Asymptotic Degree Distribution of a Kind of Asymmetric Evolving Network

Abstract

We propose a kind of evolving network which shows tree structure. The model is a combination of preferential attachment model and uniform model. We show that the proportional degree sequence {𝑝_𝑘}_𝑘>1 obeys power law, exponential distribution, and other forms according to the relation of 𝑘 and parameter 𝑚.

1. Introduction

In recent ten years, there has been much interest in understanding the properties of real large-scale complex networks which describe a wide range of systems in nature and society. Examples of such networks appear in communications, biology, social science, economics, and so forth [1]. In pursuit of such understanding, mathematicians and physicists usually use random graphs to model all these real-life networks. In the investigation of various complex networks, the degree distribution is always the main concern because it characterizes the fundamental topological properties of complex networks which show importance in network control, estimation, and sensor [2–8]. Several models were introduced to explain the properties. Bollobás [9] proposed a model with 𝑛 vertices and 𝑚 edges. In this model, the degree distribution is approximately Poisson distribution. Later, Barabási and Albert [10] proposed the following model: at each time step, add a new vertex 𝑣 and a fixed number 𝑟 of edges originating at 𝑣 and directed towards vertices chosen at random with probability proportional to their degrees. Based on simulation and heuristic approximation, they predicted that the degree distribution behaves like 𝑑⁻³ for all 𝑟≥1. The result was confirmed by Barabási et al. [11, 12]. In order to generate power laws with arbitrary exponents, Dorogovtsev et al. [13] and Drine et al. [14] introduced the following natural generalization of the above model: the destination of the 𝑟 new edges added at each time step is chosen with probability proportional to the degree plus an initial attractiveness 𝛼𝑟; they gave a nonrigorous argument that the degree distribution 𝑝_𝑑 behaves like 𝑑^−2−𝛼 for large 𝑑.

In some real networks, experiments show that the distribution obeys neither power law nor exponential. To explain the phenomenon, we propose a model as follows: starting with a single vertex, at each time step, a new vertex is added and linked to one of the existing vertices, which is chosen according the following rule: at time 𝑚,2𝑚,3𝑚,…, where 𝑚 is integer, we choose one of the existing vertices with probability proportional to the degree; that is, we have probability 𝑘/𝑠_𝑛, where 𝑘 is the degree of the vertex chosen and 𝑠_𝑛 is the total degree of vertices; at another time step, we choose one of the existing vertices with equal probability. Related models were also proposed by Krapivsky and Redner [15] and Li [16] to describe the organization of growing networks. In this paper, we will focus on the distribution of evolving network and the distribution of the number of vertices with given degree will be considered in Section 2. In Section 3, we will consider the asymptotic degree distribution.

2. The Number of Vertices with Given Degrees

Let 𝐷_𝑛(𝑘) denote the number of vertices with degree 𝑘 at time 𝑛. We will consider the case 𝑘=1,2 in this section and the case 𝑘>2 will be considered in the next section. As 𝑘=1, we obtain the following result.

Lemma 1. In the evolving network, the expectation of the number of degree 1 satisfies 𝐸𝐷_𝑛(1)=2𝑚4𝑚−1𝑛. (1)

Proof. Herein after, = denotes asymptotic equivalence as 𝑛→∞. From the way the network is formed, we can see that, for 𝑛>1, the number of vertices of degree 1 does not change if we attach a new vertex 𝑣_𝑛 to a vertex with degree 1 and increases by 1 if we attach 𝑣_𝑛 to vertices of degree larger than 1 after joining the vertex 𝑣_𝑛. Assuming 𝑛 is multiple of 𝑚, that is, 𝑛=𝑘𝑚, where 𝑘 is integer number, and taking expectation of 𝐷_𝑛(1), we obtain ED_𝑘𝑚(1)=11−2(𝑘𝑚−1)ED_𝑘𝑚−1(1)=1+11−2()×𝑘𝑚−1ED_𝑘𝑚−2(11)1−=1𝑘𝑚−2+1+11−2()1𝑘𝑚−11−𝑘𝑚−2ED_𝑘𝑚−2(1)+11−2(𝑘𝑚−1)+1. (2) The first equation shows that when we add a new vertex and link it to one of existing vertices with preferential attachment, the number of vertex increases by 1, while the second equation comes from the uniform attachment. Continuing the iteration and noticing the boundary condition ED₁(1)=0, we have ED_𝑘𝑚(1)=^𝑘𝑚_𝑗=1^[𝑗/𝑚]_𝑖=011−2)]×[(𝑘−𝑖𝑚−1_𝑗𝑣=211−×𝑘𝑚−𝑣_{[𝑗/𝑚]𝑠=0}11−(𝑘−𝑖)𝑚−1⁻¹. (3) Considering the term ^[𝑗/𝑚]_𝑖=011−2[(𝑘−𝑖)𝑚−1]^𝑗_𝑣=211−×𝑘𝑚−𝑣_{[𝑗/𝑚]𝑠=0}11−(𝑘−𝑖)𝑚−1⁻¹, (4) we have 𝑒^{ln∏[𝑗/𝑚]𝑖=0(1−(1/2[(𝑘−𝑖)𝑚−1]))∏𝑗𝑣=2(1−(1/(𝑘𝑚−𝑣)))−ln∏[𝑗/𝑚]𝑠=0(1−(1/((𝑘−𝑖)𝑚−1)))}=𝑒^{−∑[𝑗/𝑚]𝑖=0(1/2[(𝑘−𝑖)𝑚−1])−∑𝑗𝑣=2(1/(𝑘𝑚−𝑣))+∑[𝑗/𝑚]𝑖=0(1/((𝑘−𝑖)𝑚−1))}=𝑒^{−(1/(2𝑘−1))∫[𝑗/𝑚]𝑡=0(1/(1−(𝑚𝑡/(2𝑘−1))))𝑑𝑡−(1/𝑘𝑚)∫𝑗𝑣=2(1/(1−(𝑣/𝑘𝑚)))𝑑𝑣+(1/(𝑘𝑚−1))∫[𝑗/𝑚]𝑡=0(1/(1−(𝑚𝑡/(𝑘𝑚−1))))𝑑𝑡}=𝑗1−𝑘𝑚^{(1−(1/2𝑚))}. (5)We obtain that ED_𝑘𝑚(1)=^𝑘𝑚_𝑗=1𝑗1−𝑘𝑚^{(1−(1/2𝑚))}=2𝑚4𝑚−1⋅𝑘𝑚 (6) when 𝑛 is not a multiple of 𝑚, assuming 𝑛=𝑘^′𝑚+𝑠,1<𝑠<𝑚, where 𝑘^′ is an integer number; we also obtain ED_{𝑘′𝑚+𝑠}(1)=ED_{𝑘′𝑚+𝑠−1}(1)11−𝑘^′=1𝑚+𝑠−1+1=⋯1−𝑘^′×1𝑚+𝑠−11−𝑘^′⋯1𝑚+𝑠−21−𝑘^′𝑚ED^𝑚_𝑘′𝑚(1)+11−𝑘^′×1𝑚+𝑠−11−𝑘^′⋯1𝑚+𝑠−21−𝑘^′1𝑚+1+⋯+1−𝑘^′1𝑚+𝑠−11−𝑘^′+1𝑚+𝑠−21−𝑘^′.𝑚+𝑠−1 (7) When 𝑛 is large enough, we can see that ED_𝑘′𝑚(1)=ED_{𝑘′𝑚+𝑠}(1)(1+𝑜(1)). (8) As a result, we have ED_𝑛(1)=2𝑚4𝑚−1𝑛. (9)

Now we discuss the number of degree 2 in the network; we have the following.

Lemma 2. For 𝑛>2, ED_𝑛(2)=2𝑚−18𝑚−2𝑛. (10)

Proof. We prove the case that 𝑛 is a multiple of 𝑚 and assume 𝑛=𝑘𝑚, where 𝑘 is an integer number; considering the expectation of 𝐷_𝑛(2), we have ED_𝑘𝑚(2)=21−2(𝑘𝑚−1)ED_𝑘𝑚−1(2)+ED_𝑘𝑚−1(1)2()=1𝑘𝑚−11−𝑘𝑚−1ED_𝑘𝑚−2(2)11−+𝑘𝑚−2ED_𝑘𝑚−2(1)+𝑘𝑚−2ED_𝑘𝑚−1(1)2().𝑘𝑚−1 (11) Noticing the boundary condition ED₂(2)=0 and Lemma 1, we have ED_𝑘𝑚(2)=^𝑘𝑚𝑗_𝑗=1𝑖=111−−1𝑘𝑚−𝑖2^𝑘_𝑗=1^𝑗𝑚_𝑣=111−𝑘𝑚−𝑣2𝑚.4𝑚−1 (12) By the estimation ln(1+𝑥)=𝑥 and the fact that ^𝑗_𝑖=111−𝑗𝑘𝑚−𝑖=1−,𝑘𝑚_{𝑗𝑚𝑣=0}11−𝑗𝑘𝑚−𝑣=1−𝑘, (13) we obtain that ED_𝑘𝑚(2)=2𝑚4𝑚−1^𝑘𝑚_𝑗=1𝑗1−−1𝑘𝑚2^𝑘_𝑗=0𝑗1−𝑘=𝑘𝑚2𝑚4𝑚−1¹₀()11−𝑥𝑑𝑥−2¹₀()=1−𝑥𝑑𝑥2𝑚−18𝑚−2⋅𝑘𝑚. (14) The case 𝑛, which is not a multiple of 𝑚, is the same as Lemma 1, just a little tedious.

3. Asymptotic Degree Distribution of Network

Let 𝑝_𝑘(𝑛)=𝐷_𝑛(𝑘)𝑛 (15) denote the proportion of vertices with degree 𝑘 at time 𝑛. Considering the expectation of 𝐷_𝑛(𝑘), we have the following theorem.

Theorem 3. For arbitrary 𝑘>1 and 𝑛, the expectation of the number of degree 𝑘 satisfies 𝐸𝐷_𝑛(𝑘)=2𝑚4𝑚−1^𝑘_𝑖=22𝑚+𝑖−34𝑚+𝑗−2𝑛. (16)

Proof. The case 𝑘=1,2 is just the result of Lemmas 1 and 2. Assume the result is true for 𝑘; that is, ED_𝑛(𝑘)=2𝑚4𝑚−1^𝑘_𝑖=22𝑚+𝑖−34𝑚+𝑗−2𝑛. (17) We will prove the result is true for 𝑘+1. We just prove the case 𝑛 is a multiple of 𝑚; that is, 𝑛=𝑙𝑚, where 𝑙 is integer number. From the network constructed, we have ED_𝑚𝑙()=𝑘+11−𝑘+12(𝑚𝑙−1)ED_𝑚𝑙−1()+𝑘𝑘+12(𝑚𝑙−1)ED_𝑚𝑙−1(𝑘)=1−𝑘+12()×𝑚𝑙−1ED_𝑚𝑙−2(1𝑘+1)1−+𝑚𝑙−2ED_𝑚𝑙−2(𝑘)+𝑘𝑚𝑙−22(𝑚𝑙−1)ED_𝑚𝑙−1(𝑘). (18) Continuing the iteration and noticing the boundary condition ED_𝑠(𝑘+1)=0,𝑠<𝑘, we obtain that ED_𝑚𝑙()=𝑘+12𝑚4𝑚−1^𝑘_𝑖=22𝑚+𝑖−3×4𝑚+𝑗−2_{𝑙𝑚𝑗=1[𝑗/𝑚]𝑖=0}1−𝑘+12(𝑙−𝑖)𝑚−1^𝑗_𝑣=111−×𝑙𝑚−𝑣_{[𝑗/𝑚]𝑖=0}11−(𝑙−𝑖)𝑚−1⁻¹+𝑘−222𝑚×4𝑚−1_𝑘𝑖=22𝑚+𝑖−34𝑚+𝑗−2^𝑙𝑗_𝑗=1𝑖=01−𝑘+12()×𝑙−𝑖𝑚−1_𝑗𝑣=111−×𝑙𝑚−𝑣^𝑗_𝑖=011−(𝑙−𝑖)𝑚−1⁻¹. (19) Noticing the fact that ^[𝑗/𝑚]_𝑖=01−𝑘+12(𝑙−𝑖)𝑚−1^𝑗_𝑣=111−×𝑙𝑚−𝑣_{[𝑗/𝑚]𝑖=0}11−(𝑙−𝑖)𝑚−1⁻¹=𝑗1−𝑙𝑚^{(((𝑘−1)/2𝑚)+1)}_𝑗𝑖=0,1−𝑘+12(𝑙−𝑖)𝑚−1^𝑗𝑚_𝑣=111−×𝑙𝑚−𝑣_𝑗𝑖=011−()𝑗𝑙−𝑖𝑚−11−𝑙𝑚^{(((𝑘−1)/2𝑚)+1)−1}=𝑗1−𝑙^{(((𝑘−1)/2𝑚)+1)}. (20)(21) We obtain ED_𝑙𝑚()=𝑘+12𝑚4𝑚−1^𝑘_𝑖=22𝑚+𝑖−3×4𝑚+𝑗−2_{𝑙𝑚𝑗=1}𝑗1−𝑙𝑚^{(((𝑘−1)/2𝑚)+1)}+𝑘−222𝑚×4𝑚−1_𝑘𝑖=22𝑚+𝑖−34𝑚+𝑗−2^𝑙_𝑗=1𝑗1−𝑙^{(((𝑘−1)/2𝑚)+1)}=2𝑚4𝑚−1^𝑘_𝑖=22𝑚+𝑖−34𝑚+𝑗−2¹₀(1−𝑥)^{(((𝑘−1)/2𝑚)+1)}+𝑑𝑥⋅𝑙𝑚𝑘−222𝑚×4𝑚−1_𝑘𝑖=22𝑚+𝑖−34𝑚+𝑗−2¹₀(1−𝑥)^{(((𝑘−1)/2𝑚)+1)}=𝑑𝑥⋅𝑙2𝑚4𝑚−1^𝑘_𝑖=22𝑚+𝑖−34𝑚+𝑗−21+𝑘−2⋅2𝑚2𝑚=4𝑚+𝑘−1⋅𝑙𝑚2𝑚4𝑚−1^𝑘+1_𝑖=22𝑚+𝑖−34𝑚+𝑗−2⋅𝑙𝑚. (22) The result is true for 𝑘+1.

From Theorem 3, we can see that lim_𝑛→∞(ED_𝑛(𝑘)/𝑛) exists; we denote it by 𝑝_𝑘. Now we consider the relation of 𝑝_𝑘 and 𝑝_𝑘(𝑛); we introduce the following lemma.

Lemma 4. There exists a bound constant 𝐶(𝑘) such that for arbitrary 𝑎>0, 𝑃𝐷_𝑛(𝑘)−𝐸𝐷_𝑛(𝑘)≥𝑎≤2𝑒^{−𝑎2/2𝐶(𝑘)2𝑛}. (23)

Proof. Let 𝔉_𝑛=𝜎(𝐷₁(1),…,𝐷_𝑘(1),𝐷_𝑘(2),…𝐷_𝑘(𝑘),…,𝐷_𝑛(1),…𝐷_𝑛(𝑘)…,𝐷_𝑛(𝑛)) denote the 𝜎-algbra. For 𝑚=0,1,…,𝑛, we define 𝑀_𝑚=𝐸𝐷_𝑘(𝑛)∣𝔉_𝑚. (24) By the tower property of conditional expectation and the fact that the 𝜎-algbra 𝔉_𝑛 can be deduced from 𝔉_𝑛+1, we obtain that, for 𝑚<𝑛, 𝐸𝑀_𝑚+1∣𝔉_𝑚=𝐸𝐸𝐷_𝑛(𝑘)∣𝔉_𝑚+1∣𝔉_𝑚=𝐸𝐷_𝑛(𝑘)∣𝔉_𝑚=𝑀_𝑚. (25) Noticing the fact that 𝐸𝑀_𝑚=𝐸𝑀_𝑚=ED_𝑛(𝑘)<𝑛<∞, (26) we have {𝑀_𝑚}^𝑛_𝑚=0 as a martingale sequence. According to the definition of the 𝜎-algbra, we know the 𝔉₀ has no information of the network and 𝔉_𝑛 has the whole information, so we have 𝑀₀=𝐸𝐷_𝑛(𝑘)∣𝔉₀=ED_𝑛(𝑘),𝑀_𝑛=𝐸𝐷_𝑛(𝑘)∣𝔉_𝑛=𝐷_𝑛(𝑘). (27) Therefore, we have 𝐷_𝑛(𝑘)−ED_𝑛(𝑘)=𝑀_𝑛−𝑀₀=^𝑛_𝑗=0𝑀_𝑗+1−𝑀_𝑗. (28)
Now we prove that there exists a bound constant 𝐶(𝑘), such that |𝑀_𝑗+1−𝑀_𝑗|≤𝐶(𝑘). We will prove the result by induction. For the case 𝑘=1, we have 𝑀_𝑗+1−𝑀_𝑗=𝐸𝐷_𝑛(1)∣𝔉_𝑗+1−𝐸𝐷_𝑛(1)∣𝔉_𝑗=𝐸𝐷_𝑛(1)−𝐷_𝑛−1(1)∣𝔉_𝑗+1−𝐸𝐷_𝑛(1)−𝐷_𝑛−1(1)∣𝔉_𝑗+𝐸𝐷_𝑛−1(1)∣𝔉_𝑗+1−𝐸𝐷_𝑛−1(1)∣𝔉_𝑗=𝐸𝐸𝐷_𝑛(1)−𝐷_𝑛−1(1)∣𝔉_𝑛−1∣𝔉_𝑗+1−𝐸𝐸𝐷_𝑛(1)−𝐷_𝑛−1(1)∣𝔉_𝑛−1∣𝔉_𝑗+𝐸𝐷_𝑛−1(1)∣𝔉_𝑗+1−𝐸𝐷_𝑛−1(1)∣𝔉_𝑗=11−×𝐸𝐷𝑛−1_𝑛−1(1)∣𝔉_𝑗+1−𝐸𝐷_𝑛−1(1)∣𝔉_𝑗. (29)
Continuing the iteration and noticing the fact that 𝐸(𝐷_𝑚(1)∣𝔉_𝑗+1)−𝐸(𝐷_𝑚(1)∣𝔉_𝑗)=0, for 𝑚<𝑗, we obtain that 𝑀_𝑗+1−𝑀_𝑗=^𝑛−1_𝑖=𝑗11−𝑖^[𝑛/𝑚]_𝑠=111−2()×𝑠𝑚−1_{[𝑛/𝑚]𝑠=1}11−𝑠𝑚−1⁻¹⋅𝐸𝐷_𝑗+1(1)∣𝔉_𝑗+1−𝐸𝐷_𝑗+1(1)∣𝔉_𝑗=^𝑛−1_𝑖=𝑗11−𝑖^[𝑛/𝑚]_𝑠=111−2()×𝑠𝑚−1_{[𝑛/𝑚]𝑠=1}11−𝑠𝑚−1⁻¹⋅𝐷_𝑗+1(1)−𝐷_𝑗(1)−𝐸𝐷_𝑗+1(1)−𝐷_𝑗(1)∣𝔉_𝑗. (30) Obviously, 𝐷_𝑗+1(1)−𝐷_𝑗(1)≤1,_{𝑛−1𝑖=𝑗}11−𝑖^[𝑛/𝑚]_𝑠=111−2()×𝑠𝑚−1_{[𝑛/𝑚]𝑠=1}11−𝑠𝑚−1⁻¹≤1, (31) so we have 𝑀_𝑗+1−𝑀_𝑗≤2. (32) Assume the result is true for 𝑘; that is, there exists a bound constant 𝐶(𝑘), such that 𝑀_𝑗+1−𝑀_𝑗≤𝐶(𝑘). (33) For 𝑘+1, by the definition of 𝑀_𝑗+1, we have 𝑀_𝑗+1−𝑀_𝑗=𝐸𝐷_𝑛(𝑘+1)∣𝔉_𝑗+1−𝐸𝐷_𝑛(𝑘+1)∣𝔉_𝑗=11−×𝐸𝐷𝑛−1_𝑛(𝑘+1)∣𝔉_𝑗+1−𝐸𝐷_𝑛−1(𝑘+1)∣𝔉_𝑗+1𝐸𝐷𝑛−1^𝑛−1(𝑘)∣𝔉_𝑗+1−𝐸𝐷_𝑛−1(𝑘)∣𝔉_𝑗. (34) Continuing the iteration and using the assumption for 𝑘, we obtain that 𝑀_𝑗+1−𝑀_𝑗≤^𝑛−1_𝑣=𝑗+111−𝑣^[𝑛/𝑚]_{𝑖=[(𝑗+1)/𝑚]}1−𝑘+12()×𝑖𝑚−1_{[𝑛/𝑚]𝑖=[(𝑗+1)/𝑚]}11−(𝑖𝑚−1)⁻¹⋅𝐸𝐷_𝑗+1(𝑘+1)∣𝔉_𝑗+1−𝐸𝐷_𝑗+1(𝑘+1)∣𝔉_𝑗+^{𝑛−𝑗−1𝑣}_𝑣=1𝑠=111−1𝑛−𝑠𝐶(𝑘)+𝑛−𝑣−1_{[(𝑛−𝑗−1)/𝑚]𝑣=0𝑛−([𝑛/𝑚]−𝑣)𝑚𝑠=1}11−⋅𝑛−𝑠_𝑣𝑠=01−𝑘+12]))×(([𝑛/𝑚−𝑠𝑚−1_𝑗𝑠=011−([𝑛/𝑚]−𝑠)𝑚−1𝐶(𝑘)⁻¹. (35) Noticing the fact that 1−((𝑘+1)/2𝑗)<1−(1/𝑗) and ^𝑛−1_𝑣=𝑗+111−𝑣^[𝑛/𝑚]_{𝑖=[(𝑗+1)/𝑚]}1−𝑘+12()×𝑖𝑚−1_{[𝑛/𝑚]𝑖=[(𝑗+1)/𝑚]}11−(𝑖𝑚−1)⁻¹≤1, (36) we obtain that 𝑀_𝑗+1−𝑀_𝑗≤2+𝑛−𝑗−1𝐶(𝑘)+𝑛−1𝑛−𝑗−1/𝑚𝐶(𝑘)(𝑘)1𝑛−1≤2+𝐶1+𝑚. (37) We just let 𝐶(𝑘+1)=2+𝐶(𝑘)(1+(1/𝑚)) and the result for 𝑘+1 is proved. By Asume-Hoeffding’s inequality, we have the following for arbitrary 𝑎>0: 𝑃𝐷_𝑛(𝑘)−ED_𝑛(𝑘)≥𝑎≤2𝑒^{−𝑎2/2𝐶(𝑘)2𝑛}. (38)

Theorem 5. For a fixed 𝑘, one has lim_𝑛→∞𝑝_𝑘(𝑛)=𝑝_𝑘𝑎.𝑒. (39)

Proof. By the Borel-Cantelli Lemma, we need to prove the following for arbitrary 𝜀: ^∞_𝑛=1𝑃𝑝_𝑘(𝑛)−𝑝_𝑘>𝜀<∞. (40) We have ^∞_𝑛=1𝑃𝑝_𝑘(𝑛)−𝑝_𝑘=>𝜀_∞𝑛=1𝑃𝐷_𝑛(𝑘)𝑛−ED_𝑛(𝑘)𝑛+ED_𝑛(𝑘)𝑛−𝑝_𝑘≤>𝜀_∞𝑛=1𝑃𝐷_𝑛(𝑘)𝑛−ED_𝑛(𝑘)𝑛≥𝜀2+^∞_𝑛=1𝑃ED_𝑛(𝑘)𝑛−𝑝_𝑘≥𝜀2. (41) Noticing that lim_𝑛→∞ED_𝑘(𝑛)/𝑛=𝑝_𝑘 and using Lemma 4, we obtain that there exists 𝑁, such that ^∞_𝑛=1𝑃𝑝_𝑘(𝑛)−𝑝_𝑘≤>𝜀_∞𝑛=1𝑃𝐷_𝑛(𝑘)−ED_𝑛(𝑘≥𝜀)2𝑛≤+𝑁_∞𝑛=12𝑒^{−(𝜀2/4)𝑛}+𝑁<∞. (42)

Remark 6. As a result, we can see that the distribution 𝑝_𝑘 obeys the following rule.
When 𝑚≪𝑘, 𝑝_𝑘∝𝑘^−(2𝑚+1), the degree distribution obeys power law; when 𝑚≫𝑘, 𝑝_𝑘∝2^−𝑘, the degree distribution obeys exponential distribution; otherwise, 𝑝_𝑘=(2𝑚/(4𝑚−1))∏^𝑘_𝑗=2((2𝑚+𝑗−3)/(4𝑚+𝑗−2)).

Conflict of Interests

The authors declare that there is no conflict of interests.

Acknowledgments

Zhimin Li was partially supported by National Natural Science Foundation of China (71171003 and 71271003), Anhui Natural Science Foundation (nos. 10040606Q03 and 1208085QA04), and Key University Science Research Project of Anhui Province (KJ2013A044).