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

Authors: Huilin Huang
Journal: Abstract and Applied Analysis
 (2014)

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## 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 types for this process is power law with exponent 2+1+δqs+β1-qs/αqs, 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.


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## Body

## 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 N, 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 space𝒮={0,1}. Let {Yt;t≥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 {Yt;t≥2} is given by
(1)P(Yt=0)=q0,P(Yt=1)=q1,
where q0+q1=1. Let 0<α, β<1, and α+β=1. Here and thereafter we let
(2)Y0≡0,Y1≡1.Now we can define our model based on the information of{Yt;t≥0}. Consider the following process which generates a sequence of graphs {Gt}t≥1.Time-Step 1. Let G(1) be a graph consisting of two vertices v0 and v1 with one edge connecting them, which are associated by two type variables Y0 and Y1, respectively.Time-Step
t
≥
2
. The graph G(t) is constructed from G(t-1) in such a way that a vertex vt associated with one new edge, whose type is controlled by a random variable Yt, is added to the graph G(t-1). Denote the degrees of the vertices {v0,v1,…,vt-1} by d0(r),d1(r),…,dt-1(r) at time r≥t-1. For simplicity of the notations, we also denote di(r)I{Yi=s} by di(s)(r), and for i≥0, we write δ(i,s) for δI{Yi=s}, where δ is a nonnegative real number, and I{·} is the indicator function. When the new vertex arrives at the system, the endpoint of the new edge emanating from vertex vt is chosen independently from {v0,v1,…,vt-1} according to their different types.The attachment procedures of the new edge proceeded as follows.(a)
With probabilityα, it is preferential to attach to an old vertex whose type is the same as the new one. When the vertices vi and vt are of the same type s, the probability that vi is chosen as the endpoint of the new edge associated with the new vertex vt is equal to α((di(s)(t-1)+δ(i,s))/∑j=0t-1‍(dj(s)(t-1)+δ(j,s))), so that, for i=0,1,…,t-1 and s=0,1,
(3)Pr(vt⟶vi,Yi=s,Yt=s∣ℱt-1)=αqsdi(s)(t-1)+δ(i,s)∑j=0t-1(dj(s)(t-1)+δ(j,s)).(b)
With probabilityβ=1-α, it is equitable to connect to an old vertex whose type is different from the new one and the probability that vi is chosen is
(4)Pr(vt⟶vi,Yi=1-s,Yt=s∣ℱt-1)=βqsI{Yi=1-s}∑j=0t-1I{Yj=1-s};here and thereafter we let ℱt be the σ-algebra associated with the probability space up to time t.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 β=0 and q0=1, that is, Pr(Yt=0)=1 for all t≥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.


### 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 graphG(t). Let Nk(s)(t) be the number of vertices of type s with degree k in graph G(t). Define Pk(s)(t)=Nk(s)(t)/(t+1) as the fraction of vertices of type s with degree k. What we are concerned about is the limiting behavior of Pk(s)(t) as t tends to infinity.Theorem 3.
If0<α<1, for k≥1 and s=0,1, respectively, one has
(5)Pk(s)=lim⁡t→∞E[Nk(s)(t)]t+1=qs×(1+α(1+δ)qs(1+δ+α)qs+β(1-qs)+β(1-qs)qs)-1=·∏j=2k‍(+[qs+β(qs)]αqs2)-1(β[(1+δ+α)qs]αqs2j-1+δ=β(1-)qs)+β(1-qs)[(1+δ+α)qs+β(1-qs)]αqs2)=w∏j=2k‍×([(1+δ+α)qs]αqs2j+δ+(1+δ+α)qs+β(1-qs)αqs=w∏j=2k‍we+β(1-qs)[(1+δ+α)qs+β(1-qs)]αqs2)-1)(6)=cs(+βqs+β(qs)]αqs2))-1Γ(k+δ+β(1-qs)[(1+δ+α)qs+β(1-qs)]αqs2)cse×=×(Γ(k+1+δ+(1+δ+α)qs+β(1-qs)αqscs×==csw×+β(1-qs)[(1+δ+α)qs+β(1-qs)]αqs2))-1)(7)≈csk-(2+((1+δ)qs+β(1-qs))/αqs),
where Γ(·) is the Gamma function and
(8)cs=qs×(1+α(1+δ)qs(1+δ+α)qs+β(1-qs)+β(1-qs)qs)-1×Γ(2+δ+(1+δ+α)qs+β(1-qs)αqswww+β(1-qs)[(1+δ+α)qs+β(1-qs)]αqs2)×([(1)qs(1-qs)]αqs2))Γ([(α)qs(1-qs)]αqs21+δ×(Γ(1δeww+β(1-qs)[(1+δ+α)qs+β(1-qs)]αqs2))-1;
the empty product, arising when k=1, is defined to be equal to one.Remark 4.
(i) Ifα=0, the result graph G(t) is a bipartite graph. Moreover, the degree distributions of different types are as follows: for k≥1 and s=0,1,
(9)Pk(s)=(1-qs)k-1qs2.
(ii) Forα=1 and t≥1, the result graph G(t) 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 s=0,1, the degree distributions of different types of vertices are
(10)Pk(s)≈cs′k-(3+δ),
where cs′ is equal to the value of cs when α=1, β=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.For allk≥1, let Nk(t) be the number of vertices with degree k in graph G(t). Define Pk(t)=Nk(t)/(t+1) as the fraction of vertices with degree k. Since Nk(t)=Nk(0)(t)+Nk(1)(t), the following corollary is a direct result of Theorem 3.Corollary 5.
In our model, one has(11)Pk=lim⁡t→∞E[Nk(t)]t+1=Pk(0)+Pk(1),
where Pk(s) for s=0,1 are defined as in Theorem 3.Note the following fact:(12)Γ(k+a)Γ(k+b)=1b-a-1[Γ(k+a)Γ(k-1+b)-Γ(k+1+a)Γ(k+b)].
By using a telescope sum identity we can deduce the following corollary easily.Corollary 6.
For allk≥1 and s=0,1, Pk(s) are defined by (6); then one has the total fractions of vertices of types 0 and 1 in our limit graph which are equal to qs(s=0,1), respectively; that is,
(13)∑k=1∞‍Pk(s)=qs,s=0,1.Remark 7.
(
i
) We know that {Yi;i≥2} is a sequence of i.i.d. random variables; thus by strong law of large numbers, Tt-1(s)/t→qs, (s=0,1) as t→∞. Thus Corollary 6 is exactly consistent with it.
(
ii
) From Corollary 5, for 0<α, β<1, we can easily find that if q0=q1=1/2, the total degree sequence Pk follows power law with the exponent 2+(1+δ+β)/α; otherwise, Pk is a linear combination of two different power law distributions.
(
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.
Fors∈𝒮 and fixed k≥1, one has
(14)lim⁡t→∞Pk(s)(t)=Pk(s),a.s.At last, it is also interesting to find out an expression for the joint-probability distribution for degrees of adjacent vertices. Fors∈𝒮, write Nk,l(s,s)(t) for the number of adjacent pairs of vertices with type s whose degrees are k and l, respectively, at time t and Nk,l(s,1-s)(t) for the number of vertices of degree k with type s which attach to a vertex of degree l with type 1-s at time t. Nk(s)(t) is defined as before. For δ=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
(15)Pk,l(s,s)=lim⁡t→∞E[Nk,l(s,s)(t)]t+1Pk,l(s,s)=Γ(k+β(1-qs)[(1+α)qs+β(1-qs)]αqs2)Pk,l(s,s)×Γ(l+β(1-qs)[(1+α)qs+β(1-qs)]αqs2)Pk,l(s,s)×(Γ(k+l+1+(1+α)qs+β(1-qs)αqs×wwPk,l(s,s)wew+2β(1-qs)[(1+α)qs+β(1-qs)]αqs2))-1Pk,l(s,s)·γ(s,s)∑j=0l-2‍Γ(k+l-j-2)Γ(l-j-1)Γ(k)π(s,s)(j+2);
for k≥1, l≥2, the vertex of type s with degree k is younger than the vertex of type s with degree l, where
(16)γ(s,s)=Γ(2+(1+α)qs+β(1-qs)αqs=w=+β(1-qs)[(1+α)qs+β(1-qs)]αqs2)×([Γ(1+β(1-qs)[(1+α)qs+β(1-qs)]αqs2)]2)-1×P1(s),
and for all l≥2(17)π(s,s)(l)=Γ(1+l+(1+α)qs+β(1-qs)αqswwew+2β(1-qs)[(1+α)qs+β(1-qs)]αqs2)×(Γ(l+(1+α)qs+β(1-qs)αqswwwweww+β(1-qs)[(1+α)qs+β(1-qs)]αqs2))-1·(l-1)×(l-1+β(1-qs)[(1+α)qs+β(1-qs)]αqs2)-1.
(
i
i
) For q0=q1=1/2, one also has
(18)Pk,l(s,1-s)=lim⁡t→∞E[Nk,l(s,1-s)(t)]t+1=Γ(k+2β/α)Γ(l+2β/α)Γ(k+l+1+2/α+4β/α)·γ(s,1-s)∑j=0l-2‍Γ(k+l-j-2)Γ(l-j-1)Γ(k)π(s,1-s)(j+2),
where
(19)γ(s,1-s)=Γ(2+2(1+β)/α)[Γ(1+2β/α)]2·2βα[α+2(1+β)],π(s,1-s)(l)=1l-1+2β/α·Γ(l+1+2(1+2β)/α)Γ(l+2(1+2β)/α),
for k≥1, l≥2, when the vertex of type s with degree k is younger than the vertex of type 1-s with degree l.The rest of this paper is organized as follows. In Section2, 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 s and the total number of vertices with type s, respectively. In Lemma 13 we give the moment inequalities for the total degree of vertices with type s and the total number of vertices with type s. 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.


## 1.1. Definition of Our Model

At first, we introduce a type space𝒮={0,1}. Let {Yt;t≥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 {Yt;t≥2} is given by
(1)P(Yt=0)=q0,P(Yt=1)=q1,
where q0+q1=1. Let 0<α, β<1, and α+β=1. Here and thereafter we let
(2)Y0≡0,Y1≡1.Now we can define our model based on the information of{Yt;t≥0}. Consider the following process which generates a sequence of graphs {Gt}t≥1.Time-Step 1. Let G(1) be a graph consisting of two vertices v0 and v1 with one edge connecting them, which are associated by two type variables Y0 and Y1, respectively.Time-Step
t
≥
2
. The graph G(t) is constructed from G(t-1) in such a way that a vertex vt associated with one new edge, whose type is controlled by a random variable Yt, is added to the graph G(t-1). Denote the degrees of the vertices {v0,v1,…,vt-1} by d0(r),d1(r),…,dt-1(r) at time r≥t-1. For simplicity of the notations, we also denote di(r)I{Yi=s} by di(s)(r), and for i≥0, we write δ(i,s) for δI{Yi=s}, where δ is a nonnegative real number, and I{·} is the indicator function. When the new vertex arrives at the system, the endpoint of the new edge emanating from vertex vt is chosen independently from {v0,v1,…,vt-1} according to their different types.The attachment procedures of the new edge proceeded as follows.(a)
With probabilityα, it is preferential to attach to an old vertex whose type is the same as the new one. When the vertices vi and vt are of the same type s, the probability that vi is chosen as the endpoint of the new edge associated with the new vertex vt is equal to α((di(s)(t-1)+δ(i,s))/∑j=0t-1‍(dj(s)(t-1)+δ(j,s))), so that, for i=0,1,…,t-1 and s=0,1,
(3)Pr(vt⟶vi,Yi=s,Yt=s∣ℱt-1)=αqsdi(s)(t-1)+δ(i,s)∑j=0t-1(dj(s)(t-1)+δ(j,s)).(b)
With probabilityβ=1-α, it is equitable to connect to an old vertex whose type is different from the new one and the probability that vi is chosen is
(4)Pr(vt⟶vi,Yi=1-s,Yt=s∣ℱt-1)=βqsI{Yi=1-s}∑j=0t-1I{Yj=1-s};here and thereafter we let ℱt be the σ-algebra associated with the probability space up to time t.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 β=0 and q0=1, that is, Pr(Yt=0)=1 for all t≥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.


## 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 graphG(t). Let Nk(s)(t) be the number of vertices of type s with degree k in graph G(t). Define Pk(s)(t)=Nk(s)(t)/(t+1) as the fraction of vertices of type s with degree k. What we are concerned about is the limiting behavior of Pk(s)(t) as t tends to infinity.Theorem 3.
If0<α<1, for k≥1 and s=0,1, respectively, one has
(5)Pk(s)=lim⁡t→∞E[Nk(s)(t)]t+1=qs×(1+α(1+δ)qs(1+δ+α)qs+β(1-qs)+β(1-qs)qs)-1=·∏j=2k‍(+[qs+β(qs)]αqs2)-1(β[(1+δ+α)qs]αqs2j-1+δ=β(1-)qs)+β(1-qs)[(1+δ+α)qs+β(1-qs)]αqs2)=w∏j=2k‍×([(1+δ+α)qs]αqs2j+δ+(1+δ+α)qs+β(1-qs)αqs=w∏j=2k‍we+β(1-qs)[(1+δ+α)qs+β(1-qs)]αqs2)-1)(6)=cs(+βqs+β(qs)]αqs2))-1Γ(k+δ+β(1-qs)[(1+δ+α)qs+β(1-qs)]αqs2)cse×=×(Γ(k+1+δ+(1+δ+α)qs+β(1-qs)αqscs×==csw×+β(1-qs)[(1+δ+α)qs+β(1-qs)]αqs2))-1)(7)≈csk-(2+((1+δ)qs+β(1-qs))/αqs),
where Γ(·) is the Gamma function and
(8)cs=qs×(1+α(1+δ)qs(1+δ+α)qs+β(1-qs)+β(1-qs)qs)-1×Γ(2+δ+(1+δ+α)qs+β(1-qs)αqswww+β(1-qs)[(1+δ+α)qs+β(1-qs)]αqs2)×([(1)qs(1-qs)]αqs2))Γ([(α)qs(1-qs)]αqs21+δ×(Γ(1δeww+β(1-qs)[(1+δ+α)qs+β(1-qs)]αqs2))-1;
the empty product, arising when k=1, is defined to be equal to one.Remark 4.
(i) Ifα=0, the result graph G(t) is a bipartite graph. Moreover, the degree distributions of different types are as follows: for k≥1 and s=0,1,
(9)Pk(s)=(1-qs)k-1qs2.
(ii) Forα=1 and t≥1, the result graph G(t) 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 s=0,1, the degree distributions of different types of vertices are
(10)Pk(s)≈cs′k-(3+δ),
where cs′ is equal to the value of cs when α=1, β=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.For allk≥1, let Nk(t) be the number of vertices with degree k in graph G(t). Define Pk(t)=Nk(t)/(t+1) as the fraction of vertices with degree k. Since Nk(t)=Nk(0)(t)+Nk(1)(t), the following corollary is a direct result of Theorem 3.Corollary 5.
In our model, one has(11)Pk=lim⁡t→∞E[Nk(t)]t+1=Pk(0)+Pk(1),
where Pk(s) for s=0,1 are defined as in Theorem 3.Note the following fact:(12)Γ(k+a)Γ(k+b)=1b-a-1[Γ(k+a)Γ(k-1+b)-Γ(k+1+a)Γ(k+b)].
By using a telescope sum identity we can deduce the following corollary easily.Corollary 6.
For allk≥1 and s=0,1, Pk(s) are defined by (6); then one has the total fractions of vertices of types 0 and 1 in our limit graph which are equal to qs(s=0,1), respectively; that is,
(13)∑k=1∞‍Pk(s)=qs,s=0,1.Remark 7.
(
i
) We know that {Yi;i≥2} is a sequence of i.i.d. random variables; thus by strong law of large numbers, Tt-1(s)/t→qs, (s=0,1) as t→∞. Thus Corollary 6 is exactly consistent with it.
(
ii
) From Corollary 5, for 0<α, β<1, we can easily find that if q0=q1=1/2, the total degree sequence Pk follows power law with the exponent 2+(1+δ+β)/α; otherwise, Pk is a linear combination of two different power law distributions.
(
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.
Fors∈𝒮 and fixed k≥1, one has
(14)lim⁡t→∞Pk(s)(t)=Pk(s),a.s.At last, it is also interesting to find out an expression for the joint-probability distribution for degrees of adjacent vertices. Fors∈𝒮, write Nk,l(s,s)(t) for the number of adjacent pairs of vertices with type s whose degrees are k and l, respectively, at time t and Nk,l(s,1-s)(t) for the number of vertices of degree k with type s which attach to a vertex of degree l with type 1-s at time t. Nk(s)(t) is defined as before. For δ=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
(15)Pk,l(s,s)=lim⁡t→∞E[Nk,l(s,s)(t)]t+1Pk,l(s,s)=Γ(k+β(1-qs)[(1+α)qs+β(1-qs)]αqs2)Pk,l(s,s)×Γ(l+β(1-qs)[(1+α)qs+β(1-qs)]αqs2)Pk,l(s,s)×(Γ(k+l+1+(1+α)qs+β(1-qs)αqs×wwPk,l(s,s)wew+2β(1-qs)[(1+α)qs+β(1-qs)]αqs2))-1Pk,l(s,s)·γ(s,s)∑j=0l-2‍Γ(k+l-j-2)Γ(l-j-1)Γ(k)π(s,s)(j+2);
for k≥1, l≥2, the vertex of type s with degree k is younger than the vertex of type s with degree l, where
(16)γ(s,s)=Γ(2+(1+α)qs+β(1-qs)αqs=w=+β(1-qs)[(1+α)qs+β(1-qs)]αqs2)×([Γ(1+β(1-qs)[(1+α)qs+β(1-qs)]αqs2)]2)-1×P1(s),
and for all l≥2(17)π(s,s)(l)=Γ(1+l+(1+α)qs+β(1-qs)αqswwew+2β(1-qs)[(1+α)qs+β(1-qs)]αqs2)×(Γ(l+(1+α)qs+β(1-qs)αqswwwweww+β(1-qs)[(1+α)qs+β(1-qs)]αqs2))-1·(l-1)×(l-1+β(1-qs)[(1+α)qs+β(1-qs)]αqs2)-1.
(
i
i
) For q0=q1=1/2, one also has
(18)Pk,l(s,1-s)=lim⁡t→∞E[Nk,l(s,1-s)(t)]t+1=Γ(k+2β/α)Γ(l+2β/α)Γ(k+l+1+2/α+4β/α)·γ(s,1-s)∑j=0l-2‍Γ(k+l-j-2)Γ(l-j-1)Γ(k)π(s,1-s)(j+2),
where
(19)γ(s,1-s)=Γ(2+2(1+β)/α)[Γ(1+2β/α)]2·2βα[α+2(1+β)],π(s,1-s)(l)=1l-1+2β/α·Γ(l+1+2(1+2β)/α)Γ(l+2(1+2β)/α),
for k≥1, l≥2, when the vertex of type s with degree k is younger than the vertex of type 1-s with degree l.The rest of this paper is organized as follows. In Section2, 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 s and the total number of vertices with type s, respectively. In Lemma 13 we give the moment inequalities for the total degree of vertices with type s and the total number of vertices with type s. 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{at} satisfies the recurrence relation
(20)at+1=(1-btt+t1)at+ct
for t≥t0. Furthermore, suppose that lim⁡t→∞bt=b>0 and lim⁡t→∞ct=c. Then lim⁡t→∞(at/t) exists and
(21)lim⁡t→∞att=c1+b.Lemma 11 (see [11]).
Letp∈[1,2] and suppose that {Xi;i≥1} are i.i.d. sequence with E[X1]=0 and E[|X1|p]<∞. Then there exists a constant cp depending only on p, such that
(22)E[|∑i=1t‍Xi|p]≤cptE[|X1|p].In the following, in graphG(t-1), we denote the total degree of vertices with type s by St-1(s), that is,
(23)St-1(s)=∑i=0t-1‍di(s)(t-1),
and the number of vertices of type s by Tt-1(s); that is,
(24)Tt-1(s)=∑i=0t-1‍I{Yi=s}.Now we conclude this section by stating the following two lemmas whose proofs are proposed in the appendix.Lemma 12.
Fors=0,1, one has the following:
(25)(i)E[St-1(s)]=1+qs(t-2)+[αqs+β(1-qs)](t-2).(ii)E[Tt-1(s)]=1+qs(t-2).Lemma 13.
LetSt-1(s) and Tt-1(s) be defined as before; then(i) for∀0<ɛ<1, there exists a constant C which is independent of the parameter t such that(ii) one has the following:(26)E|Tt-1(s)-E[Tt-1(s)]|≤Ct1/(1+ɛ),E|St-1(s)-E[St-1(s)]|≤2t.


## 3. Proof of Theorems3 and 8

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

Fort large enough, it is easy to obtain two master equations of the degree sequences of different types as follows:
(27)E[Nk(s)(t)-Nk(s)(t-1)∣ℱt-1]=-[α(k+δ)qsSt-1(s)+δTt-1(s)+β(1-qs)Tt-1(s)]Nk(s)(t-1)+[α(k-1+δ)qsSt-1(s)+δTt-1(s)+β(1-qs)Tt-1(s)]Nk-1(s)(t-1)+δk,1I{Yt=s}s=0,1.Now let us come to solve (27). Taking expectation of both sides of above equation and rewriting it, we get
(28)E[Nk(s)(t)]={1-α(k+δ)qsE[St-1(s)+δTt-1(s)]-β(1-qs)E[Tt-1(s)]}E[Nk(s)(t-1)]+{α(k-1+δ)qsE[St-1(s)+δTt-1(s)]+β(1-qs)E[Tt-1(s)]}E[Nk-1(s)(t-1)]+δk,1qs+αqs(k+δ)×E{Nk(s)(t-1)St-1(s)+δTt-1(s)-E[St-1(s)+δTt-1(s)]E[St-1(s)+δTt-1(s)]·(St-1(s)+δTt-1(s))}-αqs(k-1+δ)×E{Nk-1(s)(t-1)St-1(s)+δTt-1(s)-E[St-1(s)+δTt-1(s)]E[St-1(s)+δTt-1(s)]·(St-1(s)+δTt-1(s))}+β(1-qs)E{Nk(s)(t-1)Tt-1(s)-E[Tt-1(s)]Tt-1(s)·E[Tt-1(s)]}-β(1-qs)E{Nk-1(s)(t-1)Tt-1(s)-E[Tt-1(s)]Tt-1(s)·E[Tt-1(s)]}.Note that, for anyk≥1 and s=0,1, we have the following inequalities:
(29)|Nk(s)(t-1)St-1(s)+δTt-1(s)|≤1,(30)|Nk(s)(t-1)Tt-1(s)|≤1.Combining Lemma12, Lemma 13, triangle inequality, and (28)–(30), the last four sets of terms of (28) tend to zero as t→∞; then for t large enough, we have
(31)E[Nk(s)(t)]=[1-α(k+δ)qsE[St-1(s)+δTt-1(s)]-β(1-qs)E[Tt-1(s)]]E[Nk(s)(t-1)]+[α(k-1+δ)qsE[St-1(s)+δTt-1(s)]+β(1-qs)E[Tt-1(s)]]E[Nk-1(s)(t-1)]+δk,1qs+o(1)=[1-α(k+δ)qs1+δ+[(1+δ+α)qs+β(1-qs)](t-2)-β(1-qs)1+qs(t-2)]E[Nk(s)(t-1)]+δk,1qs+[α(k-1+δ)qs1+δ+[(1+δ+α)qs+β(1-qs)](t-2)wwwew+β(1-qs)1+qs(t-2)]E[Nk-1(s)(t-1)]+o(1).Fork=1, all t>0, note that we have N0(s)(t)=0 for s∈𝒮; thus combining (31) we obtain
(32)E[N1(s)(t)]=(1-btt)E[N1(s)(t-1)]+qs+o(1),
where
(33)bt=α(1+δ)qs(1+δ)(1/t)+[(1+δ+α)qs+β(1-qs)]((t-2)/t)+β(1-qs)1/t+qs((t-2)/t).Fork>1 and all t>0, according to (31), and for s∈𝒮, respectively, we also have
(34)E[Nk(s)(t)]=[1-α(k+δ)qs1+δ+[(1+δ+α)qs+β(1-qs)](t-2)-β(1-qs)1+qs(t-2)]E[Nk(s)(t-1)]+[α(k-1+δ)qs1+δ+[(1+δ+α)qs+β(1-qs)](t-2)+β(1-qs)1+qs(t-2)]E[Nk-1(s)(t-1)]+o(1).To solve the above two equations ((32) and (34)), we want to show that the expected values E[Nk(s)(t)]/(t+1) follow power laws as t goes to infinity. To see it, we proceed by induction on k to show that the limits lim⁡t→∞(E[Nk(s)(t)]/(t+1)) exist for each k≥1.


### 3.2. Proof of Theorem3

Proof of Theorem3.
Fork=1, according to (33), we have
(35)bt⟶α(1+δ)qs(1+δ+α)qs+β(1-qs)+β(1-qs)qs=b,000000000000000000000000000000000ast⟶∞.
Thus we apply Lemma 10 with ct→qs=c (as t→∞) to obtain that lim⁡t→∞(E[N1(s)(t)]/(t+1)) exists and
(36)P1(s)=lim⁡t→∞E[N1(s)(t)]t+1=qs×(1+α(1+δ)qs(1+δ+α)qs+β(1-qs)=(qs)×w+β(1-qs)qsα(1+δ)qs(1)qs(qs))-1.
We suppose thatlim⁡t→∞(E[Nk-1(s)(t)]/(t+1))=Pk-1(s) exists; we use Lemma 10 again with
(37)bt=α(k+δ)qs(1+δ)(1/t)+[(1+δ+α)qs+β(1-qs)]((t-2)/t)+β(1-qs)1/t+qs((t-2)/t)⟶α(k+δ)qs(1+δ+α)qs+β(1-qs)+β(1-qs)qs=b⟶α(k+δ)qs(1+δ+α)qs+β(1-qs)⟶⟶ast→∞,ct=[α(k-1+δ)qs(1+δ)(1/t)+[(1+δ+α)qs+β(1-qs)]((t-2)/t)+β(1-qs)1/t+qs((t-2)/t)]E[Nk-1(s)(t-1)]t⟶[α(k-1+δ)qs(1+δ+α)qs+β(1-qs)+β(1-qs)qs]Pk-1(s)=c[α(k-1+δ)qs(1+δ+α)qs+β(1-qs)++β(1-qs)qs]ast⟶∞.
Then we can arrive at the limit lim⁡t→∞(E[Nk(s)(t)]/(t+1)) which exists and is equal to
(38)Pk(s)=(+(qs)[(α)qsβ(qs)]αqs2)-1k-1+δ+β(1-qs)[(1+δ+α)qs+β(1-qs)]αqs2×(k+δ+(1+δ+α)qs+β(1-qs)αqse×+β(1-qs)[(1+δ+α)qs+β(1-qs)]αqs2)-1)Pk-1(s).
Thus we can get (5) immediately, and (6) is a direct result of (5).
By Stirling's formula, we have thatΓ(k+a)/Γ(k)~ka as k→∞, from which it follows that Pk(s)~csk-(2+((1+δ)qs+β(1-qs))/αqs)  (s=0,1) for some constants cs>0. Thus the power law exponent is 2+((1+δ)qs+β(1-qs))/αqs.


### 3.3. Proof of Theorem8

In our model, Azuma's inequality no longer works because there is no uniform bound on the change in the number ofNk(s)(t) when we investigate the influence of the extra information contained in ℱn compared to the information contained in ℱn-1; that is, we have to bound the difference |E[Nk(s)(t)|ℱn]-E[Nk(s)(t)|ℱn-1]|. 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 Theorem8.
At first we note a basic fact as follows:(39)|Nk(s)(t+1)-Nk(s)(t)|≤2.
At first, we consider the case k=1 for s=0,1, respectively. Consider
(40)Δ1(s)(t+1)=E[N1(s)(t+1)-E[N1(s)(t+1)]]2=E{[N1(s)(t+1)-N1(s)(t)]+[N1(s)(t)-E[N1(s)(t)]]=Eee-[E[N1(s)(t+1)]-E[N1(s)(t)]]}2=E[N1(s)(t+1)-N1(s)(t)]2+E[N1(s)(t)-E[N1(s)(t)]]2+[E[N1(s)(t+1)]-E[N1(s)(t)]]2+2E{[N1(s)(t+1)-N1(s)(t)]·[N1(s)(t)-E[N1(s)(t)]]}-2{E[N1(s)(t+1)-N1(s)(t)]}2≤E[N1(s)(t+1)-N1(s)(t)]2+Δ1(s)(t)+2E{[N1(s)(t+1)-N1(s)(t)]·[N1(s)(t)-E[N1(s)(t)]]}≤4+Δ1(s)(t)+2E[(N1(s)(t)-E[N1(s)(t)])+2Eie·E[N1(s)(t+1)-N1(s)(t)∣ℱt]]=4+Δ1(s)(t)-2E[(qsN1(s)St(s)+δTt(s)+N1(s)Tt(s))(N1(s)(t)-E[N1(s)(t)])-2Ew·(αqs(1+δ)N1(s)(t)St(s)+δTt(s)+β(1-qs)N1(s)(t)Tt(s))]=4+Δ1(s)(t)(1-2αqs(1+δ)E[St(s)+δTt(s)]-2β(1-qs)E[Tt(s)])-2αqs(1+δ)×E[(N1(s)St(s)Tt(s)[N1(s)][Tt(s)])(N1(s)(t)-E[N1(s)(t)])e-2qs·(N1(s)(t)St(s)+δTt(s)-N1(s)(t)-E[N1(s)(t)]E[St(s)+δTt(s)])]-2β(1-qs)E[(N1(s)Tt(s)E[N1(s)]E[Tt(s)])(N1(s)(t)-E[N1(s)(t)])-2β(1-eqs)Ee·(N1(s)(t)Tt(s)-N1(s)(t)-E[N1(s)(t)]E[Tt(s)])]=4+Δ1(s)(t)(1-2αqs(1+δ)E[St(s)+δTt(s)]-2β(1-qs)E[Tt(s)])+I1+I2,
where
(41)I1=-2αqs(1+δ)×E[(N1(s)(t)E[St(s)+δTt(s)])(N1(s)(t)-E[N1(s)(t)])×weE·(N1(s)(t)St(s)+δTt(s)-N1(s)(t)E[St(s)+δTt(s)])],(42)I2=-2β(1-qs)E[(N1(s)(t)Tt(s)-N1(s)(t)E[Tt(s)])(N1(s)(t)-E[N1(s)(t)])=-2β(1-qs)Eew·(N1(s)(t)Tt(s)-N1(s)(t)E[Tt(s)])].
Now let us come to estimate (41) and (42), respectively. For (41), we note the basic fact that
(43)0<(1+δ)N1(s)(t)St(s)+δTt(s)≤1.
By Hölder’s inequality, letting ɛ=1 in (A.8) and combining with (A.16) we can easily obtain
(44)I1≤2αqsE(N1(s)(t)-E[N1(s)(t)])2·E(St(s)+δTt(s)-E[St(s)+δTt(s)])2(E[St(s)+δTt(s)])2≤2αqsE(N1(s)(t)-E[N1(s)(t)])2·C1t≤αqs[E(N1(s)(t)-E[N1(s)(t)])2A1t+C12A1]=αqsΔ1(s)(t)A1t+αqsC12A1,
where the constant C1 is independent of the parameter t, and we can take a constant A1 such that
(45)0<2(1+δ)max⁡{1+δ,(1+δ+α)qs+β(1-qs)}-1A1<1.
Similarly there also exist constants C2 (independent of the parameter t) and A2 such that
(46)I2≤β(1-qs)Δ1(s)(t)A2t+β(1-qs)C22A2,
where 0<2-(1/A2)<1. We denote
(47)φ1(s)=φ1(s)(δ,α,C1,A1,C2,A2)=αqs[2(1+δ)max⁡⁡{1+δ,(1+δ+α)qs+β(1-qs)}-1A1]+β(1-qs)(2-1A2),θ1(s)=4+αqsC12A1+β(1-qs)C22A2,
noting the initial condition that N1(s)(1)=1 so that Δ1(s)(1)=0; then for t large enough we have
(48)Δ1(s)(t+1)≤(1-φ1(s)t)Δ1(s)(t)+θ1(s)≤∑k=2t+1‍∏j=kt‍(1-φ1(s)j)θ1(s)(weletthevacantproductbeequalto1)=∑k=2t+1‍exp⁡(∑j=kt‍ln⁡(1-φ1(s)j))θ1(s)≈∑k=2t+1‍(kt)φ1(s)·θ1(s)≈tθ1(s)∫01‍xφ1(s)dx≤θ1(s)1+φ1(s)(t+1).
Thus it follows that
(49)E[N1(s)(t)t-E[N1(s)(t)]t]2≤θ1(s)1+φ1(s)1t;
then we arrive at
(50)∑n=1∞‍P(|N1(s)(n2)n2-E[N1(s)(n2)]n2|≥ɛ)≤1ε2∑n=1∞‍E[N1(s)(n2)n2-E[N1(s)(n2)]n2]2≤1ε2θ1(s)1+φ1(s)∑n=1∞‍1n2<∞.
Thus Theorem 3 and the Borel-Cantelli lemma imply that
(51)N1(s)(n2)n2⟶P1(s)a.s.,
where P1(s) is defined by (6). For all t, we take n2≤t≤(n+1)2; we have
(52)|N1(s)(t)-N1(s)(n2)|≤2(t-n2)≤4n+2.
Then we have
(53)|N1(s)(t)t-N1(s)(n2)n2|≤|N1(s)(t)-N1(s)(n2)t|+|N1(s)(n2)(t-n2)t·n2|⟶0≤|N1(s)(t)-N1(s)(n2)t|++we|(t-n2)t·n2|(asn⟶∞);
thus we get
(54)N1(s)(t)t⟶P1(s)a.s.
Similarly to the case k>1 and s=0,1, we can also prove that
(55)Nk(s)(t)t⟶Pk(s)a.s.,
where Pk(s) is defined as in Theorem 3.


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

Fort large enough, it is easy to obtain two master equations of the degree sequences of different types as follows:
(27)E[Nk(s)(t)-Nk(s)(t-1)∣ℱt-1]=-[α(k+δ)qsSt-1(s)+δTt-1(s)+β(1-qs)Tt-1(s)]Nk(s)(t-1)+[α(k-1+δ)qsSt-1(s)+δTt-1(s)+β(1-qs)Tt-1(s)]Nk-1(s)(t-1)+δk,1I{Yt=s}s=0,1.Now let us come to solve (27). Taking expectation of both sides of above equation and rewriting it, we get
(28)E[Nk(s)(t)]={1-α(k+δ)qsE[St-1(s)+δTt-1(s)]-β(1-qs)E[Tt-1(s)]}E[Nk(s)(t-1)]+{α(k-1+δ)qsE[St-1(s)+δTt-1(s)]+β(1-qs)E[Tt-1(s)]}E[Nk-1(s)(t-1)]+δk,1qs+αqs(k+δ)×E{Nk(s)(t-1)St-1(s)+δTt-1(s)-E[St-1(s)+δTt-1(s)]E[St-1(s)+δTt-1(s)]·(St-1(s)+δTt-1(s))}-αqs(k-1+δ)×E{Nk-1(s)(t-1)St-1(s)+δTt-1(s)-E[St-1(s)+δTt-1(s)]E[St-1(s)+δTt-1(s)]·(St-1(s)+δTt-1(s))}+β(1-qs)E{Nk(s)(t-1)Tt-1(s)-E[Tt-1(s)]Tt-1(s)·E[Tt-1(s)]}-β(1-qs)E{Nk-1(s)(t-1)Tt-1(s)-E[Tt-1(s)]Tt-1(s)·E[Tt-1(s)]}.Note that, for anyk≥1 and s=0,1, we have the following inequalities:
(29)|Nk(s)(t-1)St-1(s)+δTt-1(s)|≤1,(30)|Nk(s)(t-1)Tt-1(s)|≤1.Combining Lemma12, Lemma 13, triangle inequality, and (28)–(30), the last four sets of terms of (28) tend to zero as t→∞; then for t large enough, we have
(31)E[Nk(s)(t)]=[1-α(k+δ)qsE[St-1(s)+δTt-1(s)]-β(1-qs)E[Tt-1(s)]]E[Nk(s)(t-1)]+[α(k-1+δ)qsE[St-1(s)+δTt-1(s)]+β(1-qs)E[Tt-1(s)]]E[Nk-1(s)(t-1)]+δk,1qs+o(1)=[1-α(k+δ)qs1+δ+[(1+δ+α)qs+β(1-qs)](t-2)-β(1-qs)1+qs(t-2)]E[Nk(s)(t-1)]+δk,1qs+[α(k-1+δ)qs1+δ+[(1+δ+α)qs+β(1-qs)](t-2)wwwew+β(1-qs)1+qs(t-2)]E[Nk-1(s)(t-1)]+o(1).Fork=1, all t>0, note that we have N0(s)(t)=0 for s∈𝒮; thus combining (31) we obtain
(32)E[N1(s)(t)]=(1-btt)E[N1(s)(t-1)]+qs+o(1),
where
(33)bt=α(1+δ)qs(1+δ)(1/t)+[(1+δ+α)qs+β(1-qs)]((t-2)/t)+β(1-qs)1/t+qs((t-2)/t).Fork>1 and all t>0, according to (31), and for s∈𝒮, respectively, we also have
(34)E[Nk(s)(t)]=[1-α(k+δ)qs1+δ+[(1+δ+α)qs+β(1-qs)](t-2)-β(1-qs)1+qs(t-2)]E[Nk(s)(t-1)]+[α(k-1+δ)qs1+δ+[(1+δ+α)qs+β(1-qs)](t-2)+β(1-qs)1+qs(t-2)]E[Nk-1(s)(t-1)]+o(1).To solve the above two equations ((32) and (34)), we want to show that the expected values E[Nk(s)(t)]/(t+1) follow power laws as t goes to infinity. To see it, we proceed by induction on k to show that the limits lim⁡t→∞(E[Nk(s)(t)]/(t+1)) exist for each k≥1.


## 3.2. Proof of Theorem3

Proof of Theorem3.
Fork=1, according to (33), we have
(35)bt⟶α(1+δ)qs(1+δ+α)qs+β(1-qs)+β(1-qs)qs=b,000000000000000000000000000000000ast⟶∞.
Thus we apply Lemma 10 with ct→qs=c (as t→∞) to obtain that lim⁡t→∞(E[N1(s)(t)]/(t+1)) exists and
(36)P1(s)=lim⁡t→∞E[N1(s)(t)]t+1=qs×(1+α(1+δ)qs(1+δ+α)qs+β(1-qs)=(qs)×w+β(1-qs)qsα(1+δ)qs(1)qs(qs))-1.
We suppose thatlim⁡t→∞(E[Nk-1(s)(t)]/(t+1))=Pk-1(s) exists; we use Lemma 10 again with
(37)bt=α(k+δ)qs(1+δ)(1/t)+[(1+δ+α)qs+β(1-qs)]((t-2)/t)+β(1-qs)1/t+qs((t-2)/t)⟶α(k+δ)qs(1+δ+α)qs+β(1-qs)+β(1-qs)qs=b⟶α(k+δ)qs(1+δ+α)qs+β(1-qs)⟶⟶ast→∞,ct=[α(k-1+δ)qs(1+δ)(1/t)+[(1+δ+α)qs+β(1-qs)]((t-2)/t)+β(1-qs)1/t+qs((t-2)/t)]E[Nk-1(s)(t-1)]t⟶[α(k-1+δ)qs(1+δ+α)qs+β(1-qs)+β(1-qs)qs]Pk-1(s)=c[α(k-1+δ)qs(1+δ+α)qs+β(1-qs)++β(1-qs)qs]ast⟶∞.
Then we can arrive at the limit lim⁡t→∞(E[Nk(s)(t)]/(t+1)) which exists and is equal to
(38)Pk(s)=(+(qs)[(α)qsβ(qs)]αqs2)-1k-1+δ+β(1-qs)[(1+δ+α)qs+β(1-qs)]αqs2×(k+δ+(1+δ+α)qs+β(1-qs)αqse×+β(1-qs)[(1+δ+α)qs+β(1-qs)]αqs2)-1)Pk-1(s).
Thus we can get (5) immediately, and (6) is a direct result of (5).
By Stirling's formula, we have thatΓ(k+a)/Γ(k)~ka as k→∞, from which it follows that Pk(s)~csk-(2+((1+δ)qs+β(1-qs))/αqs)  (s=0,1) for some constants cs>0. Thus the power law exponent is 2+((1+δ)qs+β(1-qs))/αqs.


## 3.3. Proof of Theorem8

In our model, Azuma's inequality no longer works because there is no uniform bound on the change in the number ofNk(s)(t) when we investigate the influence of the extra information contained in ℱn compared to the information contained in ℱn-1; that is, we have to bound the difference |E[Nk(s)(t)|ℱn]-E[Nk(s)(t)|ℱn-1]|. 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 Theorem8.
At first we note a basic fact as follows:(39)|Nk(s)(t+1)-Nk(s)(t)|≤2.
At first, we consider the case k=1 for s=0,1, respectively. Consider
(40)Δ1(s)(t+1)=E[N1(s)(t+1)-E[N1(s)(t+1)]]2=E{[N1(s)(t+1)-N1(s)(t)]+[N1(s)(t)-E[N1(s)(t)]]=Eee-[E[N1(s)(t+1)]-E[N1(s)(t)]]}2=E[N1(s)(t+1)-N1(s)(t)]2+E[N1(s)(t)-E[N1(s)(t)]]2+[E[N1(s)(t+1)]-E[N1(s)(t)]]2+2E{[N1(s)(t+1)-N1(s)(t)]·[N1(s)(t)-E[N1(s)(t)]]}-2{E[N1(s)(t+1)-N1(s)(t)]}2≤E[N1(s)(t+1)-N1(s)(t)]2+Δ1(s)(t)+2E{[N1(s)(t+1)-N1(s)(t)]·[N1(s)(t)-E[N1(s)(t)]]}≤4+Δ1(s)(t)+2E[(N1(s)(t)-E[N1(s)(t)])+2Eie·E[N1(s)(t+1)-N1(s)(t)∣ℱt]]=4+Δ1(s)(t)-2E[(qsN1(s)St(s)+δTt(s)+N1(s)Tt(s))(N1(s)(t)-E[N1(s)(t)])-2Ew·(αqs(1+δ)N1(s)(t)St(s)+δTt(s)+β(1-qs)N1(s)(t)Tt(s))]=4+Δ1(s)(t)(1-2αqs(1+δ)E[St(s)+δTt(s)]-2β(1-qs)E[Tt(s)])-2αqs(1+δ)×E[(N1(s)St(s)Tt(s)[N1(s)][Tt(s)])(N1(s)(t)-E[N1(s)(t)])e-2qs·(N1(s)(t)St(s)+δTt(s)-N1(s)(t)-E[N1(s)(t)]E[St(s)+δTt(s)])]-2β(1-qs)E[(N1(s)Tt(s)E[N1(s)]E[Tt(s)])(N1(s)(t)-E[N1(s)(t)])-2β(1-eqs)Ee·(N1(s)(t)Tt(s)-N1(s)(t)-E[N1(s)(t)]E[Tt(s)])]=4+Δ1(s)(t)(1-2αqs(1+δ)E[St(s)+δTt(s)]-2β(1-qs)E[Tt(s)])+I1+I2,
where
(41)I1=-2αqs(1+δ)×E[(N1(s)(t)E[St(s)+δTt(s)])(N1(s)(t)-E[N1(s)(t)])×weE·(N1(s)(t)St(s)+δTt(s)-N1(s)(t)E[St(s)+δTt(s)])],(42)I2=-2β(1-qs)E[(N1(s)(t)Tt(s)-N1(s)(t)E[Tt(s)])(N1(s)(t)-E[N1(s)(t)])=-2β(1-qs)Eew·(N1(s)(t)Tt(s)-N1(s)(t)E[Tt(s)])].
Now let us come to estimate (41) and (42), respectively. For (41), we note the basic fact that
(43)0<(1+δ)N1(s)(t)St(s)+δTt(s)≤1.
By Hölder’s inequality, letting ɛ=1 in (A.8) and combining with (A.16) we can easily obtain
(44)I1≤2αqsE(N1(s)(t)-E[N1(s)(t)])2·E(St(s)+δTt(s)-E[St(s)+δTt(s)])2(E[St(s)+δTt(s)])2≤2αqsE(N1(s)(t)-E[N1(s)(t)])2·C1t≤αqs[E(N1(s)(t)-E[N1(s)(t)])2A1t+C12A1]=αqsΔ1(s)(t)A1t+αqsC12A1,
where the constant C1 is independent of the parameter t, and we can take a constant A1 such that
(45)0<2(1+δ)max⁡{1+δ,(1+δ+α)qs+β(1-qs)}-1A1<1.
Similarly there also exist constants C2 (independent of the parameter t) and A2 such that
(46)I2≤β(1-qs)Δ1(s)(t)A2t+β(1-qs)C22A2,
where 0<2-(1/A2)<1. We denote
(47)φ1(s)=φ1(s)(δ,α,C1,A1,C2,A2)=αqs[2(1+δ)max⁡⁡{1+δ,(1+δ+α)qs+β(1-qs)}-1A1]+β(1-qs)(2-1A2),θ1(s)=4+αqsC12A1+β(1-qs)C22A2,
noting the initial condition that N1(s)(1)=1 so that Δ1(s)(1)=0; then for t large enough we have
(48)Δ1(s)(t+1)≤(1-φ1(s)t)Δ1(s)(t)+θ1(s)≤∑k=2t+1‍∏j=kt‍(1-φ1(s)j)θ1(s)(weletthevacantproductbeequalto1)=∑k=2t+1‍exp⁡(∑j=kt‍ln⁡(1-φ1(s)j))θ1(s)≈∑k=2t+1‍(kt)φ1(s)·θ1(s)≈tθ1(s)∫01‍xφ1(s)dx≤θ1(s)1+φ1(s)(t+1).
Thus it follows that
(49)E[N1(s)(t)t-E[N1(s)(t)]t]2≤θ1(s)1+φ1(s)1t;
then we arrive at
(50)∑n=1∞‍P(|N1(s)(n2)n2-E[N1(s)(n2)]n2|≥ɛ)≤1ε2∑n=1∞‍E[N1(s)(n2)n2-E[N1(s)(n2)]n2]2≤1ε2θ1(s)1+φ1(s)∑n=1∞‍1n2<∞.
Thus Theorem 3 and the Borel-Cantelli lemma imply that
(51)N1(s)(n2)n2⟶P1(s)a.s.,
where P1(s) is defined by (6). For all t, we take n2≤t≤(n+1)2; we have
(52)|N1(s)(t)-N1(s)(n2)|≤2(t-n2)≤4n+2.
Then we have
(53)|N1(s)(t)t-N1(s)(n2)n2|≤|N1(s)(t)-N1(s)(n2)t|+|N1(s)(n2)(t-n2)t·n2|⟶0≤|N1(s)(t)-N1(s)(n2)t|++we|(t-n2)t·n2|(asn⟶∞);
thus we get
(54)N1(s)(t)t⟶P1(s)a.s.
Similarly to the case k>1 and s=0,1, we can also prove that
(55)Nk(s)(t)t⟶Pk(s)a.s.,
where Pk(s) is defined as in Theorem 3.


## 4. Proof of Theorem9

### 4.1. Master Equations for Joint Degree SequencesNk,l(s,s)(t) and Nk,l(s,1-s)(t)

Letδ=0; we can easily get the master equations as follows: for s∈𝒮(56)E[Nk,l(s,s)(t)-Nk,l(s,s)(t-1)∣ℱt-1]=[αqs(k-1)St-1(s)+β(1-qs)Tt-1(s)]Nk-1,l(s,s)(t-1)+[αqs(l-1)St-1(s)+β(1-qs)Tt-1(s)]Nk,l-1(s,s)(t-1)-[αqs(k+l)St-1(s)+2β(1-qs)Tt-1(s)]Nk,l(s,s)(t-1)+αqs(l-1)St-1(s)Nl-1(s)(t-1)δk,1,
where we suppose that the vertex of type s with degree k is younger than the vertex of type s with degree l. The first two sets of terms on the right-hand side account for the change in Nk,l(s,s)(t) due to the addition of the new edge hitting a vertex of degree k-1 or l-1 (both gain) with type s, while the third set of terms gives the change in Nk,l(s,s)(t-1) due to the addition of the new edge onto the ancestor vertex of degree k or l (both loss) with type s. Finally, the last term accounts for the gain in N1,l(s,s)(t-1) due to the addition on the new vertex.Similarly, fors∈𝒮 we also have
(57)E[Nk,l(s,1-s)(t)-Nk,l(s,1-s)(t-1)∣ℱt]=[αqs(k-1)St-1(s)+β(1-qs)Tt-1(s)]Nk-1,l(s,1-s)(t-1)+[α(1-qs)(l-1)St-1(1-s)+βqsTt-1(1-s)]Nk,l-1(s,1-s)(t-1)-[αqskSt-1(s)+β(1-qs)Tt-1(s)]Nk,l(s,1-s)(t-1)-[α(1-qs)lSt-1(1-s)+βqsTt-1(1-s)]Nk,l(s,1-s)(t-1)+βqsTt-1(1-s)Nl-1(1-s)(t-1)δk,1,
where the vertex of type s with degree k is younger than the vertex of type 1-s with degree l.We also notice a fact that the total number of pairs of adjacent vertices is equal to the number of edges at timet. The total number of edges is t in the resulting graph G(t); obviously, there exist constants θi(i=1,2,3) such that
(58)|Nk,l(s,s)(t-1)St-1(s)|≤1,|Nk,l(s,1-s)(t-1)St-1(s)|≤1,|Nk,l(s,1-s)(t)St-1(1-s)|≤1.|Nk,l(s,s)(t-1)Tt-1(s)|≤θ1,|Nk,l(s,1-s)(t)Tt-1(s)|≤θ2,|Nk,l(s,1-s)(t)Tt-1(1-s)|≤θ3.
Similarly to the analysis of (27), for t large enough, combining Lemmas 12 and 13 and (58), we can get by (56) and (57), respectively,
(59)E[Nk,l(s,s)(t)-Nk,l(s,s)(t-1)]=[αqs(k-1)1+[(1+α)qs+β(1-qs)](t-2)+β(1-qs)1+qs(t-2)]×E[Nk-1,l(s,s)(t-1)]+[αqs(l-1)1+[(1+α)qs+β(1-qs)](t-2)+β(1-qs)1+qs(t-2)]×E[Nk,l-1(s,s)(t-1)]-[αqs(k+l)1+[(1+α)qs+β(1-qs)](t-2)+2β(1-qs)1+qs(t-2)]×E[Nk,l(s,s)(t-1)]+αqs(l-1)1+[(1+α)qs+β(1-qs)](t-2)E[Nl-1(s)(t-1)]×δk,1+O(t-1/2),(60)E[Nk,l(s,1-s)(t)-Nk,l(s,1-s)(t-1)]=[αqs(k-1)1+[(1+α)qs+β(1-qs)](t-2)+β(1-qs)1+qs(t-2)]×E[Nk-1,l(s,1-s)(t-1)]+[α(1-qs)(l-1)1+[(1+α)(1-qs)+βqs](t-2)w++βqs1+(1-qs)(t-2)]E[Nk,l-1(s,1-s)(t-1)]-[αqsk1+[(1+α)qs+β(1-qs)](t-2)+β(1-qs)1+qs(t-2)]×E[Nk,l(s,1-s)(t-1)]-[α(1-qs)l1+[(1+α)(1-qs)+βqs](t-2)w++βqs1+(1-qs)(t-2)]E[Nk,l(s,1-s)(t-1)]+βqs1+(1-qs)(t-2)E[Nl-1(1-s)(t-1)]δk,1+O(t-1/2).


### 4.2. Proof of Theorem9

To solve the above equations ((59), (60), resp.), we come to prove Theorem 9 as follows.Proof of Theorem9.
At first we have(61)Nk,1(s,s)(t)=0(fork≥1),N0,l(s,s)(t)=0(forl≥2),forallt≥1
so that
(62)Pk,1(s,s)=0(fork≥1),P0,l(s,s)=0(forl≥2).
Thus we have
(63)E[N1,2(s,s)(t)]=[2β(1-qs)1+qs(t-2)1-3αqs1+[(1+α)qs+β(1-qs)](t-2)wwwe-2β(1-qs)1+qs(t-2)]E[N1,2(s,s)(t-1)]+αqs1+[(1+α)qs+β(1-qs)](t-2)E[N1(s)(t-1)];
combining Lemma 10 with Theorem 3, it follows that
(64)P1,2(s,s)=αqs/((1+α)qs+β(1-qs))1+3αqs/((1+α)qs+β(1-qs))+2β(1-qs)/qs×P1(s);
here and thereafter P1(s) is defined as (5) in Theorem 3 when δ=0.
Fork≥2, l≥2, by induction hypothesis, we suppose that
(65)lim⁡t→∞E[Nk-1,l(s,s)(t)]t=Pk-1,l(s,s),lim⁡t→∞E[Nk,l-1(s)(t)]t=Pk,l-1(s,s)exist.
Thus by using Lemma 10 and Theorem 3 again, we arrive at lim⁡t→∞(E[Nk,l(s,s)(t)]/t)=Pk,l(s,s) existence and satisfy the following time-independent recursion relation:
(66)[1+αqs(k+l)(1+α)qs+β(1-qs)+2β(1-qs)qs]Pk,l(s,s)=[αqs(k-1)(1+α)qs+β(1-qs)+β(1-qs)qs]Pk-1,l(s,s)+[αqs(l-1)(1+α)qs+β(1-qs)+β(1-qs)qs]Pk,l-1(s,s)+αqs(l-1)(1+α)qs+β(1-qs)Pl-1(s)δk,1,
which is equivalent to
(67)[k+l+(1+α)qs+β(1-qs)αqs+2β(1-qs)[(1+α)qs+β(1-qs)]αqs2]Pk,l(s,s)=[k-1+β(1-qs)[(1+α)qs+β(1-qs)]αqs2]Pk-1,l(s,s)+[l-1+β(1-qs)[(1+α)qs+β(1-qs)]αqs2]Pk,l-1(s,s)+(l-1)Pl-1(s)δk,1.
Now let us come to solve the difference equation (67). Letting
(68)Pk,l(s,s)=Γ(k+β(1-qs)[(1+α)qs+β(1-qs)]αqs2)×Γ(l+β(1-qs)[(1+α)qs+β(1-qs)]αqs2)×(Γ(k+l+1+(1+α)qs+β(1-qs)αqs=====e=+2β(1-qs)[(1+α)qs+β(1-qs)]αqs2))-1×Mk,l(s,s),
substituting the above equation (68) in (67), and combining (6), we arrive at
(69)Mk,l(s,s)=Mk-1,l(s,s)+Mk,l-1(s,s)+Γ(1+l+(1+α)qs+β(1-qs)αqswwwww+2β(1-qs)[(1+α)qs+β(1-qs)]αqs2)×(Γ(1+β(1-qs)[(1+α)qs+β(1-qs)]αqs2)wwewe×Γ(l+β(1-qs)[(1+α)qs+β(1-qs)]αqs2))-1×(l-1)Pl-1(s)δk,1=Mk-1,l(s,s)+Mk,l-1(s,s)+γ(s,s)π(s,s)(l)δk,1,
where γ(s,s), π(s,s)(l) are defined as (16) and (17) for all l≥2, respectively.
Moreover, by (69) we have
(70)Mk,1(s,s)=0(fork≥1),M0,l(s,s)=0(forl≥2).
Now we define 2-dimensional generating function as follows:
(71)G(s,s)(x,y)=∑k=1∞‍∑l=2∞‍Mk,l(s,s)xkyl|x|<1,|y|<1;
it follows that
(72)G(s,s)(x,y)=∑k-1=1∞‍∑l=2∞‍Mk-1,l(s,s)xkyl+∑k=1∞‍∑l-1=2∞‍Mk,l-1(s,s)xkyl+∑l=2∞‍γ(s,s)π(s,s)(l)xyl=xG(s,s)(x,y)+yG(s,s)(x,y)+γ(s,s)x∑l=2∞‍π(s,s)(l)yl,
so that
(73)G(s,s)(x,y)=γ(s,s)x1-x-y∑l=2∞‍π(s,s)(l)yl;
furthermore, taking 0<r<1, for |x+y|≤r, and expanding G(s,s)(x,y), then
(74)G(s,s)(x,y)=γ(s,s)x∑n=0∞‍(x+y)n∑l=2∞‍π(s,s)(l)yl=γ(s,s)x∑n=0∞‍∑i=0n‍Cnixn-iyi∑j=0∞‍π(s,s)(j+2)yj+2=γ(s,s)∑k=1∞‍∑u=0∞‍∑i+j=u‍Ck+i-1iπ(s,s)(j+2)xkyu+2=γ(s,s)∑k=1∞‍∑l=2∞‍∑j=0l-2‍Ck+l-j-3l-j-2π(s,s)(j+2)xkyl=γ(s,s)∑k=1∞‍∑l=2∞‍∑j=0l-2‍Γ(k+l-j-2)Γ(l-j-1)Γ(k)π(s,s)(j+2)xkyl.
Comparing (71) with (74), it follows that
(75)Mk,l(s,s)=γ(s,s)∑j=0l-2‍Γ(k+l-j-2)Γ(l-j-1)Γ(k)π(s,s)(j+2),
so we arrive at (10) immediately. Furthermore, if q0=q1=1/2, 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.


## 4.1. Master Equations for Joint Degree SequencesNk,l(s,s)(t) and Nk,l(s,1-s)(t)

Letδ=0; we can easily get the master equations as follows: for s∈𝒮(56)E[Nk,l(s,s)(t)-Nk,l(s,s)(t-1)∣ℱt-1]=[αqs(k-1)St-1(s)+β(1-qs)Tt-1(s)]Nk-1,l(s,s)(t-1)+[αqs(l-1)St-1(s)+β(1-qs)Tt-1(s)]Nk,l-1(s,s)(t-1)-[αqs(k+l)St-1(s)+2β(1-qs)Tt-1(s)]Nk,l(s,s)(t-1)+αqs(l-1)St-1(s)Nl-1(s)(t-1)δk,1,
where we suppose that the vertex of type s with degree k is younger than the vertex of type s with degree l. The first two sets of terms on the right-hand side account for the change in Nk,l(s,s)(t) due to the addition of the new edge hitting a vertex of degree k-1 or l-1 (both gain) with type s, while the third set of terms gives the change in Nk,l(s,s)(t-1) due to the addition of the new edge onto the ancestor vertex of degree k or l (both loss) with type s. Finally, the last term accounts for the gain in N1,l(s,s)(t-1) due to the addition on the new vertex.Similarly, fors∈𝒮 we also have
(57)E[Nk,l(s,1-s)(t)-Nk,l(s,1-s)(t-1)∣ℱt]=[αqs(k-1)St-1(s)+β(1-qs)Tt-1(s)]Nk-1,l(s,1-s)(t-1)+[α(1-qs)(l-1)St-1(1-s)+βqsTt-1(1-s)]Nk,l-1(s,1-s)(t-1)-[αqskSt-1(s)+β(1-qs)Tt-1(s)]Nk,l(s,1-s)(t-1)-[α(1-qs)lSt-1(1-s)+βqsTt-1(1-s)]Nk,l(s,1-s)(t-1)+βqsTt-1(1-s)Nl-1(1-s)(t-1)δk,1,
where the vertex of type s with degree k is younger than the vertex of type 1-s with degree l.We also notice a fact that the total number of pairs of adjacent vertices is equal to the number of edges at timet. The total number of edges is t in the resulting graph G(t); obviously, there exist constants θi(i=1,2,3) such that
(58)|Nk,l(s,s)(t-1)St-1(s)|≤1,|Nk,l(s,1-s)(t-1)St-1(s)|≤1,|Nk,l(s,1-s)(t)St-1(1-s)|≤1.|Nk,l(s,s)(t-1)Tt-1(s)|≤θ1,|Nk,l(s,1-s)(t)Tt-1(s)|≤θ2,|Nk,l(s,1-s)(t)Tt-1(1-s)|≤θ3.
Similarly to the analysis of (27), for t large enough, combining Lemmas 12 and 13 and (58), we can get by (56) and (57), respectively,
(59)E[Nk,l(s,s)(t)-Nk,l(s,s)(t-1)]=[αqs(k-1)1+[(1+α)qs+β(1-qs)](t-2)+β(1-qs)1+qs(t-2)]×E[Nk-1,l(s,s)(t-1)]+[αqs(l-1)1+[(1+α)qs+β(1-qs)](t-2)+β(1-qs)1+qs(t-2)]×E[Nk,l-1(s,s)(t-1)]-[αqs(k+l)1+[(1+α)qs+β(1-qs)](t-2)+2β(1-qs)1+qs(t-2)]×E[Nk,l(s,s)(t-1)]+αqs(l-1)1+[(1+α)qs+β(1-qs)](t-2)E[Nl-1(s)(t-1)]×δk,1+O(t-1/2),(60)E[Nk,l(s,1-s)(t)-Nk,l(s,1-s)(t-1)]=[αqs(k-1)1+[(1+α)qs+β(1-qs)](t-2)+β(1-qs)1+qs(t-2)]×E[Nk-1,l(s,1-s)(t-1)]+[α(1-qs)(l-1)1+[(1+α)(1-qs)+βqs](t-2)w++βqs1+(1-qs)(t-2)]E[Nk,l-1(s,1-s)(t-1)]-[αqsk1+[(1+α)qs+β(1-qs)](t-2)+β(1-qs)1+qs(t-2)]×E[Nk,l(s,1-s)(t-1)]-[α(1-qs)l1+[(1+α)(1-qs)+βqs](t-2)w++βqs1+(1-qs)(t-2)]E[Nk,l(s,1-s)(t-1)]+βqs1+(1-qs)(t-2)E[Nl-1(1-s)(t-1)]δk,1+O(t-1/2).


## 4.2. Proof of Theorem9

To solve the above equations ((59), (60), resp.), we come to prove Theorem 9 as follows.Proof of Theorem9.
At first we have(61)Nk,1(s,s)(t)=0(fork≥1),N0,l(s,s)(t)=0(forl≥2),forallt≥1
so that
(62)Pk,1(s,s)=0(fork≥1),P0,l(s,s)=0(forl≥2).
Thus we have
(63)E[N1,2(s,s)(t)]=[2β(1-qs)1+qs(t-2)1-3αqs1+[(1+α)qs+β(1-qs)](t-2)wwwe-2β(1-qs)1+qs(t-2)]E[N1,2(s,s)(t-1)]+αqs1+[(1+α)qs+β(1-qs)](t-2)E[N1(s)(t-1)];
combining Lemma 10 with Theorem 3, it follows that
(64)P1,2(s,s)=αqs/((1+α)qs+β(1-qs))1+3αqs/((1+α)qs+β(1-qs))+2β(1-qs)/qs×P1(s);
here and thereafter P1(s) is defined as (5) in Theorem 3 when δ=0.
Fork≥2, l≥2, by induction hypothesis, we suppose that
(65)lim⁡t→∞E[Nk-1,l(s,s)(t)]t=Pk-1,l(s,s),lim⁡t→∞E[Nk,l-1(s)(t)]t=Pk,l-1(s,s)exist.
Thus by using Lemma 10 and Theorem 3 again, we arrive at lim⁡t→∞(E[Nk,l(s,s)(t)]/t)=Pk,l(s,s) existence and satisfy the following time-independent recursion relation:
(66)[1+αqs(k+l)(1+α)qs+β(1-qs)+2β(1-qs)qs]Pk,l(s,s)=[αqs(k-1)(1+α)qs+β(1-qs)+β(1-qs)qs]Pk-1,l(s,s)+[αqs(l-1)(1+α)qs+β(1-qs)+β(1-qs)qs]Pk,l-1(s,s)+αqs(l-1)(1+α)qs+β(1-qs)Pl-1(s)δk,1,
which is equivalent to
(67)[k+l+(1+α)qs+β(1-qs)αqs+2β(1-qs)[(1+α)qs+β(1-qs)]αqs2]Pk,l(s,s)=[k-1+β(1-qs)[(1+α)qs+β(1-qs)]αqs2]Pk-1,l(s,s)+[l-1+β(1-qs)[(1+α)qs+β(1-qs)]αqs2]Pk,l-1(s,s)+(l-1)Pl-1(s)δk,1.
Now let us come to solve the difference equation (67). Letting
(68)Pk,l(s,s)=Γ(k+β(1-qs)[(1+α)qs+β(1-qs)]αqs2)×Γ(l+β(1-qs)[(1+α)qs+β(1-qs)]αqs2)×(Γ(k+l+1+(1+α)qs+β(1-qs)αqs=====e=+2β(1-qs)[(1+α)qs+β(1-qs)]αqs2))-1×Mk,l(s,s),
substituting the above equation (68) in (67), and combining (6), we arrive at
(69)Mk,l(s,s)=Mk-1,l(s,s)+Mk,l-1(s,s)+Γ(1+l+(1+α)qs+β(1-qs)αqswwwww+2β(1-qs)[(1+α)qs+β(1-qs)]αqs2)×(Γ(1+β(1-qs)[(1+α)qs+β(1-qs)]αqs2)wwewe×Γ(l+β(1-qs)[(1+α)qs+β(1-qs)]αqs2))-1×(l-1)Pl-1(s)δk,1=Mk-1,l(s,s)+Mk,l-1(s,s)+γ(s,s)π(s,s)(l)δk,1,
where γ(s,s), π(s,s)(l) are defined as (16) and (17) for all l≥2, respectively.
Moreover, by (69) we have
(70)Mk,1(s,s)=0(fork≥1),M0,l(s,s)=0(forl≥2).
Now we define 2-dimensional generating function as follows:
(71)G(s,s)(x,y)=∑k=1∞‍∑l=2∞‍Mk,l(s,s)xkyl|x|<1,|y|<1;
it follows that
(72)G(s,s)(x,y)=∑k-1=1∞‍∑l=2∞‍Mk-1,l(s,s)xkyl+∑k=1∞‍∑l-1=2∞‍Mk,l-1(s,s)xkyl+∑l=2∞‍γ(s,s)π(s,s)(l)xyl=xG(s,s)(x,y)+yG(s,s)(x,y)+γ(s,s)x∑l=2∞‍π(s,s)(l)yl,
so that
(73)G(s,s)(x,y)=γ(s,s)x1-x-y∑l=2∞‍π(s,s)(l)yl;
furthermore, taking 0<r<1, for |x+y|≤r, and expanding G(s,s)(x,y), then
(74)G(s,s)(x,y)=γ(s,s)x∑n=0∞‍(x+y)n∑l=2∞‍π(s,s)(l)yl=γ(s,s)x∑n=0∞‍∑i=0n‍Cnixn-iyi∑j=0∞‍π(s,s)(j+2)yj+2=γ(s,s)∑k=1∞‍∑u=0∞‍∑i+j=u‍Ck+i-1iπ(s,s)(j+2)xkyu+2=γ(s,s)∑k=1∞‍∑l=2∞‍∑j=0l-2‍Ck+l-j-3l-j-2π(s,s)(j+2)xkyl=γ(s,s)∑k=1∞‍∑l=2∞‍∑j=0l-2‍Γ(k+l-j-2)Γ(l-j-1)Γ(k)π(s,s)(j+2)xkyl.
Comparing (71) with (74), it follows that
(75)Mk,l(s,s)=γ(s,s)∑j=0l-2‍Γ(k+l-j-2)Γ(l-j-1)Γ(k)π(s,s)(j+2),
so we arrive at (10) immediately. Furthermore, if q0=q1=1/2, 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.



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*Source: 402821-2014-05-12.xml*