Physica A 413 (2014) 230–239
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Physica A journal homepage: www.elsevier.com/locate/physa
Synchronizability in complex ad hoc dynamical networks with accelerated growth Sen Qin ∗ , Xufeng Chen, Weigang Sun, Jingyuan Zhang School of Science, Hangzhou Dianzi University, Hangzhou 310018, China
highlights • We investigate the synchronizability and topological periodicity of the dynamical networks with accelerated growth and ad-hoc property.
• The accelerated parameter is closely related to the synchronizability of the new model, as well as the non-periodicity of network topology.
• Preferential attachment plays a key role to enhance the synchronization of the new model and to easily change its non-periodical topology.
• Topological periodicity has robustness and fragility against random and specific removal of the nodes, respectively.
Article history: Received 17 March 2014 Received in revised form 8 June 2014 Available online 7 July 2014 Keywords: Synchronization Ad hoc network Accelerated growth Robustness
abstract Recent research works have been pursued in connection with network synchronizability under various constraints for different topological structures and evolving mechanisms. However, the fundamental question of how the synchronizability of the networks relates to accelerated growth and ad hoc property in the evolving processes remains underexplored. Here we study the ad hoc dynamical models with accelerated growth where the total number of edges increases faster than linearly with network size. By adopting three attachment mechanisms: random, rewired, and preferential attachment, we investigate the second-largest eigenvalues and the ratios of the extremal eigenvalues of coupled matrices in the accelerated models with different evolving parameters. For the new models, we demonstrate the robustness and fragility of synchronization against random and specific attacks by numerical simulations. We find that accelerated growth represents a convenient tool for improving the synchronizability of an evolving network. Furthermore, we show that not only network synchronization but also topological periodicity has robustness and fragility against random failures and specifical removal of the nodes, respectively. In particular, when ad hoc property is suggested in the evolving networks, we find that the deletion of nodes is easier to change network synchronizability and robustness compared to the addition of nodes. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Complex networks widely abound in the real world. The internet, power grid, citation network, collaboration network, food web are all examples ranging from society, technological field to biological world, which consist of a lot of agents (or
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http://dx.doi.org/10.1016/j.physa.2014.07.009 0378-4371/© 2014 Elsevier B.V. All rights reserved.
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nodes) interacting with each other over those complex networked systems [1–3]. Since the discovery of two important characteristics: small-world  and scale-free properties , many researchers have paid strikingly increasing attention to the studies of complex networks and their extension, such as topological evolution, synchronization, information spreading, and percolation over some particular networks [2,3]. Among the main research issues on complex networks, the combined problems of network evolution and synchronization have received a great deal of attention from various fields of science and engineering in recent years [6–22]. In fact, an important purpose of studying complex networks is to understand the collective behavior of all nodes and edges. This particular behavior is difficult to explain in terms of the simple addition of the individual behavior of each node. Obviously, synchronization is possibly one of the simplest and most frequent behaviors. Salvo of flocks in summer evenings, simultaneous glow of fireflies, synchrony of myocardial cells are examples to show its ubiquitous characteristic . Earlier works on synchronization of coupling networks focus on the completely regular or randomly coupling configuration [7,8]. However, when considering the complexity of network topological structure and evolving mechanism, one can find some significant differences of network synchronization compared with regular or random coupling networks. Ongoing research has recently focused on synchronization with small-world [9–12] and scale-free topological structures [13–15] in complex dynamical networks. And more recently, the understanding of what mechanism controls the synchronization of dynamical networks has uncovered some particular evolving mechanisms ranging from rewired attachment in small-world networks to preferential attachment in scale-free networks. The changes in the network topology and evolution, such as small-world structure , degree correlation , and hub nodes  will distinctly enhance or weaken the network synchronization. In the evolving processes, some representative operations such as the deletion of some nodes  and discontinuous coupling strategy  have an obvious influence on coupling strength and synchronous time. Meanwhile, some research interests focus on the particular evolutionary mechanisms under the constraint of network synchronization. An amazing work in this aspect is to propose a new entangled network model with optimal topology . And network growth under the constraint of synchronization stability has recently been discussed . Since the topological characteristics and dynamical behavior of a complex network are manifested after its topological structure has evolved to a stabilized state, an interesting and significant question is how to synchronize in the whole evolutionary process. In this paper, we propose the ad hoc network models with accelerated growth for analyzing the functional relationships of network synchronization and topological periodicity in terms of acceleration parameter, addition probability of new nodes, and deletion probability of the existing nodes. Moreover, according to three attachment mechanisms of new adding edges: random, rewired, and preferential attachments, we explore the relevant factors of the attachment mechanism affecting network synchronizability. Finally, we discuss the robustness and fragility of network synchronization with the above-mentioned parameters. Those results allow us to draw novel relationships between network synchronization and the evolution mechanism including the adding and deleting rules of the nodes. The organization of the remaining parts is as follows: in Section 2, the synchronizability of a complex dynamical network is defined and the ratio of the extremal eigenvalues of the coupled matrix for this model is suggested. The description of the ad hoc network model with accelerated growth and its corresponding evolving parameters are also given. In Section 3, numerical results about network synchronizability and topological periodicity for the new models with accelerated growth are derived. Additionally, the robustness and fragility of network synchronization with random and specific attacks of the nodes are investigated in Section 4. Finally, Section 5 concludes the whole paper. 2. Ad hoc dynamical network model with accelerated growth 2.1. Synchronizability of complex dynamical networks Suppose that a dynamical network consists of N identical linearly and diffusively coupled nodes. The state equations of the coupled network whose each node has an n-dimensional dynamical system are x˙ i = f (xi ) + c
aij Γ xj ,
i = 1, 2, . . . , N ,
where xi = (xi1 , xi2 , . . . , xin ) ∈ Rn are the state variable of node i and the coupling strength is represented by the constant c > 0 [7,8,14]. We assume that Γ = diag (r1 , r2 , . . . , rn ) ∈ Rn×n with ri = 1 for a particular i and rj = 0 for j ̸= i. The coupling matrix A = (aij ) ∈ RN ×N characterizes the coupling configuration of the network, and the elements of this matrix can be valued by aij = aji = 1 if there is an edge between node i and node j (i ̸= j); otherwise, aij = aji = 0 (i ̸= j). The
diagonal elements of the coupling matrix are aii = −ki = − j=1,j̸=i aij where ki is the degree of node i. Dynamical network (1) is (asymptotically) synchronized if x1 (t ) = x2 (t ) = · · · = xN (t ) = s(t ), as t → ∞, where s(t ) is a solution of an isolated node . Consider a connected network without isolated nodes or clusters; the coupling matrix of the network A = (aij )N ×N is symmetric and irreducible. Let 0 = λ1 > λ2 ≥ λ3 ≥ · · · ≥ λN be the eigenvalues of A. As we know, given the dynamics of an isolated node, the coupling strength c characterizes the synchronizability of the network with respect to a specific coupling configuration A. And a small coupling strength implies a strong network synchronization
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capability. As a simple and effective criterion, Wang and Chen  have proved that its second-largest eigenvalue λ2 of A can characterize the synchronizability of network (1). That is, if c ≥ |d/λ2 |,
the synchronized state of the network is exponentially stable under certain assumptions, where d is the maximal Lyapunov exponent of an individual n-dimensional chaotic dynamical system . Therefore, according to inequality (2), the smaller the second-largest eigenvalue, the stronger the synchronizability of a dynamical network. Further, Farkas et al. have suggested that the eigenvalue spectrum, especially the extremal eigenvalues, of a complex dynamical network contains useful information on the topological structure of the network . To measure the distance of the first eigenvalue from the remaining bulk part of the spectrum normalized by the extension of the rest part, the ratio R=
λ1 − λ2 λ2 − λN
is proposed. It is shown that, compared with the uncorrelated random network, both the small-world network and scalefree network show a lower order of magnitude of R . So the quantity R is appropriate for measuring the periodical level of a complex network. 2.2. A dynamical network model with accelerated growth and ad hoc property Previous research on complex networks, such as regular graph , random network , and small-world network , focuses on topological properties with the fixed network size. Since the scale-free model is proposed to fit perfectly the degree distributions of lots of real-world complex systems , network growth, as one of two important ingredients of forming the special property of scale free (the other is preferential attachment), has attracted much attention over the recent years. The growth of networks is ubiquitous, and the geometric increase of web-pages in the WWW, continuous addition of routers in the Internet, new scientists joining to the scientific collaboration network, are all examples to characterize the significant increasing tendency of network size. However, not all nodes persist in many real-world evolving networks, that is to say, network nodes attached with some edges have the characteristics of aging  or disappear [26,27]. For example, all predators and preys in the food network are constantly aging until death, which is the same as scientists in the scientific collaboration network. The ad hoc network model is generally proposed describing the above-mentioned cases of node adding or removing with the corresponding probabilities, which is a specific network with stochastic entrance and leave of new nodes [28–31]. Presently, according to the features of real-world networks, several ad hoc network models were suggested to investigate the topological structures such as degree distribution, average shortest path length, clustering coefficient, and some dynamical behaviors such as disease transmission and network synchronization [26,32]. A trend of accelerated growth is shown in the evolving processes of real-world complex systems via exploring the numbers of new adding edges. For example, the speed of the interconnection between the existing nodes is faster than the new web-sites in the WWW. And a newly published paper has always been accompanied with lots of references in the scientific citation network. An increasing function of network size is generally introduced to characterize the phenomenon that the total number of edges increases faster than linearly with network size. In particular, we denote by θ (0 ≤ θ ≤ 1) the parameter of accelerating growth, and then a new node with m(t ) edges is added into the network, where m(t ) = N (t )θ and N (t ) is the number of nodes at time t. In fact, linear growth in the scale-free model can be understood as a special case of accelerated growth as θ = 0. Accordingly, recent research works have been pursued in connection with network degree distribution of the network with accelerating growth [29,30], as well as new accelerated mode associated with evolving periods . We assume that a new node with m(t ) = m0 t θ edges is added into the model with addition probability c1 at each time step. Here, the accelerated parameter θ ranges between 0 and 1 and m0 is a small number denoting the initial network size. Then, an existing node is randomly deleted and its associated edges disappear simultaneously with deletion probability c2 (
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Fig. 1. (a) The second-largest eigenvalues λ2 of the coupling matrices of the dynamical network for m0 = m = 3, 5, 7, and accelerated parameter θ = 0.2. The new nodes are preferentially attached to the existing nodes with high degree. To focus on exploring the relationship between λ2 and m, ad hoc property is not introduced in the evolving processes, which means the addition probability c1 = 1.0 and the deletion probability c2 = 0.0. (b) The ratio R of the dynamical network for m0 = m = 3, 5, 7.
3. Synchronizability in new dynamical networks 3.1. Network synchronization with accelerated growth To understand the factors that determine the synchronizability of the complex dynamical network with accelerated growth in terms of the number of original edges, the accelerated parameter θ , and the original configuration, we perform a simulation-based analysis on the second-largest eigenvalues of the network (1). Although the strict criterion expressed by inequality (2) is deduced from a network with fixed size or with equilibrium growth [7,8], it has been adopted to analyze network topological structures and dynamics, such as synchronization-optimal networks  and synchronous preference networks . Since the second-largest eigenvalues of those networks have a faster convergence rate, the criterion is still appropriate to analyze the synchronization of a non-equilibrium network. In the following numerical simulations, the eigenvalues are obtained by averaging the results of 10 runs. Starting with a small number of nodes (m0 ) with random original topology, we set the termination value of the growing process as 3000, which means the network size is N = 3000. For clarity, we take m = m0 and the number of new edges at time step t is m(t ) = mt θ . Fig. 1(a) shows the values of λ2 as functions of N, with m = 3, 5, 7, respectively. For a fixed value of m, it is seen that λ2 is moving downward with the increase of N and asymptotically converges to a negative constant. In particular, the values of λ2 in these three cases substantially decay with the similar rate. This implies that there is no apparent relationship between the synchronized speed of the complex network and the number of new edges, although a large value of m indicates strong synchronization of the network. It has been suggested that the ratio R of a network represents a useful tool for distinguishing the level of correlated periodical topology for different types of complex networks . The smaller the value of R, the higher the level of correlated periodicity of a complex network. As shown in Fig. 1(b), as the network size increases, the values of R in all three cases of networks decay as a power law. And the trends of R correlate with the values of m and it shows an obvious inverse relationship between them. The reason behind the special relationship is the increasing correlated nonperiodical topology in order to rapidly construct a power-law degree distribution as m increases. Hence, despite the fact that a power-law exponent of an accelerated network in the scheme of preferential attachment is independent of the number of original edges [28,29], Fig. 1(b) implies that the formation of a strongly correlated nonperiodical topology is closely related to m. Then, we check the trends of λ2 with three different original configurations (empty graph, complete graph, and random graph), which are shown in Fig. 2 with three gradual declined curves. Clearly, there is no significant difference in the synchronizability of the evolving network for the specific original topology. Fig. 3(a) shows the relative change of λ2 for accelerated parameter θ = 0.1, 0.2, 0.3, 0.4, and 0.5. It is shown that, as the network size increases, the values of λ2 of the networks with all five accelerated parameters decay rapidly. Note also that the values of λ2 decrease as the accelerated parameter θ increases for a fixed N. For example, the values of the second-largest eigenvalues sharply decrease from −3.565 to −146.714 when θ increases from 0.1 to 0.5 for N = 3000. This feature of the networks with accelerated growth is rooted in their extremely inhomogeneous degree distributions. When the accelerated parameter increases, the number of edges in the whole network is increased; correspondingly, the synchronizability of the network is enhanced. This implies that, meanwhile, in order to quickly stabilize a dynamical network with accelerated growth, one can appropriately increase the accelerated parameter. For five different accelerated parameters, we have the values of R of the evolving dynamical networks and their linear fitting lines on the log–log scale, as shown in Fig. 3(b). It is noticed that the values of R of all five accelerated evolving networks perfectly decay as a power law, as N increases. By fitting the data of R with least-squares regression lines, a surprising finding is that the slopes γ of the fitting lines decrease as θ increases, which indicates that γ in such networks only depends on
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Fig. 2. The second-largest eigenvalues of the coupling matrices of the dynamical networks for three original configurations: random graph, complete graph, and empty graph. Here the accelerated parameter is θ = 0.2.
Fig. 3. (a) The second-largest eigenvalues of the coupling matrices of the dynamical networks for five accelerated parameters θ = 0.1, 0.2, 0.3, 0.4, 0.5. The remaining parameters are m0 = m = 3, c1 = 1.0, and c2 = 0.0. Also, the preferential attachment mechanism is adopted to connect the new edges for a new adding node. (b) Log–log plot of the ratios R of the dynamical networks as functions of the network size for θ = 0.1, 0.2, 0.3, 0.4, and 0.5. Five linear fitting lines with least-squares regression are also shown for log(R) = γ log(N ) + C , where γ and C are the slope and intercept of the fitting line, respectively.
the accelerated parameter, independent of all other details of the network topology. This implies that a strongly correlated nonperiodical topology forms rapidly due to the preferential attachment mechanism as new edges are inserted into the whole network in an accelerated scheme, which indirectly affects the synchronizability of the dynamical network. It has been shown that the final topology of an evolving network is sensitive to the attachment mechanism of new edges. Preferential attachment, for example, is an important ingredients of self-organization of a network in a scale-free structure [5,34]; while a small-world network is formed by the rewiring rule with a small probability for a regular network . For an accelerated dynamical network, it is worth asking how the attachment mechanism affects the synchronizability of the network. Therefore, we take the relative change of λ2 in the accelerated (θ = 0.2) networks with preferential attachment, rewired attachment, and uniform attachment for the new edges (Fig. 4(a)). It can be observed that λ2 decays with the increasing network size in the networks with preferential attachment and uniform attachment; however, for the network with rewired attachment, it remains constant. Also, Fig. 4(a) shows that the values of λ2 for the former two attachment mechanisms have slight difference as the network size N increases. This implies that the rewired attachment is inappropriate to enhance the synchronizability of the dynamical network with accelerated growth, comparing to the other two attachments. It is in good agreement with the fact that the rewiring scheme cannot change the periodicity of the network topology (Fig. 4(b)), while a very small number of shortcuts based on this mechanism can severely change the whole network topology. On the other hand, for a small-world network the rewired probability pr is an important indicator of small-world characteristics—highly clustered and small averaging shortest path lengths, and determines the level of the interpolation between a regular lattice and a random network. Fig. 5 shows the influence of the rewired probability pr on the synchronizability and the topological periodicity of the networks with accelerated growth. As can be seen from Fig. 5(a), for a fixed pr = 0.5, the values of λ2 decay very slowly as the network size N increases. It is also noticed in this figure that there is not an abrupt change in the values of λ2 from a low rewired probability (pr = 0.1) to a high one (pr = 0.5). In particular, we have λ2 = −3.566, −2.989, −3.484, −2.586, and -2.847 for pr = 0.1, 0.2, 0.3, 0.4, and 0.5 with a fixed N = 3000, respectively. A similar phenomenon is also found in Fig. 5(b), which means that all five slopes of the least-squares regression lines
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Fig. 4. (a) The second-largest eigenvalues of the coupling matrices of the dynamical networks for three different attachment modes of the new adding edges: preferential attachment (PA), uniform attachment (UA), and rewired attachment (RA) with the probability pr = 0.1. (b) Log–log plot of the ratios R of the dynamical network as functions of the network size for those three schemes of the new adding edges. A clear feature is that the three network types differ significantly in their magnitudes of R. Meanwhile, for the dynamical network with the rewired attachment mechanism, the values of R decay faster than in the other two network types. Thus, one can identify the three main attachment types with the help of the relative change of R in an accelerated evolving network.
Fig. 5. (a) The second-largest eigenvalues of the coupling matrices of the dynamical networks for five different rewired probabilities pr = 0.1, 0.2, 0.3, 0.4, and 0.5. The remaining parameters are m = m0 = 3 and θ = 0.2. (b) Log–log plot of the ratios R of the dynamical network as functions of the network size for five different rewired probabilities pr = 0.1, 0.2, 0.3, 0.4, and 0.5. Correspondingly, the slopes of those fitting lines are shown as γ 1 = −0.800 for pr = 0.1, γ 2 = −0.939 for pr = 0.2 and the rest may be deduced by analogy, and γ 5 = −0.891 for pr = 0.5.
have not a substantial difference as pr increases. Similarly, on the log–log scale the values of R are well fitted by a straight line comparing with the case of the dynamical networks with preferential attachment (Fig. 3(b)). Consequently, it is difficult to dramatically enhance the synchronization of the evolving dynamical network using the rewired attachment of the new adding edges, even if the rewired probability is significantly increased. 3.2. Network synchronizability with ad hoc property To investigate the effect of both accelerated growth and ad hoc property upon the topological structure of a complex network, we have studied the degree distribution of the accelerated evolving network with ad hoc property . It has been shown that the degree distribution of the network still obeys a power-law distribution, and the exponent of the powerlaw distribution is a function of three relative parameters: accelerated parameter θ , addition probability c1 , and deletion probability c2 , i.e., γ = [2 + (c1 − c2 )(1 − θ )]/[1 − c2 − θ (c1 − c2 )]. Here, inspired by interesting theoretical results and numerical findings that the synchronizability of the network can be considerably modified by the rule of accelerated growth (Fig. 3), we are curious to see what happens to the synchronizability of the evolving network with accelerated growth if ad hoc property is introduced. To check it out, a new node with addition probability c1 is inserted into the existing network at each time step, while the attachment mechanism and the other evolving parameters are the same as the aforementioned network models. The values of λ2 of the accelerated networks with ad hoc property are plotted in Fig. 6, where the decay and intermittent features are observed again when preferential attachment is adopted in their evolving processes. For a fixed c1 , there is a significantly declined curve of λ2 as the network size increases. It is in good agreement with the results for the accelerated evolving networks without ad hoc property, which implies that the combination between accelerated growth and preferential
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Fig. 6. The second-largest eigenvalues of the coupling matrices of the dynamical networks for the addition probability c1 = 1.0, 0.9, 0.8, 0.7, and 0.6. The remaining parameters are m = m0 = 3 and θ = 0.2, and preferential attachment is adopted when the new edges are connected with the existing nodes. In order to focus on the impact of growth with addition probability on network synchronization, we take the deletion probability c2 = 0.
Fig. 7. The second-largest eigenvalues of the coupling matrices of the dynamical networks for the deletion probability c2 = 0.0, 0.1, 0.2, and 0.3. The remaining parameters are the same as in Fig. 6. And for the analogical reason we take the addition probability c1 = 1.
attachment dominates the synchronizability of the dynamical network. Also, when the addition probability c1 decreases, it is found that the values of λ2 increase as long as the network size is sufficiently large (for example, N = 3000). That is to say, roughly, a smaller addition probability c1 corresponds to a larger second-largest eigenvalue λ2 , which implies that network synchronizability can be modified by controlling the possibility of the new nodes entered into the network. We next check the synchronizability of the accelerated network under the constraint of the removal of the existing nodes with deletion probability c2 at each time step. The numerical results are plotted in Fig. 7. In contrast with the second-largest eigenvalues observed in the case of the networks with addition probability (Fig. 6), it is seen that the values of λ2 in this case have a more obvious and orderly change. For a fixed N = 3000, when the deletion probability c2 increases from 0 to 0.3, the values of λ2 increase from −10.932 to −3.625. This indicates that the topological structure has deteriorated sharply when more and more nodes are artificially removed, inducing the worse synchronizability of the dynamical network. 4. Robustness and fragility Although great efforts are being made to design the appropriate evolving process and the corresponding attachment rule to enhance the robustness and reduce the fragility of a complex network [35–42], little is known about the effect of accelerated growth and ad hoc property combined against random and specific attacks on the synchronization of the network. We consider the robustness and fragility of synchronization in the accelerated dynamical networks against either random or specific removal of a small fraction f (0 < f ≪ 1) of nodes in the network. The removal of one or more nodes in most cases affects both the global and local properties of the remaining nodes, since it makes some edges disappear and the topology and dynamics of the whole network are consequently changed. It has been shown that the scale-free network displays an unexpected degree of robustness against random attack (or ‘‘error’’) and is extremely vulnerable to specific attack (or ‘‘target attack’’) [35–37]. Generally, to simulate a specific attack one first removes the node with the most degree, and continues selecting and removing nodes in decreasing order of their degrees . Next, we investigate the tolerance of the random and specific attacks for the dynamical networks with accelerated growth. Since a small fraction of nodes in the network are removed, the network size is reduced to N − ⌈fN ⌉. For clarity, let A ∈ RN ×N and Aˆ ∈ R(N −⌈fN ⌉)×(N −⌈fN ⌉) be the coupling matrices of the network with N nodes and the new network after removal of
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Fig. 8. The changes in the second-largest eigenvalues of the coupling matrices against random attack (a) and specific attack (b) for the accelerated parameter θ = 0.1, 0.2, 0.3, 0.4, and 0.5. Here N = 3000, while the remaining parameters are the same as in Fig. 6. And we take the addition probability c1 = 1 and the deletion probability c2 = 0.
Fig. 9. Comparison of the ratios R of the dynamical networks as functions of the percentage of random (a) and specific (b) removal of the nodes for the accelerated parameter θ = 0.1, 0.2, 0.3, 0.4, and 0.5.
⌈fN ⌉ nodes, respectively. Denote λ2 , λ2R and λ2S as the second-largest eigenvalues of A, Aˆ with random attack, Aˆ with specific attack, respectively. It was found that, for a small parameter of accelerated growth (for example, θ = 0.1), the values of λ2R are almost unaffected by the random removal of as high as 5% of the nodes (as shown in Fig. 8(a), λ2R increases from −3.604 to −2.538, which implies there is no significant reduction of the second-largest eigenvalues of the coupling matrices), whereas if the same percentage of the nodes with the first ⌈fN ⌉ maximum degree are removed for specific attack, and the values of λ2S increase from −3.604 to −0.809 (Fig. 8(b)). Moreover, when θ increases fivefold, there is an obvious gap between random and specific attacks for the corresponding second-largest eigenvalues. If nodes are removed in the specific attack mode, λ2S increases faster than that in the random attack mode. Therefore, a dynamical network with accelerated growth is more robust against random attack and is more vulnerable to specific removal when the accelerated parameter increases. Let us denote R2R and R2S as the ratios of extremal eigenvalues for the dynamical network under random failure and specifical removal, respectively. Fig. 9 shows the comparison of R between two attacks for the network with accelerated growth. It was found that, for a fixed θ = 0.5 (similar trends are found for the other accelerated parameters), the values of R2R are almost constant as the percentage of the random removal of the nodes increases from 0 to 5% (Fig. 9(a)). However, the values of R2S have a distinct upward trend as more and more nodes are deliberately removed. Also, the greater the accelerated parameter, the more obvious the increasing trend of R2S (Fig. 9(b)). This implies that specific attack can effectively enhance the non-periodicity of the network with accelerated growth. To better understand the impact of random and specific attacks on the synchronization of the accelerated network using three different attachment mechanisms: preferential attachment, uniform attachment, and rewired attachment, we next check the differences between λ2R and λ2S by numerical simulations. Fig. 10(a) shows that the second-largest eigenvalues λ2R of the coupling matrices for the networks with accelerated growth (θ = 0.2) are functions of the percentage of the nodes with random removal. It is noticed that there is no remarkable fluctuation in the values of λ2R for these three attachment mechanisms. Additionally, the former two network types (the networks with preferential attachment and uniform attachment) differ from the rewired network significantly in their magnitudes of λ2R . These results imply that the three networks have similar robustness against random attack, and then the synchronization of the former two networks is stronger than that of the latter network. When considering the fragility of these networks against specific failure, as shown
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Fig. 10. Comparison of the second-largest eigenvalues λ2 of the coupling matrices as functions of the percentage of random (a) and specific (b) removal of the nodes for preferential attachment (PA), uniform attachment (UA), and rewired attachment (RA) with the probability pr = 0.1, 0.3, and 0.5.
Fig. 11. Comparison of the second-largest eigenvalues λ2 of the coupling matrices as functions of the percentage of random (a) and specific (b) removal of the nodes for ad hoc property.
in Fig. 10(b), the network with rewired attachment is more vulnerable than the other two networks, since the values of λ2S in this network converge rapidly to zero as f increases. Therefore, the attachment mechanism of new edges for the network with accelerated growth has a profound impact on the robustness and fragility of network synchronization. Fig. 11 shows a special relationship between ad hoc property of an accelerated dynamical network and the robustness and fragility of network synchronization. One can observe rather specific features even though all five networks have the same parameter of accelerated growth (θ = 0.2). For a fixed deletion probability c2 = 0 and the percentage of random removal of the nodes f = 0.05, the values of λ2R show very little difference as the addition probability c1 decreases from 1.0 to 0.8. Also, for a fixed c1 = 0.9, λ2R increases apparently as c2 increases from 0 to 0.2 (Fig. 11(a)). On the other hand, for all five networks λ2R shows no fundamental change as f increases. Therefore, it is easier for the removal of the nodes than the entrance of new nodes to weaken the synchronization of the network with accelerated growth. As far as the robustness of network synchronization is concerned, there is no significant difference between these two operations of network nodes. Comparing specifical attack with random attack, when f increases, the values of λ2S have a continuously increasing tendency no matter what the values of c1 and c2 one chooses, as shown in Fig. 11(b). This implies that all networks are vulnerable to specifical removal of the nodes for the dynamical network with accelerated growth and ad hoc property, and then intentional removal of those highest-degree nodes is more likely to cause a collapse of the connected network. 5. Conclusions Although the number of nodes has an increasing trend in the process of network evolution, it does not mean that at each time step a fixed number of new edges connected with the existing nodes. To explore the complex network model with an adjustable exponent of scale-free feature, a feasible way is to suggest the nonlinear growth of the number of new edges, especially accelerated growth. Another reason is that linear growth in many evolving networks is a special case for the network with accelerated growth when the accelerated parameter θ = 0. Hence, we focus on the dynamics of the network with accelerated growth in this paper. We have performed a detailed analysis of the synchronization and the extremal eigenvalues’ ratios for some types of networks with accelerated growth and ad hoc property. Connecting the topological evolving rules of these networks to the eigenvalue spectrum, we have demonstrated that (i) the accelerated parameter is
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closely related to the synchronizability of a dynamical network with accelerated growth, as well as the non-periodicity of network topology; (ii) when a new node with m(t ) edges is inserted into a dynamical network, preferential attachment, not rewired attachment, of new edges plays a key role to enhance the synchronization of the network with accelerated growth and to easily change its non-periodical topology; (iii) network synchronization can be modified by controlling the possibilities of new nodes entered into the network and the existing nodes removed from the whole network. Furthermore, we have presented the robustness and fragility of the synchronizability of the accelerated dynamical networks with respect to random and specific attacks of the nodes. Although the dynamical network with accelerated growth is not separated from the characteristics of ‘‘robust yet fragile’’ of scale-free networks, those adjustable parameters, such as accelerated parameter, addition probability, and deletion probability, contribute to the achievement of the desired network topology with the appropriate synchronizability. Moreover, we have suggested an interesting finding that the periodicity of network topology for a complex network has robustness and fragility against random and specifical removal of the nodes, respectively. These results will improve the understanding of causing the ‘‘Achilles’ heel’’ of complex networks, also, provide useful insight into the relevant dynamical properties of real-world complex systems. Acknowledgments We are grateful to the anonymous reviewers for their valuable suggestions which have led to the improved presentation of this paper. We thank Dr. Ming Fan for his help. This research was supported by the National Natural Science Foundation of China under Grant No. 61203155, the Zhejiang Provincial Natural Science Foundations of China under Grant No. LY13F030016, and the Research Foundation of Education Bureau of Zhejiang Province of China under Grant No. Y201223107. References                                          
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