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Journal of Probability and Statistics
Volume 2013, Article ID 707960, 12 pages
http://dx.doi.org/10.1155/2013/707960
Research Article

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

Faculty of Informatics, University of Debrecen, P.O. Box 12, Debrecen 4010, Hungary

Received 8 April 2013; Accepted 1 August 2013

Academic Editor: Shein-chung Chow

Copyright © 2013 István Fazekas and Bettina Porvázsnyik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 can be achieved. The proofs are based on martingale methods.

1. Introduction

During the last 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, , as . 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 and , for the Internet , for the movie actor network , and for the collaboration graph of mathematicians (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 a new vertex with edges is added so that the edges link the new vertex to existing vertices. The probability that the new vertex will be connected to the old vertex depends on the degree of vertex , so that . 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 is one tetragon. The initial weight of this graph is one. This graph contains vertices, edges, and triangles. Each of these objects has initial weight . After the initial step we start to increase the size of the graph. The main feature of the procedure is that at each step 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 . (That is, we increase the weights of vertices, edges, triangles, and tetragon.) The choice of the four vertices is the following.

There are two possibilities at each step. With probability we add a new vertex that interacts with three old vertices, on the other hand, with probability four old vertices interact. Here is fixed.

When we add a new vertex, then we choose old vertices and they together will form a tetragon. However, to choose the three old vertices we have two possibilities. With probability we choose a triangle from the existing triangles according to the weights of the triangles. It means that a triangle of weight is chosen with probability (preferential attachment rule). On the other hand, with probability , 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 , we choose among the existing tetragons according to their weights. It means that a tetragon of weight is chosen with probability (preferential attachment rule). On the other hand, with probability , 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 denote the number of vertices of weight and degree after the th step. Let denote the number of vertices after the th step. Let denote the -algebra of observable events after the th step. First we calculate the conditional expectation , see Lemma 2. Then we prove (Theorem 3) that almost surely (a.s.) as , where 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 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 , , , is a proper two-dimensional discrete probability distribution (Lemma 4). Then we turn to the scale-free property for the weights. Let denote the number of vertices of weight after the th step. Then for all we have almost surely and , as (Theorem 5). To derive the above results from Theorem 3, we need only some known facts about the -function, see Lemma 4 and Theorem 5. Finally, we obtain the scale-free property for the degrees. Let us denote by the number of vertices of degree after the th step. For any we have almost surely as , where are positive numbers. Furthermore, as , where , , and are appropriate constants (see Theorem 8). In both cases the exponent is . We can see that its value can be any number in .

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 -vertice model are much longer than those of the -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 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 -vertices model, too. Therefore the asymptotic results of the - and -vertice models can be extended to the -vertices model. Moreover, several versions of the -vertices model can be constructed and their fit to real-life data can be studied.

2. The Evolution of the Graph

Throughout the paper , , are fixed numbers. Let denote the number of vertices of weight and degree after the th step. Let denote the number of vertices after the th step.

Remark 1. Each vertex has initial weight and initial degree . When a vertex takes part in an interaction, then its weight is increased by and its degree may be increased by 0, 1, 2, or . Therefore can be positive only for and .
Let denote the -algebra of observable events after the th step. We compute the conditional expectation of with respect to for .
Let The following lemma contains the basic equation of the paper.

Lemma 2. We have for and . Here denotes the Dirac-delta.

Proof. The total weight of tetragons after steps is . The total weight of triangles after steps is . The total weight of the triangles having a fixed common vertex of weight is . Moreover, after steps, we have the following. When we choose three vertices randomly, then the probability that a given vertex is chosen is When we choose four vertices randomly, then the probability that a given vertex is chosen is Therefore the probability that an old vertex of weight takes part in the interaction at step is where and are defined by (4). A new vertex always takes part in the interaction. At each step with probability a new vertex with weight 1 and with degree 3 is born. This explains term in (5).
Consider a fixed vertex with weight and degree . The probability of the event that in the th step (i) neither its degree nor its weight change is (ii) its degree does not change but its weight is increased by 1 is (iii) both its degree and its weight are increased by 1 is (iv) its degree is increased by 2 and its weight is increased by 1 is (v) its degree is increased by 3 and its weight is increased by 1 is Using the above formulae, we obtain (5).

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

Theorem 3. Let , , and . For any fixed and with and one has almost surely as , where are fixed nonnegative numbers. Furthermore, the numbers satisfy the following recurrence: for , , where , , , and are given by (4). One has if , unless . If , then one has ; moreover, in this case where is a positive number which may depend on and . If does not satisfied, then .

Proof. During the proof we take care of the case which does not appear in [7]. Introduce notation is an -measurable random variable. Applying the Marcinkiewicz strong law of large numbers to the number of vertices, we have almost surely, for any .
Using (18) and the Taylor expansion for , we obtain where the error term is convergent as . Therefore almost surely, as , where is a positive random variable.
Let Using (5), we can see that is a nonnegative submartingale for any fixed , . Define for . Applying the Doob-Meyer decomposition to , we can write where is a martingale and is a predictable increasing process. The general form of and is the following: where is the trivial -algebra. Using (5) and (24), we obtain Let denote the sum of the conditional variances of the process . Now we will give an upper bound of as follows: Above we used that is -measurable and at each step four vertices can interact. Jensen’s inequality implies that is a (nonnegative) submartingale if is a martingale. Now we can apply the Doob-Meyer decomposition to . It is known that , that is the sum of the conditional variances of terms 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 . Therefore the Doob-Meyer decomposition is where is a martingale and the predictable increasing process is given by (26).
We use induction on . Let . We can see that a vertex of weight could take part in an interaction when it was born. Therefore its degree must be equal to . By (25) and (20), almost surely as . By (26), and therefore . It follows from Proposition VII-2-4 of [8] (see Proposition 10 in the Appendix) that As, by (28), almost surely, therefore using (18) and (20), relation (29) implies almost surely. So (14) is valid for .
Now let . In this case the degree of the vertex must be . If and , then we will show . By (18), (20), and (25), we can find the asymptotic behaviour of as Moreover, by (26), ; thus, for , , and . Therefore in these cases almost surely on as . It implies that with appropriate . In particular, we have as . So (14) is valid for and .
However, the case , is different from the previous cases. As , the relation would not be enough to proceed the induction. By (25) and using Remark 1, we have Using (18), (34), and the limit of , we obtain Now we have We denoted by the increasing predictable process in the Doob-Meyer decomposition of . We know, by (26), that and so with arbitrary small positive .
Applying Propositions VII-2-3 and VII-2-4 of [8] (see Proposition 9 in the Appendix), we have Moreover, on the set , the sequence is almost surely. convergent. So almost surely. Therefore, using (18), (20), and (35), we obtain if where . So the proposition is valid for and .
Now, suppose that the statement is true for all weights less than and for all possible degrees. First we study the positive limits. Consider in (25) and assume that at least one of the coefficients , , is positive. Then by (18), (20), and using the induction hypothesis, we see that In the above computation we deleted all terms having asymptotically smaller degree than the largest one.
Consider the case when and . In this case, by the induction hypothesis, there is at least one positive term in (39). Therefore (39) implies (because ). In this case . So, using Proposition VII-2-4 of [8] (see Proposition 10 in the Appendix), we have . Therefore where, by (39), with , , , and defined by (4).
To handle the case when the limit is , we argue as follows. Consider the case when . By (25) and using the induction hypothesis, we have
On the other hand, . 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 So we have obtained the desired result for the case of limit as well.

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

Lemma 4. Let and define for . Then , , are positive numbers satisfying the following recurrence: where and are defined by (4). , , is a discrete probability distribution. Moreover, , , , is a two-dimensional discrete probability distribution.

Proof. If , then the statement is obvious. Now assume . As is defined as for , therefore . From the recurrence (15) for, we obtain Using this recursive formula for , we obtain Moreover, by [9], we have the following formula: Therefore, by some calculation, we obtain , as . So . As , so and therefore , , , is a (proper) two-dimensional discrete probability distribution.

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

Theorem 5. Let , , and . Then for all one has almost surely, as , where , , are positive numbers satisfying the recurrence (45). Moreover, as , with .

Proof. We have Therefore, by Theorem 3, almost surely, as . Here each is positive.
Using formula (47) and the Stirling-formula for the Gamma function, we obtain where and .

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

Let be a random variable with distribution . Let and be independent random variables being independent of , too. For let have the following distribution: Introduce notation .

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

Theorem 6. for all , .

Proof. It is easy to see that if and , then we have If and we have Generally, for all and we have because one of should be equal to 2, which is of zero probability. Furthermore, using the recursion (45) and the assumption that are independent random variables which are independent of , we have Therefore, the sequence satisfies the same recursion (15) as .

Theorem 7. Suppose that and . Then where the error term does not depend on .

Proof. We can follow the ideas of Theorem  4.2 in [7]. For we have hence, as . Similarly, by simple computation, we have as .
Now, we can apply Theorem VII.2.5 in [10] (see Proposition 11 in the Appendix) for . The conditions of that theorem are satisfied; therefore, we have Using (62) and (63), we obtain . Therefore, it follows from (63) that The independence of and implies that . 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 interactions. The theorem shows the scale-free property for the degrees.

Theorem 8. Let , , and . Let us denote by the number of vertices of degree after steps; that is, . Then, for any one has almost surely as , where are positive numbers. Furthermore, as .

Proof. By Theorems 3 and 6, converges almost surely to the distribution . But the cardinalities of terms in are not bounded when . However, using that , , 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 with some fixed .
Using (61) and Hoeffding’s exponential inequality (Theorem  2 in [11], see Proposition 12 in the Appendix) we obtain for Here implies that Therefore in the case when we have Using this, we can obtain that Similarly, if , again by Hoeffding’s inequality, we have Using that and , we obtain . Therefore Hence
Now consider the case when . First we need some general facts. Consider the set It is easy to see that when then if and only if . More precisely, As , so we have . Then (with arbitrarily small) as . Here the error term does not depend on . We will apply Theorem 7, that is, formula (59). The asymptotic behaviour of is known from (50). Using these facts and (77), we obtain as and , where . Therefore Thus we have as . Finally, (71), (74), and (80) imply as . The proof is complete.

Appendix

We use the following results on discrete time martingales. Let be a submartingale. Its Doob-Meyer decomposition is