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Competition and fitness in one-mode collaboration network
Commun Nonlinear Sci Numer Simulat 25 (2015) 136–144
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Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Competition and fitness in one-mode collaboration network q Long Wang, Yinghong Ma ⇑ School of Management Science & Engineering, Shandong Normal University, Jinan, China
a r t i c l e
i n f o
Article history: Received 22 July 2014 Received in revised form 1 December 2014 Accepted 28 January 2015 Available online 7 February 2015 Keywords: Collaboration network Degree distribution Fitness-driven preferential attachment Mean-field approach One-mode RDP model
a b s t r a c t The fitness to compete for links is a very critical factor to decide the rate that nodes increase their connectivity in a network. In this paper, the node degree distribution of one-mode collaboration model (one-mode RDP model) is researched by mean-field approach and we obtain the node degree distribution of this model is a power-law distribution for large enough node degree k. Some numerical simulations are made to verify the feasibility of the node degree distribution for this model. Then we improve the one-mode RDP model for the competitive evolving network and come up with a model that is onemode RDP model based on fitness-driven preferential attachment (we call this model one-mode RDP model with fitness). We discover that the fitter nodes can acquire more connectivity and the dynamic exponent depends on the fitness g. By calculating the dynamic exponent aðgÞ, a general expression for the node degree distribution of one-mode RDP network with fitness is acquired. Given the fitness distribution qðgÞ, the explicit form of the node degree distribution can also be obtained. The analytical predictions are found to be in good agreement with the experimental results derived by numerical simulations. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction Complex networks in which the nodes stand for the units consisting of the network, and the edges represent the interrelation between pairs of units can describe plenty of real complex systems, such as social network [22], World Wide Web [2], biological interacting networks [18]. Complex network research upsurge first stems from Watts–Strogatz ‘small-world’ network model [23] and Barabási–Albert model (BA model) [3,4]. For a long time, these models have been regarded as completely random [3,4,23], but in real-world networks, there are many phenomena are not completely random [2,8,11]. BA Q model displays that the probability that a new node chooses to connect an ‘old’ node i (that have existed in the network) Q is ðki Þ ¼ Pki called preferential attachment mechanism, so the older nodes have higher degree. But BA model neglects such k j j
a question that not all nodes are equally successful in competing for links in real networks [1]. This question indicates that classical random networks [6,10] can not stand for real-world networks very correctly [2,8,11]. There are many examples that can demonstrate that in real-world networks the connectivity among nodes is not completely random and the connectivity and growth rate of nodes does not also only rely on their age. For instance, some new webpages with good and popular content can obtain more connectivity than some older webpages on the internet. In the science citation network, some new research papers containing high academic level content can be cited by numerous q Supported by National Natural Science Foundation of China (71471106), and Special Research Fund for the Doctoral Program of Higher Education (20133704110003). ⇑ Corresponding author. E-mail addresses: [email protected] (L. Wang), [email protected] (Y. Ma).
http://dx.doi.org/10.1016/j.cnsns.2015.01.019 1007-5704/Ó 2015 Elsevier B.V. All rights reserved.
L. Wang, Y. Ma / Commun Nonlinear Sci Numer Simulat 25 (2015) 136–144
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research papers in a short time. These facts indicate that nodes have different ability to compete for links. In [12], Bianconi and Barabási develop the BA model and propose a model that can help us to study this competitive aspect of real networks. They use the node’s fitness to describe the node’s ability to compete for links, such as the content of a webpage, or the content of a research paper. They add the node’s fitness into the preferential attachment and find the rate at which nodes increase their connectivity also depends on the fitness of the node. In [16], Lee, Chan and Hui improve the model in [12] and propose a model that shows a hybrid of the growing networks based on popularity-driven and fitness-driven preferential attachment. In the evolving process of this network, a new node with m new links is added into the network and connects to m ‘old’ nodes with a probability p based on popularity of the ‘old’ nodes and a probability 1 p based on fitness of the ‘old’ nodes. The two papers both acquire the general expression for the connectivity distribution. Collaboration network [7,9,17,19,21] is an important social network [22]. It is generally represented as a bipartite graph that contains ‘participant’ and ‘project’ denoted by two disjoint node sets. If participants work on a common project together, they connect to each other. There are many real application examples of collaboration network, such as scientific collaboration network [19], trade network [13], music network [15,20]. In [21], a more influential collaboration network that is the self-organization evolving bipartite graph model (RDP model) is proposed. In the RDP model, a new project with n participants is added at each time step. Among the n participants, m participants are ‘new’ participants who do not have previous experience, and the rest n m participants are chosen from the ‘old’ participants that have existed in the network with a probability proportional to the number k of projects that they previously participant in. Using master equation method, Ramasco et al. [21] find that the participant nodes’ degree distribution of RDP model approximately manifests a powerlaw distribution. Although collaboration network is represented as a bipartite graph, the one-mode projections of these bipartite graphs are empirically studied. In these projections, the project nodes are excluded, and participants that collaborate in a common project are connected by edges. In [24], Wang and Ma translate two-mode collaboration network model (RDP) into one-mode RDP model. By rate-equation approach, they gain the conclusion that for large enough node degree k, the participant nodes’ degree distribution of one-mode RDP model is approximately a power-law distribution. In our paper, we also obtain the similar conclusion by mean-field approach. We find that there are few articles considering the competitive aspect of collaboration network [5,14]. So we propose a model that is one-mode RDP model based on fitness-driven preferential attachment (we call this model one-mode RDP model with fitness). In this model, we also employ the fitness to denote the node’s ability to compete for links. We add the fitness Q to the preferential attachment mechanism of one-model collaboration network [24], that is, ðki Þ ¼ Pgi kgi k , called fitness-drij j j Q ven preferential attachment mechanism [12,16]. We find i depends on the connectivity ki and the fitness gi . That indicates in one-mode RDP model with fitness, the node who is young but has higher fitness (or has stronger ability to compete for links) can also have higher number of links. So one-mode RDP model with fitness can help us to better understand the competitive aspect of collaboration network. The main motivation of our study contains two aspects. First, in [21,24], they do not consider the competitive aspect of collaboration network. Second, in [12,16], although they consider the competitive aspect of the network, their models are that at each time step, only one new node with l edges (that will be connected to l nodes already present in the system) is added; however, in our paper, we consider the one-mode RDP model with fitness is that at each time step, m new nodes are added, where m P 1 and they connect with each other, and each new node connects to n m ‘old’ nodes that already exist in the network. In real world, there are generally more than one new participants join in a new collaboration project. In our model, the number of new participant nodes is m and m P 1. Based on the above points, we find that one-mode RDP model with fitness is closer to the actual situation and can better reflect ‘‘Competition exists in cooperation’’ in real world. The model can more widely be used in real life, such as scientific collaboration network and trade network, so it is very meaningful and important for real world. When m ¼ 1, our model reduces to the model in [12] and the model of [16] with p ¼ 0. In [12,16], they acquire a general expression for the node degree distribution and given the fitness distribution qðgÞ, the explicit form of node degree distribution is derived. In our paper, when m ¼ 1, we also obtain the same explicit form of node degree distribution in [12,16] with p ¼ 0. Then we extend the conclusion and consider the more general situation. The main contribution of this paper is that when m P 1, we acquire the general expression for the node degree distribution, and when the fitness distribution functions are given by qðgÞ ¼ dðg 1Þ, when m ¼ 1 and qðgÞ ¼ 1; g 2 ½0; 1, when m P 1, we also derive the explicit forms of node degree distribution. In this paper, we mainly analyze the node degree distribution of one-mode RDP model based on fitness-driven preferential attachment. For calculating the node degree distribution of one-mode RDP model with fitness, we firstly need to consider the node degree distribution of one-mode RDP model by mean-field approach. We organize this study as follows. In Section 2, the one-mode RDP model is given. By mean-field approach [4], we derive the node degree distribution of this model is a power-law distribution for large enough node degree k. And we make some numerical simulations to proof the conclusion. In Section 3, we propose the one-mode RDP model with fitness. We find that the dynamic exponent depends on the fitness g. By calculating the dynamic exponent aðgÞ, we acquire a general expression for the node degree distribution through continuum theory and some theoretical results gained in Section 2. In Section 4, we give two fitness distribution functions that are qðgÞ ¼ dðg 1Þ i.e. all fitness are equal and qðgÞ ¼ 1; g 2 ½0; 1 i.e. qðgÞ is chosen uniformly from the interval [0, 1]. By the given fitness distributions, the explicit forms of node degree distribution are obtained. We find that the analytical predictions are in good agreement with the results derived from numerical simulations. Finally, we give the conclusion.
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2. One-mode RDP model and its degree distribution 2.1. One-mode RDP model In [24], Wang and Ma propose the one-mode RDP model. This model is defined as the following. Initially, start with a complete network with n m nodes. 1. At each time step, m new nodes are added into the network. Each new node has n m edges that connect to n m ‘old’ nodes, and m 1 edges that connect to m 1 new nodes. They assume that the n m ‘old’ nodes have connected with each other. Q 2. When the ‘old’ nodes are chosen to connect the new node, the probability of a new node connecting to an ‘old’ node i is Q proportional to the degree ki of i, that is, ðki Þ ¼ Pki called preferential attachment mechanism. k j j
In one-mode RDP model, we assume n and m are constants and n m > 1. 2.2. Power-law distribution of one-mode RDP model and relevant numerical simulations The node degree distribution PðkÞ is a basic characteristic of one-mode RDP model. The node degree k is defined as the number of nodes which connect to this node. At each time step, mðn mÞ new edges that express m new nodes connect to n m ‘old’ nodes and
mðm1Þ 2
new edges that show m new nodes connect with each other are added into the network.
and the total degree Therefore, the total number of new links added into the network at each time step is mðn mÞ þ mðm1Þ 2 P mðm1Þ t ¼ mð2n m 1Þt at time t. In the process that mðn mÞ new links select to of this network is j kj ¼ 2 mðn mÞ þ 2 connect to n m ‘old’ nodes, the probability that there are m links connect to ‘old’ node i is Q Q Q m nm Cm Cm ðki Þ ðki Þ. According to the continuum theory, the degree ki of node i should approximatemðnmÞ ½ ðki Þ ½1 mðnmÞ ly satisfy the following dynamic equation:
Y @ki k ki P i ¼ Cm ¼ Cm ; ðki Þ ¼ C m mðnmÞ mðnmÞ mðnmÞ @t mð2n m 1Þt j kj
ki ðti Þ ¼ n 1:
ð2:1Þ
The initial condition ki ðti Þ ¼ n 1, expresses that the node i is added to the network at time ti and has n 1 links. Set k ¼ mð2nm1Þ , then (2.1) can be denoted by Cm mðnmÞ
@ki ki ¼ : @t kt
ð2:2Þ
The solution of (2.2), with the initial condition ki ðti Þ ¼ n 1, is
ki ðtÞ ¼ ðn 1Þ
1k t : ti
ð2:3Þ
We assume that the nodes are added to the system at equal time intervals, so the probability density of ti should be
pðt i Þ ¼
1 : nmþt
ð2:4Þ
By (2.3) and (2.4), we have
Pðki ðtÞ < kÞ ¼ P ti >
ðn 1Þk t k
k
! ¼ 1 P ti 6
ðn 1Þk t k
k
! ¼1
ðn 1Þk t k
k ðn m þ tÞ
:
ð2:5Þ
The probability density can be derived by
Pðk; tÞ ¼
@Pðki ðtÞ < kÞ t ðkþ1Þ ¼ kðn 1Þk k : @k nmþt
ð2:6Þ
Set t ! 1. Then we can obtain the network stability degree distribution is ðkþ1Þ
PðkÞ ¼ kðn 1Þk k
;
ð2:7Þ
where k ¼ mð2nm1Þ . Cm mðnmÞ
By mean-field approach, we acquire the node degree distribution of one-mode RDP model is (2.7). To verify the feasibility that (2.7) serves as the node degree distribution of one-mode RDP model, we make the numerical simulations of the onemode RDP model and (2.7) in Fig. 1.
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L. Wang, Y. Ma / Commun Nonlinear Sci Numer Simulat 25 (2015) 136–144 Numerical simulation result of one−mode RDP model (b)
Numerical simulation result of one−mode RDP model (a)
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Fig. 1. The four figures show the numerical simulation results for the node degree distribution of one-mode RDP model. The x-axis represents the node degree k and the y-axis represents the node degree distribution PðkÞ. The solid line is the theoretical result (2.7) and the asterisk is the experimental result of one-mode RDP model. The insets show that the experimental results of one-mode RDP model are in the log–log scale. In the insets, the x-axis represents logk and the y-axis represents logðPðkÞÞ. (a) shows the fitting situation of (2.7) and the experimental result in the case n ¼ 3; m ¼ 1; (b) shows the fitting 3 situation of (2.7) and the experimental result in the case n ¼ 4; m ¼ 2. When n ¼ 3; m ¼ 1, (2.7) is PðkÞ ¼ 8k and when n ¼ 4; m ¼ 2, (2.7) is 5 ð8Þ PðkÞ ¼ 53 3ð3Þ k 3 . (c) shows the fitting situation of (2.7) and the experimental result for higher degree k, in the case n ¼ 5; m ¼ 3; (d) shows the fitting 9 ð19Þ 9 situation of (2.7) and the experimental result for higher degree k, in the case n ¼ 6; m ¼ 4. When n ¼ 5; m ¼ 3, (2.7) is PðkÞ ¼ 10 4ð10Þ k 10 and when ð25Þ ð75Þ 2 n ¼ 6; m ¼ 4, (2.7) is PðkÞ ¼ 5 5 k .
In Fig. 1(a) and (b), we find the analysis predictions are in good agreement with the experimental results obtained by numerical simulations. In Fig. 1(c) and (d), we find that for higher degree k, the analysis predictions are also in good agreement with the results derived by numerical simulations. Because (2.7) is a power-law distribution, by (2.7) and the numerical simulation results in Fig. 1, we obtain the conclusion that for large enough node degree k, the node degree distribution of one-mode RDP model is a power-law distribution and the specific expression of the node degree distribution is (2.7). In [24], by rate-equation approach Wang and Ma prove the node degree distribution of one-mode RDP model is approximately a power-law distribution for large enough node degree k. In our paper, by mean-field approach we also get the similar conclusion. We find that when m ¼ 1, the one-mode RDP model turns into scale-free model [3,4]. By (2.7), we have the node degree 3
distribution is PðkÞ ¼ 2ðn 1Þ2 k . It is same as the node degree distribution in [3,4].
3. One-mode RDP model with fitness and general expression for its node degree distribution 3.1. One-mode RDP model with fitness In real systems, the connectivity among nodes is not completely random and the connectivity and growth rate of nodes does not also only rely on their age [12,16]. This fact indicates that nodes have different ability to compete for links. In [12], Bianconi and Barabási introduce a fitness parameter gi to denote the different ability. Considering the competitive aspect of
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collaboration network, we propose a model that is one-mode RDP model based on fitness-driven preferential attachment (we call this model one-mode RDP model with fitness). This model is defined as follows. Initially, start with a complete network with n m nodes. 1. At each time step, m new nodes are added into the network. Each new node has n m edges that connect to n m ‘old’ nodes, and m 1 edges that connect to m 1 new nodes. We assume that the n m ‘old’ nodes have connected with each other. Q 2. When the ‘old’ nodes are chosen to connect the new node, the probability of a new node connecting to an ‘old’ node i is proportional to the degree ki of i, that is,
Y g ki ; ðki Þ ¼ P i j gj kj
ð3:1Þ
called fitness-driven preferential attachment mechanism. Similar to one-mode RDP model (in Section 2.1), we assume n and m are constants and n m > 1 in one-mode RDP with fitness. In this model, we also assume the fitness gi that is allocated to each node is unchanged in time [12]. By the fitnessdriven preferential attachment mechanism, we find that the rate at which new links are added to a given node depends on the connectivity ki and the fitness gi . That indicates in one-mode RDP model with fitness, the relatively young nodes who possess higher fitness can also have higher number of links. 3.2. General expression for node degree distribution of one-mode RDP model with fitness By the narration in Section 3.1, we know that in one-mode RDP model with fitness, the rate at which a node i increases its connectivity ki is proportional to the probability (3.1). According to the analysis process of one-mode RDP model (in Section 2.2) and the continuum theory, in one-mode RDP model with fitness, the degree ki of node i should approximately satisfy the following dynamic equation:
@ki gk Pi i ; ¼ Cm mðnmÞ @t j g j kj
ð3:2Þ
If qðgÞ ¼ dðg 1Þ i.e. all fitness are equal, (3.2) reduces to the dynamic equation of one-mode RDP model in which ki satisfies 1k (2.3), that is ki ðtÞ ¼ ðn 1Þ tt (in Section 2.2). Similarly to the one-mode RDP model, we also assume the time evolution of i
ki follows a power law and the dynamic exponent depends on the fitness gi in one-mode RDP model with fitness. We have
aðgi Þ t kgi ðt; t i Þ ¼ ðn 1Þ ; ti
ð3:3Þ
where t i is the time at which the new node i is added into the network. A node always increases its degree in time so
aðgÞ > 0. Since ki can at most be increased by m at each time step, by our assumption that is n 1 > m (in Section 3.1), we have aðgÞ < 1. So the dynamic exponent aðgÞ is bounded, and 0 < aðgÞ < 1. Because the fitness g is selected from a disP tribution qðgÞ, the mean of the sum j gj kj can be written as * X
+
gj kj ¼
Z
dgqðgÞg
Z
t
dt i kg ðt; ti Þ ¼ 1
j
Z
dgqðgÞgðn 1Þ
t t aðgÞ : 1 aðgÞ
ð3:4Þ
Since 0 < aðgÞ < 1, for large enough t; taðgÞ =t ! 0. So when t ! 1, we can neglect the contribution of taðgÞ . Then we set
C¼
Z
dgqðgÞ
g 1 aðgÞ
:
ð3:5Þ
By (3.4) and (3.5), we have
* X
+
gj kj ¼ Cðn 1Þt:
ð3:6Þ
j
By substituting (3.6) into (3.2) and the notation kg ¼ kgi ðt; ti Þ, we can acquire
@kg gkg ¼ Cm : mðnmÞ @t Cðn 1Þt
ð3:7Þ
By solving (3.7), we find (3.7) has a solution of form (3.3), and the dynamic exponent aðgÞ is given by
aðgÞ ¼
g Ck0
;
where k0 ¼ C mn1 :. mðnmÞ
ð3:8Þ
L. Wang, Y. Ma / Commun Nonlinear Sci Numer Simulat 25 (2015) 136–144
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Hence we confirm the self-consistent nature of the assumption (3.3). Then we calculate the cumulative probability that expresses a node has a connectivity kg ðtÞ larger than k; Pðkg ðtÞ > kÞ, can be represented as
! g0 Ck0 ! t Ck n1 g : Pðkg ðtÞ > kÞ ¼ P ðn 1Þ > k ¼ P ti < t ti k
ð3:10Þ
Similar to one-mode RDP model, we also set the probability density of ti is (2.4). Substituting (2.4) into (3.10), we have
Pðkg ðtÞ > kÞ ¼
n1 k
For sufficiently long time,
Pðkg ðtÞ > kÞ ¼
Ckg 0
t : nmþt
t nmþt
n1 k
ð3:11Þ
! 1. By (3.11), we can obtain the cumulative probability is
Ckg 0 ð3:12Þ
:
To obtain the node degree distribution PðkÞ for the whole network, the general expression for the node degree distribution of one-mode RDP model with fitness is given by the integral
PðkÞ ¼
Z
gmax
0
@Pðkg > kÞ 1 dg ¼ n1 @k
Z 0
gmax
Ck0 þ1 Ck0 n 1 g dg: k g
ð3:13Þ P
where k0 ¼ C mn1
and C is determined by (3.6). Eq. (3.6) predicts that CðtÞ ¼
mðnmÞ :
gk j j j
ðn1Þt
! C, when t ! 1.
4. Explicit form of node degree distribution and relevant numerical simulations Given the fitness distribution qðgÞ, we can calculate the dynamic exponent aðgÞ analytically by (3.8) and obtain the explicit expression for the node degree distribution by (3.13). To reveal the feasibility of our predictions, in the following we give two different fitness distribution functions that are qðgÞ ¼ dðg 1Þ i.e. all fitness are equal and qðgÞ ¼ 1; g 2 ½0; 1 i.e. qðgÞ is chosen uniformly from the interval [0, 1]. By the given fitness distribution functions, we obtain the explicit forms of the node degree distribution and make the relevant numerical simulations. 4.1. Scale-free model Setting the fitness distribution qðgÞ ¼ dðg 1Þ, we have that all fitness are equal in the one-mode RDP model with fitness. So when m ¼ 1 and qðgÞ ¼ dðg 1Þ, the one-mode RDP model with fitness reduces to the scale-free model. Substituting qðgÞ ¼ dðg 1Þ into (3.5), we obtain C ¼ 2 which stands forD the Elargest possible value of C. Note also that, since P P gk gmax kj j j j Rj gj kj 6 gmax Rj kj ¼ ð2n m 1Þmtgmax , by (3.6), we have C ¼ ðn1Þt 6 ðn1Þtj ¼ ð2nm1Þm gmax . When m ¼ 1 and C ¼ 2, n1 we have k0 ¼ 1 and C ¼ 2 ¼ 2gmax . And because all fitness are equal, g ¼ gmax ¼ 1. Using (3.8), we acquire að1Þ ¼ 1=2. Substituting k0 ¼ 1; C ¼ 2 and g ¼ gmax ¼ 1 into (3.13), we have the node degree distribution of one-mode RDP model 3
with fitness is PðkÞ ¼ 2ðn 1Þ2 k when m ¼ 1 and qðgÞ ¼ dðg 1Þ. It’s same as the node degree distribution of scale-free model in [3,4]. We also acquire the same conclusion in [12]. 4.2. Uniform fitness distribution When nodes have different fitness, let us consider the case that qðgÞ is chosen uniformly from the interval [0, 1] i.e. qðgÞ ¼ 1; g 2 ½0; 1. Since g 2 ½0; 1, we have gmax ¼ 1. Substituting gmax ¼ 1 into (3.13), (3.13) can be written as
1 PðkÞ ¼ k
Z
1
0
Ck0 Ck0 n 1 g dg: k g
By making the substitution x ¼ ln 0
PðkÞ ¼
Ck k
Z
1
x0
k n1
ð4:1Þ Ckg 0
ex dx; x
where the lower limit of the integral is
, (4.1) can be represented as
ð4:2Þ
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L. Wang, Y. Ma / Commun Nonlinear Sci Numer Simulat 25 (2015) 136–144 Numerical simulation for asymptotic convergence of C(t) to limit C=1.255
Numerical simulation for asymptotic convergence of C(t) to limit C=1.983
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Fig. 2. The four figures show the numerical simulations for asymptotic convergence of CðtÞ ¼ ðn1Þt to the limit C. The x-axis represents t and the y-axis P P gk gj kj j j j . When n ¼ 3 and m ¼ 1; CðtÞ ¼ 2tj ; (a) shows CðtÞ asymptotically represents CðtÞ. The red horizontal line is the limit C and the blue curve is CðtÞ ¼ ðn1Þt P gj kj converges to the limit C ¼ 1:255 in the case n ¼ 3; m ¼ 1. When n ¼ 4 and m ¼ 2; CðtÞ ¼ 3tj ; (b) shows CðtÞ asymptotically converges to the limit P g k j j C ¼ 1:983 in the case n ¼ 4; m ¼ 2. When n ¼ 5 and m ¼ 3; CðtÞ ¼ 4tj ; (c) shows CðtÞ asymptotically converges to the limit C ¼ 2:573 in the case P gk j j j n ¼ 5; m ¼ 3. When n ¼ 6 and m ¼ 4; CðtÞ ¼ 5t ; (d) shows CðtÞ asymptotically converges to the limit C ¼ 3:138 in the case n ¼ 6; m ¼ 4.
x0 ¼ Ck0 ln
k : n1
ð4:3Þ
Next, we start to calculate (4.2). By integration by parts, (4.2) can be given by
PðkÞ ¼
Ck0 k
Z
1
x0
Z 1 x ex Ck0 ex0 e dx : dx ¼ x k x0 x2 x0
ð4:4Þ
1 is the maximum value of 1x. And by (4.3), we have x10 1 for large enough k. So for large enough k, we get x R01 ex R 1 x R 1 x R 1 x the fact that is x0 x2 dx 6 x10 x0 ex dx x0 ex dx. Thus the factor x0 ex2 dx in (4.4) can be omitted. So Eq. (4.4) can
Since x 2 ½x0 ; 1Þ;
approximately be written as
PðkÞ ¼
Ck0 ex0 : k x0
ð4:5Þ
Substituting (4.3) into (4.5), we obtain the explicit expression for the node degree distribution of one-mode RDP model with fitness when qðgÞ ¼ 1; g 2 ½0; 1 is
PðkÞ ¼
1 n1
0 k ð1þCk Þ n1 k ln n1
where k0 ¼ C mn1 and C is determined by (3.6). Eq. (3.6) predicts that CðtÞ ¼ mðnmÞ
ð4:6Þ
; P
gk j j j
ðn1Þt
! C, when t ! 1.
143
L. Wang, Y. Ma / Commun Nonlinear Sci Numer Simulat 25 (2015) 136–144 Numerical simulation for the nodes degree distribution of one−mode RDP model with fitness (b)
Numerical simulation for the nodes degree distribution of one−mode RDP model with fitness(a) 0.8
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Nodes degree k
(b) t=5000,n=4,m=2,λ =
(a) t=5000,n=3,m=1,λ = 1,C=1.255
1 2 ,C=1.983
Numerical simulation for the nodes degree distribution of one−mode RDP model with fitness (d)
Numerical simulation for the nodes degree distribution of one−mode RDP model with fitness (c) 0.9
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(c) t=5000,n=5,m=3,λ =
0
600
(d) t=10000,n=6,m=4,λ =
1 5 ,C=2.573
1 14 ,C=3.138
Fig. 3. The four figures show the numerical simulation results for the node degree distribution of one-mode RDP model with fitness when the fitness distribution qðgÞ is chosen uniformly from the interval [0, 1] i.e. qðgÞ ¼ 1; g 2 ½0; 1. The x-axis represents the node degree k and the y-axis represents the node degree distribution PðkÞ. The solid line is the theoretical result (4.6) and the asterisk is the experimental result of one-mode RDP model with fitness when qðgÞ ¼ 1; g 2 ½0; 1. The insets show that the experimental results are in the log–log scale. In the insets, the x-axis represents logk and the y-axis ð2kÞ
represents logðPðkÞÞ. When n ¼ 3; m ¼ 1, (4.6) is PðkÞ ¼ 12 n ¼ 3; m ¼ 1. When n ¼ 4; m ¼ 2, (4.6) is PðkÞ ¼ 13 n ¼ 4; m ¼ 2. When n ¼ 5; m ¼ 3, (4.6) is PðkÞ ¼ 14 n ¼ 5; m ¼ 3. When n ¼ 5; m ¼ 3, (4.6) is PðkÞ ¼ 15
2:255
lnð2kÞ
k 1:992 3 lnð3kÞ k 1:515 4 ln 4k
ðÞ
; (a) shows the fitting situation of (4.6) and the experimental simulation results in the case
; (b) shows the fitting situation of (4.6) and the experimental simulation results in the case
ðÞ
ðÞ
k 1:224 5 ln 5k
ðÞ
ðÞ
; (c) shows the fitting situation of (4.6) and the experimental simulation results in the case
; (d) shows the fitting situation of (4.6) and the experimental simulation results in the case n = 6,
m = 4.
To check the feasibility of the node degree distribution in Eq. (4.6), we perform the numerical simulations of the node degree distribution of one-mode RDP model with fitness when the fitness distribution is qðgÞ ¼ 1; g 2 ½0; 1 i.e. qðgÞ is chosen uniformly from the interval [0, 1] (see Fig. 3. Fig. 2 shows the value of C that is obtained by the numerical simulation P gk j j j to the limit C in different cases. By the corresponding value of C, Fig. 3 shows the fitting convergence of equation CðtÞ ¼ ðn1Þt situation of (4.6) and the experimental simulation results. When n ¼ 3; m ¼ 1, we have k0 ¼ 1. By (3.6) and the numerical simulation result in Fig. 2(a), we have C 1:255. It is worth noting that we acquire the same conclusion in [12,16]. Substituting n ¼ 3; m ¼ 1; k0 ¼ 1 and C ¼ 1:255 into (4.6), 2:255
we obtain the node degree distribution is PðkÞ ¼ 12
ð2kÞ
lnð2kÞ
in the case n ¼ 3; m ¼ 1. Fig. 3(a) shows the fitting situation of
(4.6) and the experimental simulation result in the case n ¼ 3; m ¼ 1. Next, let us consider some cases when m > 1. We make the numerical simulations including three cases n ¼ 4; m ¼ 2; n ¼ 5; m ¼ 3 and n ¼ 6; m ¼ 4. When n ¼ 4; m ¼ 2, we have k0 ¼ 12. As Fig. 2(b) shows, the numerical simulaP tion result indicates that CðtÞ ¼ ð j gj kj Þ=ð3tÞ converges to the limit C 1:983. Substituting n ¼ 4; m ¼ 2; k0 ¼ 12 and
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L. Wang, Y. Ma / Commun Nonlinear Sci Numer Simulat 25 (2015) 136–144
C ¼ 1:983 into (4.6), the node degree distribution is PðkÞ ¼ 13
ð3kÞ
1:992
. Fig. 3(b) shows the fitting situation of (4.6) and the lnð3kÞ experimental simulation result in the case n ¼ 4; m ¼ 2. When n ¼ 5; m ¼ 3, we have k0 ¼ 15, and by the numerical simula-
1 , and Fig. 2(d) shows C 3:138. Fig. 3(c) and tion result in Fig. 2(c), we have C 2:573. When n ¼ 6; m ¼ 4, we have k0 ¼ 14 (d) shows the fitting situation of (4.6) and the experimental simulation results in two cases n ¼ 5; m ¼ 3 and n ¼ 6; m ¼ 4 respectively. We find that the analytical predictions are in good agreement with the results obtained from the numerical simulations (see Fig. 3). By (4.6), we know that when qðgÞ ¼ 1; g 2 ½0; 1, the node degree distribution of one-mode RDP model with fitness is not a power-law distribution. But by Fig. 3, we find the degree of most of nodes is low, and a few of nodes have very high degree. So the node degree distribution approximately possesses the power-law characteristic. The node degree distribution of onemode RDP model with fitness can also approximately be seen as a power-law distribution when qðgÞ ¼ 1; g 2 ½0; 1. So the collaboration networks with competitive factor sometimes have the generic feature of collaboration network and general complex network for some given fitness distribution functions.
5. Conclusion In real systems a node’s connectivity and growth rate does not only depend on its age. It also depends on their fitness to compete for links. The main purpose of this paper is that by one-mode RDP model with fitness, we can better understand the competitive aspect of collaboration network. It reveals the fitter-gets-richer phenomenon in collaboration network. We also call this phenomenon ‘‘the survival of the fittest’’. In this paper, we firstly calculate the node degree distribution in one-mode RDP model by mean-field approach and obtain the node degree distribution of this model is a power-law distribution for large enough node degree k. Then we propose the one-mode RDP model based on fitness-driven preferential attachment (we call this model one-mode RDP model with fitness) and acquire a general expression for the node degree distribution. When the fitness distribution functions are given by qðgÞ ¼ dðg 1Þ, when m ¼ 1 and qðgÞ ¼ 1; g 2 ½0; 1, when m P 1, we also derive the explicit forms of node degree distribution. The analytical predictions are in good agreement with results obtained by numerical simulations. By numerical simulations, we also find the node degree distribution approximately possesses the power-law characteristic in one-mode RDP model with fitness, when qðgÞ is chosen uniformly from the interval [0, 1]. So the collaboration networks with competitive factor sometimes also have the generic feature of collaboration network and general complex network for some given fitness distribution functions. In summary, the fitness is a very critical factor to decide the rate at which nodes increase their connectivity in a network. The one-mode RDP model with fitness can better reflect competitive factor of collaboration network and help us to better comprehend ‘‘Competition exists in cooperation’’ in real world. So researching one-mode RDP model with fitness is very meaningful and important for real world. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
Adamic LA, Huberman BA. Power-law distribution of the world wide web. Science 2000;287:2115. Albert R, Jeong H, Barabási AL. Internet: diameter of the world-wide web. Nature 1999;401:130–1. Barabási AL, Albert R. Emergence of scaling in random networks. Science 1999;286:509–12. Barabási AL, Albert R, Jeong H. Mean-field theory for scale-free random networks. Physica A 1999;272(1–2):173–87. Beir MG, Busch JR, Grynberg SP, et al. Obtaining communities with a fitness growth process. Phys A Stat Mech Appl 2013;392(9):2278–93. Bollobás B. Random graphs. London: Academic Press; 1985. Borrett SR, Moody J, Edelmann A. The rise of network ecology: maps of the topic diversity and scientific collaboration. Ecol Modell 2014;293:111–27. Caldarelli G, Marchetti R, Pietronero L. The fractal properties of internet. Europhys Lett 2000;52(4):386–91. Camarinha-Matos LM, Afsarmanesh H. The emerging discipline of collaborative networks. J Intell Manuf 2005;16(4–5):439–52. Erdös P, Rényi A. On the strength of connectedness of a random graph. Acta Math Hungarica 1961;12(1):261–7. Faloutsos M, Faloutsos P, Faloutsos C. On power-law relationships of the Internet topology. Comput Commun Rev 1999;29(4):251–62. Bianconi G, Barabási A-L. Competition and multiscaling in evolving networks. Europhys Lett 2001;54(4):436–42. White HC. Where do markets come from. AJS 1981;87(3):517. Ke W. A fitness model for scholarly impact analysis. Scientometrics 2013;94(3):981–98. Lambiotte R, Ausloos M. Uncovering collective listening habits and music genres in bipartite networks. Phys Rev E 2005;72(6):66107. Lee HY, Chan HY, Hui PM. Scale-free networks with tunable degree-distribution exponents. Phys Rev E 2004;69(6):067102. Mangel M, Satterthwaite WH, Pirolli P, et al. Invasion biology and the success of social collaboration networks, with application to Wikipedia. Israel J Ecol Evol 2013;59(1):17–26. Matthew Hahn W, Kern AD. Mol. Biol. Evol: comparative genomics of centrality and essentiality in three eukaryotic protein-interaction networks. Oxford J. 2005;22(4):803–6. Newman MEJ, The structure of scientific collaboration networks. PNAS 2001;98(2):404–9. Park J, Celma O, Koppoenberger M, Buldu JM. The social network of contemporary popular musicians. Bifurcation Chaos 2007;17(7):2281. Ramasco JJ, Dorogovtsev SN, Pastor-Satorras R. Self-organization of collaboration networks. Phys Rev E 2004;70(3):036106. Wasserman S, Faust K. Social network analysis: methods and applications. Am Ethnol 1997;24(1):219–20. Watts DJ, Strogatz SH. Collective dynamics of ‘small-world’ networks. Nature 1998;393:440–2. Wang L, Ma Y. Structure properties of one-mode collaboration network model based on rate equation approach. Commun Nonlinear Sci Numer Simul 2014;19(12):4068–79.
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Competition and fitness in one-mode collaboration network q Long Wang, Yinghong Ma ⇑ School of Management Science & Engineering, Shandong Normal University, Jinan, China
a r t i c l e
i n f o
Article history: Received 22 July 2014 Received in revised form 1 December 2014 Accepted 28 January 2015 Available online 7 February 2015 Keywords: Collaboration network Degree distribution Fitness-driven preferential attachment Mean-field approach One-mode RDP model
a b s t r a c t The fitness to compete for links is a very critical factor to decide the rate that nodes increase their connectivity in a network. In this paper, the node degree distribution of one-mode collaboration model (one-mode RDP model) is researched by mean-field approach and we obtain the node degree distribution of this model is a power-law distribution for large enough node degree k. Some numerical simulations are made to verify the feasibility of the node degree distribution for this model. Then we improve the one-mode RDP model for the competitive evolving network and come up with a model that is onemode RDP model based on fitness-driven preferential attachment (we call this model one-mode RDP model with fitness). We discover that the fitter nodes can acquire more connectivity and the dynamic exponent depends on the fitness g. By calculating the dynamic exponent aðgÞ, a general expression for the node degree distribution of one-mode RDP network with fitness is acquired. Given the fitness distribution qðgÞ, the explicit form of the node degree distribution can also be obtained. The analytical predictions are found to be in good agreement with the experimental results derived by numerical simulations. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction Complex networks in which the nodes stand for the units consisting of the network, and the edges represent the interrelation between pairs of units can describe plenty of real complex systems, such as social network [22], World Wide Web [2], biological interacting networks [18]. Complex network research upsurge first stems from Watts–Strogatz ‘small-world’ network model [23] and Barabási–Albert model (BA model) [3,4]. For a long time, these models have been regarded as completely random [3,4,23], but in real-world networks, there are many phenomena are not completely random [2,8,11]. BA Q model displays that the probability that a new node chooses to connect an ‘old’ node i (that have existed in the network) Q is ðki Þ ¼ Pki called preferential attachment mechanism, so the older nodes have higher degree. But BA model neglects such k j j
a question that not all nodes are equally successful in competing for links in real networks [1]. This question indicates that classical random networks [6,10] can not stand for real-world networks very correctly [2,8,11]. There are many examples that can demonstrate that in real-world networks the connectivity among nodes is not completely random and the connectivity and growth rate of nodes does not also only rely on their age. For instance, some new webpages with good and popular content can obtain more connectivity than some older webpages on the internet. In the science citation network, some new research papers containing high academic level content can be cited by numerous q Supported by National Natural Science Foundation of China (71471106), and Special Research Fund for the Doctoral Program of Higher Education (20133704110003). ⇑ Corresponding author. E-mail addresses: [email protected] (L. Wang), [email protected] (Y. Ma).
http://dx.doi.org/10.1016/j.cnsns.2015.01.019 1007-5704/Ó 2015 Elsevier B.V. All rights reserved.
L. Wang, Y. Ma / Commun Nonlinear Sci Numer Simulat 25 (2015) 136–144
137
research papers in a short time. These facts indicate that nodes have different ability to compete for links. In [12], Bianconi and Barabási develop the BA model and propose a model that can help us to study this competitive aspect of real networks. They use the node’s fitness to describe the node’s ability to compete for links, such as the content of a webpage, or the content of a research paper. They add the node’s fitness into the preferential attachment and find the rate at which nodes increase their connectivity also depends on the fitness of the node. In [16], Lee, Chan and Hui improve the model in [12] and propose a model that shows a hybrid of the growing networks based on popularity-driven and fitness-driven preferential attachment. In the evolving process of this network, a new node with m new links is added into the network and connects to m ‘old’ nodes with a probability p based on popularity of the ‘old’ nodes and a probability 1 p based on fitness of the ‘old’ nodes. The two papers both acquire the general expression for the connectivity distribution. Collaboration network [7,9,17,19,21] is an important social network [22]. It is generally represented as a bipartite graph that contains ‘participant’ and ‘project’ denoted by two disjoint node sets. If participants work on a common project together, they connect to each other. There are many real application examples of collaboration network, such as scientific collaboration network [19], trade network [13], music network [15,20]. In [21], a more influential collaboration network that is the self-organization evolving bipartite graph model (RDP model) is proposed. In the RDP model, a new project with n participants is added at each time step. Among the n participants, m participants are ‘new’ participants who do not have previous experience, and the rest n m participants are chosen from the ‘old’ participants that have existed in the network with a probability proportional to the number k of projects that they previously participant in. Using master equation method, Ramasco et al. [21] find that the participant nodes’ degree distribution of RDP model approximately manifests a powerlaw distribution. Although collaboration network is represented as a bipartite graph, the one-mode projections of these bipartite graphs are empirically studied. In these projections, the project nodes are excluded, and participants that collaborate in a common project are connected by edges. In [24], Wang and Ma translate two-mode collaboration network model (RDP) into one-mode RDP model. By rate-equation approach, they gain the conclusion that for large enough node degree k, the participant nodes’ degree distribution of one-mode RDP model is approximately a power-law distribution. In our paper, we also obtain the similar conclusion by mean-field approach. We find that there are few articles considering the competitive aspect of collaboration network [5,14]. So we propose a model that is one-mode RDP model based on fitness-driven preferential attachment (we call this model one-mode RDP model with fitness). In this model, we also employ the fitness to denote the node’s ability to compete for links. We add the fitness Q to the preferential attachment mechanism of one-model collaboration network [24], that is, ðki Þ ¼ Pgi kgi k , called fitness-drij j j Q ven preferential attachment mechanism [12,16]. We find i depends on the connectivity ki and the fitness gi . That indicates in one-mode RDP model with fitness, the node who is young but has higher fitness (or has stronger ability to compete for links) can also have higher number of links. So one-mode RDP model with fitness can help us to better understand the competitive aspect of collaboration network. The main motivation of our study contains two aspects. First, in [21,24], they do not consider the competitive aspect of collaboration network. Second, in [12,16], although they consider the competitive aspect of the network, their models are that at each time step, only one new node with l edges (that will be connected to l nodes already present in the system) is added; however, in our paper, we consider the one-mode RDP model with fitness is that at each time step, m new nodes are added, where m P 1 and they connect with each other, and each new node connects to n m ‘old’ nodes that already exist in the network. In real world, there are generally more than one new participants join in a new collaboration project. In our model, the number of new participant nodes is m and m P 1. Based on the above points, we find that one-mode RDP model with fitness is closer to the actual situation and can better reflect ‘‘Competition exists in cooperation’’ in real world. The model can more widely be used in real life, such as scientific collaboration network and trade network, so it is very meaningful and important for real world. When m ¼ 1, our model reduces to the model in [12] and the model of [16] with p ¼ 0. In [12,16], they acquire a general expression for the node degree distribution and given the fitness distribution qðgÞ, the explicit form of node degree distribution is derived. In our paper, when m ¼ 1, we also obtain the same explicit form of node degree distribution in [12,16] with p ¼ 0. Then we extend the conclusion and consider the more general situation. The main contribution of this paper is that when m P 1, we acquire the general expression for the node degree distribution, and when the fitness distribution functions are given by qðgÞ ¼ dðg 1Þ, when m ¼ 1 and qðgÞ ¼ 1; g 2 ½0; 1, when m P 1, we also derive the explicit forms of node degree distribution. In this paper, we mainly analyze the node degree distribution of one-mode RDP model based on fitness-driven preferential attachment. For calculating the node degree distribution of one-mode RDP model with fitness, we firstly need to consider the node degree distribution of one-mode RDP model by mean-field approach. We organize this study as follows. In Section 2, the one-mode RDP model is given. By mean-field approach [4], we derive the node degree distribution of this model is a power-law distribution for large enough node degree k. And we make some numerical simulations to proof the conclusion. In Section 3, we propose the one-mode RDP model with fitness. We find that the dynamic exponent depends on the fitness g. By calculating the dynamic exponent aðgÞ, we acquire a general expression for the node degree distribution through continuum theory and some theoretical results gained in Section 2. In Section 4, we give two fitness distribution functions that are qðgÞ ¼ dðg 1Þ i.e. all fitness are equal and qðgÞ ¼ 1; g 2 ½0; 1 i.e. qðgÞ is chosen uniformly from the interval [0, 1]. By the given fitness distributions, the explicit forms of node degree distribution are obtained. We find that the analytical predictions are in good agreement with the results derived from numerical simulations. Finally, we give the conclusion.
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2. One-mode RDP model and its degree distribution 2.1. One-mode RDP model In [24], Wang and Ma propose the one-mode RDP model. This model is defined as the following. Initially, start with a complete network with n m nodes. 1. At each time step, m new nodes are added into the network. Each new node has n m edges that connect to n m ‘old’ nodes, and m 1 edges that connect to m 1 new nodes. They assume that the n m ‘old’ nodes have connected with each other. Q 2. When the ‘old’ nodes are chosen to connect the new node, the probability of a new node connecting to an ‘old’ node i is Q proportional to the degree ki of i, that is, ðki Þ ¼ Pki called preferential attachment mechanism. k j j
In one-mode RDP model, we assume n and m are constants and n m > 1. 2.2. Power-law distribution of one-mode RDP model and relevant numerical simulations The node degree distribution PðkÞ is a basic characteristic of one-mode RDP model. The node degree k is defined as the number of nodes which connect to this node. At each time step, mðn mÞ new edges that express m new nodes connect to n m ‘old’ nodes and
mðm1Þ 2
new edges that show m new nodes connect with each other are added into the network.
and the total degree Therefore, the total number of new links added into the network at each time step is mðn mÞ þ mðm1Þ 2 P mðm1Þ t ¼ mð2n m 1Þt at time t. In the process that mðn mÞ new links select to of this network is j kj ¼ 2 mðn mÞ þ 2 connect to n m ‘old’ nodes, the probability that there are m links connect to ‘old’ node i is Q Q Q m nm Cm Cm ðki Þ ðki Þ. According to the continuum theory, the degree ki of node i should approximatemðnmÞ ½ ðki Þ ½1 mðnmÞ ly satisfy the following dynamic equation:
Y @ki k ki P i ¼ Cm ¼ Cm ; ðki Þ ¼ C m mðnmÞ mðnmÞ mðnmÞ @t mð2n m 1Þt j kj
ki ðti Þ ¼ n 1:
ð2:1Þ
The initial condition ki ðti Þ ¼ n 1, expresses that the node i is added to the network at time ti and has n 1 links. Set k ¼ mð2nm1Þ , then (2.1) can be denoted by Cm mðnmÞ
@ki ki ¼ : @t kt
ð2:2Þ
The solution of (2.2), with the initial condition ki ðti Þ ¼ n 1, is
ki ðtÞ ¼ ðn 1Þ
1k t : ti
ð2:3Þ
We assume that the nodes are added to the system at equal time intervals, so the probability density of ti should be
pðt i Þ ¼
1 : nmþt
ð2:4Þ
By (2.3) and (2.4), we have
Pðki ðtÞ < kÞ ¼ P ti >
ðn 1Þk t k
k
! ¼ 1 P ti 6
ðn 1Þk t k
k
! ¼1
ðn 1Þk t k
k ðn m þ tÞ
:
ð2:5Þ
The probability density can be derived by
Pðk; tÞ ¼
@Pðki ðtÞ < kÞ t ðkþ1Þ ¼ kðn 1Þk k : @k nmþt
ð2:6Þ
Set t ! 1. Then we can obtain the network stability degree distribution is ðkþ1Þ
PðkÞ ¼ kðn 1Þk k
;
ð2:7Þ
where k ¼ mð2nm1Þ . Cm mðnmÞ
By mean-field approach, we acquire the node degree distribution of one-mode RDP model is (2.7). To verify the feasibility that (2.7) serves as the node degree distribution of one-mode RDP model, we make the numerical simulations of the onemode RDP model and (2.7) in Fig. 1.
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L. Wang, Y. Ma / Commun Nonlinear Sci Numer Simulat 25 (2015) 136–144 Numerical simulation result of one−mode RDP model (b)
Numerical simulation result of one−mode RDP model (a)
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5 3
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(d) t=10000,n=6,m=4,λ =
2 5
Fig. 1. The four figures show the numerical simulation results for the node degree distribution of one-mode RDP model. The x-axis represents the node degree k and the y-axis represents the node degree distribution PðkÞ. The solid line is the theoretical result (2.7) and the asterisk is the experimental result of one-mode RDP model. The insets show that the experimental results of one-mode RDP model are in the log–log scale. In the insets, the x-axis represents logk and the y-axis represents logðPðkÞÞ. (a) shows the fitting situation of (2.7) and the experimental result in the case n ¼ 3; m ¼ 1; (b) shows the fitting 3 situation of (2.7) and the experimental result in the case n ¼ 4; m ¼ 2. When n ¼ 3; m ¼ 1, (2.7) is PðkÞ ¼ 8k and when n ¼ 4; m ¼ 2, (2.7) is 5 ð8Þ PðkÞ ¼ 53 3ð3Þ k 3 . (c) shows the fitting situation of (2.7) and the experimental result for higher degree k, in the case n ¼ 5; m ¼ 3; (d) shows the fitting 9 ð19Þ 9 situation of (2.7) and the experimental result for higher degree k, in the case n ¼ 6; m ¼ 4. When n ¼ 5; m ¼ 3, (2.7) is PðkÞ ¼ 10 4ð10Þ k 10 and when ð25Þ ð75Þ 2 n ¼ 6; m ¼ 4, (2.7) is PðkÞ ¼ 5 5 k .
In Fig. 1(a) and (b), we find the analysis predictions are in good agreement with the experimental results obtained by numerical simulations. In Fig. 1(c) and (d), we find that for higher degree k, the analysis predictions are also in good agreement with the results derived by numerical simulations. Because (2.7) is a power-law distribution, by (2.7) and the numerical simulation results in Fig. 1, we obtain the conclusion that for large enough node degree k, the node degree distribution of one-mode RDP model is a power-law distribution and the specific expression of the node degree distribution is (2.7). In [24], by rate-equation approach Wang and Ma prove the node degree distribution of one-mode RDP model is approximately a power-law distribution for large enough node degree k. In our paper, by mean-field approach we also get the similar conclusion. We find that when m ¼ 1, the one-mode RDP model turns into scale-free model [3,4]. By (2.7), we have the node degree 3
distribution is PðkÞ ¼ 2ðn 1Þ2 k . It is same as the node degree distribution in [3,4].
3. One-mode RDP model with fitness and general expression for its node degree distribution 3.1. One-mode RDP model with fitness In real systems, the connectivity among nodes is not completely random and the connectivity and growth rate of nodes does not also only rely on their age [12,16]. This fact indicates that nodes have different ability to compete for links. In [12], Bianconi and Barabási introduce a fitness parameter gi to denote the different ability. Considering the competitive aspect of
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collaboration network, we propose a model that is one-mode RDP model based on fitness-driven preferential attachment (we call this model one-mode RDP model with fitness). This model is defined as follows. Initially, start with a complete network with n m nodes. 1. At each time step, m new nodes are added into the network. Each new node has n m edges that connect to n m ‘old’ nodes, and m 1 edges that connect to m 1 new nodes. We assume that the n m ‘old’ nodes have connected with each other. Q 2. When the ‘old’ nodes are chosen to connect the new node, the probability of a new node connecting to an ‘old’ node i is proportional to the degree ki of i, that is,
Y g ki ; ðki Þ ¼ P i j gj kj
ð3:1Þ
called fitness-driven preferential attachment mechanism. Similar to one-mode RDP model (in Section 2.1), we assume n and m are constants and n m > 1 in one-mode RDP with fitness. In this model, we also assume the fitness gi that is allocated to each node is unchanged in time [12]. By the fitnessdriven preferential attachment mechanism, we find that the rate at which new links are added to a given node depends on the connectivity ki and the fitness gi . That indicates in one-mode RDP model with fitness, the relatively young nodes who possess higher fitness can also have higher number of links. 3.2. General expression for node degree distribution of one-mode RDP model with fitness By the narration in Section 3.1, we know that in one-mode RDP model with fitness, the rate at which a node i increases its connectivity ki is proportional to the probability (3.1). According to the analysis process of one-mode RDP model (in Section 2.2) and the continuum theory, in one-mode RDP model with fitness, the degree ki of node i should approximately satisfy the following dynamic equation:
@ki gk Pi i ; ¼ Cm mðnmÞ @t j g j kj
ð3:2Þ
If qðgÞ ¼ dðg 1Þ i.e. all fitness are equal, (3.2) reduces to the dynamic equation of one-mode RDP model in which ki satisfies 1k (2.3), that is ki ðtÞ ¼ ðn 1Þ tt (in Section 2.2). Similarly to the one-mode RDP model, we also assume the time evolution of i
ki follows a power law and the dynamic exponent depends on the fitness gi in one-mode RDP model with fitness. We have
aðgi Þ t kgi ðt; t i Þ ¼ ðn 1Þ ; ti
ð3:3Þ
where t i is the time at which the new node i is added into the network. A node always increases its degree in time so
aðgÞ > 0. Since ki can at most be increased by m at each time step, by our assumption that is n 1 > m (in Section 3.1), we have aðgÞ < 1. So the dynamic exponent aðgÞ is bounded, and 0 < aðgÞ < 1. Because the fitness g is selected from a disP tribution qðgÞ, the mean of the sum j gj kj can be written as * X
+
gj kj ¼
Z
dgqðgÞg
Z
t
dt i kg ðt; ti Þ ¼ 1
j
Z
dgqðgÞgðn 1Þ
t t aðgÞ : 1 aðgÞ
ð3:4Þ
Since 0 < aðgÞ < 1, for large enough t; taðgÞ =t ! 0. So when t ! 1, we can neglect the contribution of taðgÞ . Then we set
C¼
Z
dgqðgÞ
g 1 aðgÞ
:
ð3:5Þ
By (3.4) and (3.5), we have
* X
+
gj kj ¼ Cðn 1Þt:
ð3:6Þ
j
By substituting (3.6) into (3.2) and the notation kg ¼ kgi ðt; ti Þ, we can acquire
@kg gkg ¼ Cm : mðnmÞ @t Cðn 1Þt
ð3:7Þ
By solving (3.7), we find (3.7) has a solution of form (3.3), and the dynamic exponent aðgÞ is given by
aðgÞ ¼
g Ck0
;
where k0 ¼ C mn1 :. mðnmÞ
ð3:8Þ
L. Wang, Y. Ma / Commun Nonlinear Sci Numer Simulat 25 (2015) 136–144
141
Hence we confirm the self-consistent nature of the assumption (3.3). Then we calculate the cumulative probability that expresses a node has a connectivity kg ðtÞ larger than k; Pðkg ðtÞ > kÞ, can be represented as
! g0 Ck0 ! t Ck n1 g : Pðkg ðtÞ > kÞ ¼ P ðn 1Þ > k ¼ P ti < t ti k
ð3:10Þ
Similar to one-mode RDP model, we also set the probability density of ti is (2.4). Substituting (2.4) into (3.10), we have
Pðkg ðtÞ > kÞ ¼
n1 k
For sufficiently long time,
Pðkg ðtÞ > kÞ ¼
Ckg 0
t : nmþt
t nmþt
n1 k
ð3:11Þ
! 1. By (3.11), we can obtain the cumulative probability is
Ckg 0 ð3:12Þ
:
To obtain the node degree distribution PðkÞ for the whole network, the general expression for the node degree distribution of one-mode RDP model with fitness is given by the integral
PðkÞ ¼
Z
gmax
0
@Pðkg > kÞ 1 dg ¼ n1 @k
Z 0
gmax
Ck0 þ1 Ck0 n 1 g dg: k g
ð3:13Þ P
where k0 ¼ C mn1
and C is determined by (3.6). Eq. (3.6) predicts that CðtÞ ¼
mðnmÞ :
gk j j j
ðn1Þt
! C, when t ! 1.
4. Explicit form of node degree distribution and relevant numerical simulations Given the fitness distribution qðgÞ, we can calculate the dynamic exponent aðgÞ analytically by (3.8) and obtain the explicit expression for the node degree distribution by (3.13). To reveal the feasibility of our predictions, in the following we give two different fitness distribution functions that are qðgÞ ¼ dðg 1Þ i.e. all fitness are equal and qðgÞ ¼ 1; g 2 ½0; 1 i.e. qðgÞ is chosen uniformly from the interval [0, 1]. By the given fitness distribution functions, we obtain the explicit forms of the node degree distribution and make the relevant numerical simulations. 4.1. Scale-free model Setting the fitness distribution qðgÞ ¼ dðg 1Þ, we have that all fitness are equal in the one-mode RDP model with fitness. So when m ¼ 1 and qðgÞ ¼ dðg 1Þ, the one-mode RDP model with fitness reduces to the scale-free model. Substituting qðgÞ ¼ dðg 1Þ into (3.5), we obtain C ¼ 2 which stands forD the Elargest possible value of C. Note also that, since P P gk gmax kj j j j Rj gj kj 6 gmax Rj kj ¼ ð2n m 1Þmtgmax , by (3.6), we have C ¼ ðn1Þt 6 ðn1Þtj ¼ ð2nm1Þm gmax . When m ¼ 1 and C ¼ 2, n1 we have k0 ¼ 1 and C ¼ 2 ¼ 2gmax . And because all fitness are equal, g ¼ gmax ¼ 1. Using (3.8), we acquire að1Þ ¼ 1=2. Substituting k0 ¼ 1; C ¼ 2 and g ¼ gmax ¼ 1 into (3.13), we have the node degree distribution of one-mode RDP model 3
with fitness is PðkÞ ¼ 2ðn 1Þ2 k when m ¼ 1 and qðgÞ ¼ dðg 1Þ. It’s same as the node degree distribution of scale-free model in [3,4]. We also acquire the same conclusion in [12]. 4.2. Uniform fitness distribution When nodes have different fitness, let us consider the case that qðgÞ is chosen uniformly from the interval [0, 1] i.e. qðgÞ ¼ 1; g 2 ½0; 1. Since g 2 ½0; 1, we have gmax ¼ 1. Substituting gmax ¼ 1 into (3.13), (3.13) can be written as
1 PðkÞ ¼ k
Z
1
0
Ck0 Ck0 n 1 g dg: k g
By making the substitution x ¼ ln 0
PðkÞ ¼
Ck k
Z
1
x0
k n1
ð4:1Þ Ckg 0
ex dx; x
where the lower limit of the integral is
, (4.1) can be represented as
ð4:2Þ
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L. Wang, Y. Ma / Commun Nonlinear Sci Numer Simulat 25 (2015) 136–144 Numerical simulation for asymptotic convergence of C(t) to limit C=1.255
Numerical simulation for asymptotic convergence of C(t) to limit C=1.983
1.4
2.3
1.35
2.25
1.3
2.2
1.25
2.15 2.1 C(t)
C(t)
1.2 1.15
2.05
1.1
2
1.05
1.95
1
1.9
0.95
1.85
0.9
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
1.8
10000
0
1000
2000
3000
4000
5000
6000
7000
t
t
(a) t=10000,n=3,m=1
(b) t=20000,n=4,m=2
Numerical simulation for asymptotic convergence of C(t) to limit C=2.573
8000
9000
10000
Numerical simulation for asymptotic convergence of C(t) to limit C=3.138
3.4
3.4
3.2
3.3 3
C(t)
C(t)
3.2 2.8
3.1 2.6 3
2.4
2.2
2.9
0
1000
2000
3000
4000
5000
6000
2.8
0
1000
2000
t
3000
4000
5000
6000
7000
t
(c) t=20000,n=5,m=3
(d) t=30000,n=6,m=4 P
gk j j j
Fig. 2. The four figures show the numerical simulations for asymptotic convergence of CðtÞ ¼ ðn1Þt to the limit C. The x-axis represents t and the y-axis P P gk gj kj j j j . When n ¼ 3 and m ¼ 1; CðtÞ ¼ 2tj ; (a) shows CðtÞ asymptotically represents CðtÞ. The red horizontal line is the limit C and the blue curve is CðtÞ ¼ ðn1Þt P gj kj converges to the limit C ¼ 1:255 in the case n ¼ 3; m ¼ 1. When n ¼ 4 and m ¼ 2; CðtÞ ¼ 3tj ; (b) shows CðtÞ asymptotically converges to the limit P g k j j C ¼ 1:983 in the case n ¼ 4; m ¼ 2. When n ¼ 5 and m ¼ 3; CðtÞ ¼ 4tj ; (c) shows CðtÞ asymptotically converges to the limit C ¼ 2:573 in the case P gk j j j n ¼ 5; m ¼ 3. When n ¼ 6 and m ¼ 4; CðtÞ ¼ 5t ; (d) shows CðtÞ asymptotically converges to the limit C ¼ 3:138 in the case n ¼ 6; m ¼ 4.
x0 ¼ Ck0 ln
k : n1
ð4:3Þ
Next, we start to calculate (4.2). By integration by parts, (4.2) can be given by
PðkÞ ¼
Ck0 k
Z
1
x0
Z 1 x ex Ck0 ex0 e dx : dx ¼ x k x0 x2 x0
ð4:4Þ
1 is the maximum value of 1x. And by (4.3), we have x10 1 for large enough k. So for large enough k, we get x R01 ex R 1 x R 1 x R 1 x the fact that is x0 x2 dx 6 x10 x0 ex dx x0 ex dx. Thus the factor x0 ex2 dx in (4.4) can be omitted. So Eq. (4.4) can
Since x 2 ½x0 ; 1Þ;
approximately be written as
PðkÞ ¼
Ck0 ex0 : k x0
ð4:5Þ
Substituting (4.3) into (4.5), we obtain the explicit expression for the node degree distribution of one-mode RDP model with fitness when qðgÞ ¼ 1; g 2 ½0; 1 is
PðkÞ ¼
1 n1
0 k ð1þCk Þ n1 k ln n1
where k0 ¼ C mn1 and C is determined by (3.6). Eq. (3.6) predicts that CðtÞ ¼ mðnmÞ
ð4:6Þ
; P
gk j j j
ðn1Þt
! C, when t ! 1.
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L. Wang, Y. Ma / Commun Nonlinear Sci Numer Simulat 25 (2015) 136–144 Numerical simulation for the nodes degree distribution of one−mode RDP model with fitness (b)
Numerical simulation for the nodes degree distribution of one−mode RDP model with fitness(a) 0.8
0.7 0
0
10
10 0.7
0.6
experimental result theoretical result
experimental result theoretical result
−1
10
−1
10
−2
10
0.4 −3
10 0.3
−4
10
0
1
10
2
10 log k
10
0.5
−2
10
−3
10 0.4
−4
10
0
1
10
2
10 log k
0.3
10
0.2
0.2
0.1
0.1
0 0
log(P(k))
0.5
Nodes degree distribution P(k)
log(P(k))
Nodes degree distribution P(k)
0.6
0
50
100
150
200
250
300
350
400
450
0
50
100
150
200
250
300
500 Nodes degree k
Nodes degree k
(b) t=5000,n=4,m=2,λ =
(a) t=5000,n=3,m=1,λ = 1,C=1.255
1 2 ,C=1.983
Numerical simulation for the nodes degree distribution of one−mode RDP model with fitness (d)
Numerical simulation for the nodes degree distribution of one−mode RDP model with fitness (c) 0.9
0.8
0
10
0
10
0.8
0.7 experimental result theoretical result
−2
10
−3
10 0.4
−4
10
0
1
10
2
10
0.3
10
log k
−1
10 log(P(k))
0.7 Nodes degree distribution P(k)
0.5
experimental result theoretical result
−1
10 log(P(k))
Nodes degree distribution P(k)
0.6
0.6 0.5
−2
10
−3
10
0.4
−4
10
0
1
10
2
10
0.3
10
log k
0.2
0.2
0.1 0.1 0 0
0
100
200
300
400
500
50
100
150
200
250
300
350
400
450
500
Nodes degree k
Nodes degree k
(c) t=5000,n=5,m=3,λ =
0
600
(d) t=10000,n=6,m=4,λ =
1 5 ,C=2.573
1 14 ,C=3.138
Fig. 3. The four figures show the numerical simulation results for the node degree distribution of one-mode RDP model with fitness when the fitness distribution qðgÞ is chosen uniformly from the interval [0, 1] i.e. qðgÞ ¼ 1; g 2 ½0; 1. The x-axis represents the node degree k and the y-axis represents the node degree distribution PðkÞ. The solid line is the theoretical result (4.6) and the asterisk is the experimental result of one-mode RDP model with fitness when qðgÞ ¼ 1; g 2 ½0; 1. The insets show that the experimental results are in the log–log scale. In the insets, the x-axis represents logk and the y-axis ð2kÞ
represents logðPðkÞÞ. When n ¼ 3; m ¼ 1, (4.6) is PðkÞ ¼ 12 n ¼ 3; m ¼ 1. When n ¼ 4; m ¼ 2, (4.6) is PðkÞ ¼ 13 n ¼ 4; m ¼ 2. When n ¼ 5; m ¼ 3, (4.6) is PðkÞ ¼ 14 n ¼ 5; m ¼ 3. When n ¼ 5; m ¼ 3, (4.6) is PðkÞ ¼ 15
2:255
lnð2kÞ
k 1:992 3 lnð3kÞ k 1:515 4 ln 4k
ðÞ
; (a) shows the fitting situation of (4.6) and the experimental simulation results in the case
; (b) shows the fitting situation of (4.6) and the experimental simulation results in the case
ðÞ
ðÞ
k 1:224 5 ln 5k
ðÞ
ðÞ
; (c) shows the fitting situation of (4.6) and the experimental simulation results in the case
; (d) shows the fitting situation of (4.6) and the experimental simulation results in the case n = 6,
m = 4.
To check the feasibility of the node degree distribution in Eq. (4.6), we perform the numerical simulations of the node degree distribution of one-mode RDP model with fitness when the fitness distribution is qðgÞ ¼ 1; g 2 ½0; 1 i.e. qðgÞ is chosen uniformly from the interval [0, 1] (see Fig. 3. Fig. 2 shows the value of C that is obtained by the numerical simulation P gk j j j to the limit C in different cases. By the corresponding value of C, Fig. 3 shows the fitting convergence of equation CðtÞ ¼ ðn1Þt situation of (4.6) and the experimental simulation results. When n ¼ 3; m ¼ 1, we have k0 ¼ 1. By (3.6) and the numerical simulation result in Fig. 2(a), we have C 1:255. It is worth noting that we acquire the same conclusion in [12,16]. Substituting n ¼ 3; m ¼ 1; k0 ¼ 1 and C ¼ 1:255 into (4.6), 2:255
we obtain the node degree distribution is PðkÞ ¼ 12
ð2kÞ
lnð2kÞ
in the case n ¼ 3; m ¼ 1. Fig. 3(a) shows the fitting situation of
(4.6) and the experimental simulation result in the case n ¼ 3; m ¼ 1. Next, let us consider some cases when m > 1. We make the numerical simulations including three cases n ¼ 4; m ¼ 2; n ¼ 5; m ¼ 3 and n ¼ 6; m ¼ 4. When n ¼ 4; m ¼ 2, we have k0 ¼ 12. As Fig. 2(b) shows, the numerical simulaP tion result indicates that CðtÞ ¼ ð j gj kj Þ=ð3tÞ converges to the limit C 1:983. Substituting n ¼ 4; m ¼ 2; k0 ¼ 12 and
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L. Wang, Y. Ma / Commun Nonlinear Sci Numer Simulat 25 (2015) 136–144
C ¼ 1:983 into (4.6), the node degree distribution is PðkÞ ¼ 13
ð3kÞ
1:992
. Fig. 3(b) shows the fitting situation of (4.6) and the lnð3kÞ experimental simulation result in the case n ¼ 4; m ¼ 2. When n ¼ 5; m ¼ 3, we have k0 ¼ 15, and by the numerical simula-
1 , and Fig. 2(d) shows C 3:138. Fig. 3(c) and tion result in Fig. 2(c), we have C 2:573. When n ¼ 6; m ¼ 4, we have k0 ¼ 14 (d) shows the fitting situation of (4.6) and the experimental simulation results in two cases n ¼ 5; m ¼ 3 and n ¼ 6; m ¼ 4 respectively. We find that the analytical predictions are in good agreement with the results obtained from the numerical simulations (see Fig. 3). By (4.6), we know that when qðgÞ ¼ 1; g 2 ½0; 1, the node degree distribution of one-mode RDP model with fitness is not a power-law distribution. But by Fig. 3, we find the degree of most of nodes is low, and a few of nodes have very high degree. So the node degree distribution approximately possesses the power-law characteristic. The node degree distribution of onemode RDP model with fitness can also approximately be seen as a power-law distribution when qðgÞ ¼ 1; g 2 ½0; 1. So the collaboration networks with competitive factor sometimes have the generic feature of collaboration network and general complex network for some given fitness distribution functions.
5. Conclusion In real systems a node’s connectivity and growth rate does not only depend on its age. It also depends on their fitness to compete for links. The main purpose of this paper is that by one-mode RDP model with fitness, we can better understand the competitive aspect of collaboration network. It reveals the fitter-gets-richer phenomenon in collaboration network. We also call this phenomenon ‘‘the survival of the fittest’’. In this paper, we firstly calculate the node degree distribution in one-mode RDP model by mean-field approach and obtain the node degree distribution of this model is a power-law distribution for large enough node degree k. Then we propose the one-mode RDP model based on fitness-driven preferential attachment (we call this model one-mode RDP model with fitness) and acquire a general expression for the node degree distribution. When the fitness distribution functions are given by qðgÞ ¼ dðg 1Þ, when m ¼ 1 and qðgÞ ¼ 1; g 2 ½0; 1, when m P 1, we also derive the explicit forms of node degree distribution. The analytical predictions are in good agreement with results obtained by numerical simulations. By numerical simulations, we also find the node degree distribution approximately possesses the power-law characteristic in one-mode RDP model with fitness, when qðgÞ is chosen uniformly from the interval [0, 1]. So the collaboration networks with competitive factor sometimes also have the generic feature of collaboration network and general complex network for some given fitness distribution functions. In summary, the fitness is a very critical factor to decide the rate at which nodes increase their connectivity in a network. The one-mode RDP model with fitness can better reflect competitive factor of collaboration network and help us to better comprehend ‘‘Competition exists in cooperation’’ in real world. So researching one-mode RDP model with fitness is very meaningful and important for real world. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
Adamic LA, Huberman BA. Power-law distribution of the world wide web. Science 2000;287:2115. Albert R, Jeong H, Barabási AL. Internet: diameter of the world-wide web. Nature 1999;401:130–1. Barabási AL, Albert R. Emergence of scaling in random networks. Science 1999;286:509–12. Barabási AL, Albert R, Jeong H. Mean-field theory for scale-free random networks. Physica A 1999;272(1–2):173–87. Beir MG, Busch JR, Grynberg SP, et al. Obtaining communities with a fitness growth process. Phys A Stat Mech Appl 2013;392(9):2278–93. Bollobás B. Random graphs. London: Academic Press; 1985. Borrett SR, Moody J, Edelmann A. The rise of network ecology: maps of the topic diversity and scientific collaboration. Ecol Modell 2014;293:111–27. Caldarelli G, Marchetti R, Pietronero L. The fractal properties of internet. Europhys Lett 2000;52(4):386–91. Camarinha-Matos LM, Afsarmanesh H. The emerging discipline of collaborative networks. J Intell Manuf 2005;16(4–5):439–52. Erdös P, Rényi A. On the strength of connectedness of a random graph. Acta Math Hungarica 1961;12(1):261–7. Faloutsos M, Faloutsos P, Faloutsos C. On power-law relationships of the Internet topology. Comput Commun Rev 1999;29(4):251–62. Bianconi G, Barabási A-L. Competition and multiscaling in evolving networks. Europhys Lett 2001;54(4):436–42. White HC. Where do markets come from. AJS 1981;87(3):517. Ke W. A fitness model for scholarly impact analysis. Scientometrics 2013;94(3):981–98. Lambiotte R, Ausloos M. Uncovering collective listening habits and music genres in bipartite networks. Phys Rev E 2005;72(6):66107. Lee HY, Chan HY, Hui PM. Scale-free networks with tunable degree-distribution exponents. Phys Rev E 2004;69(6):067102. Mangel M, Satterthwaite WH, Pirolli P, et al. Invasion biology and the success of social collaboration networks, with application to Wikipedia. Israel J Ecol Evol 2013;59(1):17–26. Matthew Hahn W, Kern AD. Mol. Biol. Evol: comparative genomics of centrality and essentiality in three eukaryotic protein-interaction networks. Oxford J. 2005;22(4):803–6. Newman MEJ, The structure of scientific collaboration networks. PNAS 2001;98(2):404–9. Park J, Celma O, Koppoenberger M, Buldu JM. The social network of contemporary popular musicians. Bifurcation Chaos 2007;17(7):2281. Ramasco JJ, Dorogovtsev SN, Pastor-Satorras R. Self-organization of collaboration networks. Phys Rev E 2004;70(3):036106. Wasserman S, Faust K. Social network analysis: methods and applications. Am Ethnol 1997;24(1):219–20. Watts DJ, Strogatz SH. Collective dynamics of ‘small-world’ networks. Nature 1998;393:440–2. Wang L, Ma Y. Structure properties of one-mode collaboration network model based on rate equation approach. Commun Nonlinear Sci Numer Simul 2014;19(12):4068–79.