- Email: [email protected]
A local-world evolving network model
Available online at www.sciencedirect.com
Physica A 328 (2003) 274 – 286
www.elsevier.com/locate/physa
A local-world evolving network model Xiang Lia; b , Guanrong Chenb;∗ a Department
of Automation, Shanghai Jiaotong University, Shanghai 200030, People’s Republic of China b Department of Electronic Engineering, City University of Hong Kong, People’s Republic of China Received 8 January 2003; received in revised form 26 May 2003
Abstract We propose and study a novel evolving network model with the new concept of local-world connectivity, which exists in many physical complex networks. The local-world evolving network model represents a transition between power-law and exponential scaling, while the Barab3asi– Albert scale-free model is only one of its special (limiting) cases. We found that this local-world evolving network model can maintain the robustness of scale-free networks and can improve the network reliance against intentional attacks, which is the inherent fragility of most scale-free networks. c 2003 Elsevier B.V. All rights reserved. Keywords: Local-world; Scale-free; Exponential scaling; Synchronizability; Robustness
1. Introduction What a complex network! While we are confronting the Internet and World Wide Web, which are still expanding in an accelerated growth speed, it is the natural reaction that we usually have. In fact, we live in a sea of complex networks: the telecommunication hub-like network, the aircraft web-link lines, the ;ourished scienti
c 2003 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/S0378-4371(03)00604-6
X. Li, G. Chen / Physica A 328 (2003) 274 – 286
275
in 1960, to model the random-like complexity of various networks [1–3]. Motivated by the signi
276
X. Li, G. Chen / Physica A 328 (2003) 274 – 286
connections with other countries. On the other hand, however, many countries are accelerating their economy cooperation in various regional economy cooperative organizations, such as EU, ASEAN, and NAFTA [20], which indicates the existence of preferential attachment mechanisms within local economy regions of the WTW. Similarly, in the scale-free Internet, the computer network is arranged in a domain-router structure, where a host only has connections with the other hosts in the same domain while a router links other routers on behalf of its domain hosts. Another example in point is the protein family and superfamily, as described by DayhoQ [21], where a family is de
X. Li, G. Chen / Physica A 328 (2003) 274 – 286
277
Fig. 1. Degree distribution P(k) of the Barab3asi–Albert model, with N = 10 000 and m = 1; 3; 5; 7, where the straight lines are the theoretical power-law scaling factors, 2m2 =k 3 ; m = 1; 3; 5; 7, respectively.
After t time steps this procedure results in a network with N = t + m0 nodes and mt edges. Numerical simulations and theoretical analysis based on the continuum theory [9], master-equation [22] and rate-equation [12] approaches, all indicate that this network evolves into a scale-invariant state, for which the probability P(k) that a node has k edges satis
3. The new local-world evolving network model Using Eq. (1), the Barab3asi–Albert scale-free model calculates each node’s preferential attachment probability, yielding an overall network’s average degree value. However, in many real-life networks, owing to the existence of the local-world connectivity discussed above, each node in a network only has local connections therefore only owns local information about the entire network. To model such a local-world eQect, a local-world evolving network model is now proposed, to be generated by the following algorithm: (i) Start with a small number m0 of nodes and small number e0 of edges. (ii) Select M nodes randomly from the existing network, referred to as the “local world” of the new coming node. (iii) Add a new node with m edges, linking to m nodes in its local world determined in (ii), using a preferential attachment with probability local (ki ) de
278
X. Li, G. Chen / Physica A 328 (2003) 274 – 286
time step t by Local (ki ) = (i ∈ Local-world)
ki
j Local
kj
(2)
where (i ∈ Local-world) = M=(m0 + t), and “Local-world ” here refers to all the nodes in interest with respect to the new coming node at time step t. Thus, at every time step t, the newly coming node connects to m nodes, which are selected from its local world with preferential attachment, but does not connect to the whole system as in the Barab3asi–Albert scale-free model. The map from the whole system to a local-world structure can vary, depending on the actual (local) connectivity. Here, we only consider a simple case with random selection. It is obvious that at every time step t; m 6 M 6 m0 + t, and there are two limiting cases in the above-proposed local-world evolving network model: M =m and M =t+m0 , which are further discussed below. Case A: M = m In this limiting case, the new node connects to every node in its local world. It means that the preferential attachment selection is not eQective in the network growing process. This is the same as the case of model A in the Barab3asi–Albert scale-free model [9,11], which keeps the growing manner without preferential attachment. Therefore, the rate of change of the ith node’s degree is 9ki m = : 9t m0 + t
(3)
In Ref. [9], it has been proved that Eq. (3) results in an exponentially decayed degree distribution in the limiting case, as P(k) ∼ e−k=m . Case B: M = t + m0 This limiting case, with M = t + m0 , means that the local world is the same as the whole network, which keeps growing with the time evolution. Hence, the local-world model in this limiting case is exactly the same as the Barab3asi–Albert scale-free model [9], where the rate of change of the ith node’s degree is 9ki ki = 9t 2t
(4)
and the degree distribution follows the power law P(k) ∼ 2m2 =k 3 . From the above two limiting cases, we can see that if M ≈ m, the degree distribution is very close to that of Case A above, with M = m, and it also decays exponentially. While if M ≈ m0 +t, the degree distribution is similar to that of Case B, which follows a power law distribution. Therefore, if m ¡ M ¡ m0 +t, the local-world model represents a transition for the degree distribution between the exponential and the power-law distributions, as further illustrated by Figs. 2–4. In Fig. 4, we can see that by increasing the local-world scale M from 4 to 30 with a
X. Li, G. Chen / Physica A 328 (2003) 274 – 286
279
Fig. 2. Degree distribution comparison in log–log scale of Case A with M = m = 3, and a local-world evolving network with M = 4 and m = 3. Both networks have N = 10 000. The inset is in the log-linear scale of the same curves.
Fig. 3. Degree distribution comparison in the log–log scale, for Case B with M = m0 + t; m = 3, and for a local-world evolving network with M = 30 and m = 3. Both networks have N = 10 000.
as in Fig. 3, the local-world evolving network also has a scale-free feature, similar to the Barab3asi–Albert scale-free network. An explanation based on the continuum theory is as follows. Owing to the random selection of M nodes in establishing a local-world connectivity in the existing network with t + m0 nodes at time step t, the probability that node i
280
X. Li, G. Chen / Physica A 328 (2003) 274 – 286
Fig. 4. Degree distribution comparison in the log–log scale of the local-world evolving networks with M = 4; 10; 30, m = 3, respectively. All networks have N = 10 000.
is selected into the M -node local-world network is (i ∈ Local-world) = M=(t + m0 ). Hence, Eq. (2) can be written as Local (ki ) =
k M i : t + m0 j Local kj
(5)
Assuming that ki is a continuous real variable, as in Ref. [9], the rate change of ki is expected to be proportional to Local (ki ). Hence, ki satis
(6)
Because the random selection of the M nodes contributes to a local world connection at every time step t, the cumulative degree of the local world depends on the random selection. To simplify the following analysis, we assume that kj = ki M ; (7) j Local
where the average degree ki = (2mt + 2e0 )=(m0 + t), with e0 being the number of edges that were initially linked to m0 nodes. Substituting Eq. (7) into Eq. (6) leads to 9ki mM = 9t m0 + t
ki 2(mt+e0 )M m0 +t
=
ki mki ≈ 2(mt + e0 ) 2t
(8)
which means, with the assumption (7), that the distribution follows the same power-law as the Barab3asi–Albert scale-free model, that is, P(k) ∼ 2m2 =k 3 . This reveals the main
X. Li, G. Chen / Physica A 328 (2003) 274 – 286
281
Fig. 5. Heterogeneous characters ki =ki comparison of the local-world evolving networks with M = 3; 10; 30, m = 3, respectively. All networks have N = 3000. The white horizontal line indicates the homogeneous value of ki = ki . The inset shows the heterogeneous characters comparison in a fraction of network nodes.
reason why a local-world evolving network, with mM m0 + t, can also follow a power-law degree distribution with the exponent equal to 3, as shown in Fig. 3. It can be veri
282
X. Li, G. Chen / Physica A 328 (2003) 274 – 286
purposeful attack of node removal. This feature is mainly due to its signi
N
aij xj ;
i = 1; 2; : : : ; N :
(10)
j=1
Here, f(·) is a given nonlinear vector-valued function describing the dynamics of the nodes, xi = (xi1 ; xi2 ; : : : ; xin ) ∈ Rn are the state variables of node i, and the constant coupling strength c is assumed positive. Moreover, ∈ Rn×n is a constant 0 –1 diagonal matrix denoted as = diag(r1 ; r2 ; : : : ; rn ). If there is a connection between node i and node j (j = i), aij =aji =1; and if there is no connection between them, aij =aji =0 (j = i). The degree ki of node i is de
N j=1 j=i
aij = −
N
aji = −ki ;
i = 1; 2; : : : ; N :
(11)
j=1 j=i
Hence, the coupling matrix A = (aij ) ∈ RN ×N represents the coupling con
as t → ∞ ;
(12)
where s(t) ∈ Rn is a solution of an isolate node satisfying s(t) ˙ = f(s(t)). Here, s(t) can be an equilibrium point, a periodic orbit, or even a chaotic attractor. We need the following synchronization stability result from [23].
X. Li, G. Chen / Physica A 328 (2003) 274 – 286
283
Lemma (Wang and Chen [23]): Consider network (10), and suppose that there exist an n × n diagonal matrix D ¿ 0 and two constants dW ¡ 0 and $ ¿ 0, such that [J (s(t)) + d ]T D + D[J (s(t)) + d ] 6 − $In
(13)
W where In ∈ Rn×n is the identity matrix, and J (s(t)) ∈ Rn×n is the Jacofor all d 6 d, bian of f(·) at s(t). If c!2 6 dW
(14)
then the synchronized states (12) of network (10) are exponentially stable. 4.2. Synchronizability against random errors and intentional attacks As usual, random error or intentional attack in a network means random or speci
284
X. Li, G. Chen / Physica A 328 (2003) 274 – 286
Fig. 6. Synchronization robustness comparison of local-world evolving networks as m = 3, and M = 3; 4; 10; 30; m0 + t, respectively. All networks have N = 3000. Each curve in the
robustness and fragility of the local-world evolving network also show a transition between that of the scale-free network and of the exponential network, as shown in Figs. 6 and 7.
5. Conclusions In this paper, we have proposed an evolving network model with local-world connectivity. The concept comes from the phenomena of human’s local world-view, locally collective information, and local relationships in one’s friendship network development. The degree distribution of this new local-world evolving network model represents a transition between that of an exponential network and of a power-law scaling network, and the synchronization robustness and fragility of the local-world network model also display a transition between the exponential and the scale-free ones. To some extent, a local-world evolving network can maintain the robustness with an improved reliance as compared to the severe synchronization fragility of a scale-free network. It should be mentioned that the selection of a local-world connection in a real-life complex network is much more complicated and ;exible, while here we only consider the random selection for the most generic case. This random selection is good enough for the present study, in the sense that it has clearly shown the eQect on scaling
X. Li, G. Chen / Physica A 328 (2003) 274 – 286
285
Fig. 7. Synchronization fragility comparison of local-world evolving networks with m = 3, and M = 3; 4; 10; 30; m0 + t, respectively. All networks have N = 3000. Each curve in the
exponent of complex networks, indicating a transition between power-law and exponential scaling, as discussed in detail above. Some evolving features such as gradual aging [14],
286
X. Li, G. Chen / Physica A 328 (2003) 274 – 286
to use a local-world network model when synchronization is a main concern in the modeling. Acknowledgements This research was supported by the Hong Kong Research Grants Council under the CERG Grants CityU 1018/01E and 1004/02E. The authors would like to thank R. Albert and X.F. Wang for their valuable discussions during the development of this research work. The authors also are grateful to the anonymous reviewers for their valuable comments and suggestions, which have led to a better presentation of this paper. 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] [25] [26] [27] [28] [29] [30]
P. ErdDos, A. R3enyi, Publ. Math. 6 (1959) 290. P. ErdDos, A. R3enyi, Publ. Math. Inst. Hung. Acad. Sci. 5 (1960) 17. M.E.J. Newman, Santa Fe Institute working paper 02-02-005. M.E.J. Newman, J. Statist. Phys. 101 (2000) 819. S.H. Strogatz, Nature 410 (2001) 268. D.J. Watts, Small Worlds, Princeton Univ. Press, Princeton, 1999. D.J. Watts, S.H. Strogatz, Nature 393 (1998) 440. A.L. Barab3asi, R. Albert, Science 285 (1999) 509. A.L. Barab3asi, R. Albert, H. Jeong, Physica A 272 (1999) 173. A.L. Barab3asi, R. Albert, H. Jeong, G. Bianconi, Science 287 (2000) 2115 a. R. Albert, A.L. Barab3asi, Rev. Mod. Phys. 74 (2002) 47. P.L. Krapivsky, S. Redner, F. Leyvraz, Phys. Rev. Lett. 85 (2000) 4629. S.N. Dorogovstsev, J.F.F. Mendes, Phys. Rev. E 63 (2001) 025 101. S.N. Dorogovstsev, J.F.F. Mendes, Phys. Rev. E 62 (2000) 1842. R. Albert, A.L. Barab3asi, Phys. Rev. Lett. 85 (2000) 5234. S.N. Dorogovstsev, J.F.F. Mendes, Europhys. Lett. 52 (2000) 33. G. Bianconi, A.L. Barab3asi, Europhys. Lett. 54 (2001) 436. J. Jost, M.P. Joy, Preprint Cond-mat/0202343. M.A. Serrano, M. Boguna, Preprint Cond-mat/0301015. X. Li, Y.Y. Jin, G. Chen, Physica A 328 (2003) 287. M.O. DayhoQ, Fed. Proc. 35 (1976) 2132. S.N. Dorogovstsev, J.F.F. Mendes, A.N. Samukhim, Phys. Rev. Lett. 85 (2000) 4633. X.F. Wang, G. Chen, IEEE Trans. Circuits Syst. I 49 (2002) 54. R.M. May, A.L. Lloyd, Phys. Rev. E 64 (2001) 066 112. D. Volchenkov, L. Volchenkov, Ph. Blanchard, Phys. Rev. E 66 (2002) 046 137. R. Albert, H. Jeong, A.L. Barab3asi, Nature 406 (2000) 378. C.W. Wu, L.O. Chua, IEEE Trans. Circuits Syst. I 42 (1995) 430. E.V. Koonin, Y.I. Wolf, G.P. Karev, Nature 420 (2002) 218. S.N. Dorogovstsev, J.F.F. Mendes, Preprint Cond-mat/0105093. H.J. Kim, I.M. Kim, J. Korean Phys. Soc. 40 (2002) 1105.
Physica A 328 (2003) 274 – 286
www.elsevier.com/locate/physa
A local-world evolving network model Xiang Lia; b , Guanrong Chenb;∗ a Department
of Automation, Shanghai Jiaotong University, Shanghai 200030, People’s Republic of China b Department of Electronic Engineering, City University of Hong Kong, People’s Republic of China Received 8 January 2003; received in revised form 26 May 2003
Abstract We propose and study a novel evolving network model with the new concept of local-world connectivity, which exists in many physical complex networks. The local-world evolving network model represents a transition between power-law and exponential scaling, while the Barab3asi– Albert scale-free model is only one of its special (limiting) cases. We found that this local-world evolving network model can maintain the robustness of scale-free networks and can improve the network reliance against intentional attacks, which is the inherent fragility of most scale-free networks. c 2003 Elsevier B.V. All rights reserved. Keywords: Local-world; Scale-free; Exponential scaling; Synchronizability; Robustness
1. Introduction What a complex network! While we are confronting the Internet and World Wide Web, which are still expanding in an accelerated growth speed, it is the natural reaction that we usually have. In fact, we live in a sea of complex networks: the telecommunication hub-like network, the aircraft web-link lines, the ;ourished scienti
c 2003 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/S0378-4371(03)00604-6
X. Li, G. Chen / Physica A 328 (2003) 274 – 286
275
in 1960, to model the random-like complexity of various networks [1–3]. Motivated by the signi
276
X. Li, G. Chen / Physica A 328 (2003) 274 – 286
connections with other countries. On the other hand, however, many countries are accelerating their economy cooperation in various regional economy cooperative organizations, such as EU, ASEAN, and NAFTA [20], which indicates the existence of preferential attachment mechanisms within local economy regions of the WTW. Similarly, in the scale-free Internet, the computer network is arranged in a domain-router structure, where a host only has connections with the other hosts in the same domain while a router links other routers on behalf of its domain hosts. Another example in point is the protein family and superfamily, as described by DayhoQ [21], where a family is de
X. Li, G. Chen / Physica A 328 (2003) 274 – 286
277
Fig. 1. Degree distribution P(k) of the Barab3asi–Albert model, with N = 10 000 and m = 1; 3; 5; 7, where the straight lines are the theoretical power-law scaling factors, 2m2 =k 3 ; m = 1; 3; 5; 7, respectively.
After t time steps this procedure results in a network with N = t + m0 nodes and mt edges. Numerical simulations and theoretical analysis based on the continuum theory [9], master-equation [22] and rate-equation [12] approaches, all indicate that this network evolves into a scale-invariant state, for which the probability P(k) that a node has k edges satis
3. The new local-world evolving network model Using Eq. (1), the Barab3asi–Albert scale-free model calculates each node’s preferential attachment probability, yielding an overall network’s average degree value. However, in many real-life networks, owing to the existence of the local-world connectivity discussed above, each node in a network only has local connections therefore only owns local information about the entire network. To model such a local-world eQect, a local-world evolving network model is now proposed, to be generated by the following algorithm: (i) Start with a small number m0 of nodes and small number e0 of edges. (ii) Select M nodes randomly from the existing network, referred to as the “local world” of the new coming node. (iii) Add a new node with m edges, linking to m nodes in its local world determined in (ii), using a preferential attachment with probability local (ki ) de
278
X. Li, G. Chen / Physica A 328 (2003) 274 – 286
time step t by Local (ki ) = (i ∈ Local-world)
ki
j Local
kj
(2)
where (i ∈ Local-world) = M=(m0 + t), and “Local-world ” here refers to all the nodes in interest with respect to the new coming node at time step t. Thus, at every time step t, the newly coming node connects to m nodes, which are selected from its local world with preferential attachment, but does not connect to the whole system as in the Barab3asi–Albert scale-free model. The map from the whole system to a local-world structure can vary, depending on the actual (local) connectivity. Here, we only consider a simple case with random selection. It is obvious that at every time step t; m 6 M 6 m0 + t, and there are two limiting cases in the above-proposed local-world evolving network model: M =m and M =t+m0 , which are further discussed below. Case A: M = m In this limiting case, the new node connects to every node in its local world. It means that the preferential attachment selection is not eQective in the network growing process. This is the same as the case of model A in the Barab3asi–Albert scale-free model [9,11], which keeps the growing manner without preferential attachment. Therefore, the rate of change of the ith node’s degree is 9ki m = : 9t m0 + t
(3)
In Ref. [9], it has been proved that Eq. (3) results in an exponentially decayed degree distribution in the limiting case, as P(k) ∼ e−k=m . Case B: M = t + m0 This limiting case, with M = t + m0 , means that the local world is the same as the whole network, which keeps growing with the time evolution. Hence, the local-world model in this limiting case is exactly the same as the Barab3asi–Albert scale-free model [9], where the rate of change of the ith node’s degree is 9ki ki = 9t 2t
(4)
and the degree distribution follows the power law P(k) ∼ 2m2 =k 3 . From the above two limiting cases, we can see that if M ≈ m, the degree distribution is very close to that of Case A above, with M = m, and it also decays exponentially. While if M ≈ m0 +t, the degree distribution is similar to that of Case B, which follows a power law distribution. Therefore, if m ¡ M ¡ m0 +t, the local-world model represents a transition for the degree distribution between the exponential and the power-law distributions, as further illustrated by Figs. 2–4. In Fig. 4, we can see that by increasing the local-world scale M from 4 to 30 with a
X. Li, G. Chen / Physica A 328 (2003) 274 – 286
279
Fig. 2. Degree distribution comparison in log–log scale of Case A with M = m = 3, and a local-world evolving network with M = 4 and m = 3. Both networks have N = 10 000. The inset is in the log-linear scale of the same curves.
Fig. 3. Degree distribution comparison in the log–log scale, for Case B with M = m0 + t; m = 3, and for a local-world evolving network with M = 30 and m = 3. Both networks have N = 10 000.
as in Fig. 3, the local-world evolving network also has a scale-free feature, similar to the Barab3asi–Albert scale-free network. An explanation based on the continuum theory is as follows. Owing to the random selection of M nodes in establishing a local-world connectivity in the existing network with t + m0 nodes at time step t, the probability that node i
280
X. Li, G. Chen / Physica A 328 (2003) 274 – 286
Fig. 4. Degree distribution comparison in the log–log scale of the local-world evolving networks with M = 4; 10; 30, m = 3, respectively. All networks have N = 10 000.
is selected into the M -node local-world network is (i ∈ Local-world) = M=(t + m0 ). Hence, Eq. (2) can be written as Local (ki ) =
k M i : t + m0 j Local kj
(5)
Assuming that ki is a continuous real variable, as in Ref. [9], the rate change of ki is expected to be proportional to Local (ki ). Hence, ki satis
(6)
Because the random selection of the M nodes contributes to a local world connection at every time step t, the cumulative degree of the local world depends on the random selection. To simplify the following analysis, we assume that kj = ki M ; (7) j Local
where the average degree ki = (2mt + 2e0 )=(m0 + t), with e0 being the number of edges that were initially linked to m0 nodes. Substituting Eq. (7) into Eq. (6) leads to 9ki mM = 9t m0 + t
ki 2(mt+e0 )M m0 +t
=
ki mki ≈ 2(mt + e0 ) 2t
(8)
which means, with the assumption (7), that the distribution follows the same power-law as the Barab3asi–Albert scale-free model, that is, P(k) ∼ 2m2 =k 3 . This reveals the main
X. Li, G. Chen / Physica A 328 (2003) 274 – 286
281
Fig. 5. Heterogeneous characters ki =ki comparison of the local-world evolving networks with M = 3; 10; 30, m = 3, respectively. All networks have N = 3000. The white horizontal line indicates the homogeneous value of ki = ki . The inset shows the heterogeneous characters comparison in a fraction of network nodes.
reason why a local-world evolving network, with mM m0 + t, can also follow a power-law degree distribution with the exponent equal to 3, as shown in Fig. 3. It can be veri
282
X. Li, G. Chen / Physica A 328 (2003) 274 – 286
purposeful attack of node removal. This feature is mainly due to its signi
N
aij xj ;
i = 1; 2; : : : ; N :
(10)
j=1
Here, f(·) is a given nonlinear vector-valued function describing the dynamics of the nodes, xi = (xi1 ; xi2 ; : : : ; xin ) ∈ Rn are the state variables of node i, and the constant coupling strength c is assumed positive. Moreover, ∈ Rn×n is a constant 0 –1 diagonal matrix denoted as = diag(r1 ; r2 ; : : : ; rn ). If there is a connection between node i and node j (j = i), aij =aji =1; and if there is no connection between them, aij =aji =0 (j = i). The degree ki of node i is de
N j=1 j=i
aij = −
N
aji = −ki ;
i = 1; 2; : : : ; N :
(11)
j=1 j=i
Hence, the coupling matrix A = (aij ) ∈ RN ×N represents the coupling con
as t → ∞ ;
(12)
where s(t) ∈ Rn is a solution of an isolate node satisfying s(t) ˙ = f(s(t)). Here, s(t) can be an equilibrium point, a periodic orbit, or even a chaotic attractor. We need the following synchronization stability result from [23].
X. Li, G. Chen / Physica A 328 (2003) 274 – 286
283
Lemma (Wang and Chen [23]): Consider network (10), and suppose that there exist an n × n diagonal matrix D ¿ 0 and two constants dW ¡ 0 and $ ¿ 0, such that [J (s(t)) + d ]T D + D[J (s(t)) + d ] 6 − $In
(13)
W where In ∈ Rn×n is the identity matrix, and J (s(t)) ∈ Rn×n is the Jacofor all d 6 d, bian of f(·) at s(t). If c!2 6 dW
(14)
then the synchronized states (12) of network (10) are exponentially stable. 4.2. Synchronizability against random errors and intentional attacks As usual, random error or intentional attack in a network means random or speci
284
X. Li, G. Chen / Physica A 328 (2003) 274 – 286
Fig. 6. Synchronization robustness comparison of local-world evolving networks as m = 3, and M = 3; 4; 10; 30; m0 + t, respectively. All networks have N = 3000. Each curve in the
robustness and fragility of the local-world evolving network also show a transition between that of the scale-free network and of the exponential network, as shown in Figs. 6 and 7.
5. Conclusions In this paper, we have proposed an evolving network model with local-world connectivity. The concept comes from the phenomena of human’s local world-view, locally collective information, and local relationships in one’s friendship network development. The degree distribution of this new local-world evolving network model represents a transition between that of an exponential network and of a power-law scaling network, and the synchronization robustness and fragility of the local-world network model also display a transition between the exponential and the scale-free ones. To some extent, a local-world evolving network can maintain the robustness with an improved reliance as compared to the severe synchronization fragility of a scale-free network. It should be mentioned that the selection of a local-world connection in a real-life complex network is much more complicated and ;exible, while here we only consider the random selection for the most generic case. This random selection is good enough for the present study, in the sense that it has clearly shown the eQect on scaling
X. Li, G. Chen / Physica A 328 (2003) 274 – 286
285
Fig. 7. Synchronization fragility comparison of local-world evolving networks with m = 3, and M = 3; 4; 10; 30; m0 + t, respectively. All networks have N = 3000. Each curve in the
exponent of complex networks, indicating a transition between power-law and exponential scaling, as discussed in detail above. Some evolving features such as gradual aging [14],
286
X. Li, G. Chen / Physica A 328 (2003) 274 – 286
to use a local-world network model when synchronization is a main concern in the modeling. Acknowledgements This research was supported by the Hong Kong Research Grants Council under the CERG Grants CityU 1018/01E and 1004/02E. The authors would like to thank R. Albert and X.F. Wang for their valuable discussions during the development of this research work. The authors also are grateful to the anonymous reviewers for their valuable comments and suggestions, which have led to a better presentation of this paper. 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] [25] [26] [27] [28] [29] [30]
P. ErdDos, A. R3enyi, Publ. Math. 6 (1959) 290. P. ErdDos, A. R3enyi, Publ. Math. Inst. Hung. Acad. Sci. 5 (1960) 17. M.E.J. Newman, Santa Fe Institute working paper 02-02-005. M.E.J. Newman, J. Statist. Phys. 101 (2000) 819. S.H. Strogatz, Nature 410 (2001) 268. D.J. Watts, Small Worlds, Princeton Univ. Press, Princeton, 1999. D.J. Watts, S.H. Strogatz, Nature 393 (1998) 440. A.L. Barab3asi, R. Albert, Science 285 (1999) 509. A.L. Barab3asi, R. Albert, H. Jeong, Physica A 272 (1999) 173. A.L. Barab3asi, R. Albert, H. Jeong, G. Bianconi, Science 287 (2000) 2115 a. R. Albert, A.L. Barab3asi, Rev. Mod. Phys. 74 (2002) 47. P.L. Krapivsky, S. Redner, F. Leyvraz, Phys. Rev. Lett. 85 (2000) 4629. S.N. Dorogovstsev, J.F.F. Mendes, Phys. Rev. E 63 (2001) 025 101. S.N. Dorogovstsev, J.F.F. Mendes, Phys. Rev. E 62 (2000) 1842. R. Albert, A.L. Barab3asi, Phys. Rev. Lett. 85 (2000) 5234. S.N. Dorogovstsev, J.F.F. Mendes, Europhys. Lett. 52 (2000) 33. G. Bianconi, A.L. Barab3asi, Europhys. Lett. 54 (2001) 436. J. Jost, M.P. Joy, Preprint Cond-mat/0202343. M.A. Serrano, M. Boguna, Preprint Cond-mat/0301015. X. Li, Y.Y. Jin, G. Chen, Physica A 328 (2003) 287. M.O. DayhoQ, Fed. Proc. 35 (1976) 2132. S.N. Dorogovstsev, J.F.F. Mendes, A.N. Samukhim, Phys. Rev. Lett. 85 (2000) 4633. X.F. Wang, G. Chen, IEEE Trans. Circuits Syst. I 49 (2002) 54. R.M. May, A.L. Lloyd, Phys. Rev. E 64 (2001) 066 112. D. Volchenkov, L. Volchenkov, Ph. Blanchard, Phys. Rev. E 66 (2002) 046 137. R. Albert, H. Jeong, A.L. Barab3asi, Nature 406 (2000) 378. C.W. Wu, L.O. Chua, IEEE Trans. Circuits Syst. I 42 (1995) 430. E.V. Koonin, Y.I. Wolf, G.P. Karev, Nature 420 (2002) 218. S.N. Dorogovstsev, J.F.F. Mendes, Preprint Cond-mat/0105093. H.J. Kim, I.M. Kim, J. Korean Phys. Soc. 40 (2002) 1105.