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Asymptotic Degree Distribution of a Kind of Asymmetric Evolving Network
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 obeys power law, exponential distribution, and other forms according to the relation of and parameter .
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 [2–8]. Several models were introduced to explain the properties. Bollobás  proposed a model with vertices and 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 and a fixed number of edges originating at 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 for all . 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 new edges added at each time step is chosen with probability proportional to the degree plus an initial attractiveness ; they gave a nonrigorous argument that the degree distribution behaves like for large .
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 , where is integer, we choose one of the existing vertices with probability proportional to the degree; that is, we have probability , where is the degree of the vertex chosen and 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 denote the number of vertices with degree at time . We will consider the case in this section and the case will be considered in the next section. As , we obtain the following result.
Lemma 1. In the evolving network, the expectation of the number of degree 1 satisfies
Proof. Herein after, denotes asymptotic equivalence as . From the way the network is formed, we can see that, for , the number of vertices of degree 1 does not change if we attach a new vertex to a vertex with degree 1 and increases by 1 if we attach to vertices of degree larger than 1 after joining the vertex . Assuming is multiple of , that is, , where is integer number, and taking expectation of , we obtain 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 , we have Considering the term we have We obtain that when is not a multiple of , assuming , where is an integer number; we also obtain When is large enough, we can see that As a result, we have
Now we discuss the number of degree 2 in the network; we have the following.
Lemma 2. For ,
Proof. We prove the case that is a multiple of and assume , where is an integer number; considering the expectation of , we have Noticing the boundary condition and Lemma 1, we have By the estimation and the fact that we obtain that The case , which is not a multiple of , is the same as Lemma 1, just a little tedious.
3. Asymptotic Degree Distribution of Network
Let denote the proportion of vertices with degree at time . Considering the expectation of , we have the following theorem.
Theorem 3. For arbitrary and , the expectation of the number of degree satisfies
Proof. The case is just the result of Lemmas 1 and 2. Assume the result is true for ; that is, We will prove the result is true for . We just prove the case is a multiple of ; that is, , where is integer number. From the network constructed, we have Continuing the iteration and noticing the boundary condition , we obtain that Noticing the fact that We obtain The result is true for .
From Theorem 3, we can see that exists; we denote it by . Now we consider the relation of and ; we introduce the following lemma.
Lemma 4. There exists a bound constant such that for arbitrary ,
Proof. Let denote the -algbra. For , we define
By the tower property of conditional expectation and the fact that the -algbra can be deduced from , we obtain that, for ,
Noticing the fact that
we have as a martingale sequence. According to the definition of the -algbra, we know the has no information of the network and has the whole information, so we have
Therefore, we have
Now we prove that there exists a bound constant , such that . We will prove the result by induction. For the case , we have
Continuing the iteration and noticing the fact that , for , we obtain that Obviously, so we have Assume the result is true for ; that is, there exists a bound constant , such that For , by the definition of , we have Continuing the iteration and using the assumption for , we obtain that Noticing the fact that and we obtain that We just let and the result for is proved. By Asume-Hoeffding’s inequality, we have the following for arbitrary :
Theorem 5. For a fixed , one has
Proof. By the Borel-Cantelli Lemma, we need to prove the following for arbitrary : We have Noticing that and using Lemma 4, we obtain that there exists , such that
Remark 6. As a result, we can see that the distribution obeys the following rule.
When , , the degree distribution obeys power law; when , , the degree distribution obeys exponential distribution; otherwise, .
Conflict of Interests
The authors declare that there is no conflict of interests.
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).
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