Limit theorems for the weights and the degrees in anN-interactions random graph model

Abstract A random graph evolution based on interactions of N vertices is studied. During the evolution both the preferential attachment rule and the uniform choice of vertices are allowed. The weight of an M-clique means the number of its interactions. The asymptotic behaviour of the weight of a fixed M-clique is studied. Asymptotic theorems for the weight and the degree of a fixed vertex are also presented. Moreover, the limits of the maximal weight and the maximal degree are described. The proofs are based on martingale methods.


Introduction
Network theory is currently one of the most popular research topics. Random graphs are used to describe real-life networks. For overviews of random graph models and their properties see [1][2][3]. It is known that many real-life networks (e.g. the WWW, biological and social networks) are scale-free, that is their asymptotic degree distributions follow power laws. To describe the evolution of such networks, in [4], the preferential attachment model was suggested. However, in [4], the description of the evolution of the graph was informal. A rigorous definition of the preferential attachment model was given in [5], where a mathematical proof of the power law degree distribution was presented for d Ä n 1=15 (d is the degree, n is the number of steps). For the recent development of the topic see [3,6].
Besides the degree distribution, other characteristics are also worth studying. The degree of a fixed vertex and the maximal degree in some preferential attachment models were investigated in [3,7,8]. In [9,10] the degree of a given vertex and the maximal degree were studied in a 2-parameter scale-free random graph model. A well-known technique to analyse the growth of the maximal degree is the martingale method (see [1,3,11]).
There are several versions of the preferential attachment model (see [3,12]). In [12] a general graph evolution procedure was introduced. In that procedure both the preferential attachment rule and the uniform choice of vertices are allowed, moreover, new links can be created between old vertices. The evolution method introduced in [13] in some sense resembles the one in [12]. However, the main feature of the model applied in [13] is the interaction of three vertices. Power law degree distribution in the three-interactions model was proved in [13,14]. The asymptotic behaviour of the weight and the degree of a fixed vertex, as well as the limits of the maximal weight and the maximal degree, were also described in [13,14]. Scale-free weight distributions, both for the edges and the triangles in the

Main results
Let 1 Ä M Ä N be a fixed integer. We introduce the following notation.
We see that when M D 1, then˛0 D˛andˇ0 Dˇ. First we list some results concerning the scale-free property.
Remark 2.1. The scale-free property of our model means the following (see [16]). Let N 3 be fixed. Let X .n; w/ denote the number of vertices of weight w after n steps. Let 0 < p < 1, q > 0, r > 0 and .1 r/.1 q/ > 0. Then for all w D 1; 2; : : : we have almost surely, as n ! 1, where x w , w D 1; 2; : : : , are positive numbers satisfying . Let U .n; d / denote the number of vertices of degree d after n steps. Then, for any d N 1 we have a.s. as n ! 1 , where u d , d D N 1; N; : : : , are positive numbers with Remark 2.2. In our model, besides the vertices, the cliques also have weights. It turns out that the weight distribution of the M -cliques is also power law. Let N 3 be fixed and let M be fixed with 1 < M Ä N and denote by X M .n; w/ the number of M -cliques having weight w after n steps. If p > 0 and either r > 0 or .1 p/q > 0, then X M .n; w/ n ! x M;w (5) almost surely, as n ! 1, where x M;w , w D 1; 2; : : : , are numbers satisfying This result was presented in [15] for the case of M D 2, N D 3 and also for the case of M D N for arbitrary N > 2. The general case can be obtained by using the ideas of [18]. To this end one has to consider a slight modification of the general model of [18] and to prove appropriate versions of Theorems 2 and 3 in [18]. First, we study the weight of a fixed M -clique. At time n D 0, the initial complete graph on N vertices is symmetric. Therefore and by the evolution mechanism of our graph, it is enough to describe W OEn; M; j and DOEn; j for j D 0; 1; 2; : : : . The following theorem describes the asymptotic behaviour of the weight of a fixed M -clique. almost surely as n ! 1, where M;j is a positive random variable.
As a particular case with M D 1, the asymptotic behaviour of the weight of a fixed vertex is the following. almost surely as n ! 1, where 1;j is a positive random variable.
We turn to the limit of the degree sequence of a fixed vertex. almost surely as n ! 1, where the positive random variable 1;j is given in (8).
Now, we turn to the maximal weight and the maximal degree. Let us denote by W n the maximum of the weights of the vertices after n steps, that is Theorem 2.6. Let˛> 0. Then where D supf 1;j W j .N 1/g is a finite positive random variable with 1;j defined in (8).
Let us denote by D n the maximal degree after n steps, that is Theorem 2.7. Let˛> 0. Then D n

.C˛/˛2
n˛almost surely as n ! 1; where D supf 1;j W j .N 1/g is the positive random variable defined in Theorem 2.6. Remark 2.8. We see that the parameters˛1 and˛2 belong to the preferential attachment part of the model, whilě 1 andˇ2 are connected to the uniform choice part. Moreover, in the above Theorems 2.3-2.7 only˛1 and˛2 play role. Therefore the asymptotic behaviour in those theorems is not influenced by the uniform choice parameters. This phenomenon can be explained as follows. If j is fixed and n is large, then the j th vertex is 'old' among the V n np vertices. Therefore the degree and the weight of the j th vertex are relatively high compared to those of the 'young' vertices. As there are lot of 'young' vertices, therefore the uniform choice has minor influence on the degree and the weight of the j th vertex. Remark 2.9. If we compare Corollary 2.4 and Theorem 2.5, we see that the asymptotic ratio of the weight and the degree of vertex j is W OEn; 1; j DOEn; j ˛2 D .N 1/P .
Step NEW and Choice PA/ C N P .
Step OLD and Choice PA/ .N 1/P .
Step NEW and Choice PA/ : (14) And the last expression in (14) is nothing else but the ratio of the expected growth of the weights of the 'old' vertices to the expected growth of the degrees of the 'old' vertices during one step when the choice is PA. Similar observation is true for the asymptotic ratio of the maximal weight to the maximal degree. That is, by Theorems 2.6 and 2.7, we have W n =D n ˛=˛2.
Remark 2.10. Let n denote the maximum of the labels of those vertices where the maximal weight is attained, that is let Then the sequence n .!/, n D 1; 2; : : : is bounded for almost all fixed elementary events !. It is a simple consequence of Theorem 2.6 and its proof. A similar statement is valid for the degrees.

Proofs and auxiliary lemmas
Introduce the following notation. Let F n 1 denote the -algebra of observable events after the .n 1/th step. Let V n denote the number of vertices after the nth step. Let j 0 and 1 Ä M Ä N be fixed integers. Let I OEn; M; j be the indicator of the event that the j th M -clique exists after n steps, that is We can see that bOEn; M; k and e n;M are deterministic, while d OEn; M; k; j is an F n 1 -measurable random variable for any n, M , k and j . Using the definition of bOEn; M; k and the Stirling-formula for the Gamma function, we can show that bOEn; M; k b M;k n k˛0 as n ! 1; where b M;k D .1 C˛0k/ > 0, k and M are fixed. Moreover, we can easily see that e n;M 1 ˛0 n˛0 as n ! 1: In the following lemma we introduce a martingale which will play an important role in the paper.
Then .ZOEn; M; k; j ; F n / is a martingale for n l.
Proof. At each step, the weight of a fixed M -clique is increased by 1 if and only if it takes part in an interaction. The total weight of N -cliques after n steps is n C 1. The total weight of .N 1/-cliques after n steps is N.n C 1/. When a new vertex is born and we choose N 1 vertices uniformly, the probability that the vertices of a given M -clique are selected is When we choose N vertices uniformly at random, the probability that the vertices of a given M -clique are chosen is Therefore, as in [15], it is easy to show that the probability that the j th M -clique takes part in interaction at step Using that d OEn C 1; M; k; j is F n -measurable, we obtain the desired result. Multiplying both sides of (24) by e nC1;M , we obtain the result.
Proof of Theorem 2.3. The proof contains two parts. First, we will show that the result is valid with non-negative M;j . Then we will show that M;j is positive with probability 1.
Let B nC1 D fW OEn C 1; M; j D W OEn; M; j C 1g. Consider the event that the j th M -clique exists after n steps. On this event, by (21), The sequence .B n ; n 2 N/ is adapted to the sequence of -algebras .F n ; n 2 N/. Using Corollary VII-2-6 of [19] and (25)    where 0 Ä R n Ä .N 1/ pV n .
Proof. Consider the conditional expectation EfDOEn C 1; j DOEn; j jF n g provided that the j th vertex exists after k steps. When the model evolves according to the preferential attachment rule the degree of a fixed vertex can be increased by 0 or 1 at each step. Therefore, the expected growth of the degree of the j th vertex in the .n C 1/th step when the choice is PA is˛2 W OEn;1;j nC1 . At steps when the choice is UNI the degree of a fixed vertex can be increased at most by N 1. Moreover, using (21), the probability that the growth of the degree of the j th vertex is positive when the choice is UNI is not greater than pV n . Therefore, the expected growth of the degree of the j th vertex in the .n C 1/th step when the choice is UNI is less than or equal to .N 1/ pV n . Proof of Theorem 2.5. Consider the following bounded random variable: n D I OEk;1;j N 1 .DOEn; j DOEn 1; j /. By the Remark 3.3, we have 0 Ä n Ä 1. Applying an appropriate version of Corollary VII-2-6 of [19] (see Proposition 2.4 of [20]), then using Lemma 3.4 and (8), we have provided that the j th vertex exists after k steps. As lim k!1 W OEk; 1; j D 1 a.s., we obtain the statement.
The following lemma will be used to study the maximal weight. It is an extension of Lemma 5.2 in [14]. Suppose that the statement is true for k 1. By Lemma 3.1, ZOEn; 1; k; j is a martingale. The difference of two martingales is also a martingale. So, in the definition of ZOEn; 1; k; j changing I OEl; 1; j for J OEl; 1; j , we obtain again a martingale. Using the definitions of J OEn; 1; j and ZOEn; 1; k; j , we have S OEm; n; k D In the last step we used that W .l; 1; j / D 1 if J OEl; 1; j D 1. Now, we give upper bounds for the two terms in (32) separately. We have already seen that, for a fixed k, the sequence bOEn; 1; k is decreasing. Therefore, applying also (18), is a submartingale. For non-negative numbers the maximum is majorized by the sum. Therefore, and by Lemma 3.5, we obtain E bOEn; 1; k QOEm; n C k 1 k !! Ä S OEm; n; k Ä C k n X j Dm j ˛k : Since 0 Ä .bOEn; 1; 1QOEm; n/ k Ä bOEn; 1; 1 k bOEn; 1; k kŠbOEn; 1; k QOEm; n C k 1 k ! ; we see that the submartingale bOEn; 1; 1QOEm; n is bounded in L k for all k˛> 1. Hence, this submartingale converges almost surely and in L k for every k > 1 . Moreover, by (18)  This relation and (38) imply (11). By relation (41), is a.s. finite.
Proof of Theorem 2.7. The evolution mechanism of the graph implies that DOEn; j Ä .N 1/W OEn; 1; j . Therefore we have maxfDOEn; j W .N 1/ Ä j < mg Ä D n Ä Ä maxfDOEn; j W .N 1/ Ä j < mg C maxf.N 1/W OEn; 1; j W m Ä j Ä ng: Multiplying both sides by n ˛a nd then considering the limit as n ! 1, Theorem 2. n ˛Q OEm; n as n ! 1. As m ! 1, by (42), we obtain the desired result.