The Degree Analysis of an Inhomogeneous Growing Network with Two Types of Vertices

Abstract

We consider an inhomogeneous growing network with two types of vertices. The degree sequences of two different types of vertices are investigated, respectively. We not only prove that the asymptotical degree distribution of type for this process is power law with exponent , but also give the strong law of large numbers for degree sequences of two different types of vertices by using a different method instead of Azuma’s inequality. Then we determine asymptotically the joint probability distribution of degree for pairs of adjacent vertices with the same type and with different types, respectively.

1. Introduction

Recently there has been much interest in studying inhomogeneous large-scale networks and attempting to model their topological properties. The classical random graph models are generally homogeneous, in the sense that all the vertices come in the same type (for static random graphs, see [1]; for growing random graphs, see [2, 3]). In contrast, many large real-world graphs are highly 0 inhomogeneous. In fact, vertices of many real networks are born of difference, and this difference by birth may influence the evolving of the networks to some extent. In order to depict such phenomenon, Söderberg [4] presented a class of inhomogeneous random graph models of order , by means of a straightforward generalization of the classic E-R model to a situation where vertices may come in different types, such that the probability that an edge arises depends on the types of its pair of terminal vertices. Bollobás et al. [5], based on the work of Söderberg [4], introduce a model of an inhomogeneous random graph with conditional independence between the edges; moreover various results have been proved. Their model also includes some special cases such as the CHKNS model [6] and Turova’s model [7–10]. Recently, van der Hofstad [11] studies the critical behavior of inhomogeneous random graphs where edges are present independently but with unequal edge occupation probabilities and shows that this critical behavior depends sensitively on the asymptotic properties of their degree sequence. For more substantial details about such inhomogeneous random graph, we can see van der Hofstad [12]. Most of those models are static and do not involve the effect of preferential attachment. Papadopoulos et al. [13] point out that it is very important to study inhomogeneous growing networks which combine the effects of popularity and similarity. There are also many papers studying synchronization control of dynamical networks; see [14] and the references therein.

In this paper, our main purpose is to define an inhomogeneous model which can combine the effects of preferential attachment and difference by birth and then to provide quantitative descriptions of its properties of degree sequence. Our model inherits certain features from homogeneous growing model such that it is capable of producing asymptotic degree distributions such as power law distributions and exponential distributions by choosing proper parameters.

1.1. Definition of Our Model

At first, we introduce a type space . Let be a sequence of independent random variables with identical distribution (i.i.d. for short), which take values in . The distribution of the random variables is given by where . Let , , and . Here and thereafter we let

Now we can define our model based on the information of . Consider the following process which generates a sequence of graphs .

Time-Step 1. Let be a graph consisting of two vertices and with one edge connecting them, which are associated by two type variables and , respectively.

Time-Step . The graph is constructed from in such a way that a vertex associated with one new edge, whose type is controlled by a random variable , is added to the graph . Denote the degrees of the vertices by at time . For simplicity of the notations, we also denote by , and for , we write for , where is a nonnegative real number, and is the indicator function. When the new vertex arrives at the system, the endpoint of the new edge emanating from vertex is chosen independently from according to their different types.

The attachment procedures of the new edge proceeded as follows.(a)With probability , it is preferential to attach to an old vertex whose type is the same as the new one. When the vertices and are of the same type , the probability that is chosen as the endpoint of the new edge associated with the new vertex is equal to , so that, for and , (b)With probability , it is equitable to connect to an old vertex whose type is different from the new one and the probability that is chosen is here and thereafter we let be the -algebra associated with the probability space up to time .

Remark 1. In our model, (3) depicts preferential competitive mechanism among the vertices with the same type. On the other hand, (4) describes the fair competitive effect on the vertices which are of different types. In our model, we can consider the degree of a vertex as an indication of its success, so that vertices with large degree correspond to successful vertices. Naturally, in reality, both the previous success of a vertex and its initial type may play important roles in the final success of the vertex. In our model individuals arrive at the network with different types and one initial edge, which form the basis for their future success. Heuristically, the larger the group is, the more the individuals which are successful exist; on the other hand, the rich will be richer as time increases.

Remark 2. Our model is different from the inhomogeneous model [5] in that it is dynamic; more precisely, a new vertex is added to the graph at each integer time. If and , that is, for all , our model reduces to the original preferential model which is very similar to the one from Barabási and Albert [2], once the types are ignored.

1.2. States of Our Main Results

What we are interested in is the limit distribution of the degree sequences of different types in the resulting graph . Let be the number of vertices of type with degree in graph . Define as the fraction of vertices of type with degree . What we are concerned about is the limiting behavior of as tends to infinity.

Theorem 3. If  , for and , respectively, one has where is the Gamma function and the empty product, arising when , is defined to be equal to one.

Remark 4. (i) If , the result graph is a bipartite graph. Moreover, the degree distributions of different types are as follows: for and ,
(ii) For and , the result graph has two different connected branches such that one branch is composed of the vertices of type , and the vertices of type consist of the other one. Moreover, there is only one initial edge that connects the two branches. Furthermore, for , the degree distributions of different types of vertices are where is equal to the value of when , .

We want to say that the proof of (9) and (10) is very similar to the proof of Theorem 3, so we omitted it.

For all , let be the number of vertices with degree in graph . Define as the fraction of vertices with degree . Since , the following corollary is a direct result of Theorem 3.

Corollary 5. In our model, one has where for are defined as in Theorem 3.

Note the following fact: By using a telescope sum identity we can deduce the following corollary easily.

Corollary 6. For all and , are defined by (6); then one has the total fractions of vertices of types and in our limit graph which are equal to , respectively; that is,

Remark 7. We know that is a sequence of i.i.d. random variables; thus by strong law of large numbers, , as . Thus Corollary 6 is exactly consistent with it.
From Corollary 5, for , , we can easily find that if , the total degree sequence follows power law with the exponent ; otherwise, is a linear combination of two different power law distributions.
From our analysis of the degree distribution of different types, we can find that our model grasps two heuristic phenomena as follows: one is “rich-get-richer” effect; the other is that the larger the group is, the more the successful individuals in that group are.

We are also concerned about the strong law of large numbers of degree sequences for different types, respectively, as follows.

Theorem 8. For and fixed , one has

At last, it is also interesting to find out an expression for the joint-probability distribution for degrees of adjacent vertices. For , write for the number of adjacent pairs of vertices with type whose degrees are and , respectively, at time and for the number of vertices of degree with type which attach to a vertex of degree with type at time . is defined as before. For , we have the following.

Theorem 9. In our model, for , the joint degree distributions of pairs of adjacent vertices with the same type are for , , the vertex of type with degree is younger than the vertex of type with degree , where and for all
For , one also has where for , , when the vertex of type with degree is younger than the vertex of type with degree .

The rest of this paper is organized as follows. In Section 2, we state four lemmas which are useful to prove our main results. Especially, in Lemma 12 we compute the expectation of the total degree of vertices with type and the total number of vertices with type , respectively. In Lemma 13 we give the moment inequalities for the total degree of vertices with type and the total number of vertices with type . In Section 3, we give the proofs of our main results in Theorems 3 and 8. In Section 4, we prove our main result about joint-probability distribution for degrees of adjacent vertices with the same type and different types, respectively. In the appendix, we give the proof of Lemmas 12 and 13.

2. Preliminaries

The following two lemmas are useful to prove our main results. The readers who are interested in their details can refer to the associated materials.

Lemma 10 (see [15]). Suppose that a sequence satisfies the recurrence relation for . Furthermore, suppose that and . Then exists and

Lemma 11 (see [11]). Let and suppose that are i.i.d. sequence with and . Then there exists a constant depending only on , such that

In the following, in graph , we denote the total degree of vertices with type by , that is, and the number of vertices of type by ; that is,

Now we conclude this section by stating the following two lemmas whose proofs are proposed in the appendix.

Lemma 12. For , one has the following:

Lemma 13. Let and be defined as before; then (i) for , there exists a constant which is independent of the parameter such that (ii) one has the following:

3. Proof of Theorems 3 and 8

3.1. The Master Equations for the Degree Sequences of Vertices with Two Different Types

For large enough, it is easy to obtain two master equations of the degree sequences of different types as follows:

Now let us come to solve (27). Taking expectation of both sides of above equation and rewriting it, we get

Note that, for any and , we have the following inequalities:

Combining Lemma 12, Lemma 13, triangle inequality, and (28)–(30), the last four sets of terms of (28) tend to zero as ; then for large enough, we have

For , all , note that we have for ; thus combining (31) we obtain where

For and all , according to (31), and for , respectively, we also have

To solve the above two equations ((32) and (34)), we want to show that the expected values follow power laws as goes to infinity. To see it, we proceed by induction on to show that the limits exist for each .

3.2. Proof of Theorem 3

Proof of Theorem 3. For , according to (33), we have Thus we apply Lemma 10 with (as ) to obtain that exists and
We suppose that exists; we use Lemma 10 again with Then we can arrive at the limit which exists and is equal to Thus we can get (5) immediately, and (6) is a direct result of (5).
By Stirling's formula, we have that as , from which it follows that    for some constants . Thus the power law exponent is .

3.3. Proof of Theorem 8

In our model, Azuma's inequality no longer works because there is no uniform bound on the change in the number of when we investigate the influence of the extra information contained in compared to the information contained in ; that is, we have to bound the difference . It is very difficult to do it for our model, so we use our method instead of Azuma’s inequality, in which there is no need to use such a uniform bound.

Proof of Theorem 8. At first we note a basic fact as follows: At first, we consider the case for , respectively. Consider where
Now let us come to estimate (41) and (42), respectively. For (41), we note the basic fact that By Hölder’s inequality, letting in and combining with we can easily obtain where the constant is independent of the parameter , and we can take a constant such that Similarly there also exist constants (independent of the parameter ) and such that where . We denote noting the initial condition that so that ; then for large enough we have Thus it follows that then we arrive at Thus Theorem 3 and the Borel-Cantelli lemma imply that where is defined by (6). For all , we take ; we have Then we have thus we get Similarly to the case and , we can also prove that where is defined as in Theorem 3.

4. Proof of Theorem 9

4.1. Master Equations for Joint Degree Sequences and

Let ; we can easily get the master equations as follows: for where we suppose that the vertex of type with degree is younger than the vertex of type with degree . The first two sets of terms on the right-hand side account for the change in due to the addition of the new edge hitting a vertex of degree or (both gain) with type , while the third set of terms gives the change in due to the addition of the new edge onto the ancestor vertex of degree or (both loss) with type . Finally, the last term accounts for the gain in due to the addition on the new vertex.

Similarly, for we also have where the vertex of type with degree is younger than the vertex of type with degree .

We also notice a fact that the total number of pairs of adjacent vertices is equal to the number of edges at time . The total number of edges is in the resulting graph ; obviously, there exist constants such that Similarly to the analysis of (27), for large enough, combining Lemmas 12 and 13 and (58), we can get by (56) and (57), respectively,

4.2. Proof of Theorem 9

To solve the above equations ((59), (60), resp.), we come to prove Theorem 9 as follows.

Proof of Theorem 9. At first we have so that Thus we have combining Lemma 10 with Theorem 3, it follows that here and thereafter is defined as (5) in Theorem 3 when .
For , , by induction hypothesis, we suppose that Thus by using Lemma 10 and Theorem 3 again, we arrive at existence and satisfy the following time-independent recursion relation: which is equivalent to Now let us come to solve the difference equation (67). Letting substituting the above equation (68) in (67), and combining (6), we arrive at where , are defined as (16) and (17) for all , respectively.
Moreover, by (69) we have Now we define -dimensional generating function as follows: it follows that so that furthermore, taking , for , and expanding , then Comparing (71) with (74), it follows that so we arrive at (10) immediately. Furthermore, if , similarly to the analysis of (59), by doing the same procedure to (60), (13) can be easily derived. Thus we complete the proof of Theorem 9.

Appendix

Proof of Lemma 12. For , , and , we define a sequence of random variables as follows: It is simple to see that For , Moreover combining (3) and (4), we have so for the case , Similarly we can get Thus (i) holds. Assertion (ii) is obvious since the are i.i.d. sequence.

Proof of Lemma 13. (i) Applying Lemma 11 to and combining with Hölder's inequality we have for and where .
(ii) We introduce for where is -field generated by the empty graph. By Lemma 12 we have ; the process is a Doob martingale with respect to . Moreover, we have and , so that Furthermore, we claim that To see this, we at first consider the case , noticing that so that It follows that Similarly we get Combining this fact and the martingale property we arrive at Applying Hölder’s inequality again, we can get assertion (ii). Thus we complete the proof of Lemma 13.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author also thanks the anonymous reviewers for valuable comments and suggestions that improved the presentation of this paper. This work is supported by the National Natural Science Foundation of China no. 11201344.