Exact Solutions of a Generalized Weighted Scale Free Network

Abstract

We investigate a class of generalized weighted scale-free networks, where the new vertex connects to m pairs of vertices selected preferentially. The key contribution of this paper is that, from the standpoint of random processes, we provide rigorous analytic solutions for the steady state distributions, including the vertex degree distribution, the vertex strength distribution and the edge weight distribution. Numerical simulations indicate that this network model yields three power law distributions for the vertex degrees, vertex strengths and edge weights, respectively.

1. Introduction

Recently, complex networks have become a hot topic due to the two most important discoveries of real networks, that is, the small world phenomenon raised by Watts and Strogatz [1], and the scale-free property presented by Barabási and Albert [2]. For most real-world complex systems, the proportion of vertices with degree 𝑘 obeys a power-law form, that is, 𝑃(𝑘)∝𝑘^−𝛾, where 𝛾 is a constant (see Albert et al. [3] for detail). To describe such a property, Barabási et al. [4] proposed a model (BA model for short) with growth and preferential attachment. The mean field method was used in [4] to get approximate expressions of the degree distribution.

Since then, Krapivsky et al. [5] expanded the preferential attachment of the BA model to be nonlinear. With the rate equation method, it is found that power-law degree distribution can only be obtained in the case of linear preferential attachment. Meanwhile, Dorogovtsev et al. [6] generalized the BA model to include initial attractiveness. Except for preferential selection of vertices, Dorogovtsev et al. [7] introduced a simple model based on selection of edges. This model starts with three vertices connected to each other; at each time step, an edge is randomly selected, and both its ends are connected to the new vertex. With the master equation method, the degree distribution of this model was shown to behave as a power-law form [7].

All the models of [4–7] share a common property that all links are equivalent. However, it is widely known that interaction strengths can vary widely, and such variations are essential to the network’s ability to carry on its basic functions [8]. Barrat et al. [9] proposed a model for the growth of weighted networks that couples the establishment of new edges and vertices, and the weights’ dynamical evolution. Tanaka and Aoyagi [10] generalized the model of [9] through the weight-driven preferential attachment of new vertices to existing vertices and the growth of the weights of existing links, generating scale-free networks with variable power-law exponents.

As one of the most important characteristics of networks, the degree distribution is always a big consideration. However, there is no systematic and rigorous method for solving degree distribution, the methods in [4–7] have to assume the existence of 𝑃(𝑘) or assume the vertex number 𝑁_𝑘(𝑡) with degree 𝑘 be continuous. But Bollobás et al. [11] gave a rigorous method, but this only applies to networks with multiple edges and loops. Stem from this consideration, in this paper, we try to provide a systematic and rigorous method to solve the steady state distribution from a new perspective.

Firstly, we shall give a new weighted network model. The model starts with a complete graph, at each step, the new vertex is connected to 𝑚(𝑚≥1) pairs of vertices which are selected preferentially; and meantime, the weights between the selected pairs of vertices are strengthened. Based on the Stolz theorem, rigorous analytic derivations to the steady state degree distribution of the model is provided. What’s more, the method of solving the degree distribution can be applied to study the steady state of both the vertex strength distribution and the edge weight distribution. Both theoretical derivations and numerical simulations show that the model displays power-law behavior for the degree, the strength and the weight distribution. Lastly, we show that the clustering coefficient is larger than that of the BA model.

The rest of the paper is organized as follows. Section 2 is the description of the model, and Section 3 is devoted to discuss some properties of the network. The analytical results are demonstrated by numerical simulations in the last section.

2. The Model

The detailed model construction algorithm is described as follows: (i)Initialization. The initial network, denoted as 𝐺₀, is a complete graph with 𝑛 vertices, which are denoted by 𝑣₀₁,𝑣₀₂,…,𝑣_0𝑛. The weight of each link is assigned 𝑤₀(𝑤₀≥𝑣₀₁). (ii)Growth. To construct 𝐺_𝑡+1 from 𝐺_𝑡, we add a new vertex 𝑣_𝑡+1 and then add 2𝑚(2𝑚≤𝑛−1) edges between 𝑣_𝑡+1 and vertices of 𝐺_𝑡. We choose 𝑚 pairs of vertices 𝑊¹_𝑡,𝑊^1′_𝑡,𝑊²_𝑡,𝑊^2′_𝑡,…,𝑊^𝑚_𝑡,𝑊^𝑚′_𝑡 according to a preferential attachment rule. If 𝑤_𝑖𝑖′(𝑡) is the weight between vertices 𝑣_𝑖 and 𝑣_𝑖′ in 𝐺_𝑡, then the vertices 𝑣_𝑖 and 𝑣_𝑖′ are selected with probability 𝑤_𝑖𝑖′(𝑡)∑_{𝐺𝑞,𝑙∈𝑉𝑡}𝑤_𝑞𝑙(𝑡) (1)independently for each vertex pair 𝑖,𝑖^′∈1,2,…,𝑚.

Note that this allows the possibility that we choose the same vertex more than once, and hence the graphs may have multiple edges.

The weight of each new edge is 𝑤₀. Meanwhile, the weight between each selected pair of vertices is strengthened by adding 𝑤₀ to it. If there are multiple edges added between two vertices, then the weight between them will increase multiple times. For example, if there are 𝑎 new edges added between vertices 𝑣_𝑡 and 𝑣_𝑖, then the weight between them will increase 𝑎𝑤₀.

In this case, the strength of vertex 𝑣_𝑖 can be expressed as 𝑠_𝑖(𝑡)=∑_{𝑗∈𝑉(𝑖)}𝑤_𝑖𝑗(𝑡), where 𝑉(𝑖) is the neighbor set of vertex 𝑣_𝑖.

The number of vertices in 𝐺_𝑡 is obviously 𝑁_𝑡=𝑛+𝑡. Because 2𝑚 edges are added at each stage, the total number of edges in 𝐺_𝑡 is 𝐿_𝑡=𝑛(𝑛−1)/2+2𝑚𝑡. Moreover, the total strength of vertices in 𝐺_𝑡 is 𝑆_𝑡=𝑛(𝑛−1)𝑤₀+6𝑚𝑡𝑤₀.

Before analyzing the steady state distribution of the network, let us firstly introduce some concepts and symbols. Let the random variable 𝑁_𝑘(𝑡) be the number of vertices with degree 𝑘 in 𝐺_𝑡, and moreover let the network degree distribution be the average over all its vertices at time 𝑡, namely 𝐸(𝑁_𝑘(𝑡))/𝑁_𝑡. If lim_𝑡→∞(𝐸(𝑁_𝑘(𝑡))/𝑁_𝑡) exists, we say that the steady state degree distribution of the network exists and it can be quantified by 𝑃_𝐷(𝑘)=lim_𝑡→∞(𝐸(𝑁_𝑘(𝑡))/𝑁_𝑡).

Let 𝑀_𝑠(𝑡) represent the number of vertices with strength 𝑠 in 𝐺_𝑡 Similarly, the steady state distribution of vertex strength is denoted as 𝑃_𝑆(𝑠)=lim_𝑡→∞(𝐸(𝑀_𝑠(𝑡))/𝑁_𝑡).

2.1. Network Analysis

To analytically obtain the statistical properties of the network generated by the above algorithm, we give strict proofs from the perspective of random process.

Lemma 1. Let 𝑝_𝑡>0,𝑞_𝑡≥0,𝑡≥1, then the difference equation 𝑥_𝑡+1=𝑝_𝑡𝑥_𝑡+𝑞_𝑡 has the solution 𝑥_𝑡=^𝑡−1_𝑖−1𝑝_𝑖𝑥₁+^𝑡−1_𝑖=1𝑞_𝑙∏^𝑙_𝑗=1𝑝_𝑗. (2)

Lemma 2 (Stolz Theorem, [12]). In sequence {𝑥_𝑡/𝑦_𝑡}, assume that {𝑦_𝑡} is a monotone increasing sequence with 𝑦_𝑡→∞, if lim_𝑡→∞((𝑥_𝑡+1−𝑥_𝑡)/(𝑦_𝑡+1−𝑦_𝑡))=𝑙 exists, where −∞≤𝑙≤∞, then lim_𝑡→∞(𝑥_𝑡/𝑦_𝑡)=𝑙.

Theorem 3 (Degree Distribution). For any positive integer 𝑘≥2𝑚, the steady state degree distribution of the model exists and is given by 𝑃_𝐷(𝑘)=lim_𝑡→∞𝐸𝑁_𝑘(𝑡)𝑁_𝑡=Γ(𝑘−𝑚−1)Γ((Γ((𝑘−𝑚+3/2))𝑚+3/2))Γ()3𝑚−1,2𝑚+3 (3) where Γ(𝑥)=(𝑥−1)!.

Proof.  For use in this proof, as well as later proofs, we define the 𝜎-algebra ℱ_𝑡=𝜎(𝐺_𝑖;0≤𝑖≤𝑡).
For 0≤𝑑≤𝑚, the probability that an old vertex 𝑣_𝑖, with degree 𝑘 and strength 𝑠, to get 𝑑 edges from the new vertex 𝑣_𝑡+1 is 𝑚𝑑∑_𝑗𝑤_𝑖𝑗(𝑡)∑_𝑞,𝑙𝑤_𝑞𝑙(𝑡)^𝑑∑1−^𝑗𝑤_𝑖𝑗(𝑡)∑_𝑞,𝑙𝑤_𝑞,𝑙(𝑡)^𝑚−𝑑. (4)
Clearly, the above equation can also be rewritten as 𝑚𝑑2𝑠_𝑖(𝑡)𝑆_𝑡^𝑑1−2𝑠_𝑖(𝑡)𝑆_𝑡^𝑚−𝑑. (5)
Note that for vertices 𝑣₀₁,𝑣₀₂,…,𝑣_0𝑛, the initial strength is (𝑛−1)𝑤₀. If the degree of 𝑣_0𝑟(𝑟=1,2,…,𝑛) at time 𝑡 is 𝑘_0𝑟(𝑡), then it means that there are 𝑘_0𝑟(𝑡)−(𝑛−1) new edges linked to vertex 𝑣_0𝑟 in the process of network growth, thus the strength of 𝑣_0𝑟 at time 𝑡 is 𝑠_0𝑟(𝑡)=(𝑛−1)𝑤₀+2𝑤₀𝑘_0𝑟(𝑡)−()=𝑛−12𝑘_0𝑟(𝑡)−(𝑛−1)𝑤₀,𝑟=1,2,…,𝑛. (6)
Similarly, for vertices 𝑣₁ to 𝑣_𝑡, we have 𝑠_𝑖(𝑡)=2𝑚𝑤₀+2𝑤₀𝑘_𝑖(𝑡)=−2𝑚2𝑘_𝑖(𝑡)−2𝑚𝑤₀, (7) where 𝑘_𝑖(𝑡) is the degree of vertex 𝑣_𝑖 in 𝐺_𝑡.
The above two cases can be rewritten together as 𝑠_𝑖(𝑡)=𝐾₀𝑤₀+2𝑤₀𝑘_𝑖(𝑡)−𝐾₀=2𝑘_𝑖(𝑡)−𝐾₀𝑤₀,𝑖=01,…,0𝑛,1,…,𝑛, (8) where 𝐾₀=𝑛−1,𝑖=01,…,0𝑛,2𝑚,𝑖=1,…,𝑛. (9)
For variable 𝑡,𝑁_𝑘(𝑡) is a random process. Thus, given ℱ_𝑡, the expected number of vertices with degree 𝑘 in 𝐺_𝑡+1 is 𝐸𝑁_𝑘(𝑡+1)∣ℱ_𝑡=^𝑚_𝑑=0𝑁_𝑘−𝑑(𝑡)𝑚𝑑4(𝑘−𝑑)−2𝐾₀𝑤₀𝑆_𝑡^𝑑×4()1−𝑘−𝑑−2𝐾₀𝑤₀𝑆_𝑡^𝑚−𝑑+𝐼_{𝑘=2𝑚}. (10)
In particular, for 𝑘=2𝑚, we have 𝐸𝑁_2𝑚(𝑡+1)∣ℱ_𝑡=1−4𝑚𝑤₀𝑆_𝑡^𝑚𝑁_2𝑚(𝑡)+1. (11)
Taking expectations on both sides and combing the property of conditional probability, 𝐸𝑁_2𝑚(𝑡+1)=1−4𝑚𝑤₀𝑆_𝑡^𝑚𝐸𝑁_2𝑚(𝑡)+1. (12)
Computing by Lemma 1 iteratively, we get 𝐸𝑁_2𝑚(𝑡)=^𝑡−1_𝑖=11−4𝑚𝑤₀𝑆_𝑖^𝑚×𝐸𝑁_2𝑚(1)+^𝑡−1_𝑙=11∏^𝑙_𝑗=11−4𝑚𝑤₀/𝑆_𝑗^𝑚. (13)
Divide by 𝑁_𝑡 on both sides, let 𝑥_𝑡=𝐸𝑁_2𝑚(1)+^𝑡−1_𝑙=11∏^𝑙_𝑗=11−4𝑚𝑤₀/𝑆_𝑗^𝑚,𝑦_𝑡=𝑁_𝑡^𝑡−1_𝑖=11−4𝑚𝑤₀𝑆_𝑖^−𝑚, (14) then 𝑥_𝑡,𝑦_𝑡>0,𝑦_𝑡+1−𝑦_𝑡>0 and {𝑦_𝑡} is a monotone increasing sequence with 𝑦_𝑡→∞, lim_𝑡→∞((𝑥_𝑡+1−𝑥_𝑡)/(𝑦_𝑡+1−𝑦_𝑡))=3/(2𝑚+3), by Lemma 2, we have lim_𝑡→∞(𝑥_𝑡/𝑦_𝑡)=3/(2𝑚+3).
Thus, lim_𝑡→∞(𝐸(𝑁_2𝑚(𝑡))/𝑁_𝑡) exists and 𝑃_𝐷(2𝑚)=lim_𝑡⟶∞𝐸𝑁_2𝑚(𝑡)𝑁_𝑡=3.2𝑚+3 (15)
For 𝑘>2𝑚, 𝐸𝑁_𝑘(𝑡+1)∣ℱ_𝑡=^𝑚_𝑑=0𝑁_𝑘−𝑑(𝑡)𝑚𝑑4(𝑘−𝑑)−2𝐾₀𝑤₀𝑆_𝑡^𝑑×4()1−𝑘−𝑑−2𝐾₀𝑤₀𝑆_𝑡^𝑚−𝑑. (16)
Taking expectation on both sides and using the properties of conditional expectations, 𝐸𝑁_𝑘()=𝑡+11−4𝑘−2𝐾₀𝑤₀𝑆_𝑡^𝑚𝐸𝑁_𝑘(𝑡)+^𝑚_𝑑=1𝑚𝑑4(𝑘−𝑑)−2𝐾₀𝑤₀𝑆_𝑡^𝑑⋅4()1−𝑘−𝑑−2𝐾₀𝑤₀𝑆_𝑡^𝑚−𝑑𝐸𝑁_𝑘−𝑑(𝑡). (17)
Suppose lim_𝑡→∞(𝐸(𝑁_𝑘−1(𝑡))/𝑁_𝑡) exists and equals 𝑃_𝐷(𝑘−1), use the same procedure of the case for 𝑘=2𝑚, by Lemmas 1 and 2, then lim_𝑡→∞(𝐸(𝑁_𝑘(𝑡))/𝑁_𝑡) also exists and 𝑃_𝐷(𝑘)=lim_𝑡→∞𝐸𝑁_𝑘(𝑡)𝑁_𝑡=2𝑘−2𝑚−2𝑃2𝑘−2𝑚+3^𝐷(𝑘−1). (18)
Combine (15) and (18), resulting in 𝑃_𝐷(𝑘)=Γ(𝑘−𝑚−1)Γ((Γ((𝑘−𝑚+3/2))𝑚+3/2))Γ()3𝑚−1.2𝑚+3 (19)
This completes the proof.

Remark 4. For large 𝑘, the degree distribution is proportional to 𝑘^−5/2, that is, 𝑃_𝐷(𝑘)∝𝑘^−5/2. This implies that the degree distribution of our model obeys a power-law form like the BA model but the degree exponent is no longer 3.

In order to obtain the vertex strength distribution analytically, we derive it in the same way as the degree distribution

Theorem 5 (Strength Distribution). For vertex strength 𝑠≥2𝑚𝑤₀, the steady state of vertex strength distribution exists and satisfies that 𝑃_𝑆(𝑠)=lim_𝑡→∞𝐸𝑀_𝑠(𝑡)𝑁_𝑡∝𝑠^−5/2. (20)

Proof. Following the arguments of Theorem 3, given ℱ_𝑡, the expected number of vertices with strength 𝑠 in 𝐺_𝑡+1 is 𝐸𝑀_𝑠(𝑡+1)∣ℱ_𝑡=^𝑚_𝑑=0𝑀_{𝑠−2𝑑𝑤0}(𝑡)𝑚𝑑×2𝑠−2𝑑𝑤₀𝑆_𝑡^𝑑21−𝑠−2𝑑𝑤₀𝑆_𝑡^𝑚−𝑑+𝐼_{{𝑠=2𝑚𝑤0}.} (21) Use the same procedure of Theorem 3, we know that lim_𝑡→∞(𝐸(𝑀_2𝑚𝑤0(𝑡))/𝑁_𝑡) exists and 𝑃_𝑆2𝑚𝑤₀=lim_𝑡→∞𝐸𝑀_2𝑚𝑤0(𝑡)𝑁_𝑡=3.2𝑚+3 (22)
For 𝑠>2𝑚𝑤₀, suppose lim_𝑡→∞(𝐸(𝑀_𝑠−2𝑤₀(𝑡))/𝑁_𝑡) exists and equals 𝑃_𝑆(𝑠−2𝑤₀), then lim_𝑡→∞(𝐸(𝑀_𝑠(𝑡))/𝑁_𝑡) also exists and 𝑃_𝑆(𝑠)=lim_𝑡→∞𝐸𝑀_𝑠(𝑡)𝑁_𝑡=𝑠−2𝑤₀𝑠+3𝑤₀𝑃_𝑆𝑠−2𝑤₀. (23)
This, together with (22), leads to 𝑃_𝑆(𝑠)=lim_𝑡→∞𝐸𝑀_𝑠(𝑡)𝑁_𝑡∝𝑠^−5/2. (24)
Set 𝑤₀=1, the exact expression of vertex strength distribution has the form of 𝑃_𝑆(𝑠)=lim_𝑡→∞𝐸𝑀_𝑠(𝑡)𝑁_𝑡=(𝑠−2)!!(2𝑚+3)!!()()𝑃𝑠+3!!2𝑚−2!!^𝑆(2𝑚)∝𝑠^−5/2, (25) as required. This completes the proof.

In what follows, we shall discuss the edge weight distribution.

Theorem 6 (Weight Distribution). The steady state edge weight distribution of the model exists and is given by 𝑃_𝑊(𝑤)∝𝑤^−(3𝑚+1). (26)

Proof. Since the proof is similar to that of Theorems 3 and 5, here we omit it.

Remark 7. By Theorem 6, we can conclude that the edge weight distribution follows power-law behavior with degree exponent 𝛾=3𝑚+1, which is related to 𝑚.

3. Numerical Simulations

In this section, we perform several numerical simulations in regard to the vertex degree distribution, the vertex strength distribution and the edge weight distribution. The initial network is a complete graph with 10 vertices. All the simulations are conducted 200,000 steps. It is obvious that all the results shown in Figures 1–3 are consistently conformed to the analytical analysis obtained in the previous sections.

Figure 1 

The degree distribution.

Figure 2 

The strength distribution.

Figure 3 

The weight distribution.

Figure 1 is the simulation of degree distribution 𝑃_𝐷(𝑘) in log-log scale. The circles (○), the upper triangles (△) and the lower triangles (▽) represent the simulation values of 𝑚=1,2,3, respectively. The dotted lines are theoretical results predicted by (3). Moreover, the legends here are the same as those in Figures 2 and 3.

Figure 2 shows the results of the simulations and the analysis of the vertex strength distribution 𝑃_𝑆(𝑠) in log-log scale. Because of the linear relation between the degree and the strength of the vertex, their distributions are in the same form, that is, both obey a power-law form with degree exponent 𝛾=5/2.

Figure 3 clearly verifies the theoretical results obtained by (26) with 𝑚=1. However, when it comes to 𝑚=2 and 𝑚=3, there are too few simulation data to explain the theoretical formula. All of these are resulted from the attribute of the negative exponential form of (26). With this form, the percentage of the edges dramatically decreases with the increase of 𝑤 and/or 𝑚, and the number of steps required for more accurate simulation results increases exponentially. In fact, the number is at least 𝑤^(3𝑚+1)/2𝑚 without considering the random error. For example, the minimum number of required step for simulation is 174,763, when 𝑤=4 and 𝑚=3; however, we can obtain 𝑤=24 by the same number of simulation step if 𝑚=1. Despite the fact that the data of 𝑚=2 and 𝑚=3 are so few, the simulations can explain the edge weight distribution in some degree; they can at least illustrate the trend of the distribution. It is also easy to see that 𝑚 has great influence on the weight distribution.

In Figure 4, we make a comparison of the simulation time under different values of 𝑚. It is obvious that the time required increases dramatically with the increase of the network size.

Figure 4 

Time required for simulation.

4. Conclusions

A weighted network model based on preferential selection of weights between vertices is studied in this paper. Based on the Stolz theorem, rigorous and analytic proofs for the steady state distributions of the model are provided. The approach developed here is quite general, applicable to many other scale free types of complex networks. Moreover, the theoretical derivations are verified by computer simulations (see Figures 1–3 for details). Both theoretical derivations and numerical simulations show that the model displays power-law behavior for the degree, the strength and the edge weight distribution. In addition, to illustrate the complexity of generating complex networks, we compare the number of steps and the simulation time under different values of 𝑚.

Acknowledgments

The authors are very grateful to the Editor and the anonymous referees for their insightful and constructive comments and suggestions that have led to an improved version of this paper. This work was supported by the Natural Science Foundation of China (70971140), the postdoctoral research funding and fundamental research funds of the Central South University.