AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2014/761235 761235 Research Article Asymptotic Degree Distribution of a Kind of Asymmetric Evolving Network Li Zhimin 1 He Zhaolin 2 Hu Chunhua 3 Dong Hongli 1 School of Mathematics and Physics Anhui Polytechnic University Wuhu, Anhui 241000 China ahpu.edu.cn 2 School of Management Engineering Anhui Polytechnic University Wuhu, Anhui 241000 China ahpu.edu.cn 3 School of Applied Mathematics Beijing Normal University Zhuhai Zhuhai, Guangdong 519087 China bnuz.edu.cn 2014 972014 2014 19 05 2014 04 06 2014 9 7 2014 2014 Copyright © 2014 Zhimin Li et al. 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.

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 pkk>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 . 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 . Several models were introduced to explain the properties. Bollobás  proposed a model with n vertices and m edges. In this model, the degree distribution is approximately Poisson distribution. Later, Barabási and Albert  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 r1. The result was confirmed by Barabási et al. [11, 12]. In order to generate power laws with arbitrary exponents, Dorogovtsev et al.  and Drine et al.  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 pd 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/sn, where k is the degree of the vertex chosen and sn 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  and Li  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 Dn(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 (1)EDn(1)=2m4m-1n.

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 vn to a vertex with degree 1 and increases by 1 if we attach vn to vertices of degree larger than 1 after joining the vertex vn. Assuming n is multiple of m, that is, n=km, where k is integer number, and taking expectation of Dn(1), we obtain (2)EDkm(1)=(1-12(km-1))EDkm-1(1)+1=(1-12(km-1))×[EDkm-2(1)(1-1km-2)+1]+1=(1-12(km-1))(1-1km-2)EDkm-2(1)+(1-12(km-1))+1. 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 (3)EDkm(1)=j=1kmi=0[j/m](1-12[(k-i)m-1])×v=2j(1-1km-v)×(s=0[j/m](1-1(k-i)m-1))-1. Considering the term (4)i=0[j/m](1-12[(k-i)m-1])v=2j(1-1km-v)×(s=0[j/m](1-1(k-i)m-1))-1, we have (5)elni=0[j/m](1-(1/2[(k-i)m-1]))v=2j(1-(1/(km-v)))-lns=0[j/m](1-(1/((k-i)m-1)))=e-i=0[j/m](1/2[(k-i)m-1])-v=2j(1/(km-v))+i=0[j/m](1/((k-i)m-1))=e-(1/(2k-1))t=0[j/m](1/(1-(mt/(2k-1))))dt-(1/km)v=2j(1/(1-(v/km)))dv+(1/(km-1))t=0[j/m](1/(1-(mt/(km-1))))dt=(1-jkm)(1-(1/2m)).We obtain that (6)EDkm(1)=j=1km(1-jkm)(1-(1/2m))=2m4m-1·km when n is not a multiple of m, assuming n=km+s,1<s<m, where k is an integer number; we also obtain (7)EDkm+s(1)=EDkm+s-1(1)(1-1km+s-1)+1==(1-1km+s-1)×(1-1km+s-2)(1-1km)EDkmm(1)+(1-1km+s-1)×(1-1km+s-2)(1-1km+1)++(1-1km+s-1)(1-1km+s-2)+(1-1km+s-1). When n is large enough, we can see that (8)EDkm(1)=EDkm+s(1)(1+o(1)). As a result, we have (9)EDn(1)=2m4m-1n.

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

Lemma 2.

For n>2, (10)EDn(2)=2m-18m-2n.

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 Dn(2), we have (11)EDkm(2)=(1-22(km-1))EDkm-1(2)+EDkm-1(1)2(km-1)=(1-1km-1)[EDkm-2(2)(1-1km-2)EDkm-2(1)km-2hhhhhhhhhhhhh+EDkm-2(1)km-2]+EDkm-1(1)2(km-1). Noticing the boundary condition ED2(2)=0 and Lemma 1, we have (12)EDkm(2)=(j=1kmi=1j(1-1km-i)hl-12j=1kv=1jm(1-1km-v))2m4m-1. By the estimation ln(1+x)=x and the fact that (13)i=1j(1-1km-i)=1-jkm,v=0jm(1-1km-v)=1-jk, we obtain that (14)EDkm(2)=2m4m-1[j=1km(1-jkm)-12j=0k(1-jk)]=km2m4m-1[01(1-x)dx-1201(1-x)dx]=2m-18m-2·km. 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 (15)pk(n)=Dn(k)n denote the proportion of vertices with degree k at time n. Considering the expectation of Dn(k), we have the following theorem.

Theorem 3.

For arbitrary k>1 and n, the expectation of the number of degree k satisfies (16)EDn(k)=2m4m-1i=2k2m+i-34m+j-2n.

Proof.

The case k=1,2 is just the result of Lemmas 1 and 2. Assume the result is true for k; that is, (17)EDn(k)=2m4m-1i=2k2m+i-34m+j-2n. 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 (18)EDml(k+1)=(1-k+12(ml-1))EDml-1(k+1)+k2(ml-1)EDml-1(k)=(1-k+12(ml-1))×[EDml-2(k+1)(1-1ml-2)+EDml-2(k)ml-2]+k2(ml-1)EDml-1(k). Continuing the iteration and noticing the boundary condition EDs(k+1)=0,s<k, we obtain that (19)EDml(k+1)=2m4m-1i=2k2m+i-34m+j-2×j=1lmi=0[j/m](1-k+12(l-i)m-1)v=1j(1-1lm-v)×(i=0[j/m](1-1(l-i)m-1))-1+k-222m4m-1×i=2k2m+i-34m+j-2j=1li=0j(1-k+12(l-i)m-1)×v=1j(1-1lm-v)×(i=0j(1-1(l-i)m-1))-1. Noticing the fact that (20)i=0[j/m](1-k+12(l-i)m-1)v=1j(1-1lm-v)×(i=0[j/m](1-1(l-i)m-1))-1=(1-jlm)(((k-1)/2m)+1),(21)i=0j(1-k+12(l-i)m-1)v=1jm(1-1lm-v)×(i=0j(1-1(l-i)m-1)(1-jlm)(((k-1)/2m)+1))-1=(1-jl)(((k-1)/2m)+1). We obtain (22)EDlm(k+1)=2m4m-1i=2k2m+i-34m+j-2×j=1lm(1-jlm)(((k-1)/2m)+1)+k-222m4m-1×i=2k2m+i-34m+j-2j=1l(1-jl)(((k-1)/2m)+1)=2m4m-1i=2k2m+i-34m+j-201(1-x)(((k-1)/2m)+1)dx·lm+k-222m4m-1×i=2k2m+i-34m+j-201(1-x)(((k-1)/2m)+1)dx·l=2m4m-1i=2k2m+i-34m+j-2(1+k-22m)·2m4m+k-1·lm=2m4m-1i=2k+12m+i-34m+j-2·lm. The result is true for k+1.

From Theorem 3, we can see that limn(EDn(k)/n) exists; we denote it by pk. Now we consider the relation of pk and pk(n); we introduce the following lemma.

Lemma 4.

There exists a bound constant C(k) such that for arbitrary a>0, (23)P(|Dn(k)-EDn(k)|a)2e-a2/2C(k)2n.

Proof.

Let Fn=σ(D1(1),,Dk(1),Dk(2),Dk(k),,Dn(1),Dn(k),Dn(n)) denote the σ-algbra. For m=0,1,,n, we define (24)Mm=E(Dk(n)Fm). By the tower property of conditional expectation and the fact that the σ-algbra Fn can be deduced from Fn+1, we obtain that, for m<n, (25)E(Mm+1Fm)=E[E(Dn(k)Fm+1)Fm]=E(Dn(k)Fm)=Mm. Noticing the fact that (26)E[|Mm|]=EMm=EDn(k)<n<, we have {Mm}m=0n as a martingale sequence. According to the definition of the σ-algbra, we know the F0 has no information of the network and Fn has the whole information, so we have (27)M0=E[Dn(k)F0]=EDn(k),Mn=E[Dn(k)Fn]=Dn(k). Therefore, we have (28)Dn(k)-EDn(k)=Mn-M0=j=0n(Mj+1-Mj).

Now we prove that there exists a bound constant C(k), such that |Mj+1-Mj|C(k). We will prove the result by induction. For the case k=1, we have (29)|Mj+1-Mj|=|E(Dn(1)Fj+1)-E(Dn(1)Fj)|=|E(Dn(1)-Dn-1(1)Fj+1)h-E(Dn(1)-Dn-1(1)Fj)h+E(Dn-1(1)Fj+1)-E(Dn-1(1)Fj)|=|E(E(Dn(1)-Dn-1(1)Fn-1)Fj+1)h-E(E(Dn(1)-Dn-1(1)Fn-1)Fj)h+E(Dn-1(1)Fj+1)-E(Dn-1(1)Fj)|=(1-1n-1)×|E(Dn-1(1)Fj+1)-E(Dn-1(1)Fj)|.

Continuing the iteration and noticing the fact that E(Dm(1)Fj+1)-E(Dm(1)Fj)=0, for m<j, we obtain that (30)|Mj+1-Mj|=i=jn-1(1-1i)s=1[n/m](1-12(sm-1))×(s=1[n/m](1-1sm-1))-1·|E(Dj+1(1)Fj+1)-E(Dj+1(1)Fj)|=i=jn-1(1-1i)s=1[n/m](1-12(sm-1))×(s=1[n/m](1-1sm-1))-1·|(Dj+1(1)-Dj(1))-E(Dj+1(1)-Dj(1)Fj)|. Obviously, (31)|Dj+1(1)-Dj(1)|1,i=jn-1(1-1i)s=1[n/m](1-12(sm-1))×(s=1[n/m](1-1sm-1))-11, so we have (32)|Mj+1-Mj|2. Assume the result is true for k; that is, there exists a bound constant C(k), such that (33)|Mj+1-Mj|C(k). For k+1, by the definition of Mj+1, we have (34)|Mj+1-Mj|=|E(Dn(k+1)Fj+1)-E(Dn(k+1)Fj)|=(1-1n-1)×|E(Dn(k+1)Fj+1)-E(Dn-1(k+1)Fj)|+1n-1[E(Dn-1(k)Fj+1)-E(Dn-1(k)Fj)]. Continuing the iteration and using the assumption for k, we obtain that (35)|Mj+1-Mj|v=j+1n-1(1-1v)i=[(j+1)/m][n/m](1-k+12(im-1))×(i=[(j+1)/m][n/m](1-1(im-1)))-1·|E(Dj+1(k+1)Fj+1)-E(Dj+1(k+1)Fj)|+v=1n-j-1s=1v(1-1n-s)1n-v-1C(k)+v=0[(n-j-1)/m]s=1n-([n/m]-v)m(1-1n-s)·s=0v(1-k+12(([n/m]-s)m-1))×(s=0j(1-1([n/m]-s)m-1C(k)))-1. Noticing the fact that 1-((k+1)/2j)<1-(1/j) and (36)v=j+1n-1(1-1v)i=[(j+1)/m][n/m](1-k+12(im-1))×(i=[(j+1)/m][n/m](1-1(im-1)))-11, we obtain that (37)|Mj+1-Mj|2+n-j-1n-1C(k)+[(n-j-1)/m]n-1C(k)2+C(k)(1+1m). 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: (38)P(|Dn(k)-EDn(k)|a)2e-a2/2C(k)2n.

Theorem 5.

For a fixed k, one has (39)limnpk(n)=pka.e.

Proof.

By the Borel-Cantelli Lemma, we need to prove the following for arbitrary ɛ: (40)n=1P(|pk(n)-pk|>ɛ)<. We have (41)n=1P(|pk(n)-pk|>ɛ)=n=1P(|Dn(k)n-EDn(k)n+EDn(k)n-pk|>ɛ)n=1P(|Dn(k)n-EDn(k)n|ɛ2)+n=1P(|EDn(k)n-pk|ɛ2). Noticing that limnEDk(n)/n=pk and using Lemma 4, we obtain that there exists N, such that (42)n=1P(|pk(n)-pk|>ɛ)n=1P(|Dn(k)-EDn(k)|ɛ2n)+Nn=12e-(ɛ^2/4)n+N<.

Remark 6.

As a result, we can see that the distribution pk obeys the following rule.

When mk, pkk-(2m+1), the degree distribution obeys power law; when mk, pk2-k, the degree distribution obeys exponential distribution; otherwise, pk=(2m/(4m-1))j=2k((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).

Chung F. Lu L. Y. Complex Graphs and Networks 2006 American Mathematical Society Zhang X. Yang L. A fiber Bragg grating quasi-distributed sensing network with a wavelength-tunable chaotic fiber laser Systems Science and Control Engineering 2014 2 1 268 274 10.1080/21642583.2014.888962 Ahmada H. Namerikawa T. Extended Kalman filter-based mobile robot localization with intermittent measurements Systems Science and Control Engineering 2013 1 1 113 126 Ding D. Wang Z. Dong H. Shu H. Distributed H state estimation with stochastic parameters and nonlinearities through sensor networks: the finite-horizon case Automatica 2012 48 8 1575 1585 10.1016/j.automatica.2012.05.070 MR2950405 2-s2.0-84864447590 Ding D. Wang Z. Hu J. Shu H. Dissipative control for state-saturated discrete time-varying systems with randomly occurring nonlinearities and missing measurements International Journal of Control 2013 86 4 674 688 10.1080/00207179.2012.757652 MR3046933 ZBLl1278.93279 2-s2.0-84876121109 Wang Z. Dong H. Shen B. Gao H. Finite-horizon H filtering with missing measurements and quantization effects IEEE Transactions on Automatic Control 2013 58 7 1707 1718 10.1109/TAC.2013.2241492 MR3072855 2-s2.0-84879974725 Ahmada H. Namerikawa T. Extended Kalman filter-based mobile robot localization with intermittent measurements Systems Science & Control Engineering 2013 1 1 113 126 Kumar G. Kumar K. Network security—an updated perspective Systems Science & Control Engineering 2014 2 1 325 334 10.1080/21642583.2014.895969 Bollobás B. Random Graphs 2001 Cambrige, UK Combrige University Press Barabási A. L. Albert R. Emergence of scaling in random networks Science 1999 286 5439 509 512 10.1126/science.286.5439.509 MR2091634 2-s2.0-0038483826 Barabási A. L. Albert R. Jeong H. Mean-field theory for scale-free random networks Physica A: Statistical Mechanics and its Applications 1999 272 1 173 187 10.1016/S0378-4371(99)00291-5 2-s2.0-18744421488 Bollobás B. Riordan O. Spencer J. Tusnády G. The degree sequence of a scale-free random graph process Random Structures & Algorithms 2001 18 3 279 290 10.1002/rsa.1009 MR1824277 Dorogovtsev S. N. Mendes J. F. F. Samukhin A. N. Structure of growing networks with preferential linking Physical Review Letters 2000 85 21 4633 4636 10.1103/PhysRevLett.85.4633 2-s2.0-0034310713 Drine E. Enachescu M. Mitzenmacher M. Variations on random graph models for the web 2001 Havard University, Deparment of Computer Science Krapivsky P. L. Redner S. Organization of growing random networks Physical Review E—Statistical, Nonlinear, and Soft Matter Physics 2001 63 6 066123 2-s2.0-0035363060 Li Z. Branching structure and maximum degree of an evolving random tree Theoretical and Mathematical Physics 2011 166 2 270 277 10.1007/s11232-011-0021-2 2-s2.0-79953690235