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

and Applied Analysis 3 where Γ(⋅) is the Gamma function and


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][8][9][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 S = {0, 1}.Let {  ;  ≥ 2} be a sequence of independent random variables with identical distribution (i.i.d. for short), which take values in S. The distribution of the random variables {  ;  ≥ 2} is given by where  0 +  1 = 1.Let 0 < ,  < 1, and  +  = 1.Here and thereafter we let Now we can define our model based on the information of {  ;  ≥ 0}.Consider the following process which generates a sequence of graphs {  } ≥1 .Time-Step 1.Let (1) be a graph consisting of two vertices V 0 and V 1 with one edge connecting them, which are associated by two type variables  0 and  1 , respectively.

Time-
Step  ≥ 2. The graph () is constructed from ( − 1) in such a way that a vertex V  associated with one new edge, whose type is controlled by a random variable   , is added to the graph ( − 1).Denote the degrees of the vertices For simplicity of the notations, we also denote   () {  =} by  ()   (), and for  ≥ 0, 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 V  is chosen independently from {V 0 , V 1 , . . ., V −1 } 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 V  and V  are of the same type , the probability that V  is chosen as the endpoint of the new edge associated with the new vertex V  is equal to (( ()  ( − 1) +  (,) )/ ∑ −1 j=0 ( ()  ( − 1) +  (,) )), so that, for  = 0, 1, . . .,  − 1 and  = 0, 1, (b) With probability  = 1 − , it is equitable to connect to an old vertex whose type is different from the new one and the probability that V  is chosen is here and thereafter we let F  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  = 0 and  0 = 1, that is, Pr(  = 0) = 1 for all  ≥ 2, 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.

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  ()   () =  ()   ()/( + 1) 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 0 <  < 1, for  ≥ 1 and  = 0, 1, respectively, one has where Γ(⋅) is the Gamma function and the empty product, arising when  = 1, is defined to be equal to one.
Remark 4. (i) If  = 0, the result graph () is a bipartite graph.Moreover, the degree distributions of different types are as follows: for  ≥ 1 and  = 0, 1, (ii) For  = 1 and  ≥ 1, the result graph () has two different connected branches such that one branch is composed of the vertices of type 0, and the vertices of type 1 consist of the other one.Moreover, there is only one initial edge that connects the two branches.Furthermore, for  = 0, 1, the degree distributions of different types of vertices are where    is equal to the value of   when  = 1,  = 0.
We want to say that the proof of ( 9) and ( 10) is very similar to the proof of Theorem 3, so we omitted it.

Corollary 5. In our model, one has
where  ()  for  = 0, 1 are defined as in Theorem 3.
Note the following fact: By using a telescope sum identity we can deduce the following corollary easily.
(iii) 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  ∈ S and fixed  ≥ 1, one has At last, it is also interesting to find out an expression for the joint-probability distribution for degrees of adjacent vertices.For  ∈ S, write  (,) , () for the number of adjacent pairs of vertices with type  whose degrees are  and , respectively, at time  and  (,1−) , () for the number of vertices of degree  with type  which attach to a vertex of degree  with type 1 −  at time . ()   () is defined as before.For  = 0, we have the following.

Theorem 9. (𝑖) In our model, for 𝑠 ∈ S, the joint degree distributions of pairs of adjacent vertices with the same type are
for  ≥ 1,  ≥ 2, the vertex of type  with degree  is younger than the vertex of type  with degree , where and for all  ≥ 2 (ii) For  0 =  1 = 1/2, one also has where for  ≥ 1,  ≥ 2, when the vertex of type  with degree  is younger than the vertex of type 1 −  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.

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 Lemma 11 (see [11]).Let  ∈ [1,2] In the following, in graph ( − 1), we denote the total degree of vertices with type  by  ()  −1 , that is, and the number of vertices of type  by  () −1 ; that is, Now we conclude this section by stating the following two lemmas whose proofs are proposed in the appendix.

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 F  compared to the information contained in F −1 ; 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  = 1 for  = 0, 1, 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  = 1 in (A.8) and combining with (A.16) we can easily obtain where the constant  1 is independent of the parameter , and we can take a constant  1 such that Similarly there also exist constants  2 (independent of the parameter ) and  2 such that where 0 < 2 − (1/ 2 ) < 1.We denote noting the initial condition that  () 1 (1) = 1 so that Δ () 1 (1) = 0; then for  large enough we have (we let the vacant product be equal to 1) Thus it follows that then we arrive at Thus Theorem 3 and the Borel-Cantelli lemma imply that where  () 1 is defined by (6).For all , we take  2 ≤  ≤ (+1) Similarly to the case  > 1 and  = 0, 1, we can also prove that where  ()  is defined as in Theorem 3.

Proof of
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  − 1 or  − 1 (both gain) with type , while the third set of terms gives the change in  (,) , ( − 1) 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  (,) 1, ( − 1) due to the addition on the new vertex.
Similarly, for  ∈ S we also have where the vertex of type  with degree  is younger than the vertex of type 1 −  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   ( = 1, 2, 3) such that             (,) , ( − 1) 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.