- Email: [email protected]
Growing networks with temporal effect and mixed attachment mechanisms
Physica A 413 (2014) 147–152
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Growing networks with temporal effect and mixed attachment mechanisms Zhi-Gang Shao a,∗ , Tao Chen b , Bao-quan Ai a a
Laboratory of Quantum Engineering and Quantum Materials, SPTE, South China Normal University, Guangzhou 510006, China
b
Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China
highlights • We propose a novel growing network model. • The model combines the mixed attachment mechanisms and the temporal effect. • The topology of our network model follows the exponential-decaying form.
article
info
Article history: Received 12 April 2013 Received in revised form 10 May 2014 Available online 8 July 2014 Keywords: Preferential attachment Uniform attachment Temporal effect
abstract In this paper, motivated by previous research works on the evolution of the network topology, i.e., the combination of preferential attachment and temporal effect (aging of vertices), we propose a novel model which combines the mixed attachment mechanisms (preferential attachment and uniform attachment) and the temporal effect to investigate the topological properties of the network during the evolution. Our results show that the degree distribution of the model follows the exponential-decaying form. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Numerous social, biological, and communication systems can be modeled as complex networks. A great variety of complex network models have been proposed with various mechanisms [1,2]. Over the last 15 years, as a crucial attachment mechanism in the evolution of complex network models, preferential attachment has been widely studied by the researchers of a variety of fields to model different real-world complex systems [1–11]. Nevertheless, only a few research results show the combination of preferential attachment and temporal effect [12–19]. In reality, mixed attachment mechanisms (preferential attachment and uniform attachment) can better the discussions in terms of the evolution of network topology [20–23]. But there has not been any research result which is the combination of mixed attachment mechanisms and temporal effect which may be more advantageous to describing the evolution of the topology of complex networks. Based on the above discussions, in order to model some real-world complex systems with more accuracy, we propose the model of growing networks with temporal effect and mixed attachment mechanisms. And then, the degree distribution of the model is investigated by means of the master equation approach and numerical simulations.
∗
Corresponding author. Tel.: +86 02039310066. E-mail addresses: [email protected], [email protected] (Z.-G. Shao).
http://dx.doi.org/10.1016/j.physa.2014.06.070 0378-4371/© 2014 Elsevier B.V. All rights reserved.
148
Z.-G. Shao et al. / Physica A 413 (2014) 147–152
2. Model and method The algorithm of our model, which is characterized by growth, mixed attachment mechanisms, and temporal effect, is as follows: (1) growth: initially (at t = t0 , where t0 is the time where the system starts to evolve), there are a few vertices (the number is m0 ) and no links in the system. At each time step t, a new vertex t, which can generate m new links, is introduced to the system; (2) mixed attachments with temporal effect: the newly-introduced vertex links to the existing vertices with the following two attachments: (i) preferential attachment with temporal effect; preferential attachment is defined by the fact that the probability that the newly-introduced vertex links to the old vertex i depends on the degree of vertex i, i.e., ki . In addition, motivated by the results in Refs. [12–19], we add the temporal effect (the relevance of a vertex as Ri (t ) = Ri (0) exp[−β(t − ti )], where β is the decay index, for generality Ri (0) = 1). So the probability associated with the preferential attachment with temporal effect can be expressed as ki (t )Ri (t )
Π pref (i, ki ) =
kj (t )Rj (t )
,
(1)
j
where j stands for the index of all the nodes in the network except the newly-introduced one. (ii) uniform attachment: uniform attachment means that the newly-introduced vertex links to each old vertex i already in the system with equal probability, so the probability associated with the uniform attachment is
Π unif (i, ki ) =
1 m0 + t − 1
.
(2)
3. Results and discussion After t time steps, the system will evolve into a network with vertices N = m0 + t and mt links. In the evolution of the model, the proportions of preferential attachment with time decay and of uniform attachment are q(0 < q < 1) and 1 − q, respectively, so the overall probability can be expressed as follows:
Πall (i, ki ) = mqΠ pref (i, ki ) + m(1 − q)Π unif (i, ki ).
(3)
Eqs. (1) and (2) describe a complex network where the evolution of each vertex’s degree depends on two factors: (1) the vertex itself as well as the degree and the relevance of the other vertices (preferential attachment and time decay), and (2) uniform attachment. If one assumes that at any time, there are many vertices with non-negligible values of ki (t )Ri (t ), the denominator of Eq. (1) fluctuates little with time. In addition, when t → +∞, m +1t −1 ≈ 1t . This result indicates that the 0 overall selection probability Πall (i, ki ) can be written as
Πall (i, ki ) = mqΠ pref (i, ki ) + m(1 − q)Π unif (i, ki ) = mq
ki (t )Ri (t )
Ω (t )
1
+ m(1 − q) , t
(4)
1 where Ω (t ) = j kj (t )Rj (t ) is a normalization factor. If Ri (t ) decays faster than t and fulfills the condition limt →∞ Ri (t ) = 0, ∗ the initial growth of Ω (t ) will stabilize at a certain value Ω (as shown in Fig. 1), which can be calculated from the perspective of self-consistency. Due to the generality of the expression of the relevance, we take the master equation approach to deduce the degree distribution, as the evolution process of our model is a Markov chain [14,17,24]. The master equation for the degree distribution takes the following form:
p(ki , t + 1) =
1−m q
ki Ri (t )
Ω∗
+ (1 − q)
1 t
(ki − 1)Ri (t ) 1 p(ki , t ) + m q + (1 − q) p(ki − 1, t ). Ω∗ t
(5)
In the case described by Eq. (5), p(ki , t ) of the stationary states lie in diminishing transition probabilities as a result of limt →∞ Ri (t ) = 0. As the precursor of tackling the degree distribution p(ki , t ) of the model, we first turn to obtaining the expected degree of vertex i, i.e., ⟨ki ⟩. By multiplying the master equation with ki and summing it over all ki , we can obtain the following difference equation: Ri ( t )
d ⟨ki (t )⟩ dt
=
mqRi (t )
Ω∗
Ω∗
⟨ki (t )⟩ +
⟨ki (t )⟩ +
m(1 − q)
. (6) t If Ri (t ) decays sufficiently slowly, one can switch to continuous time variable and obtain an ordinary differential equation: ⟨ki (t + 1)⟩ = ⟨ki (t )⟩ + mq
m(1 − q) t
.
(7)
Z.-G. Shao et al. / Physica A 413 (2014) 147–152
149
Fig. 1. (Color online) Ω (t ) will stabilize at a certain value Ω ∗ with N = 100 000, m = 5, q = 0.5, and β = 0.1.
Solving the above differential equation, we obtain
⟨ki ⟩ = exp
mq
Ri (t )dt
Ω∗
m (1 − q)
t
−mq exp Ω∗
Ri (t )dt dt + c ,
(8)
where c is a constant. mq We define αi = exp[ Ω ∗ Ri (t )dt ], then
⟨ki ⟩ =
m (1 − q) t
dt + c αi .
(9)
Therefore,
⟨ki ⟩ = m (1 − q) ln t + c αi + c1 ,
(10)
where c1 is a constant. By taking into account the initial condition ⟨ki (ti )⟩ = m, one can solve the above equation and get the expression of the expected degree of the vertex i,
⟨ki (t )⟩ = m(1 − q) ln(t /ti ) + m.
(11)
By utilizing the inequality ⟨ki (t )⟩ < k, we get ti > t exp
m−k m(1 − q)
.
(12)
Therefore, the probability that the node i has the degree ki (t ) smaller than k can be written as P (ki (t ) < k) = P
ti > t exp
m−k m(1 − q)
m−k = 1 − P ti ≤ t exp . m(1 − q)
(13)
Assuming that we add the nodes to the system at equal time intervals, the probability density of ti is [1,21,22] Pi (ti ) =
1 m◦ + t
≈
1 t
.
(14)
Thus,
P
ti ≤ t exp
m−k m(1 − q)
= exp
m−k m(1 − q)
.
(15)
Substituting Eq. (15) into Eq. (13) we obtain P (ki (t ) < k) = 1 − exp
m−k m(1 − q)
.
(16)
150
Z.-G. Shao et al. / Physica A 413 (2014) 147–152
a
b
c
d
Fig. 2. (Color online) The degree distribution P (k) versus degree k with the given values of input parameters: (a) N = 100 000, m = 5, q = 0.50, and β = 0.10, 0.50, and 1.00; (b) N = 100 000, m = 5, q = 0.25, 0.50, and 0.75, β = 0.1; (c) N = 100 000, m = 3, 5, and 7, q = 0.50, β = 0.10; (d) N = 100 000, 300 000, and 500 000, m = 3, q = 0.50, β = 0.1.
The degree distribution P (k) can be obtained from Eq. (16) P (k) =
∂ P (ki (t ) < k) 1 m−k = exp . ∂k m(1 − q) m(1 − q)
(17)
Then,
P (k) =
1 exp (1− q)
m(1 − q)
exp −
k m(1 − q)
.
(18)
So, the degree distribution degenerates into the exponential-decaying form
P (k) ∼ exp −
k m(1 − q)
.
(19)
For q = 1, there is no uniform attachment. In this situation, the topologies of growing networks with temporal effect were found as scale-free degree distributions [12–14]. From Eq. (7), previous results with scale-free degree distributions can be deduced with q = 1. During our derivation from Eqs. (7) to (19), it is shown that the uniform attachment gives a leading contribution into the network topology despite its probability can be small in the infinite time limit. The topology of our growing network with the uniform attachment follows the exponential-decaying form in contrast with scale-free degree distributions of growing networks with temporal effect and no uniform attachment [12–14]. In order to verify the degree distribution obtained analytically, we have performed extensive numerical simulations to investigate the behaviors of the degree distribution. To reduce the effect of fluctuation on simulation results, for every system with a certain size the simulation results are averaged over ten network realizations. As is indicated in Fig. 2, in our model, there are four key parameters which exert influence on the shapes of the curves of the degree distribution P (k) of the model: the size N of the network, the number m of links generated by each newly-introduced vertex, the proportion q of preferential attachment, and the time-decaying index β . The curves of the degree distribution P (k) are sensitive to
Z.-G. Shao et al. / Physica A 413 (2014) 147–152
a
151
b
′
Fig. 3. (Color online) The rescaled degree distribution P (k) versus degree k with N = 100 000, β = 0.10, and (a) m = 5, q = 0.25, 0.50, and 0.75; (b) m = 3, 5, and 7, q = 0.50. The slopes of green dash lines are −0.9 (a) and −1.0 (b).
the change of the values of the proportion of preferential attachment q and the number of links generated by each newlyintroduced vertex m, but are insensitive to the change of the values of the size of the network N and the time-decaying index β , which is similar to the corresponding result in Ref. [14]. In the numerical simulations shown in Fig. 2, the value of ′ P (k) decreases and follows the exponential-decaying form. The rescaled degree distribution P (k) = m (1 − q) ln[P (k)] is shown in Fig. 3. It can be seen from Fig. 3, the simulation results are consistent with the previous analytical results. 4. Conclusion In this paper, we have proposed a novel network model, which is characterized by the combination of mixed attachment mechanisms and temporal effect (aging of vertices), to offer a more precise description of the topology properties of some real-world complex systems. By virtue of the analytical approach and numerical simulations, we have deduced that the degree distribution of the model follows the exponential-decaying form. Even when our model is more accurate than the previous ones to investigate the topological properties of some complex systems, this model does not take into account the following effects which might be of great significance in describing other systems: directed nature of the network, accelerating growth of the network, and the gradual fragmentation of the network. Furthermore, the future research can also focus on other network characteristics, such as clustering coefficient and degree correlation, based on our model, which may unveil more topological properties for the real-world complex networks. Last but not the least, the application of our model to better understanding the corresponding dynamical properties remains open as a fruitful question. Acknowledgments This work was supported by the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (Grant No. LYM10065) and the National Natural Science Foundation of China (Grant Nos. 11105054, 11175067 and 11274124) and the National Natural Science Foundation of Guangdong Province, China (Grant No. S201101000332) and PCSIRT (Grant No. IRT1243). References [1] A.-L. Barabái, R. Albert, Science 286 (1999) 509; A.-L. Barabái, R. Albert, H. Jeong, Physica A 272 (1999) 173; R. Albert, A.-L. Barabái, Rev. Modern Phys. 74 (2002) 47. [2] S.N. Dorogovtsev, J.F.F. Mendes, Adv. Phys. 51 (2002) 1079; M.E.J. Newman, SIAM Rev. 45 (2003) 167; S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D.-U. Hwang, Phys. Rep. 424 (2006) 175. [3] M. Faloutsos, P. Faloutsos, C. Faloutsos, Comput. Commun. Rev. 29 (1999) 251. [4] R. Govindan, H. Tangmunarunkit, Proceedings of IEEE INFOCOM 2000, Vol.3, IEEE, Piscataway, N.J., Tel Aviv, Israel, 2000, p. 1371. [5] F. Liljeros, C.R. Edling, L.A.N. Amaral, H.E. Stanley, Y. Aberg, Nature 411 (2001) 907. [6] H. Jeong, B. Tombor, R. Albert, Z.N. Oltvai, A.-L. Barabái, Nature 407 (2000) 651. [7] J. Abello, P.M. Pardalos, M.G.C. Resende, External Memory Algorithms, in: J. Abello, J. Vitter (Eds.), DIMACS Series in Discrete Mathematics Theoretical Computer Science, American Mathematical Society, 1999, p. 119. [8] W. Aiello, F. Chung, L. Lu, Proceedings of the 32nd ACM Symposium on the Theory of Computing, ACM, New York, 2000, p. 171. [9] S. Redner, Eur. Phys. J. B 4 (1998) 131. [10] G.U. Yule, Philos. Trans. R. Soc. B 213 (1925) 21. [11] D. De Solla Price, J. Am. Soc. Inf. Sci. 27 (1976) 192. [12] S.N. Dorogovtsev, J.F.F. Mendes, Phys. Rev. E 62 (2000) 1842.
152 [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
Z.-G. Shao et al. / Physica A 413 (2014) 147–152 H. Zhu, X. Wang, J.-Y. Zhu, Phys. Rev. E 68 (2003) 056121. M. Medo, G. Cimini, S. Gualdi, Phys. Rev. Lett. 107 (2011) 238701. P. Holme, J. Saramäi, Phys. Rep. 519 (2012) 97. G. Bianconi, A.-L. Barabái, Eur. Phys. Lett. 54 (2001) 436. M.E.J. Newman, Eur. Phys. Lett. 86 (2009) 68001. M. Masuda, K. Klemm, V.M. Eguíluz, Phys. Rev. Lett. 111 (2013) 188701. E. Estrada, Phys. Rev. E 88 (2013) 042811. Z. Liu, Y.-C. Lai, N. Ye, P. Dasgupta, Phys. Lett. A 303 (2002) 337. Z.-G. Shao, X.-W. Zou, Z.-J. Tan, Z.-Z. Jin, J. Phys. A: Math. Gen. 39 (2006) 2035. T. Chen, Z.-G. Shao, Physica A 391 (2012) 2778. D.M. Pennock, G.W. Flake, S. Lawrence, E.J. Glover, C.L. Giles, Proc. Natl. Acad. Sci. USA 99 (2002) 5207. D. He, Z. Liu, B. Wang, Complex Systems and Complex Networks, Higher Education Press, China.
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Growing networks with temporal effect and mixed attachment mechanisms Zhi-Gang Shao a,∗ , Tao Chen b , Bao-quan Ai a a
Laboratory of Quantum Engineering and Quantum Materials, SPTE, South China Normal University, Guangzhou 510006, China
b
Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China
highlights • We propose a novel growing network model. • The model combines the mixed attachment mechanisms and the temporal effect. • The topology of our network model follows the exponential-decaying form.
article
info
Article history: Received 12 April 2013 Received in revised form 10 May 2014 Available online 8 July 2014 Keywords: Preferential attachment Uniform attachment Temporal effect
abstract In this paper, motivated by previous research works on the evolution of the network topology, i.e., the combination of preferential attachment and temporal effect (aging of vertices), we propose a novel model which combines the mixed attachment mechanisms (preferential attachment and uniform attachment) and the temporal effect to investigate the topological properties of the network during the evolution. Our results show that the degree distribution of the model follows the exponential-decaying form. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Numerous social, biological, and communication systems can be modeled as complex networks. A great variety of complex network models have been proposed with various mechanisms [1,2]. Over the last 15 years, as a crucial attachment mechanism in the evolution of complex network models, preferential attachment has been widely studied by the researchers of a variety of fields to model different real-world complex systems [1–11]. Nevertheless, only a few research results show the combination of preferential attachment and temporal effect [12–19]. In reality, mixed attachment mechanisms (preferential attachment and uniform attachment) can better the discussions in terms of the evolution of network topology [20–23]. But there has not been any research result which is the combination of mixed attachment mechanisms and temporal effect which may be more advantageous to describing the evolution of the topology of complex networks. Based on the above discussions, in order to model some real-world complex systems with more accuracy, we propose the model of growing networks with temporal effect and mixed attachment mechanisms. And then, the degree distribution of the model is investigated by means of the master equation approach and numerical simulations.
∗
Corresponding author. Tel.: +86 02039310066. E-mail addresses: [email protected], [email protected] (Z.-G. Shao).
http://dx.doi.org/10.1016/j.physa.2014.06.070 0378-4371/© 2014 Elsevier B.V. All rights reserved.
148
Z.-G. Shao et al. / Physica A 413 (2014) 147–152
2. Model and method The algorithm of our model, which is characterized by growth, mixed attachment mechanisms, and temporal effect, is as follows: (1) growth: initially (at t = t0 , where t0 is the time where the system starts to evolve), there are a few vertices (the number is m0 ) and no links in the system. At each time step t, a new vertex t, which can generate m new links, is introduced to the system; (2) mixed attachments with temporal effect: the newly-introduced vertex links to the existing vertices with the following two attachments: (i) preferential attachment with temporal effect; preferential attachment is defined by the fact that the probability that the newly-introduced vertex links to the old vertex i depends on the degree of vertex i, i.e., ki . In addition, motivated by the results in Refs. [12–19], we add the temporal effect (the relevance of a vertex as Ri (t ) = Ri (0) exp[−β(t − ti )], where β is the decay index, for generality Ri (0) = 1). So the probability associated with the preferential attachment with temporal effect can be expressed as ki (t )Ri (t )
Π pref (i, ki ) =
kj (t )Rj (t )
,
(1)
j
where j stands for the index of all the nodes in the network except the newly-introduced one. (ii) uniform attachment: uniform attachment means that the newly-introduced vertex links to each old vertex i already in the system with equal probability, so the probability associated with the uniform attachment is
Π unif (i, ki ) =
1 m0 + t − 1
.
(2)
3. Results and discussion After t time steps, the system will evolve into a network with vertices N = m0 + t and mt links. In the evolution of the model, the proportions of preferential attachment with time decay and of uniform attachment are q(0 < q < 1) and 1 − q, respectively, so the overall probability can be expressed as follows:
Πall (i, ki ) = mqΠ pref (i, ki ) + m(1 − q)Π unif (i, ki ).
(3)
Eqs. (1) and (2) describe a complex network where the evolution of each vertex’s degree depends on two factors: (1) the vertex itself as well as the degree and the relevance of the other vertices (preferential attachment and time decay), and (2) uniform attachment. If one assumes that at any time, there are many vertices with non-negligible values of ki (t )Ri (t ), the denominator of Eq. (1) fluctuates little with time. In addition, when t → +∞, m +1t −1 ≈ 1t . This result indicates that the 0 overall selection probability Πall (i, ki ) can be written as
Πall (i, ki ) = mqΠ pref (i, ki ) + m(1 − q)Π unif (i, ki ) = mq
ki (t )Ri (t )
Ω (t )
1
+ m(1 − q) , t
(4)
1 where Ω (t ) = j kj (t )Rj (t ) is a normalization factor. If Ri (t ) decays faster than t and fulfills the condition limt →∞ Ri (t ) = 0, ∗ the initial growth of Ω (t ) will stabilize at a certain value Ω (as shown in Fig. 1), which can be calculated from the perspective of self-consistency. Due to the generality of the expression of the relevance, we take the master equation approach to deduce the degree distribution, as the evolution process of our model is a Markov chain [14,17,24]. The master equation for the degree distribution takes the following form:
p(ki , t + 1) =
1−m q
ki Ri (t )
Ω∗
+ (1 − q)
1 t
(ki − 1)Ri (t ) 1 p(ki , t ) + m q + (1 − q) p(ki − 1, t ). Ω∗ t
(5)
In the case described by Eq. (5), p(ki , t ) of the stationary states lie in diminishing transition probabilities as a result of limt →∞ Ri (t ) = 0. As the precursor of tackling the degree distribution p(ki , t ) of the model, we first turn to obtaining the expected degree of vertex i, i.e., ⟨ki ⟩. By multiplying the master equation with ki and summing it over all ki , we can obtain the following difference equation: Ri ( t )
d ⟨ki (t )⟩ dt
=
mqRi (t )
Ω∗
Ω∗
⟨ki (t )⟩ +
⟨ki (t )⟩ +
m(1 − q)
. (6) t If Ri (t ) decays sufficiently slowly, one can switch to continuous time variable and obtain an ordinary differential equation: ⟨ki (t + 1)⟩ = ⟨ki (t )⟩ + mq
m(1 − q) t
.
(7)
Z.-G. Shao et al. / Physica A 413 (2014) 147–152
149
Fig. 1. (Color online) Ω (t ) will stabilize at a certain value Ω ∗ with N = 100 000, m = 5, q = 0.5, and β = 0.1.
Solving the above differential equation, we obtain
⟨ki ⟩ = exp
mq
Ri (t )dt
Ω∗
m (1 − q)
t
−mq exp Ω∗
Ri (t )dt dt + c ,
(8)
where c is a constant. mq We define αi = exp[ Ω ∗ Ri (t )dt ], then
⟨ki ⟩ =
m (1 − q) t
dt + c αi .
(9)
Therefore,
⟨ki ⟩ = m (1 − q) ln t + c αi + c1 ,
(10)
where c1 is a constant. By taking into account the initial condition ⟨ki (ti )⟩ = m, one can solve the above equation and get the expression of the expected degree of the vertex i,
⟨ki (t )⟩ = m(1 − q) ln(t /ti ) + m.
(11)
By utilizing the inequality ⟨ki (t )⟩ < k, we get ti > t exp
m−k m(1 − q)
.
(12)
Therefore, the probability that the node i has the degree ki (t ) smaller than k can be written as P (ki (t ) < k) = P
ti > t exp
m−k m(1 − q)
m−k = 1 − P ti ≤ t exp . m(1 − q)
(13)
Assuming that we add the nodes to the system at equal time intervals, the probability density of ti is [1,21,22] Pi (ti ) =
1 m◦ + t
≈
1 t
.
(14)
Thus,
P
ti ≤ t exp
m−k m(1 − q)
= exp
m−k m(1 − q)
.
(15)
Substituting Eq. (15) into Eq. (13) we obtain P (ki (t ) < k) = 1 − exp
m−k m(1 − q)
.
(16)
150
Z.-G. Shao et al. / Physica A 413 (2014) 147–152
a
b
c
d
Fig. 2. (Color online) The degree distribution P (k) versus degree k with the given values of input parameters: (a) N = 100 000, m = 5, q = 0.50, and β = 0.10, 0.50, and 1.00; (b) N = 100 000, m = 5, q = 0.25, 0.50, and 0.75, β = 0.1; (c) N = 100 000, m = 3, 5, and 7, q = 0.50, β = 0.10; (d) N = 100 000, 300 000, and 500 000, m = 3, q = 0.50, β = 0.1.
The degree distribution P (k) can be obtained from Eq. (16) P (k) =
∂ P (ki (t ) < k) 1 m−k = exp . ∂k m(1 − q) m(1 − q)
(17)
Then,
P (k) =
1 exp (1− q)
m(1 − q)
exp −
k m(1 − q)
.
(18)
So, the degree distribution degenerates into the exponential-decaying form
P (k) ∼ exp −
k m(1 − q)
.
(19)
For q = 1, there is no uniform attachment. In this situation, the topologies of growing networks with temporal effect were found as scale-free degree distributions [12–14]. From Eq. (7), previous results with scale-free degree distributions can be deduced with q = 1. During our derivation from Eqs. (7) to (19), it is shown that the uniform attachment gives a leading contribution into the network topology despite its probability can be small in the infinite time limit. The topology of our growing network with the uniform attachment follows the exponential-decaying form in contrast with scale-free degree distributions of growing networks with temporal effect and no uniform attachment [12–14]. In order to verify the degree distribution obtained analytically, we have performed extensive numerical simulations to investigate the behaviors of the degree distribution. To reduce the effect of fluctuation on simulation results, for every system with a certain size the simulation results are averaged over ten network realizations. As is indicated in Fig. 2, in our model, there are four key parameters which exert influence on the shapes of the curves of the degree distribution P (k) of the model: the size N of the network, the number m of links generated by each newly-introduced vertex, the proportion q of preferential attachment, and the time-decaying index β . The curves of the degree distribution P (k) are sensitive to
Z.-G. Shao et al. / Physica A 413 (2014) 147–152
a
151
b
′
Fig. 3. (Color online) The rescaled degree distribution P (k) versus degree k with N = 100 000, β = 0.10, and (a) m = 5, q = 0.25, 0.50, and 0.75; (b) m = 3, 5, and 7, q = 0.50. The slopes of green dash lines are −0.9 (a) and −1.0 (b).
the change of the values of the proportion of preferential attachment q and the number of links generated by each newlyintroduced vertex m, but are insensitive to the change of the values of the size of the network N and the time-decaying index β , which is similar to the corresponding result in Ref. [14]. In the numerical simulations shown in Fig. 2, the value of ′ P (k) decreases and follows the exponential-decaying form. The rescaled degree distribution P (k) = m (1 − q) ln[P (k)] is shown in Fig. 3. It can be seen from Fig. 3, the simulation results are consistent with the previous analytical results. 4. Conclusion In this paper, we have proposed a novel network model, which is characterized by the combination of mixed attachment mechanisms and temporal effect (aging of vertices), to offer a more precise description of the topology properties of some real-world complex systems. By virtue of the analytical approach and numerical simulations, we have deduced that the degree distribution of the model follows the exponential-decaying form. Even when our model is more accurate than the previous ones to investigate the topological properties of some complex systems, this model does not take into account the following effects which might be of great significance in describing other systems: directed nature of the network, accelerating growth of the network, and the gradual fragmentation of the network. Furthermore, the future research can also focus on other network characteristics, such as clustering coefficient and degree correlation, based on our model, which may unveil more topological properties for the real-world complex networks. Last but not the least, the application of our model to better understanding the corresponding dynamical properties remains open as a fruitful question. Acknowledgments This work was supported by the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (Grant No. LYM10065) and the National Natural Science Foundation of China (Grant Nos. 11105054, 11175067 and 11274124) and the National Natural Science Foundation of Guangdong Province, China (Grant No. S201101000332) and PCSIRT (Grant No. IRT1243). References [1] A.-L. Barabái, R. Albert, Science 286 (1999) 509; A.-L. Barabái, R. Albert, H. Jeong, Physica A 272 (1999) 173; R. Albert, A.-L. Barabái, Rev. Modern Phys. 74 (2002) 47. [2] S.N. Dorogovtsev, J.F.F. Mendes, Adv. Phys. 51 (2002) 1079; M.E.J. Newman, SIAM Rev. 45 (2003) 167; S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D.-U. Hwang, Phys. Rep. 424 (2006) 175. [3] M. Faloutsos, P. Faloutsos, C. Faloutsos, Comput. Commun. Rev. 29 (1999) 251. [4] R. Govindan, H. Tangmunarunkit, Proceedings of IEEE INFOCOM 2000, Vol.3, IEEE, Piscataway, N.J., Tel Aviv, Israel, 2000, p. 1371. [5] F. Liljeros, C.R. Edling, L.A.N. Amaral, H.E. Stanley, Y. Aberg, Nature 411 (2001) 907. [6] H. Jeong, B. Tombor, R. Albert, Z.N. Oltvai, A.-L. Barabái, Nature 407 (2000) 651. [7] J. Abello, P.M. Pardalos, M.G.C. Resende, External Memory Algorithms, in: J. Abello, J. Vitter (Eds.), DIMACS Series in Discrete Mathematics Theoretical Computer Science, American Mathematical Society, 1999, p. 119. [8] W. Aiello, F. Chung, L. Lu, Proceedings of the 32nd ACM Symposium on the Theory of Computing, ACM, New York, 2000, p. 171. [9] S. Redner, Eur. Phys. J. B 4 (1998) 131. [10] G.U. Yule, Philos. Trans. R. Soc. B 213 (1925) 21. [11] D. De Solla Price, J. Am. Soc. Inf. Sci. 27 (1976) 192. [12] S.N. Dorogovtsev, J.F.F. Mendes, Phys. Rev. E 62 (2000) 1842.
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