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Scale-free property of complex ad hoc networks with accelerated growth
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Physica A 376 (2007) 679–686 www.elsevier.com/locate/physa
Scale-free property of complex ad hoc networks with accelerated growth Sen Qin, Guanzhong Dai, Lin Wang, Ming Fan School of Automation, Northwestern Polytechnical University, Xi’an, Shaanxi, 710072, China Received 16 June 2006; received in revised form 26 September 2006 Available online 15 November 2006
Abstract A complex ad hoc network model with accelerated growth is proposed in this paper. In the evolving process of the model, at each time step with the addition probability c1 , a new vertex with mðtÞ edges is added into the model, which new adding edges are connected with the old vertices according to the preferential attachment rule. And with the deletion probability c2 ðoc1 Þ, a random chosen vertex is deleted and its associated edges are disappeared simultaneously. The accelerated growth means that at each time step the number of new adding edges is the increasing function of the time t. We consider mainly mðtÞ ¼ ty , where 0pyp1, to analyze the degree distribution of the model. Using the continuous approach, we prove theoretically and numerically that the model follows the scale-free degree distribution and generates both the Baraba´si and Albert (BA) model and the model introduced by Sarshar and Roychowdhury (SR). In addition, we present the appropriate numerical range of c1 , c2 and y according to theoretical analyzes and numerical simulation. The model takes on a certain universality to complex evolving network and has the guiding significance to the application of the peer-to-peer networks and complex ad hoc networks. r 2006 Elsevier B.V. All rights reserved. Keywords: Scale-free network; Power-law; Ad hoc network; Accelerated growth
1. Introduction In the last few years, the scale-free model of Baraba´si and Albert has given rise to a burst of interest in the field of complex networks [1–3]. A variety of diverse real-world social, technological and biological networks [4], including the Internet , World Wide Web (WWW), citation, collaboration network, word web, food web, etc., have discovered to exhibit the scale-free degree distribution. That is to say, the degree distributions of those networks follow a power-law distribution with the exponent g40, i.e., PðkÞ / kg . And existed results indicate that these exponents mostly range from 2 to 3. According to mean-field theory, Baraba´si et al. [5] have demonstrated that a scale-free network can be formed by two evolving mechanisms—growth and preferential attachment. Network growth implies that the number of vertices in a network increases with the evolving time Corresponding author.
E-mail addresses: [email protected] (S. Qin), [email protected] (G. Dai), [email protected] (L. Wang), [email protected] (M. Fan). 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2006.10.068
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as a whole, which is in accord with the fact that the scale of most natural networks grow gradually in their evolving process. The mechanism of preferential attachment describes the edge-attachment rule of the evolving process of many natural networks in principle and explains the so-called ‘‘rich get richer’’ phenomenon existed in natural societies. Although the BA model depicts the evolving process of most natural networks by and large, it is incompletely reasonable in the model suppose that each vertex of a network exists in the network all the time. Three shortcomings of the BA model are considered in the paper. Firstly, some vertices maybe disappear from the network as a result of their aging or the viability for a given period of time. For instance, if a website has a few browsed quantity or its owner builds a new website to replace it, it will be closed in WWW. From the point of view of network topology, a vertex is deleted at the time. Secondly, the trend of the number of vertices for a network is increasing, but it does not mean that at each time step a new vertex will be inserted into the network certainly. In a citation network, whether a published paper is cited by an author of a new paper or not lies on its theoretical value to the new paper, the influence in the field, or other factors. Hence it is deserved deliberation that at each time step a vertex must be inserted into the network. Thirdly, in the BA model at each time step there are a fixed number of new adding edges to connect with the existed vertices, i.e., it is a linear growth model. However, the number of edges grows in a nonlinear fashion in several real-world networks, and most of them have even an accelerated growth characteristic. Sen [6] has proposed a nonlinear scheme that the number of outgoing edges mðtÞ at any time t is taken as NðtÞy where NðtÞ is the number of edges at that time. A complex ad hoc network is the network where the addition of a new vertex and the deletion of an existing vertex are stochastic at each time step. Sarshar and Roychowdhury (SR) [7] have studied the effect of random deletions of vertices for a complex ad hoc network at a fixed rate. The SR model mainly deals with the first shortcoming of the BA model, in which network a random selected vertex is deleted with the deletion probability and its associated edges are disappeared at the same time. Sen [6] discussed the accelerated growth model in evolving networks, which can be considered a modification of the BA model for the third shortcoming in mentioned above. In this paper, we consider a complex ad hoc network with accelerated growth. In the network, with the addition probability c1 , a new vertex is added to the network at each time step, which describes the uncertainty of vertex addition and makes up the second shortcoming of the BA model. And with the deletion probability c2 , a randomly chosen vertex is deleted and its associated edges are disappeared simultaneously, which deals with the first shortcoming of the BA model. If the deletion probability c2 is nearly to the addition probability c1 , the network grows very slowly or almost cease to grow. The growth acceleration of the network is in proportion to the difference between c1 and c2 . Then in order to achieve a rapid growth of the network, c1 should be much greater than c2 in the paper. In addition, we compare the difference of their topologies between the linear growth ðmðtÞ ¼ mÞ model and the accelerated growth ðmðtÞ ¼ ty Þ model. The evolving processes of linear growth model and accelerated growth model are shown in Fig. 1. In the processes of those model, new added edges are connected with the existed vertices following the preferential
Fig. 1. The evolving processes of the linear growth and accelerated growth networks. Scheme (a) shows the evolving process of the linear growth network. At each time step, with probability c1 , a new vertex (denoted by vertex ) is added to the network with m new edges (solid lines). Then with probability c2 , a random selected vertex (denoted by vertex ) is deleted and the edges (dashed lines) associated with this vertex are deleted simultaneously. Scheme (b) shows the corresponding process of accelerated growth fashion, and the number of new edges at each time step is mðtÞ ¼ ty , which is one and only difference between this model and the linear model.
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attachment rule, which is the same to the BA model. We prove that both two models have the power-law degree distributions. Since adding a new vertex and deleting an existed vertex at each time step are random with the given probabilities, the power-law exponent of the accelerated growth model has a large range of variation. The corresponding results verify it. Sarshar et al. [7] have suggested that ‘‘Additional preferential targeted links will not help’’. That is to say, at each time step, if there is a randomly chosen vertex initiates b preferential targeted edges, the rate equation of this network is same to that of the original network, so they have the power-law distribution with the same exponent. However, in the accelerated growth model, we prove that its exponent of power-law degree distribution depends on the coefficient y of edge accelerated growth, which indicates that edge accelerated growth and the additional preferential attachment are different in the influence of the structure of the model. The remainder of this paper is focused on scaling properties of complex ad hoc networks with linear and accelerated growth. We mainly prove that both two models have the scale-free degree distribution analytically and numerically. In Section 2, the network with linear growth using the continuous rate equation approach is discussed. The corresponding network of nonlinear fashion are studied in Section 3. In last section, the results are discussed and this paper is concluded. 2. Linear growth model According to the same approach as introduced in Refs. [7,8], the ith vertex in the network is the vertex that was inserted at time step i. And define ki ðtÞ as the degree of the ith vertex at time t. Let SðtÞ be the sum of the degree of all vertices appeared in the network at time t and NðtÞ ¼ ðc1 c2 Þt be the total number of vertices in the network. In addition, let Di ðtÞ be the probability that the ith vertex is not deleted until time t (t4i). So we have Z t SðtÞ ¼ Di ðtÞki ðtÞ di. (1) 0
Note that there are two foundational causes induced to increase the value of the degree ki ðtÞ according to the evolving mechanism of the linear growth model. One is that with a certain probability c1 , a new vertex t with a number of new edges m is inserted into the network. So c1 mki ðtÞ=SðtÞ, the ratio of the expected number of edges that vertex i receives to the sum of the new adding edges m, whose edges are as a result of the preferential attachment made by the newly introduced vertex t, is the first term of the rate equation for ki ðtÞ. The other is that with probability c2 , a random chosen vertex j is deleted and the associated edges of this vertex are disappeared. Then the probability that when vertex j links with vertex i, vertex i loses a edge, c2 ki ðtÞ=NðtÞ, is taken into account to the rate equation necessarily. Therefore, the rate equation of ki ðtÞ is qki ðtÞ ki ðtÞ ki ðtÞ ¼ c1 m c2 . qt SðtÞ NðtÞ
(2)
At each time step a vertex is deleted with probability c2 from the network, so the events of deleted vertex are independent each other. Therefore, it is obviously for the probability Di ðtÞ that the following equation is tenable c2 Di ðt þ 1Þ ¼ Di ðtÞ 1 . (3) NðtÞ According to the continuous hypothesis, the differential equation for Di ðtÞ is qDi ðtÞ Di ðtÞ c2 Di ðtÞ ¼ c2 ¼ , qt NðtÞ c1 c2 t and the boundary condition of the above equation is Di ðiÞ ¼ c1 , so we get tc2 =ðc2 c1 Þ . Di ðtÞ ¼ c1 i
(4)
(5)
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Using the same approach introduced in Ref. [7], with the boundary condition ki ðiÞ ¼ c1 m and Eq. (5), we obtain qSðtÞ c2 SðtÞ c21 m SðtÞ ¼ c1 m c2 . qt tðc1 c2 Þ tðc2 c1 Þ
(6)
The solution to the above equation is SðtÞ ¼ c1 mð1 þ c1 Þ
c1 c2 NðtÞ t ¼ c1 mð1 þ c1 Þ . c1 þ c2 c1 þ c2
(7)
Inserting Eq. (7) back into Eq. (2), we have qki ðtÞ ki ðtÞ ki ðtÞ ¼ c1 m c1 c2 c2 qt c1 mð1 þ c1 Þ c1 þc2 t ðc1 c2 Þt 1 ki ðtÞ c1 þ c2 c2 ¼ c1 c2 t 1 þ c1 c1 ð1 c2 Þ ki ðtÞ , ¼ ð1 þ c1 Þðc1 c2 Þ t
ð8Þ
which implies that ki ðtÞ ¼ c1 m
tb , i
(9)
where b¼
c1 ð1 c2 Þ . ð1 þ c1 Þðc1 c2 Þ
(10)
It is obvious that b is the monotone decreasing function of the addition probability c1 and the monotone increasing function of the deletion probability c2 according to Eq. (10). And when c1 ¼ 1, the value of b ¼ 12 does not depend on the deletion probability. However, when c2 ¼ 0, b ¼ 1=ð1 þ c1 Þ is still correlative with c1 . In other words, Eq. (9) indicates that the degree of a vertex mainly depends on the addition of new vertex at each time with a large probability c1 , which is reasonable and intelligible since a new vertex attaches m new edges. On the other hand, although with probability c2 a random chosen vertex is deleted and its edges are disappeared correspondingly, the degree of a vertex has hardly any change since 0pc2 oc1 . In addition, according to Eqs. (5) and (9), we obtain easily that the expected degree E i ðtÞ of the ith vertex at time t is E i ðtÞ ¼ ki ðtÞDi ðtÞ ¼ ðc1 Þ2 m
tðc1 c2 2c1 c2 Þ=ð1þc1 Þðc1 c2 Þ i
.
(11)
So we find that the critical of deletion probability c2 is c2 ¼
c1 . 1 þ 2c1
(12)
That is, for 0pc2 oc1 =ð1 þ 2c1 Þ, the expected degree of any given vertex E i ðtÞ ! 1, and for c1 =ð1 þ 2c1 Þoc2 oc1 , E i ðtÞ ! 0 when t=i ! 1. Furthermore, when c1 ¼ 1, the analysis results are the same with that of the SR model. Let I k ðtÞ be the set of all vertex i with degree k at time step t and ik be the solution to the equation ki ðtÞ ¼ k according to continuous approach. Following the same method of Ref. [7] to obtain the degree distribution of
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the network, we have ½ðNo: of vertices with degreeÞ ¼ k ðTotal number of verticesÞ qki ðtÞ1 1 X 1 ¼ Dik ðtÞ Di ðtÞ NðtÞ NðtÞ qi
Pk ðtÞ ¼
i2I k ðtÞ
i¼ik
"
#c2 =ðc2 c1 Þ
1 t ¼ ðc1 c2 Þt ðc1 mÞ1=b k1=b t
ðc1 mÞ
1=b 1=b1
k
1 t b
/ k1ðc1 =½ðc1 c2 ÞbÞ ,
ð13Þ
which is a power-law distribution with the exponent c1 . g¼1þ ðc1 c2 Þb
(14)
Inserting Eq. (10) into the above equation, we get g¼1þ
1 þ c1 . 1 c2
(15)
Now we analyze the validity of the exponent of power-law degree distribution for the linear growth model. Note that when c1 ¼ 1 and c2 ¼ 0, the model degenerates the BA model with the exponent g ¼ 3, and when c1 ¼ 1 and c2 ¼ c, the model degenerate the SR model with the exponent g ¼ ð1 þ 2Þ=ð1 cÞ. Therefore, the linear growth model generalizes both the BA model and the SR model, which is coincident with its evolving mechanism. On the other hand, we have made numerical simulations to verify the correctness of the exponent g. Fig. 2 demonstrates that the numerical results approach asymptotically the theoretical predictions. The simulation results (dashed lines) are in perfect agreement with the theory values (solid lines) for cp0:25. However, for larger values of c2 , the exponents of power-law degree distributions of this model have obvious deviations with 4.4 Simulation
4.2
Theory
Power-law exponent γ
4 3.8 3.6 3.4 3.2 3 2.8 2.6 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Deletion probability c2 Fig. 2. The power-law exponent g of degree distribution for the linear growth model with different c1 and c2 . The evolving time steps of all simulations are 1 105 and m ¼ 3. Note that the maximum of the deletion probability c2 is 0.4, the value of the addition probability c1 ranges from 0.6 to 1.0 (c1 ¼ 0:6, circles; c1 ¼ 0:7, asterisk; c1 ¼ 0:8, squares; c1 ¼ 0:9, plus signs and c1 ¼ 1:0, diamonds), which assumes that the network grows with the evolving time, i.e., c1 4c2 .
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the theory values. This phenomenon can be induced by three key reasons. Firstly, the theory value is the limit of NðtÞ ! 1, but we are restricted in computing power to simulate just the finite number of evolving time steps. Secondly, the growth rate will become slowly with a large deletion probability since the number of its edge is NðtÞ ¼ ðc1 c2 Þt. Lastly, as mentioned above, when c2 4c1 =ð1 þ c1 Þ, zero is the limit of the degree of any vertex as t=i approaches 1. Here for c1 ¼ 0:6, the threshold of c2 is 0:6=ð1 þ 1:2Þ 0:273. So when c2 40:25, the power-law exponent of the degree distribution has a great deviation. 3. Accelerated growth model In this section we consider the scaling properties, especially degree distribution, of the complex ad hoc network with accelerated growth. The evolving process of this model is similar with the linear growth model and the only difference between two models is that a new vertex with mðtÞ ¼ ty edges is inserted into the latter model, not m edges as the former model at each time step. That is, the foundational evolving process of the accelerated growth model is as follows. (i) At each time step t, a new vertex with mðtÞ ¼ ty is added into the network with probability c1 , where 0pyp1. (ii) With probability c2 , a random chosen vertex is deleted and its associated edges are cut at the same time. As the same hypotheses and denotations, we get the rate equation of the degree ki ðtÞ qki ðtÞ ki ðtÞ ki ðtÞ ¼ c1 t y c2 . qt SðtÞ NðtÞ
(16)
Since the difference between the accelerated growth model and the linear growth model is the number of new adding edges at each time step, the probability that vertex i exists in the network until time t, Di ðtÞ, does not change in this model. Hence, Eq. (5) holds good in the model. With new boundary condition ki ðiÞ ¼ c1 iy , we get the sum of the degree of all vertices appeared in the network at time t c1 ð1 þ c1 Þðc1 c2 Þ yþ1 t yðc1 c2 Þ þ ðc1 þ c2 Þ c1 ð1 þ c1 Þ ty NðtÞ. ¼ yðc1 c2 Þ þ ðc1 þ c2 Þ
SðtÞ ¼
ð17Þ
Inserting Eq. (17) into Eq. (16), we obtain qki ðtÞ ki ðtÞ ki ðtÞ ¼ c1 ty c1 ð1þc1 Þðc1 c2 Þ c2 yþ1 qt ðc c2 Þt 1 t yðc c Þþðc þc Þ 1
2
1
2
yðc1 c2 Þ þ c1 ð1 c2 Þ ki ðtÞ . ¼ ð1 þ c1 Þðc1 c2 Þ t which implies that tb0 ki ðtÞ ¼ c1 iy , i where b0 ¼
yðc1 c2 Þ þ c1 ð1 c2 Þ . ð1 þ c1 Þðc1 c2 Þ
The expected degree in the accelerated growth model is tðyðc1 c2 Þþc1 c2 2c1 c2 Þ=ð1þc1 Þðc1 c2 Þ E i ðtÞ ¼ ki ðtÞDi ðtÞ ¼ ðc1 Þ2 iy . i Then according to the exponent of i we obtain that the critical of growth coefficient y is y0 ¼
c1 c2 2c1 c2 . c1 ðc1 c2 Þ
ð18Þ
(19)
(20)
(21)
(22)
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Since the exponent of t is different with that of i, it is difficult to fix on the limit of E i ðtÞ when t=i approaches to 1. However, we will find that the critical of y influence the power-law exponent of degree distribution of the model in the following discussion. Adopting the same approach in Section 2 to get the degree distribution of the accelerated growth model, we have 0
Pk ðtÞ / k1ðc1 =ðb yÞðc1 c2 ÞÞ .
(23)
Since 0pc2 oc1 p1, it is easily proved that 1 c2 c1 c2 4 ¼ 14y, c1 c2 c1 c2 then we have b0 y40. Eq. (23) implies that the degree distribution of the model is a power-law distribution with the exponent g¼1þ
c1 2 þ ðc1 c2 Þð1 yÞ ¼ . 0 1 c2 yðc1 c2 Þ ðc1 c2 Þðb yÞ
(24)
In the above expression of g when y ¼ 0, we can find Eq. (24) turns into Eq. (15), which implies that the accelerated growth model extends the linear growth model. Combining the results of Section 2, we can suggest that the accelerated growth model is a universal model relative to the BA model, the SR model and the linear growth model. It can be also shown from the form of the power-law exponent g. For example, when c1 ¼ 1, c2 ¼ 0 and y ¼ 0, the exponent of the accelerated growth model degenerates g ¼ 3, which is consistent with the exponent of the BA model. On the other hand, when y ¼ 1, g ¼ 2=ð1 c1 Þ is not relative to the deletion probability c2 , which proves again that the degree of a vertex mainly depends on the adding vertex with a certain new edges. And in such a case, the network is a fully connected network (an Nclique) in the certain probability sense. We use numerical simulations to verify the validity of the power-law exponent of the degree distribution for this model, which is shown in Fig. 3. 6.5 c1=0.9, c2=0.2 c1=0.8, c2=0.2 c1=0.9, c2=0.1 c1=0.8, c2=0.1
Power-law exponent γ
6
5.5
5
4.5
4
3.5
3 0.1
0.2
0.3
0.4
0.5
0.6
Accelerated growth coefficient θ
Fig. 3. The power-law exponent g of degree distribution for the accelerated growth model with different coefficient y. The evolving time steps of all simulations are 2 104 for yp0:4, and 1 104 for y ¼ 0:5, y ¼ 0:6. Note that when yp0:4 the simulation values (dashed lines) are consistent with the theoretical values (solid lines) calculated by Eq. (24) with the different addition and deletion probabilities. And when y40:4, the simulation values are larger than the limits of the power-law exponents of degree distribution because of the critical of y (Eq. (22)) and the former two reasons in Fig. 2.
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4. Discussions and conclusions We have suggested the complex ad hoc network with accelerated growth, in which model the vertex addition and deletion have a certain randomness at each evolving time step. Since the addition and deletion probabilities are introduced in the evolving process of this model, its topology structure takes on a board adjustability by means of turning in corresponding probabilities. Therefore, the model can win through the three shortcomings of the BA model, which are mentioned in Section 1, and include both the BA model and the SR model as its special cases. Because c1 4c2 is always satisfied through all evolving processes and the new added edges are connected according to the rule of preferential attachment, so the two fundamental mechanisms forming the scale-free structure of a network, growth and preferential attachment, are still presented in the model. So the model has a power-law degree distribution and its exponent extends the exponents of the BA model and the SR model, as good as our analyzes and simulations. The topologies of linear growth model is comparatively researched with that of the accelerated growth model. And the degree distribution of this model also exhibits a scale-free behavior and the power-law exponent always exceeds 2 by Eq. (15). In order to ensure that gp3, which is the range of power-law exponent of most of the real-world networks, it is enough to restrict the deletion probability c2 in the range of ½0; ð1 p ffifficffi 1 Þ=2Þ. According to the critical of c2 obtained in Eq. (12), let ð1 c1 Þ=2 ¼ c1 =ð1 þ c1 Þ, we have c1 ¼ 2 1 0:414. If we suppose that the addition probability c1 40:5, there are three conditions of the topology of linear growth model for the different c2 . (i) 0pc2 oð1 c1 Þ=2. In this case, the exponent g of the model is in the range of ½2; 3 and the theoretical value calculated by Eq. (15) is accurately consistent with the simulated value. Therefore, the case describes the practical situation of most of real-world networks optimally. (ii) ð1 c1 Þ=2oc2 oc1 =ð1 þ c1 Þ. Although the theoretical value is consistent with the simulated value, the exponent is g43 in the case. It is only suitably to some complex networks with the exponents exceeding 3. (iii) c1 =ð1 þ c1 Þoc2 oc1 . Not only the theoretical value of g predicted by Eq. (15) has an obvious deviation with the simulated value, but also the exponent g is in the range of ½3; þ1Þ. Hence, there would be a great error if one makes use of the linear growth model to simulate a practical evolving process. In addition, we get the threshold of the accelerated coefficient y by Eq. (22). And the value of y has a straightforward influence of the power-law exponent of degree distribution of the accelerated growth model and the accuracy of theory results. Therefore, when c1 40:5, 0pc2 oð1 c1 Þ=2 and yoy0 , the accelerated growth model will preferably describe the structures of most of the real-world complex networks, whose model has a scale-free degree distribution with the power-law exponent not exceeding 3. And the model takes on a certain universality to complex evolving network since it includes the BA model, the SR model and the linear growth model as its special cases. When we design a complex network which satisfies a certain property, such as an ad hoc network, a peer-to-peer network, or other complex evolving network, it has some guiding significance and practical values in theory. References [1] [2] [3] [4] [5] [6] [7] [8]
A.L. Baraba´si, R. Albert, Science 286 (1999) 509. S.N. Dorogovtsev, J.F.F. Mendes, cond-mat/0106144 (2001). R. Albert, A.L. Baraba´si, Rev. Mod. Phys. 74 (2002) 47. M.E.J. Newman, SIAM Rev. 45 (2003) 167. A.L. Baraba´si, R. Albert, H. Jeong, Physica A 272 (1999) 173. P. Sen, Phys. Rev. E 69 (2004) 046107. N. Sarshar, V. Roychowdhury, Phys. Rev. E 69 (2004) 026101. S.N. Dorogovtsev, J.F.F. Mendes, Phys. Rev. E 63 (2001) 056125.
Physica A 376 (2007) 679–686 www.elsevier.com/locate/physa
Scale-free property of complex ad hoc networks with accelerated growth Sen Qin, Guanzhong Dai, Lin Wang, Ming Fan School of Automation, Northwestern Polytechnical University, Xi’an, Shaanxi, 710072, China Received 16 June 2006; received in revised form 26 September 2006 Available online 15 November 2006
Abstract A complex ad hoc network model with accelerated growth is proposed in this paper. In the evolving process of the model, at each time step with the addition probability c1 , a new vertex with mðtÞ edges is added into the model, which new adding edges are connected with the old vertices according to the preferential attachment rule. And with the deletion probability c2 ðoc1 Þ, a random chosen vertex is deleted and its associated edges are disappeared simultaneously. The accelerated growth means that at each time step the number of new adding edges is the increasing function of the time t. We consider mainly mðtÞ ¼ ty , where 0pyp1, to analyze the degree distribution of the model. Using the continuous approach, we prove theoretically and numerically that the model follows the scale-free degree distribution and generates both the Baraba´si and Albert (BA) model and the model introduced by Sarshar and Roychowdhury (SR). In addition, we present the appropriate numerical range of c1 , c2 and y according to theoretical analyzes and numerical simulation. The model takes on a certain universality to complex evolving network and has the guiding significance to the application of the peer-to-peer networks and complex ad hoc networks. r 2006 Elsevier B.V. All rights reserved. Keywords: Scale-free network; Power-law; Ad hoc network; Accelerated growth
1. Introduction In the last few years, the scale-free model of Baraba´si and Albert has given rise to a burst of interest in the field of complex networks [1–3]. A variety of diverse real-world social, technological and biological networks [4], including the Internet , World Wide Web (WWW), citation, collaboration network, word web, food web, etc., have discovered to exhibit the scale-free degree distribution. That is to say, the degree distributions of those networks follow a power-law distribution with the exponent g40, i.e., PðkÞ / kg . And existed results indicate that these exponents mostly range from 2 to 3. According to mean-field theory, Baraba´si et al. [5] have demonstrated that a scale-free network can be formed by two evolving mechanisms—growth and preferential attachment. Network growth implies that the number of vertices in a network increases with the evolving time Corresponding author.
E-mail addresses: [email protected] (S. Qin), [email protected] (G. Dai), [email protected] (L. Wang), [email protected] (M. Fan). 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2006.10.068
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as a whole, which is in accord with the fact that the scale of most natural networks grow gradually in their evolving process. The mechanism of preferential attachment describes the edge-attachment rule of the evolving process of many natural networks in principle and explains the so-called ‘‘rich get richer’’ phenomenon existed in natural societies. Although the BA model depicts the evolving process of most natural networks by and large, it is incompletely reasonable in the model suppose that each vertex of a network exists in the network all the time. Three shortcomings of the BA model are considered in the paper. Firstly, some vertices maybe disappear from the network as a result of their aging or the viability for a given period of time. For instance, if a website has a few browsed quantity or its owner builds a new website to replace it, it will be closed in WWW. From the point of view of network topology, a vertex is deleted at the time. Secondly, the trend of the number of vertices for a network is increasing, but it does not mean that at each time step a new vertex will be inserted into the network certainly. In a citation network, whether a published paper is cited by an author of a new paper or not lies on its theoretical value to the new paper, the influence in the field, or other factors. Hence it is deserved deliberation that at each time step a vertex must be inserted into the network. Thirdly, in the BA model at each time step there are a fixed number of new adding edges to connect with the existed vertices, i.e., it is a linear growth model. However, the number of edges grows in a nonlinear fashion in several real-world networks, and most of them have even an accelerated growth characteristic. Sen [6] has proposed a nonlinear scheme that the number of outgoing edges mðtÞ at any time t is taken as NðtÞy where NðtÞ is the number of edges at that time. A complex ad hoc network is the network where the addition of a new vertex and the deletion of an existing vertex are stochastic at each time step. Sarshar and Roychowdhury (SR) [7] have studied the effect of random deletions of vertices for a complex ad hoc network at a fixed rate. The SR model mainly deals with the first shortcoming of the BA model, in which network a random selected vertex is deleted with the deletion probability and its associated edges are disappeared at the same time. Sen [6] discussed the accelerated growth model in evolving networks, which can be considered a modification of the BA model for the third shortcoming in mentioned above. In this paper, we consider a complex ad hoc network with accelerated growth. In the network, with the addition probability c1 , a new vertex is added to the network at each time step, which describes the uncertainty of vertex addition and makes up the second shortcoming of the BA model. And with the deletion probability c2 , a randomly chosen vertex is deleted and its associated edges are disappeared simultaneously, which deals with the first shortcoming of the BA model. If the deletion probability c2 is nearly to the addition probability c1 , the network grows very slowly or almost cease to grow. The growth acceleration of the network is in proportion to the difference between c1 and c2 . Then in order to achieve a rapid growth of the network, c1 should be much greater than c2 in the paper. In addition, we compare the difference of their topologies between the linear growth ðmðtÞ ¼ mÞ model and the accelerated growth ðmðtÞ ¼ ty Þ model. The evolving processes of linear growth model and accelerated growth model are shown in Fig. 1. In the processes of those model, new added edges are connected with the existed vertices following the preferential
Fig. 1. The evolving processes of the linear growth and accelerated growth networks. Scheme (a) shows the evolving process of the linear growth network. At each time step, with probability c1 , a new vertex (denoted by vertex ) is added to the network with m new edges (solid lines). Then with probability c2 , a random selected vertex (denoted by vertex ) is deleted and the edges (dashed lines) associated with this vertex are deleted simultaneously. Scheme (b) shows the corresponding process of accelerated growth fashion, and the number of new edges at each time step is mðtÞ ¼ ty , which is one and only difference between this model and the linear model.
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attachment rule, which is the same to the BA model. We prove that both two models have the power-law degree distributions. Since adding a new vertex and deleting an existed vertex at each time step are random with the given probabilities, the power-law exponent of the accelerated growth model has a large range of variation. The corresponding results verify it. Sarshar et al. [7] have suggested that ‘‘Additional preferential targeted links will not help’’. That is to say, at each time step, if there is a randomly chosen vertex initiates b preferential targeted edges, the rate equation of this network is same to that of the original network, so they have the power-law distribution with the same exponent. However, in the accelerated growth model, we prove that its exponent of power-law degree distribution depends on the coefficient y of edge accelerated growth, which indicates that edge accelerated growth and the additional preferential attachment are different in the influence of the structure of the model. The remainder of this paper is focused on scaling properties of complex ad hoc networks with linear and accelerated growth. We mainly prove that both two models have the scale-free degree distribution analytically and numerically. In Section 2, the network with linear growth using the continuous rate equation approach is discussed. The corresponding network of nonlinear fashion are studied in Section 3. In last section, the results are discussed and this paper is concluded. 2. Linear growth model According to the same approach as introduced in Refs. [7,8], the ith vertex in the network is the vertex that was inserted at time step i. And define ki ðtÞ as the degree of the ith vertex at time t. Let SðtÞ be the sum of the degree of all vertices appeared in the network at time t and NðtÞ ¼ ðc1 c2 Þt be the total number of vertices in the network. In addition, let Di ðtÞ be the probability that the ith vertex is not deleted until time t (t4i). So we have Z t SðtÞ ¼ Di ðtÞki ðtÞ di. (1) 0
Note that there are two foundational causes induced to increase the value of the degree ki ðtÞ according to the evolving mechanism of the linear growth model. One is that with a certain probability c1 , a new vertex t with a number of new edges m is inserted into the network. So c1 mki ðtÞ=SðtÞ, the ratio of the expected number of edges that vertex i receives to the sum of the new adding edges m, whose edges are as a result of the preferential attachment made by the newly introduced vertex t, is the first term of the rate equation for ki ðtÞ. The other is that with probability c2 , a random chosen vertex j is deleted and the associated edges of this vertex are disappeared. Then the probability that when vertex j links with vertex i, vertex i loses a edge, c2 ki ðtÞ=NðtÞ, is taken into account to the rate equation necessarily. Therefore, the rate equation of ki ðtÞ is qki ðtÞ ki ðtÞ ki ðtÞ ¼ c1 m c2 . qt SðtÞ NðtÞ
(2)
At each time step a vertex is deleted with probability c2 from the network, so the events of deleted vertex are independent each other. Therefore, it is obviously for the probability Di ðtÞ that the following equation is tenable c2 Di ðt þ 1Þ ¼ Di ðtÞ 1 . (3) NðtÞ According to the continuous hypothesis, the differential equation for Di ðtÞ is qDi ðtÞ Di ðtÞ c2 Di ðtÞ ¼ c2 ¼ , qt NðtÞ c1 c2 t and the boundary condition of the above equation is Di ðiÞ ¼ c1 , so we get tc2 =ðc2 c1 Þ . Di ðtÞ ¼ c1 i
(4)
(5)
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Using the same approach introduced in Ref. [7], with the boundary condition ki ðiÞ ¼ c1 m and Eq. (5), we obtain qSðtÞ c2 SðtÞ c21 m SðtÞ ¼ c1 m c2 . qt tðc1 c2 Þ tðc2 c1 Þ
(6)
The solution to the above equation is SðtÞ ¼ c1 mð1 þ c1 Þ
c1 c2 NðtÞ t ¼ c1 mð1 þ c1 Þ . c1 þ c2 c1 þ c2
(7)
Inserting Eq. (7) back into Eq. (2), we have qki ðtÞ ki ðtÞ ki ðtÞ ¼ c1 m c1 c2 c2 qt c1 mð1 þ c1 Þ c1 þc2 t ðc1 c2 Þt 1 ki ðtÞ c1 þ c2 c2 ¼ c1 c2 t 1 þ c1 c1 ð1 c2 Þ ki ðtÞ , ¼ ð1 þ c1 Þðc1 c2 Þ t
ð8Þ
which implies that ki ðtÞ ¼ c1 m
tb , i
(9)
where b¼
c1 ð1 c2 Þ . ð1 þ c1 Þðc1 c2 Þ
(10)
It is obvious that b is the monotone decreasing function of the addition probability c1 and the monotone increasing function of the deletion probability c2 according to Eq. (10). And when c1 ¼ 1, the value of b ¼ 12 does not depend on the deletion probability. However, when c2 ¼ 0, b ¼ 1=ð1 þ c1 Þ is still correlative with c1 . In other words, Eq. (9) indicates that the degree of a vertex mainly depends on the addition of new vertex at each time with a large probability c1 , which is reasonable and intelligible since a new vertex attaches m new edges. On the other hand, although with probability c2 a random chosen vertex is deleted and its edges are disappeared correspondingly, the degree of a vertex has hardly any change since 0pc2 oc1 . In addition, according to Eqs. (5) and (9), we obtain easily that the expected degree E i ðtÞ of the ith vertex at time t is E i ðtÞ ¼ ki ðtÞDi ðtÞ ¼ ðc1 Þ2 m
tðc1 c2 2c1 c2 Þ=ð1þc1 Þðc1 c2 Þ i
.
(11)
So we find that the critical of deletion probability c2 is c2 ¼
c1 . 1 þ 2c1
(12)
That is, for 0pc2 oc1 =ð1 þ 2c1 Þ, the expected degree of any given vertex E i ðtÞ ! 1, and for c1 =ð1 þ 2c1 Þoc2 oc1 , E i ðtÞ ! 0 when t=i ! 1. Furthermore, when c1 ¼ 1, the analysis results are the same with that of the SR model. Let I k ðtÞ be the set of all vertex i with degree k at time step t and ik be the solution to the equation ki ðtÞ ¼ k according to continuous approach. Following the same method of Ref. [7] to obtain the degree distribution of
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the network, we have ½ðNo: of vertices with degreeÞ ¼ k ðTotal number of verticesÞ qki ðtÞ1 1 X 1 ¼ Dik ðtÞ Di ðtÞ NðtÞ NðtÞ qi
Pk ðtÞ ¼
i2I k ðtÞ
i¼ik
"
#c2 =ðc2 c1 Þ
1 t ¼ ðc1 c2 Þt ðc1 mÞ1=b k1=b t
ðc1 mÞ
1=b 1=b1
k
1 t b
/ k1ðc1 =½ðc1 c2 ÞbÞ ,
ð13Þ
which is a power-law distribution with the exponent c1 . g¼1þ ðc1 c2 Þb
(14)
Inserting Eq. (10) into the above equation, we get g¼1þ
1 þ c1 . 1 c2
(15)
Now we analyze the validity of the exponent of power-law degree distribution for the linear growth model. Note that when c1 ¼ 1 and c2 ¼ 0, the model degenerates the BA model with the exponent g ¼ 3, and when c1 ¼ 1 and c2 ¼ c, the model degenerate the SR model with the exponent g ¼ ð1 þ 2Þ=ð1 cÞ. Therefore, the linear growth model generalizes both the BA model and the SR model, which is coincident with its evolving mechanism. On the other hand, we have made numerical simulations to verify the correctness of the exponent g. Fig. 2 demonstrates that the numerical results approach asymptotically the theoretical predictions. The simulation results (dashed lines) are in perfect agreement with the theory values (solid lines) for cp0:25. However, for larger values of c2 , the exponents of power-law degree distributions of this model have obvious deviations with 4.4 Simulation
4.2
Theory
Power-law exponent γ
4 3.8 3.6 3.4 3.2 3 2.8 2.6 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Deletion probability c2 Fig. 2. The power-law exponent g of degree distribution for the linear growth model with different c1 and c2 . The evolving time steps of all simulations are 1 105 and m ¼ 3. Note that the maximum of the deletion probability c2 is 0.4, the value of the addition probability c1 ranges from 0.6 to 1.0 (c1 ¼ 0:6, circles; c1 ¼ 0:7, asterisk; c1 ¼ 0:8, squares; c1 ¼ 0:9, plus signs and c1 ¼ 1:0, diamonds), which assumes that the network grows with the evolving time, i.e., c1 4c2 .
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the theory values. This phenomenon can be induced by three key reasons. Firstly, the theory value is the limit of NðtÞ ! 1, but we are restricted in computing power to simulate just the finite number of evolving time steps. Secondly, the growth rate will become slowly with a large deletion probability since the number of its edge is NðtÞ ¼ ðc1 c2 Þt. Lastly, as mentioned above, when c2 4c1 =ð1 þ c1 Þ, zero is the limit of the degree of any vertex as t=i approaches 1. Here for c1 ¼ 0:6, the threshold of c2 is 0:6=ð1 þ 1:2Þ 0:273. So when c2 40:25, the power-law exponent of the degree distribution has a great deviation. 3. Accelerated growth model In this section we consider the scaling properties, especially degree distribution, of the complex ad hoc network with accelerated growth. The evolving process of this model is similar with the linear growth model and the only difference between two models is that a new vertex with mðtÞ ¼ ty edges is inserted into the latter model, not m edges as the former model at each time step. That is, the foundational evolving process of the accelerated growth model is as follows. (i) At each time step t, a new vertex with mðtÞ ¼ ty is added into the network with probability c1 , where 0pyp1. (ii) With probability c2 , a random chosen vertex is deleted and its associated edges are cut at the same time. As the same hypotheses and denotations, we get the rate equation of the degree ki ðtÞ qki ðtÞ ki ðtÞ ki ðtÞ ¼ c1 t y c2 . qt SðtÞ NðtÞ
(16)
Since the difference between the accelerated growth model and the linear growth model is the number of new adding edges at each time step, the probability that vertex i exists in the network until time t, Di ðtÞ, does not change in this model. Hence, Eq. (5) holds good in the model. With new boundary condition ki ðiÞ ¼ c1 iy , we get the sum of the degree of all vertices appeared in the network at time t c1 ð1 þ c1 Þðc1 c2 Þ yþ1 t yðc1 c2 Þ þ ðc1 þ c2 Þ c1 ð1 þ c1 Þ ty NðtÞ. ¼ yðc1 c2 Þ þ ðc1 þ c2 Þ
SðtÞ ¼
ð17Þ
Inserting Eq. (17) into Eq. (16), we obtain qki ðtÞ ki ðtÞ ki ðtÞ ¼ c1 ty c1 ð1þc1 Þðc1 c2 Þ c2 yþ1 qt ðc c2 Þt 1 t yðc c Þþðc þc Þ 1
2
1
2
yðc1 c2 Þ þ c1 ð1 c2 Þ ki ðtÞ . ¼ ð1 þ c1 Þðc1 c2 Þ t which implies that tb0 ki ðtÞ ¼ c1 iy , i where b0 ¼
yðc1 c2 Þ þ c1 ð1 c2 Þ . ð1 þ c1 Þðc1 c2 Þ
The expected degree in the accelerated growth model is tðyðc1 c2 Þþc1 c2 2c1 c2 Þ=ð1þc1 Þðc1 c2 Þ E i ðtÞ ¼ ki ðtÞDi ðtÞ ¼ ðc1 Þ2 iy . i Then according to the exponent of i we obtain that the critical of growth coefficient y is y0 ¼
c1 c2 2c1 c2 . c1 ðc1 c2 Þ
ð18Þ
(19)
(20)
(21)
(22)
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Since the exponent of t is different with that of i, it is difficult to fix on the limit of E i ðtÞ when t=i approaches to 1. However, we will find that the critical of y influence the power-law exponent of degree distribution of the model in the following discussion. Adopting the same approach in Section 2 to get the degree distribution of the accelerated growth model, we have 0
Pk ðtÞ / k1ðc1 =ðb yÞðc1 c2 ÞÞ .
(23)
Since 0pc2 oc1 p1, it is easily proved that 1 c2 c1 c2 4 ¼ 14y, c1 c2 c1 c2 then we have b0 y40. Eq. (23) implies that the degree distribution of the model is a power-law distribution with the exponent g¼1þ
c1 2 þ ðc1 c2 Þð1 yÞ ¼ . 0 1 c2 yðc1 c2 Þ ðc1 c2 Þðb yÞ
(24)
In the above expression of g when y ¼ 0, we can find Eq. (24) turns into Eq. (15), which implies that the accelerated growth model extends the linear growth model. Combining the results of Section 2, we can suggest that the accelerated growth model is a universal model relative to the BA model, the SR model and the linear growth model. It can be also shown from the form of the power-law exponent g. For example, when c1 ¼ 1, c2 ¼ 0 and y ¼ 0, the exponent of the accelerated growth model degenerates g ¼ 3, which is consistent with the exponent of the BA model. On the other hand, when y ¼ 1, g ¼ 2=ð1 c1 Þ is not relative to the deletion probability c2 , which proves again that the degree of a vertex mainly depends on the adding vertex with a certain new edges. And in such a case, the network is a fully connected network (an Nclique) in the certain probability sense. We use numerical simulations to verify the validity of the power-law exponent of the degree distribution for this model, which is shown in Fig. 3. 6.5 c1=0.9, c2=0.2 c1=0.8, c2=0.2 c1=0.9, c2=0.1 c1=0.8, c2=0.1
Power-law exponent γ
6
5.5
5
4.5
4
3.5
3 0.1
0.2
0.3
0.4
0.5
0.6
Accelerated growth coefficient θ
Fig. 3. The power-law exponent g of degree distribution for the accelerated growth model with different coefficient y. The evolving time steps of all simulations are 2 104 for yp0:4, and 1 104 for y ¼ 0:5, y ¼ 0:6. Note that when yp0:4 the simulation values (dashed lines) are consistent with the theoretical values (solid lines) calculated by Eq. (24) with the different addition and deletion probabilities. And when y40:4, the simulation values are larger than the limits of the power-law exponents of degree distribution because of the critical of y (Eq. (22)) and the former two reasons in Fig. 2.
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4. Discussions and conclusions We have suggested the complex ad hoc network with accelerated growth, in which model the vertex addition and deletion have a certain randomness at each evolving time step. Since the addition and deletion probabilities are introduced in the evolving process of this model, its topology structure takes on a board adjustability by means of turning in corresponding probabilities. Therefore, the model can win through the three shortcomings of the BA model, which are mentioned in Section 1, and include both the BA model and the SR model as its special cases. Because c1 4c2 is always satisfied through all evolving processes and the new added edges are connected according to the rule of preferential attachment, so the two fundamental mechanisms forming the scale-free structure of a network, growth and preferential attachment, are still presented in the model. So the model has a power-law degree distribution and its exponent extends the exponents of the BA model and the SR model, as good as our analyzes and simulations. The topologies of linear growth model is comparatively researched with that of the accelerated growth model. And the degree distribution of this model also exhibits a scale-free behavior and the power-law exponent always exceeds 2 by Eq. (15). In order to ensure that gp3, which is the range of power-law exponent of most of the real-world networks, it is enough to restrict the deletion probability c2 in the range of ½0; ð1 p ffifficffi 1 Þ=2Þ. According to the critical of c2 obtained in Eq. (12), let ð1 c1 Þ=2 ¼ c1 =ð1 þ c1 Þ, we have c1 ¼ 2 1 0:414. If we suppose that the addition probability c1 40:5, there are three conditions of the topology of linear growth model for the different c2 . (i) 0pc2 oð1 c1 Þ=2. In this case, the exponent g of the model is in the range of ½2; 3 and the theoretical value calculated by Eq. (15) is accurately consistent with the simulated value. Therefore, the case describes the practical situation of most of real-world networks optimally. (ii) ð1 c1 Þ=2oc2 oc1 =ð1 þ c1 Þ. Although the theoretical value is consistent with the simulated value, the exponent is g43 in the case. It is only suitably to some complex networks with the exponents exceeding 3. (iii) c1 =ð1 þ c1 Þoc2 oc1 . Not only the theoretical value of g predicted by Eq. (15) has an obvious deviation with the simulated value, but also the exponent g is in the range of ½3; þ1Þ. Hence, there would be a great error if one makes use of the linear growth model to simulate a practical evolving process. In addition, we get the threshold of the accelerated coefficient y by Eq. (22). And the value of y has a straightforward influence of the power-law exponent of degree distribution of the accelerated growth model and the accuracy of theory results. Therefore, when c1 40:5, 0pc2 oð1 c1 Þ=2 and yoy0 , the accelerated growth model will preferably describe the structures of most of the real-world complex networks, whose model has a scale-free degree distribution with the power-law exponent not exceeding 3. And the model takes on a certain universality to complex evolving network since it includes the BA model, the SR model and the linear growth model as its special cases. When we design a complex network which satisfies a certain property, such as an ad hoc network, a peer-to-peer network, or other complex evolving network, it has some guiding significance and practical values in theory. References [1] [2] [3] [4] [5] [6] [7] [8]
A.L. Baraba´si, R. Albert, Science 286 (1999) 509. S.N. Dorogovtsev, J.F.F. Mendes, cond-mat/0106144 (2001). R. Albert, A.L. Baraba´si, Rev. Mod. Phys. 74 (2002) 47. M.E.J. Newman, SIAM Rev. 45 (2003) 167. A.L. Baraba´si, R. Albert, H. Jeong, Physica A 272 (1999) 173. P. Sen, Phys. Rev. E 69 (2004) 046107. N. Sarshar, V. Roychowdhury, Phys. Rev. E 69 (2004) 026101. S.N. Dorogovtsev, J.F.F. Mendes, Phys. Rev. E 63 (2001) 056125.