𝑤+𝛽+1−𝑎=𝑂^{𝛼𝑤+𝛽+1−𝑎}. (42)
On the other hand, 𝐵(𝑛,𝑑,𝑤)=𝑂(𝑛^{2(𝛼𝑤+𝛽)+1}). Therefore, using (18), (20), and (42), by Propositions VII-2-3 and VII-2-4 of [8] (see Proposition 9 in the Appendix), we obtain 𝑋(𝑛,𝑑,𝑤)𝑉_𝑛=𝑍(𝑛,𝑑,𝑤)𝑐(𝑛,𝑤)𝑉_𝑛=𝑀(𝑛,𝑑,𝑤)+𝐴(𝑛,𝑑,𝑤)𝑐(𝑛,𝑤)𝑉_𝑛=𝑂𝑛^{𝛼𝑤+𝛽+1−𝑎}𝑛^{𝛼𝑤+𝛽}𝑛𝑝𝑛=𝑂^−𝑎⟶0=𝑥_{3𝑤−1,𝑤},a.s. (43) So we have obtained the desired result for the case of 0 limit as well.

3. The Scale-Free Property for the Weights and Degrees

Lemma 4. Let 𝑝>0 and define 𝑥_𝑤=𝑥_3,𝑤+𝑥_4,𝑤+⋯+𝑥_3𝑤,𝑤 (44) for 𝑤=1,2,…. Then 𝑥_𝑤, 𝑤=1,2,…, are positive numbers satisfying the following recurrence: 𝑥₁=1,𝑥𝛼+𝛽+1_𝑤=𝛼(𝑤−1)+𝛽𝑥𝛼𝑤+𝛽+1^𝑤−1,𝑖𝑓𝑤>1, (45) where 𝛼 and 𝛽 are defined by (4). 𝑥_𝑤, 𝑤=1,2,…, is a discrete probability distribution. Moreover, 𝑥_𝑑,𝑤, 𝑑=3,4,…,3𝑤, 𝑤=1,2,…, is a two-dimensional discrete probability distribution.

Proof. If 𝛼=0, then the statement is obvious. Now assume 𝛼≠0. As 𝑥_𝑑,𝑤 is defined as 𝑥_𝑑,𝑤=0 for 𝑑∉{3,4,…,3𝑤}, therefore 𝑥_𝑤=∑_𝑑𝑥_𝑑,𝑤. From the recurrence (15) for𝑥_𝑑,𝑤, we obtain 𝑥_𝑤=^3𝑤_𝑑=3𝑥_𝑑,𝑤=_𝑑𝑥_𝑑,𝑤=1𝛼𝛼𝑤+𝛽+1¹(𝑤−1)_𝑑𝑥_{𝑑,𝑤−1}+𝛼₂(𝑤−1)_𝑑𝑥_{𝑑−1,𝑤−1}+𝛽_𝑑𝑥_{𝑑−3,𝑤−1}=𝛼(𝑤−1)+𝛽𝑥𝛼𝑤+𝛽+1^𝑤−1. (46) Using this recursive formula for 𝑥_𝑤, we obtain 𝑥_𝑤=𝑥₁^𝑤_𝑗=2𝛼𝑗−1+𝛽=1𝛼𝑗+𝛽+1𝛼+𝛽+1𝛼+𝛽2𝛼+𝛽+12𝛼+𝛽⋯()3𝛼+𝛽+1𝑤−1𝛼+𝛽=1𝑤𝛼+𝛽+1𝛼𝑤+𝛽+1^𝑤−1_𝑗=1𝛽/𝛼+𝑗=1𝛽+1/𝛼+𝑗Γ𝛼𝑤+𝛽+1𝑤+𝛽/𝛼ΓΓ1+𝛽/𝛼1+𝛽+1/𝛼Γ=Γ𝑤+𝛽+1/𝛼1+𝛽+1/𝛼Γ𝛼Γ1+𝛽/𝛼𝑤+𝛽/𝛼Γ.𝑤+𝛽+1/𝛼+1 (47) Moreover, by [9], we have the following formula: ^𝑛_𝑘=0Γ(𝑘+𝑎)Γ()=1𝑘+𝑏Γ()𝑎−𝑏+1𝑛+𝑎+1Γ()−Γ(𝑎)𝑛+𝑏Γ().𝑏−1 (48) Therefore, by some calculation, we obtain ∑^𝑛_𝑤=1𝑥_𝑤→1, as 𝑛→∞. So ∑^∞_𝑤=1𝑥_𝑤=1. As ∑_𝑑𝑥_𝑑,𝑤=𝑥_𝑤, so ∑^∞_𝑤=1∑^3𝑤_𝑑=3𝑥_𝑑,𝑤=1 and therefore 𝑥_𝑑,𝑤, 𝑑=3,4,…,3𝑤, 𝑤=1,2,…, is a (proper) two-dimensional discrete probability distribution.

Let 𝑋(𝑛,𝑤) denote the number of vertices of weight 𝑤 after 𝑛 steps. Next theorem is the scale-free property for the weights. It is an extension of Theorem  3.1 in [6].

Theorem 5. Let 0<𝑝<1, 𝑞>0, 𝑟>0 and (1−𝑟)(1−𝑞)>0. Then for all 𝑤=1,2,… one has 𝑋(𝑛,𝑤)𝑉_𝑛⟶𝑥_𝑤=𝑥_3,𝑤+𝑥_4,𝑤+⋯+𝑥_3𝑤,𝑤 (49) almost surely, as 𝑛→∞, where 𝑥_𝑤, 𝑤=1,2,…, are positive numbers satisfying the recurrence (45). Moreover, 𝑥_𝑤∼𝐶𝑤^{−(1+1/𝛼)}, (50) as 𝑤→∞, with 𝐶=Γ(1+(𝛽+1)/𝛼)/(𝛼Γ(1+𝛽/𝛼)).

Proof. We have 𝑋(𝑛,𝑤)=𝑋(𝑛,3,𝑤)+𝑋(𝑛,4,𝑤)+⋯+𝑋(𝑛,3𝑤,𝑤). (51) Therefore, by Theorem 3, 𝑋(𝑛,𝑤)𝑉_𝑛⟶𝑥_3,𝑤+⋯+𝑥_3𝑤,𝑤=𝑥_𝑤 (52) almost surely, as 𝑛→∞. Here each 𝑥_𝑤 is positive.
Using formula (47) and the Stirling-formula for the Gamma function, we obtain 𝑥_𝑤=Γ1+𝛽+1/𝛼Γ𝛼Γ1+𝛽/𝛼𝑤+𝛽/𝛼Γ𝑤+𝛽+1/𝛼+1∼𝐶₀𝑤+𝛽/𝛼^{(𝑤+𝛽/𝛼)}𝑤+𝛽/𝛼+1/𝛼+1^{(𝑤+𝛽/𝛼+1/𝛼+1)}=𝐶₀𝑤+𝛽/𝛼𝑤+𝛽/𝛼+1/𝛼+1^{(𝑤+𝛽/𝛼)}×1𝑤+𝛽/𝛼+1/𝛼+1^1/𝛼+1∼𝐶𝑤^{−(1+1/𝛼)}, (53) where 𝐶₀=Γ(1+(𝛽+1)/𝛼)/(𝛼Γ(1+𝛽/𝛼))(1/(1/𝑒)^1+1/𝛼) and 𝐶=Γ(1+(𝛽+1)/𝛼)(𝛼Γ(1+𝛽/𝛼)).

Now, following [7], we construct a representation of the limiting joint distribution of degrees and weights.

Let 𝑊 be a random variable with distribution ℙ(𝑊=𝑤)=𝑥_𝑤,𝑤=1,2,…. Let 𝜉₁≡3 and 𝜉₂,𝜉₃,… be independent random variables being independent of 𝑊, too. For 𝑤≥2 let 𝜉_𝑤 have the following distribution: ℙ𝜉_𝑤=𝛼=0¹(𝑤−1)𝛼(),ℙ𝜉𝑤−1+𝛽_𝑤=𝛼=1²(𝑤−1)𝛼(),ℙ𝜉𝑤−1+𝛽_𝑤=𝛽=3𝛼().𝑤−1+𝛽 (54) Introduce notation 𝑆_𝑤=𝜉₁+𝜉₂+⋯+𝜉_𝑤.

The following representation of the joint distribution of degrees and weights is useful to obtain scale-free property for degrees.

Theorem 6. ℙ(𝑆_𝑊=𝑑,𝑊=𝑤)=𝑥_𝑑,𝑤 for all 𝑤=1,2,…, 𝑑=3,4,…,3𝑤.

Proof. It is easy to see that if 𝑤=1 and 𝑑=3, then we have ℙ𝑆_𝑊=3,𝑊=1=ℙ𝑆₁𝜉=3,𝑊=1=ℙ₁()=3,𝑊=1=ℙ𝑊=1=𝑥₁=𝑥_3,1. (55) If 𝑤=2 and 𝑑=5 we have ℙ𝑆_𝑊=5,𝑊=2=ℙ𝑆₂𝜉=5,𝑊=2=ℙ₁=3,𝜉₂=2,𝑊=2=0=𝑥_5,2. (56) Generally, for all 𝑤=1,2,… and 𝑑=3𝑤−1 we have ℙ𝑆_𝑊𝜉=3𝑤−1,𝑊=𝑤=ℙ₁=3,𝜉₂=𝑘₂,…,𝜉_𝑤=𝑘_𝑤,𝑊=𝑤=0=𝑥_{3𝑤−1,𝑤}, (57) because one of 𝑘₂,…,𝑘_𝑤 should be equal to 2, which is of zero probability. Furthermore, using the recursion (45) and the assumption that 𝜉_𝑖 are independent random variables which are independent of 𝑊, we have ℙ𝑆_𝑊𝑆=𝑑,𝑊=𝑤=ℙ_𝑤𝑆=𝑑,𝑊=𝑤=ℙ_𝑤ℙ()=ℙ𝑆=𝑑𝑊=𝑤_𝑤−1=𝑑ℙ𝜉_𝑤𝑆=0+ℙ_𝑤−1=𝑑−1ℙ𝜉_𝑤𝑆=1+ℙ_𝑤−1=𝑑−3ℙ𝜉_𝑤()𝑥=3×ℙ𝑊=𝑤−1^𝑤𝑥_𝑤−1=ℙ𝑆_𝑤−1=𝑑ℙ(𝑊=𝑤−1)𝛼₁()𝑆𝑤−1+ℙ_𝑤−1=𝑑−1ℙ(𝑊=𝑤−1)𝛼₂()𝑆𝑤−1+ℙ_𝑤−1ℙ()𝛽×1=𝑑−3𝑊=𝑤−1=1𝛼𝑤+𝛽+1×𝛼𝛼𝑤+𝛽+1₁(𝑤−1)ℙ𝑆_𝑊=𝑑,𝑊=𝑤−1+𝛼₂(𝑤−1)𝑆_𝑊𝑆=𝑑−1,𝑊=𝑤−1+𝛽ℙ_𝑊=𝑑−3,𝑊=𝑤−1. (58) Therefore, the sequence ℙ(𝑆_𝑊=𝑑,𝑊=𝑤) satisfies the same recursion (15) as 𝑥_𝑑,𝑤.

Theorem 7. Suppose that 𝛼₁>0 and 𝛼₂>0. Then 𝑥_𝑑,𝑤=𝑥_𝑤𝛼2𝜋𝛼₁𝛼₂𝑤×exp−𝑑−𝔼𝑆_𝑤²2𝔻²𝑆_𝑤+𝑂𝑤^−1/2,𝑎𝑠𝑤⟶∞, (59) where the error term 𝑂(𝑤^−1/2) does not depend on 𝑑.

Proof. We can follow the ideas of Theorem  4.2 in [7]. For 𝑤≥1 we have 𝔼𝜉_𝑤=𝛼₂(𝑤−1)𝛼()𝛽𝑤−1+𝛽+3𝛼()=𝛼𝑤−1+𝛽₂𝛼+3𝛼−𝛼₂𝛽𝛼𝛼();𝑤−1+𝛽 (60) hence, 𝔼𝑆_𝑤=𝔼𝜉₁+⋯+𝔼𝜉_𝑤𝛼=𝑤²𝛼+𝑂log𝑤, (61) as 𝑤→∞. Similarly, by simple computation, we have 𝔻²𝜉_𝑤=𝛼₁𝛼₂𝛼²1+𝑂𝑤,𝔻²𝑆_𝑤=𝛼₁𝛼₂𝛼²𝑤+𝑂log𝑤, (62) as 𝑤→∞.
Now, we can apply Theorem VII.2.5 in [10] (see Proposition 11 in the Appendix) for 𝑆_𝑤. The conditions of that theorem are satisfied; therefore, we have sup_𝑑∈ℤ𝔻𝑆_𝑤ℙ𝑆_𝑤−1=𝑑√−2𝜋exp𝑑−𝔼𝑆_𝑤²2𝔻²𝑆_𝑤1=𝑂√𝑤. (63) Using (62) and (63), we obtain |𝔻𝑆_𝑤−√𝛼₁𝛼₂𝑤/𝛼|ℙ(𝑆_𝑤=𝑑)=𝑂(𝑤^−1/2). Therefore, it follows from (63) that sup_𝑑∈ℤ√𝛼₁𝛼₂𝑤𝛼ℙ𝑆_𝑤−1=𝑑√−2𝜋exp𝑑−𝔼𝑆_𝑤²2𝔻²𝑆_𝑤1=𝑂√𝑤. (64) The independence of 𝑊 and 𝜉_𝑖 implies that 𝑥_𝑑,𝑤=ℙ(𝑆_𝑊=𝑑,𝑊=𝑤)=ℙ(𝑆_𝑤=𝑑)𝑥_𝑤. Using this in (64), we can obtain the desired result.

Our last theorem is an extension of Theorem  4.3 in [7] to the case of 4 interactions. The theorem shows the scale-free property for the degrees.

Theorem 8. Let 0<𝑝<1, 𝑞>0, 𝑟>0 and (1−𝑟)(1−𝑞)>0. Let us denote by 𝑈(𝑛,𝑑) the number of vertices of degree 𝑑 after 𝑛 steps; that is, 𝑈(𝑛,𝑑)=∑_{𝑤∶𝑑/3≤𝑤≤𝑛+1}𝑋(𝑛,𝑑,𝑤). Then, for any 𝑑≥3 one has 𝑈(𝑛,𝑑)𝑉_𝑛⟶𝑢_𝑑=_𝑤𝑥_𝑑,𝑤 (65) almost surely as 𝑛→∞, where 𝑢_𝑑 are positive numbers. Furthermore, 𝑢_𝑑∼Γ1+𝛽+1/𝛼𝛼₂Γ1+𝛽/𝛼𝛼𝑑𝛼₂^{−(1+1/𝛼)}, (66) as 𝑑→∞.

Proof. By Theorems 3 and 6, 𝑋(𝑛,𝑑,𝑤)/𝑉_𝑛 converges almost surely to the distribution 𝑥_𝑑,𝑤=ℙ(𝑆_𝑊=𝑑,𝑊=𝑤). But the cardinalities of terms in ∑_{𝑤∶𝑑/3≤𝑤≤𝑛+1}𝑋(𝑛,𝑑,𝑤) are not bounded when 𝑛→∞. However, using that 𝑥_𝑑,𝑤, 𝑑=3,4,…,3𝑤, 𝑤=1,2,… is a proper two-dimensional discrete distribution, the convergence of the marginal distributions is a consequence of the convergence of the two-dimensional distributions. So we obtain (65).
To obtain (66), we can apply the method of Theorem  4.3 in [7]. Let 𝛼𝑓=𝛼₂𝑑,𝐻=𝐻_𝑑=𝑤∶𝑓−𝑓^1/2+𝜀≤𝑤≤𝑓+𝑓^1/2+𝜀,𝐻⁻=𝐻⁻_𝑑=𝑤∶𝑤<𝑓−𝑓^1/2+𝜀,𝐻⁺=𝐻⁺_𝑑=𝑤∶𝑤>𝑓+𝑓^1/2+𝜀 (67) with some fixed 0<𝜀<1/6.
Using (61) and Hoeffding’s exponential inequality (Theorem  2 in [11], see Proposition 12 in the Appendix) we obtain for 𝑤∈𝐻⁻ℙ𝑆_𝑤=𝑑≤ℙ𝑆_𝑤𝑆≥𝑑≤ℙ_𝑤−𝔼𝑆_𝑤𝛼≥𝑑−²𝛼−2𝑤−𝑂log𝑤≤exp𝛼9𝑤𝑑−²𝛼𝑤−𝑂log𝑤²−2=exp9𝛼₂𝛼²𝑓−𝑤−𝑂log𝑤²𝑤. (68) Here 𝑤∈𝐻⁻ implies that 𝑓−𝑤−𝑂log𝑤²=𝑓−𝑤²−2𝑓−𝑤𝑂log𝑤+𝑂log𝑤²≥𝑓^1+2𝜀−𝑂𝑓log𝑓. (69) Therefore in the case when 𝑤∈𝐻⁻ we have ℙ𝑆_𝑤−2=𝑑≤exp9𝛼₂𝛼²𝑓^1+2𝜀−𝑂𝑓log𝑓𝑓−2=exp9𝛼₂𝛼²𝑓^2𝜀+𝑂log𝑓. (70) Using this, we can obtain that ℙ𝑆_𝑊=𝑑,𝑊∈𝐻⁻=_{𝑤∈𝐻−}ℙ𝑆_𝑤=𝑑,𝑊=𝑤≤_{𝑤∈𝐻−}ℙ𝑆_𝑤−2=𝑑≤𝑓exp9𝛼₂𝛼²𝑓^2𝜀𝑓+𝑂log𝑓=𝑜^{−(1+1/𝛼)}. (71) Similarly, if 𝑤∈𝐻⁺, again by Hoeffding’s inequality, we have ℙ𝑆_𝑤=𝑑≤ℙ𝑆_𝑤𝑆≤𝑑≤ℙ_𝑤−𝔼𝑆_𝑤𝛼≤𝑑−²𝛼𝑤−2≤exp𝛼9𝑤𝑑−²𝛼𝑤²−2=exp9𝛼₂𝛼²𝑓−𝑤²𝑤. (72) Using that 𝑤∈𝐻⁺ and 1/2+𝜀<1, we obtain 2(𝑤−𝑓)≥𝑓^1/2+𝜀+𝑤−𝑓≥𝑓^1/2+𝜀+(𝑤−𝑓)^1/2+𝜀≥𝑤^1/2+𝜀. Therefore ℙ𝑆_𝑤−2=𝑑≤exp9𝛼₂𝛼²𝑤^1+2𝜀−14𝑤=exp𝛼18𝛼²²𝑤^2𝜀. (73) Hence ℙ𝑆_𝑊=𝑑,𝑊∈𝐻⁺≤_{{𝑤∶𝑓<𝑤}}−1exp𝛼18𝛼²²𝑤^2𝜀=𝑜𝑓^{−(1+1/𝛼)}. (74)
Now consider the case when 𝑤∈𝐻=𝐻_𝑑. First we need some general facts. Consider the set 𝐵=(𝑑,𝑤)∶𝑤≥1,𝑑≥3,𝑤∈𝐻_𝑑. (75) It is easy to see that when (𝑑,𝑤)∈𝐵 then 𝑑→∞ if and only if 𝑤→∞. More precisely, 𝑤𝑑⟶1,if𝑑⟶∞,(𝑑,𝑤)∈𝐵. (76) As 𝑤∈𝐻, so we have 𝑤=𝑓+𝑂(𝑓^1/2+𝜀). Then (with 𝜀₁>0 arbitrarily small) −𝑑−𝔼𝑆_𝑤²2𝔻²𝑆_𝑤𝛼=−𝑑−𝑤₂/𝛼−𝑂log𝑤²2𝛼₁𝛼₂/𝛼²𝛼𝑤+𝑂log𝑤=−²𝛼₁𝑓−𝑤−𝑂log𝑤²𝛼2𝑤+𝑂log𝑤=−²𝛼₁𝑓−𝑤²+𝑂𝑓^{1/2+𝜀+𝜀1}𝛼2𝑤+𝑂log𝑤=−²𝛼₁𝑓−𝑤²+𝑂𝑓^{1/2+𝜀+𝜀1}×2𝑓2𝑓2𝑓+𝑂𝑓^1/2+𝜀𝛼=−²𝛼₁𝑓−𝑤²+𝑂𝑓^{1/2+𝜀+𝜀1}×𝑂𝑓2𝑓1−^1/2+𝜀2𝑓+𝑂𝑓^1/2+𝜀𝛼=−²𝛼₁𝑓−𝑤²𝑓2𝑓+𝑂^−1/2+3𝜀, (77) as 𝑑→∞. Here the error term does not depend on 𝑤. We will apply Theorem 7, that is, formula (59). The asymptotic behaviour of 𝑥_𝑤 is known from (50). Using these facts and (77), we obtain 𝑥_𝑑,𝑤∼𝐶𝑤^{−(1+1/𝛼)}𝛼2𝜋𝛼₁𝛼₂𝑤×−𝛼exp²𝛼₁𝑓−𝑤²𝑓2𝑓+𝑂^−1/2+3𝜀+𝑂𝑤^−1/2∼𝐶𝑓^{−(1+1/𝛼)}𝛼𝛼₂12𝜋𝛼₁/𝛼₂𝑓exp−𝑓−𝑤²2𝛼₁/𝛼₂𝑓 (78) as 𝑑→∞ and 𝑤∈𝐻, where 𝐶=Γ(1+(𝛽+1)/𝛼)/(𝛼Γ(1+𝛽/𝛼)). Therefore _𝑤∈𝐻𝑥_𝑑,𝑤∼_{𝑓−𝑓1/2+𝜀<𝑤<𝑓+𝑓1/2+𝜀}𝐶𝑓^{−(1+1/𝛼)}𝛼𝛼₂12𝜋𝛼₁/𝛼₂𝑓×exp−𝑓−𝑤²2𝛼₁/𝛼₂𝑓=𝐶𝑓^{−(1+1/𝛼)}𝛼𝛼_{2−𝑓1/2+𝜀<𝑘<𝑓1/2+𝜀}12𝜋𝛼₁/𝛼₂𝑓−𝑘×exp²2𝛼₁/𝛼₂𝑓=𝐴_{−𝑓𝜀<𝑘/𝑓<𝑓𝜀}1𝑓12𝜋𝛼₁/𝛼₂×exp−𝑘/𝑓²2𝛼₁/𝛼₂⟶𝐴^+∞_−∞12𝜋𝛼₁/𝛼₂−𝑥exp²2𝛼₁/𝛼₂𝑑𝑥=𝐴. (79) Thus we have ℙ𝑆_𝑊=Γ=𝑑,𝑊∈𝐻∼𝐴1+𝛽+1/𝛼𝛼₂Γ1+𝛽/𝛼𝛼𝑑𝛼₂^{−(1+1/𝛼)}, (80) as 𝑑→∞. Finally, (71), (74), and (80) imply 𝑢_𝑑=_𝑤𝑥_𝑑,𝑤=_{𝑤∈𝐻−}𝑥_𝑑,𝑤+_𝑤∈𝐻𝑥_𝑑,𝑤+_𝑤∈𝐻+𝑥_𝑑,𝑤∼𝑜𝑓^{−(1+1/𝛼)}𝛼+𝐶𝛼₂𝑓^{−(1+1/𝛼)}+𝑜𝑓^{−(1+1/𝛼)}∼Γ1+𝛽+1/𝛼𝛼₂Γ1+𝛽/𝛼(𝛼/𝛼₂)𝑑^{−(1+1/𝛼)}, (81) as 𝑑→∞. The proof is complete.

Appendix

We use the following results on discrete time martingales. Let {𝑍_𝑛,ℱ_𝑛} be a submartingale. Its Doob-Meyer decomposition is 𝑍_𝑛=𝑀_𝑛+𝐴_𝑛, where {𝑀_𝑛,ℱ_𝑛} is a martingale and {𝐴_𝑛,ℱ_𝑛} is an increasing predictable process. Here, up to an additive constant, 𝐴_𝑛=𝔼𝑍₁+^𝑛_𝑖=2𝔼𝑍_𝑖∣ℱ_𝑖−1−𝑍_𝑖−1. (A.1) Now, {𝑀²_𝑛,ℱ_𝑛} is again a submartingale. Let 𝑀²_𝑛=𝑌_𝑛+𝐵_𝑛 (A.2) be the Doob-Meyer decomposition of 𝑀²_𝑛. Here, up to an additive constant, 𝐵_𝑛=^𝑛_𝑖=2𝔻²𝑍_𝑖∣ℱ_𝑖−1=^𝑛_𝑖=2𝔼𝑍_𝑖−𝔼𝑍_𝑖∣ℱ_𝑖−1²∣ℱ_𝑖−1. (A.3)

Proposition 9 (Propositions VII-2-3 and VII-2-4 of [8]). Let 𝑀₁=0. On the set {𝐵_∞<∞} the martingale 𝑀_𝑛 almost surely converges to a finite limit. Moreover, 𝑀_𝑛=𝑜(𝐵_𝑛^1/2log𝐵_𝑛) almost surely on the set {𝐵_𝑛→∞}.

A consequence of the above proposition is the following.

Proposition 10 (Proposition 2.3 of [12]). Let {𝑍_𝑛,ℱ_𝑛} be a square integrable nonnegative submartingale. If 𝐵_𝑛^1/2log𝐵_𝑛=𝑂(𝐴_𝑛), then 𝑍_𝑛∼𝐴_𝑛, as 𝑛→∞, almost surely on the set {𝐴_𝑛→∞}.

We need the following local limit theorem. Let 𝑋₁,𝑋₂,… be independent, integer valued random variables. Let 𝑝_𝑗,𝑚=ℙ(𝑋_𝑗=𝑚) be the distribution, while 𝑝_{𝑗,𝑚𝑗}=max_𝑚𝑝_𝑗,𝑚 be the maximal value of the distribution. Let 𝑆_𝑛=∑^𝑛_𝑖=1𝑋_𝑖 be the partial sum, 𝑃_𝑛(𝑁)=ℙ(𝑆_𝑛=𝑁) be its distribution, 𝑀_𝑛=∑^𝑛_𝑖=1𝔼𝑋_𝑖 be the expectation, and 𝐵_𝑛=∑^𝑛_𝑖=1𝔼(𝑋_𝑖−𝔼𝑋_𝑖)² be the variance of 𝑆_𝑛.

Proposition 11 (Theorem 5 and its consequence in Section VII, 2 of [10]). Assume that the greatest common divisor of the values 1𝑚∶log𝑛^𝑛_𝑗=1𝑝_{𝑗,𝑚𝑗}𝑝_{𝑗,𝑚+𝑚𝑗}⟶∞ (A.4) is equal to 1; moreover, 𝐵liminf^𝑛𝑛1>0,limsup𝑛^𝑛_𝑖=1𝔼𝑋_𝑖−𝔼𝑋_𝑖³<∞. (A.5) Then sup_𝑁𝐵_𝑛𝑃_𝑛(𝑁)−1√2𝜋exp𝑁−𝑀_𝑛²2𝐵_𝑛1=𝑂√𝑛. (A.6)

The well-known Hoeffding’s inequality is the following.

Proposition 12 (Theorem 2 of [11]). Let 𝑋₁,𝑋₂,…,𝑋_𝑛 be independent random variables, 𝑎_𝑖≤𝑋_𝑖≤𝑏_𝑖(𝑖=1,2,…,𝑛). Let 𝑋=(𝑋₁+𝑋₂+⋯+𝑋_𝑛)/𝑛, 𝜇=𝔼𝑋. Then for any 𝑡>0ℙ𝑋−𝜇≥𝑡≤exp−2𝑛²𝑡²∑^𝑛_𝑖=1𝑏_𝑖−𝑎_𝑖². (A.7)

Acknowledgments

The publication was supported by the TÁMOP-4.2.2.C-11/1/KONV-2012-0001 project. The project has been supported by the European Union, cofinanced by the European Social Fund.