Universality of Competitive Networks in Complex Networks

Transcription

1 J Syst Sci Complex (2015) 28: Universality of Competitive Networks in Complex Networks GUO Jinli FAN Chao JI Yali DOI: /s Received: 27 February 2013 / Revised: 18 December 2013 c The Editorial Office of JSSC & Springer-Verlag Berlin Heidelberg 2015 Abstract The paper proposes a model which helps to investigate the competitive aspect of real networks in quantitative terms. Through theoretical analysis and numerical simulations, it shows that the competitive model has the universality for a weighted network. The relation between parameters in the weighted network and the competitiveness in the competitive network is obtained by theoretical analysis. Based on the expression of the degree distribution of the competitive network, the strength and degree distributions of the weighted network can be calculated. The analytical solution reveals that the degree distribution of the weighted network is correlated with the increment and initial value of edge weights, which is verified by numerical simulations. Moreover, the evolving pattern of a clustering coefficient along with network parameters such as the size of a network, an updating coefficient, an initial weight and the competitiveness are obtained by further simulations. Keywords Competitive network, complex network, scale-free network, weighted network. 1 Introduction It is remarkable that Watts, et al. proposed WS model to describe the small-world effect and Barabási, et al. proposed an evolving model named BA model to construct a scale-free network. They began a new era in the study of complex networks. So far, a number of important results have emerged in this field [1 5]. However, the BA model has some limitations, which has been improved by a lot of scientists so as to describe real networks better. Moreover, Bianconi and Barabási have proposed a fitness model in view of the competition phenomenon in evolving process [6]. In this model, the fitness refers to the capacity of nodes to obtain edges, emphasizing that the more competitive nodes can attract more edges. In addition, Dorogovtsev, et al. conducted an initial attractiveness model of directed networks [7]. In the model of Dorogovtsev, GUO Jinli FAN Chao JI Yali Business School, University of Shanghai for Science and Technology, Shanghai , China. Phd5816@163.com. This research was supported by the National Natural Science Foundation of China under Grant No , the Hujiang Foundation of China under Grant No. A14006, and the Shanghai First-Class Academic Discipline Project under Grant No. S1201YLXK. This paper was recommended for publication by Editor LÜ Jinhu.

2 UNIVERSALITY OF COMPETITIVE NETWORKS 547 et al., the attractiveness is a constant. However, in real networks, different nodes usually have different attractiveness. Both the BA model and the initial attractiveness model have limitations to describe the mechanism of competitive networks. For instance, if two countries have diplomatic relations, an edge between them can be established, which forms a national network of relationships. However, this network cannot well reflect the competition of the economic and military strength of the two countries. The comprehensive national strength reflects the country s competitiveness. In the evolution of the national network, degrees of nodes are considered together with the comprehensive national strength, based on which the competitive network is formed. A competitive model was proposed to describe the competitive mechanism of growth networks [8]. Moreover, the degree distribution of the fitness model is approximately estimated by the competitive model. In addition, weighted networks can also depict competitive advantages of real systems better than un-weighted ones. For example, weights can denote the amount of paper published cooperatively in the science collaboration network [9,10] or the number of passenger flow among stations in the transport network [11,12]. In order to explore its property and evolving process deeply, many weighted models have been proposed [2 4,13 17], of which the BBV (Barrat-Barthelemy- Vespignani) model is the most influential one [15,16]. This model combines the network topology with the dynamical weight to investigate the evolving process. More specifically, an evolving mechanism of preferential attachment based on the vertex (node) strength is proposed in this model. It is known from the review above that each model is proposed for a typical purpose or application, leading to the fact that there has been some network models with overlapped mechanism. Barzel and Barabási have studied the universality in network dynamics [18]. The universality of the competitive network for the fitness model has been discussed in a previous work [8]. In our research, we focus on the issue whether competitive networks have the universality for weighted networks. The main finding of the current study is that the competition for links translates into a competitiveness-dependent degree distribution, allowing more competitive nodes to overcome the more connected but less competitive ones. It also finds that the degree distribution of competitive networks is universal for weighted networks. Owning to the fact that the competitive network is easy to be understood and simulated, once the relation between the competitive network and the weighted network is ascertained, it will provide a new perspective for the study on weighted networks. The structure of the paper is as follows. A new model is firstly proposed which helps to investigate the competitive aspect of real networks through quantitative method. Secondly, a general expression for the connectivity distribution of the model is derived. In Section 3, it is analytically deduced that the BBV weighted network is a special case of the competitive network. In Section 4, the theoretical results are verified through numerical simulations. Furthermore, the influence of model parameters on degree distributions and clustering coefficients is analyzed as well. Finally, a summary and discussion about the universality for other kinds of weighted networks are made in Section 5.

3 548 GUO JINLI FAN CHAO JI YALI 2 A Competitive Network In social networks some individuals acquire more links than others; or on the WWW some web pages attract considerably more links than others. The rate at which nodes in a network increase their connectivity depends on their competitiveness for links. A competitive network means that a node of the network has its own competitiveness, and the evolution of the network is related to not only the degree of the node, but also the competitiveness of that node. The competitive network in which each node has its own competitiveness is discussed as follows: i) Random growth. The network starts from initial one with m 0 nodes. The arrival process of nodes is a Poisson process N(t) having constant rate λ. When a new node is added to the system at time t, this new site is connected to m N(t) (m N(t) m 0 ) different nodes already present in the system, where m N(t) is the positive integer that is sampled from the population with probability density function g(x) andm = xg(x)dx is finite. Each node entering the system is tagged with its own η i that is assumed as an independent random variable taken from a given distribution with probability density function ρ(η), characterizing the system s competitiveness, and a = ηρ(η)dη is finite. ii) Preferential attachment. We assume that the probability that a new node will be connected a node i depends on the connectivity k i (t) and on the competitiveness η i of that node, such that, (ki (t)) = k i(t)+η i i (k i(t)+η i ). (1) t i denotes the time at which the ith node is added to the system. k i (t) denotes the degree of the ith node at time t. Assuming that k i (t) is a continuous real variable, the rate at which k i (t) changes is expected to be proportional to the degree k i (t). Consequently, k i (t) satisfies the dynamical equation k i (t) k i (t)+η i = m i λ t i (k i(t)+η i ). (2) N(t) represents the total number of nodes that occur by time t. According to the Poisson process theory, E[N(t)] λt, thus,foralongtimet, the following is obtained (k i (t)+η i ) (2m + a)λt. (3) Substituting Equation (3) into Equation (2), leading to i k i (t) t = m i(k i (t)+η i ), t >> t i. (4) (2m + a)t The solution of Equation (4), with the initial condition that every node i at its introduction has k i (t i )=m i, from Equation (4), is ( ) t m i 2m+a k i (t, η) =(m i + η i ) ηi, t >> t i. (5) From Equation (5), the following is obtained { ( ) 2m+a } mi m + η i i P {k i (t, η) k} = P t i t, t >> t i. k + η i t i

4 UNIVERSALITY OF COMPETITIVE NETWORKS 549 It is noticed that the node arrival process is the Poisson process with rate λ, therefore the time t i follows a gamma distribution with parameter (i, λ). Thus P {k i (t, η) k} =e λt( m i +η 2m+a i k+η ) m i i From Equation (6), we have P {k i (t, η) =k} ( ) ( 2m + a λt mi + η i m i m i + η i k + η i ) 1+ 2m+a m i i 1 l=0 [λt( mi+ηi k+η i (λt( mi+ηi k+η i l! ) 2m+a m i (i 1)! ] i 1 ) 2m+a m i From Equation (7), we obtain the stationary average degree distribution 1 P (k) (2m + a) x g(x)dx ) l, t >> t i. (6) 2m+a e λt( m i +η i k+η ) i m i, t >> ti. (7) ( ) 1+ 2m+a 1 x + η x ρ(η)dη. (8) x + η k + η If g(x) = δ(x m) andρ(η) = δ(η a), where δ(x) denotes the δ-function, the model reduces to the initial attractiveness model. From Equation (8), the degree distribution of the initial attractiveness model is obtained P (k) 2m + a ( ) 3+ a m + a m (9) m(m + a) k + a and the degree exponent γ =3+ a m. (10) 3 The Universality of the Competitive Network for the Weighted Network In the Internet, it is easy to realize that the introduction of a new connection to a router corresponds to an increase in the traffic handled on the other router s links [16]. Indeed in many technological, large infrastructure and social networks, it is commonly believed that a reinforcement of the weights is due to the network s growth. From this perspective, Barrat, et al. developed a model for a growing weighted network. They took into account the coupled evolution in time of topology and weights [16]. The definition of the BBV model is based on two coupled mechanisms: The topological growth and the weights dynamics. Weighted networks are usually described by an adjacency matrix w ij which represent the weight on the edge connecting vertices i and j, withi, j = 1, 2,,N,whereN is the size of the network. Here undirected networks are only considered, in which the weights matrix are symmetric, that is w ij = w ji. The BBV weighted network model is defined as follows [15,16] : i) Random growth. The network starts from an initial seed of m 0 nodes connected by links with assigned weight w 0. Nodes arrive in accordance withapoissonprocesshavingrateλ. A

5 550 GUO JINLI FAN CHAO JI YALI new node n is added at time t. This new site is connected to m(m m 0 ) previously existing nodes (i.e., each new node will have initially exactly m edges, all with equal weight w 0 ). ii) Preferential attachment. The new node n preferentially chooses sites with large strength; i.e., a node i is chosen according to the probability: n i = s i j s, (11) j where s i = j Ω(i) w ij is the vertex i strength, the sum runs over the set Ω(i) of the neighbors of i. iii) Weights dynamics. The weight of each new edge (n, i) is initially set a given value w 0. A new edge on node i will trigger only local rearrangements of weights on the existing neighbors j Ω(i), according to the simple rule w ij w ij +Δw ij, (12) where Δw ij = δ wij s i. δ is called as updating coefficient and it is independent on the time t. The changed strength is composed by three parts: The original strength, the new edge weight brought by the new node and the increment of the old edge weight. When a new vertex n is added to the system, an already present vertex i can be affected in two ways: (a) It is chosen with Probability (11) to be connected to n; then its connectivity increases by 1, and its strength by w 0 + δ. (b) One of its neighbors j Ω(i) is chosen to be connected to n. Then the connectivity of i is not modified but w ij is increased according to the rule equation (12), and thus s i is increased by δ wij. This dynamical process modulated by the respective occurrence probabilities and k i (t): s i(t) l s l(t) and s i sj(t) l s l(t) is thus described by the following evolution equations for s i(t) ds i dt = m(w s i 0 +2δ) j s, j (13) dk i dt = m s i j s. j (14) Substituting Equation (14) into Equation (13), we obtain ds i dt =(w 0 +2δ) dk i dt. Since node i arrives at the network by time t i,wehavek i (t i )=m and s i (t i )=mw 0,then the above equation is integrated from t i to t, and Probability (11) is modified as s i =(w 0 +2δ)k i 2δm, (15) n i = k i 2δ w m 0+2δ j [k j 2δ (16) w 0+2δ m].

6 UNIVERSALITY OF COMPETITIVE NETWORKS 551 By comparing Probability (16) and Probability (1) with the distribution ρ(η) =δ(η+ 2δm w ), 0+2δ where δ(x) denotes the δ-function, it can be inferred that only if a = 2δ m. (17) w 0 +2δ ki+a j The probability of the preferential attachment can be modified as n i = (kj+a), which is in accord with that of the initial attractiveness model. That is to say, if the updating coefficient δ is a constant, the corresponding competitiveness a will be a constant too, which verifies the universality on the competitive network for the weighted network. Moreover, from Equation (9) and Equation (10), the degree distribution of the BBV weighted network behaves as P (k) k γ where γ = 4δ +3w 0 =2+ w 0. (18) 2δ + w 0 2δ + w 0 Therefore, the degree distribution of the weighted network can be obtained directly from the results of the competitive network without discussing the relation among the node degree, strength and the time it arrives. According to Equation (15) and Equation (6), we obtain { P {s i <x} = P k i < x +2δm } w 0 +2δ i 1 =e λt( mw 0 a x )2+ m l=0 ( 1 λt l! ( mw0 Differentiation of Equation (19) yields f si (x), the density function of s i. f si (x) ( 2+ a ) (w0 m) 2+ [ a m λt m x 3+ e λt( mw 0 a a x )2+ m 1 λt m (i 1)! x ( mw0 x ) 2+ a m ) l. (19) ) 2+ a m ] i 1. (20) From Equation (17) and Equation (20), we know that the density function of the stationary average node strength distribution of the BBV weighted network behaves as the power-law function, and the exponent is the same as the degree distribution like γ = 4δ+3w0 2δ+w 0. The density function is f(x) = 2δ +2w 0 (w 0 m) 2δ+2w0 2δ+w 0 x 4δ+3w 0 2δ+w 0. (21) 2δ + w 0 4 Numerical Simulation The degree distribution and the clustering coefficient are two basic statistics reflecting the topological structure of networks. The degree of a node in a network is the number of connections it has to other nodes and the degree distribution is the probability distribution of these degrees over the whole network. The local clustering coefficient of a node in a network quantifies how close its neighbors are to being a clique. The clustering coefficient of the whole network is the average of the local clustering coefficients of all the nodes. It is known from the previous study that the main factor influencing statistics of the competitive model as ρ(x) = δ(x a), is the competitiveness a (it is supposed here that all the nodes have the same competitiveness, i.e., the model is reduced to the initial attractiveness model) and that of the weighted network

7 552 GUO JINLI FAN CHAO JI YALI are the updating coefficient δ and the initial weight w 0. In this section, numerical simulations are carried out to verify the theoretical results above. The influence of these factors on the degree distribution and the clustering coefficient is explored. Based on that, the results of the two networks are compared. 4.1 The Comparison of Degree Distributions Firstly, the influence of parameters on the degree distribution of the competitive network and the weighted network is observed. We take N = 10000, m 0 =5andm =4forboth networks. The updating coefficient δ of the weighted network equals to 0.1, 0.2, 0.3 and 0.5, the corresponding competitiveness a equals to 2/3, 8/7, 3/2and 2, respectively according to Equation (17). The comparisons are shown in Figure Weighted network Competitive network Theoretical value P(k) 10 2 P(k) k k Figure 1(a) Figure 1(b) P(k) 10 2 P(k) k k Figure 1(c) Figure 1(d) Figure 1 The comparisons of degree distributions of the weighted network and the competitive model. The sub-plots (a) to (d) correspond to different simulations with δ = 0.1, 0.2, 0.3 and 0.5, a = 2/3, 8/7, 3/2 and 2, respectively The similarity of degree distributions between the two networks is clearly shown in Figure 1. The slope of main body keeps the same with each other and with the theoretical result. The validity of the analysis in the above section is verified.

8 UNIVERSALITY OF COMPETITIVE NETWORKS The Comparison of Clustering Coefficients Secondly, the influence of parameters on clustering coefficients of both networks is observed. We take m 0 =5,m =4,δ = w 0 =1anda = 8 3. The clustering coefficient is calculated along with the growth of networks and the results are shown in Figure Competitive network Weighted network C N Figure 2 The comparisons of clustering coefficients of the weighted network and the competitive model with different network size N It is clearly shown in Figure 2 that the clustering coefficient of the weighted network and that of the competitive network are the same tendency that the clustering coefficient gets smaller when the network grows, i.e., the larger the network is, the lower the clustering effect is. It is found that δ and w 0 have significant impact on the clustering coefficient of the weighted network. We have N = 10000, m 0 =5andm =4. Thevaluesofδ and w 0 are assigned from 0.1 to 100 respectively. When one of δ and w 0 changes, the other one is fixed to 1. The clustering coefficients are shown in Figure 3. Figure 3 exhibits an interesting shape of the relation between the parameter and the clustering coefficient in a log-log plot. The influence of the value of δ on the clustering coefficient and that of the value of w 0 on the clustering coefficient is symmetric. Since it can be easily found that δ and w 0 is symmetric in Equation (17), i.e., a = 2δ w m = 2(δ/w0) 0+2δ 1+2(δ/w m. 0) In the evolving process of the weighted network, m edges will be brought into the network by a new node. The impact of parameter m on the degree distribution and the clustering coefficient is discussed. Let N = 10000, δ =1,m 0 =10andw 0 =1,thesimulationsare performed as m decreases discretely from 9 to 2. Furthermore, values of the degree distribution and the clustering coefficient C are calculated and the results are shown in Figures 4 and 5, respectively.

9 554 GUO JINLI FAN CHAO JI YALI 10 0 δ ω 0 C Parameter Figure 3 The influence of parameter δ and w 0 on the clustering coefficient in the weighted network m=2 m=3 m=4 m=5 m=6 m=7 m=8 m=9 P(k) k Figure 4 The impact of the number of edges that a new node brings into a weighted network on the degree distribution It can be seen from Figure 4 that the distribution exponent of the weighted network is independent of m, in agreement with the theoretical results, i.e., Equation (18).

10 UNIVERSALITY OF COMPETITIVE NETWORKS 555 Obviously, when a new node is added to the system, it brings m edges, and the connectivity of the network will be more evident, leading to greater clustering coefficient. However, the number of new edges has relatively little impact on the degree exponent of the weighted network. Finally, the influence of the competitiveness a on the clustering coefficient of the competitive network is discussed. Similarly, let N = 10000, m 0 =5andm = 4. The simulation results are shown in Figure C m Figure 5 The impact of the number of edges that a new node brings into a weighted network on the clustering coefficient C Simulation Result a Figure 6 The impact of the competitiveness on the clustering coefficient. The horizontal ordinate is the opposite number of value a calculated by Equation (17)

11 556 GUO JINLI FAN CHAO JI YALI 5 Summary and Further Discussion The competitive model and the weighted network are two kinds of evolving models with theoretical and realistic significance. The evolving mechanism of them is discussed in the paper. On the one hand, the relationship between the competitiveness in the competitive model and updating coefficient, initial weight in the weighted network is given through theoretical analysis. Such analysis reveals the universality of the competitive network for the weighted network, i.e., the weighted network is a special case of the competitive model. On the other hand, numerical simulations are conducted to investigate the impact of network parameters, including updating coefficients, competitiveness, initial number of edges, network size and the initial weight especially which is neglected in previous researches on topological property such as degree distribution and clustering coefficient. Moreover, the simulation results verify the validity of theoretical analysis. The weighted network model can be divided into two categories according to the cases that weight is assigned with fixed or variable values. The former includes the weighted scalefree (WSF) model [13,14] and the Zheng-Trimper-Zheng-Hui (ZTZH) model [22], and the latter includes the BBV model [15,16] and the Dorogovtsev-Mendes(DM) [17], etc. Most of these evolving models are derived from the BA model. For example, the topological structure of the WSF model completely accord with that of the BA model. Therefore, the model can be seen as a special case of the competitive model with the distribution g(x) = δ(x m) andρ(η) = δ(η). Furthermore, by contrast, the probability of preferential attachment in the DM model is positively correlated with the edge weight. New nodes are brought in by the increase of edge weight, finally giving rise to the network growth. The probability of preferential attachment in the DM model is completely the same with that in the BA model in the case that the increment of edge weight is 0. Since the increment of the edge weight is a constant in the DM model, its preferential attachment mechanism is similar to that of the BBV model. In consideration of the relation of the edge weight and the node strength, the advantage of simple structures and widespread use, there is certain universality to take the BBV model as an example to investigate the relation between the competitive model and the weighted network. In view of its potentiality and significance to describe real networks, the study on weighted networks has aroused the interest from many scholars [19 22,28]. Most of the fruits focus on the improvement of an evolving mechanism. However, many other aspects such as structural property, dynamical feature and practical application are also worth considering. A deep understanding about the relationship between the competitive model and the weighted network is really helpful to investigate their property and represent the reality. Web community can be seen as a set of pages that are created by users with similar interests or topics [24]. It can provide some references for the Web community management in the case that we assign every page in WWW with a competitive factor according to the extent of people s interest and divide the community according to such interest. An enterprise business network can be set up, in which a node represents an enterprise and an edge denotes the trade contact between them. This is a competitive network. The competitive factor of nodes indicates the

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