Physica A 338 (2004) 367–378www.elsevier.com/locate/physa

Transition to chaos in complexdynamical networks

Xiang Lia;b;∗, Guanrong Chenb, King-Tim KobaDepartment of Automation, Shanghai Jiaotong University, Shanghai 200030,

People’s Republic of ChinabDepartment of Electronic Engineering, City University of Hong Kong, Hong Kong S.A.R.,

People’s Republic of China

Received 14 October 2003; received in revised form 22 December 2003

Abstract

The transition from a non-chaotic state to a chaotic state is a commonly concerned issuein the study of coupled dynamical networks. In this work, we consider a network consistingof nodes that are in non-chaotic states with parameters in non-chaotic regions before they arecoupled together. We show that if these non-chaotic nodes are linked together through a suitablestructural topology, positive Lyapunov exponents of the coupled network can be generated bychoosing a certain uniform coupling strength, and the threshold for this coupling strength isdetermined by the complexity of the network topology. Moreover, we show that topologicale6ects of scale-free and random networks, which are two basic types of complex network models,can be visualized based on their topological sensitivity to random failures and intentional attacks.Our simulation results on a 1000-node scale-free network and a 1000-node random network ofthe Logistic maps have veri8ed that, during the transition from non-chaotic to chaotic states, ifthe topology is more heterogenous then the coupling strength required to achieve the transitioncan be decreased.c© 2004 Elsevier B.V. All rights reserved.

PACS: 05.40.+j; 05.45.+b; 05.20.−y

Keywords: Coupled network; State transition; Scale-free network; Random graph; Topological sensitivity;Logistic map; Lyapunov exponent

∗ Corresponding author. Department of Automation, Shanghai Jiaotong University, Shanghai 200030,China.

E-mail address: xli@sjtu.edu.hk (X. Li).

0378-4371/$ - see front matter c© 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2004.02.010

368 X. Li et al. / Physica A 338 (2004) 367–378

1. Introduction

Complex dynamical behaviors of various networks have been a focal point ofresearch for more than two decades. Generally speaking, a concerned issue in this8eld is about the dynamics of coupled systems, under the assumption that all individ-ual systems (here, regarded as nodes) are connected with a regular network topology,such as a chain, a ring, a lattice, etc. Typical examples are coupled map lattices (CML)[1] and cellular neural networks (CNN) [2]. In addition to synchronization and controlof (chaotic) dynamical networks, hyperchaos and spatiotemporal chaos in dynamicalnetworks have been extensively studied [3–10, and some references cited therein].These investigations, however, seem to indicate that the focus is only on the complex-ity from the local nonlinear dynamics of each single node, leaving aside the importantconsideration of complexity from the global network topology [11].Recently, with the increasing computation power, research emphasis in this area

has seen a dramatic shift from the understanding of local and individual dynamicalcomplexity to network topological complexity. Some new concepts such as small-worldnetworks [12] and scale-free networks [13,14] have been developed, which evolvesbeyond the now-classical random graph theory that has dominated the network studiesfor more than 40 years [15]. This is motivated particularly by the existence of someuniversal small-world and scale-free features in real-life large-scale networks, such asthe Internet, the World Wide Web, the World Trade Web, food webs, linguistic webs,and some economic networks etc. [13,16–22, and some references given therein]. Alongthis line of research, many new dynamical characteristics have been discovered insynchronization and control of various complex small-world and scale-free dynamicalnetworks [19,21,23–27].A coupled dynamical network can be regarded as a high-dimensional dynamical sys-

tem. In the current literature, majority of research works have been concerning withthe transition from chaos to hyper-chaos, or vice versa, in a chain or a lattice ofdynamical nodes. In Refs. [5,6], for example, hyperchaos was experimentally ob-served in unidirectionally coupled chains of Chua’s circuits with a weak couplingstrength, showing that symmetry-increasing bifurcation was a predictor of such a chaos-hyperchaos transition [28]. In Refs. [29,30], a possible bifurcation route to hyperchaoswas discovered. Other dynamical behaviors such as unstable dimension variability incoupled chaotic systems were also studied [31,32].How does the network topology a6ect the dynamical transition of a network, from

chaos to hyperchaos, or, to our current interest, from regularity to chaos, is essentiallyan unexplored subject. In this letter, we present our study of the topological e6ectson the dynamic behaviors of a coupled complex network. In our study, we selecttwo typical types of complex network models for comparison: the ErdLos–RMenyi (E–R)random networks and the BarabMasi–Albert (B–A) scale-free networks. We show thateven when the parameters of the nodes are not in chaotic regions, a coupled large-scalenetwork can exhibit chaotic behaviors with a suitable network topology. Of particularinterest in our 8nding is that, during the nonchaos–chaos transition in such a network,the more heterogenous the network topology is, the weaker the coupling strength isneeded.

X. Li et al. / Physica A 338 (2004) 367–378 369

The remainder of this paper is organized as follows: The relationship of Lyapunovexponents in an individual node and those in the coupled network of such (identical)nodes is 8rst discussed in Section 2. The main results of this paper are presented inSection 3, where the e6ects of the network topology as well as their sensitivities torandom failures and intentional attacks are visualized for both scale-free and randomnetworks. A scale-free network and a random network of Logistic maps are used asexamples for simulation in Section 4, where the main observation is that, during thenonchaos–chaos transition, if the network topology is more heterogenous, then thecoupling strength needed to achieve the transition can be decreased. Finally, Section 5concludes the investigation.

2. Coupled dynamical networks: Chaos, Lyapunov exponents, and the couplingstrength

Consider a coupled dynamical network of N linearly and di6usively coupled identi-cal nodes, with a full and diagonal coupling, where each node is an n-dimensional dy-namical system [7,9,10,18,24–27]. The state equations of this network are speci8ed by

xi(k + 1) = f(xi(k))− cN∑

j=1

aijf(xj(k)); i = 1; 2; : : : ; N : (1)

If there is a connection between node i and node j (i �= j), then aij = aji = 1;otherwise, aij = aji = 0 (i �= j). Let aii = −di, i = 1; 2; : : : ; N , where the degree di ofnode i is de8ned to be the number of its outreaching connections, i.e.,

N∑

j=1j �=i

aij =N∑

j=1j �=i

aji = di; i = 1; 2; : : : ; N : (2)

Hence, the de8ned coupling matrix A=(aij)∈RN×N represents the coupling con8gura-tion of the entire network. Assume that A is a symmetric and irreducible matrix, whichmeans that the network is fully connected in the sense of having no isolate clusters.Based on these assumptions on the coupling matrix A, it can be veri8ed [9,10] thatzero is the largest eigenvalue, denoted as �1, of the coupling matrix A with multiplicity1, and all the other eigenvalues �2¿ · · ·¿ �N are strictly negative.The function f(·) in network (1) is a given nonlinear vector-valued map, describing

the dynamics of a node, and xi(k)=(xi1(k); xi2(k); : : : ; xin(k))∈Rn are the state variablesof node i. For simplicity, we let n=1 in this discussion, and assume that the constantcoupling strength c is positive. Furthermore, we assume that the parameters of all these(identical) one-dimensional maps are not located in chaotic regions. Consequently, theleading Lyapunov exponent h0 is non-positive or even strictly negative.Rewrite the coupled network (1) in a vector form as an N -dimensional discrete-time

map

x(k + 1) = F(x(k); c) ;

370 X. Li et al. / Physica A 338 (2004) 367–378

where x(k) = [x1(k); x2(k); : : : ; xN (k)]T ∈RN . Recall that Lyapunov exponents of anN -dimensional map are de8ned by Refs. [33,34]

�i = limp→∞

1pln |DFp(x0) · ui|; i = 1; 2; : : : ; N ; (3)

where DFp(x0) is the Jacobian matrix of the p-time iterated map starting from arandom initial state x0, and ui (i = 1; 2; : : : ; N ) is a set of orthonormal vectors in thetangent space of the map.For studying complex dynamical behaviors in nonlinear systems, Lyapunov exponent

is an important tool [4,25,35–37]. For example, transversal Lyapunov exponent (TLE)[4,7,25] helps characterize the synchronization manifold. In Ref. [36], the relation-ship between the Lyapunov-exponent spectra of di6usively coupled one-dimensionalmaps and the spectrum of the discrete SchrLodinger operator was discussed, which ledto a conclusion that when the coupling strength is larger than a critical value, theLyapunov-exponent spectrum extends to −∞. In this letter, we are interested in therelationship between the Lyapunov exponent of an individual node, h0, and the Lya-punov exponents of the coupled dynamical network (1), �i; i=1; 2; : : : ; N . To calculate�i, we di6erentiate network (1) and then evaluate the resulting derivatives at a randominitial condition x0:

�xi(k + 1) = f′(xi(0)) �xi(k)− cN∑

j=1

aijf′(xj(0))�xj(k); i = 1; 2; : : : ; N (4)

which, similar to Refs. [4,34], yields

�i(�i) = h0 + ln |1− c�i|; i = 1; 2; : : : ; N : (5)

Due to the ordering of the eigenvalues of the coupling matrix A, 0=�1¿�2¿ · · ·¿�N , and because the coupling strength c is positive, we can order the Lyapunov expo-nents �i as follows:

�N (�N ) = h0 + ln |1− c�N |¿ �N−1(�N−1)¿ · · ·¿ �2(�2)¿�1(�1) = h0(¡ 0) :(6)

In general, if the coupled network (1) is chaotic, then there is at least one positiveLyapunov exponent in Eq. (6), so �N ¿ 0.As mentioned above, our main interest here is the e6ects of the network topology

on the network dynamical behaviors. We consider the situation where the parametersof all the coupled nodes are in non-chaotic regions. Hence, we may simply assumethat every non-chaotic node, described by the map xi(k + 1) = f(xi(k)), is ‘active’ insuch a way that, with some proper coupling strength c¿ 0, the Lyapunov exponentsof coupled network (1) satisfy

�N ¿ �N−1¿ · · ·¿ �T+1¿ 0 ; (7)

h0 = �1¡�26 · · ·6 �T ¡ 0 ; (8)

where T (16T6N − 1) is an positive integer.

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Combining Eqs. (5)–(8) together, we obtain the following conditions for the positivecoupling strength c that can generate at least one positive Lyapunov exponent in thecoupled network (1)

c1 =e−h0 − 1|�N | ¡c¡

e−h0 − 1|�T | = c2 : (9)

3. Complex network topology versus coupling strength

3.1. Topological e5ects

Eq. (9) indicates a relation between the coupling strength and the Lyapunovexponents in the coupled network (1). It is clear that the network topology determinesthe eigenvalues of the network coupling matrix A, while these eigenvalues determinethe coupling thresholds c1 and c2, and 8nally this coupling strength leads to chaos inthe coupled network. Hence, di6erent network topologies have di6erent e6ects on thechaotic behaviors of the coupled networks.Consider the network topology issue in two basic complex network models: the

E–R random network and the B–A scale-free network models (for detailed descriptionsof these models, see recent reviews [13,20,21]). There are some signi8cant di6erencesbetween an E–R random network and a B–A scale-free network. The former has aPoisson connectivity distribution and is homogenous in nature: each node in such anetwork has about the same number of connections. On the contrary, the B–A scale-freenetwork shows a power-law connectivity distribution, with heterogeity mainly due toits non-homogenous topology, i.e., most nodes have very few connections but a fewnodes have many connections.The di6erent connectivity distributions between these two types of complex network

models yield di6erent spectral densities �(�), de8ned as a sum of some � functions

�(�): =1N

N∑

j=1

�(�− �j) : (10)

It should be noted that the spectral density (10) is de8ned for the coupling matrix A,and is a little di6erent from the one generally used, i.e.,

�( Q�): =1N

N∑

j=1

�(�− Q�j) ;

where Q�j is the jth largest eigenvalue of the network adjacent matrix QA=( Qaij)∈RN×N ,which is slightly di6erent from the aforementioned coupling matrix A, and is de8nedby

Qaij = aij; i �= j ;Qaij = 0; i = j :

372 X. Li et al. / Physica A 338 (2004) 367–378

In Ref. [38], the spectral density �( Q�) of the B–A scale-free network is shown tobe in a triangle-like form, with a power-law tail, and extremely far from being similarto the semicircular spectral density of the E–R random networks.If we assume T = 2 in Eqs. (7), (8), then Eq. (9) becomes

c1 =e−h0 − 1|�N | ¡c¡

e−h0 − 1|�2| = c2 (11)

and, therefore,

c2 − c1c1

=|�N | − |�2|

|�2| =1R; (12)

where R := (�1 − �2)=(�2 − �N ), which measures the distance from the 8rst eigenvalueto the rest part of the spectral density �(�) normalized by the extension of the restpart.Fig. 1 is a comparison of the R values between scale-free networks and random

networks, which are computed based on the network sizes N , ranging from 100 to3000. It can be observed that as the network size N increases, the R value of scale-freenetworks decreases drastically, which is also much less than that of random networksin the same size with the same number of connections. Hence, (c2−c1)=c1 =1=R spanswider for scale-free networks than that for random networks.In addition to the di6erent values of the spectral density �(�) and R, we concern

even more about the eigenvalue �N , because it a6ects the coupling strength thresholdc1 in generating the 8rst positive Lyapunov exponent in the coupled network (1).

Fig. 1. Comparison of the R values between scale-free and random networks: Scale-free networks are gen-erated by the B–A model with m = m0 = 2; 3; 4; 5 (solid lines with circles, stars, squares, and diamonds),respectively; random networks are generated by the E–R model with the same number of links (connections)and the same network sizes (dashed lines with circles, stars, squares, and diamonds, respectively). Everycurve in the 8gure is the average result of 10 groups of networks.

X. Li et al. / Physica A 338 (2004) 367–378 373

Fig. 2. Comparison of the ratio crn1 =csf1 =�sfN =�

rnN between scale-free networks and random networks: Scale-free

networks are generated by the B–A model with m = m0 = 1; 2; 3; 4; 5, respectively; random networks aregenerated by the E–R model with the same number of links (connections) and the same network sizes.Every curve in the 8gure is the average result of 10 groups of networks.

For a comparison, we visualize the ratio crn1 =csf1 = �sfN =�

rnN in Fig. 2, where �sfN is the

maximum eigenvalue of a B–A scale-free network, N=3000, with m=m0=1; 2; 3; 4; 5,respectively, and �rnN is the maximum eigenvalue of a corresponding random networkwith the same numbers of nodes and links (connections). It can be observed that theratio crn1 =c

sf1 = �sfN =�

rnN increases as the network size increases.

As can be seen from the comparison between scale-free networks and random net-works, the coupling strength c for generating the 8rst positive Lyapunov exponentof the coupled network (1) is a6ected drastically by the network topology. Generallyspeaking, homogenous networks, such as random networks, require a much larger cou-pling strength threshold c1 than that of non-homogenous networks, such as scale-freenetworks. Hence, for a nonchaos–chaos transition, the more heterogeous the networktopology is, the weaker coupling strength c is needed.

3.2. Topological sensitivity to random failures and intentional attacks

It has been argued that scale-free networks are robust against random removal ofa large fraction of nodes (considered as random failures), and yet fragile to speci8cremoval of a small fraction of nodes which have many connections (considered asintentional attacks), from both real-life complex network investigations and stabilitystudies of synchronization and control problems, in comparison with the E–R randomnetworks [14,17,26,27]. This di6erence is mainly due to the extreme non-homongenousconnectivity of scale-free networks, in which a very small number of higher-degreenodes (having many connections) dominate the whole network dynamics. On the

374 X. Li et al. / Physica A 338 (2004) 367–378

contrary, random networks generally do not have such higher-degree nodes, where allthe nodes have about the same number of connections with other nodes.The eigenvalue �N of the coupling matrix A, as discussed in the previous sections,

determines the coupling strength threshold c1 responsible for generating the 8rst positiveLyapunov exponent in the coupled network (1). Hence, it is natural to further studythe topological sensitivity of the threshold c1 to random failures and intentional attacksin such complex networks.As usual, random failures or intentional attacks in a network refer to random or pur-

poseful removal of a small fraction �(0¡��1) of nodes from the network,respectively. In the following, let A∈R(N−[�N ])×(N−[�N ]) be the coupling matrix ofthe new complex network obtained after the removal of [�N ] nodes from the networkdescribed by matrix A, where [�N ] stands for the integer part of a real number �N .Also, let �N be the smallest negative eigenvalue of A. If a scale-free dynamical net-work has a random failure, i.e., a small fraction � of nodes are randomly removed,then those removed nodes are most likely to be “small” nodes (with only a few con-nections) due to the signi8cant heterogeneity of the scale-free networks. Therefore,�sfN ≈ �sfN , and csf1 ˙ 1=|�sfN | keeps almost unchanged as csf1 ˙ 1=|�sfN |. However, ifthe network is attacked intentionally, i.e., a small fraction � of higher-degree nodes arepurposely removed, then the original network will be changed signi8cantly, which mayeven be broken into parts, implying that |�sfN |�|�sfN |. In this case, the coupling strengththreshold csf1 is increased signi8cantly. This is the so-called “robust and yet fragile”property of scale-free networks in the nonchaos–chaos transition process. This charac-teristic seems unique, which does not exist in random networks. Due to its homogenousconnections distribution of a random network, both random and speci8c removals ofa small fraction of nodes from the network will only result in a slight change of thewhole network [14]; therefore, in this case, �rnN ≈ �rnN .

We visualize the ratio crn1 =csf1 = �sfN =�

rnN in cases of random and speci8c removals

of nodes from these two di6erent networks by simulation results shown in Fig. 3. Inthe simulation, scale-free networks were generated by the B–A scale-free model, withN=3000 and m0=m=1; 3, respectively, and random networks were generated by the E–R random model containing 3000 nodes with 3000 and 9000 connections, respectively.From the 8gure, it can be observed that random removal of a small fraction of nodesdoes not have much e6ects on both �sfN and �rnN , while speci8c removal of a smallfraction of nodes makes �sfN decrease much faster than �rnN .

4. Chaos in a complex network of coupled Logistic maps

Consider a Logistic map, described in the dimensionless form by

x(k + 1) = f(x(k)) = px(k)(1− x(k)); 0¡�6 4 : (13)

This map, when is uncoupled (i.e., isolated), has dynamical behaviors determined bythe parameter p. Here, p is speci8cally set as p=2:5, at which the Logistic map thenhas a stable 8xed point, with Lyapunov exponent h0 =−0:6936.

X. Li et al. / Physica A 338 (2004) 367–378 375

Fig. 3. Comparison of the ratio crn1 =csf1 = �sfN =�

rnN versus � between scale-free networks and random networks,

with random and speci8c removals of a small fraction of nodes. Solid lines are for random removal, anddashed lines are for speci8c removal of a fraction of nodes. Lines with squares and upper triangles are fornetworks with m= m0 = 1; 3, respectively, where N = 3000. Every curve in the 8gure is the average resultof 10 groups of networks.

Now, consider the following network of coupled Logistic maps:

xi(k + 1) =p(xi(k))(1− xi(k))− cN∑

j=1

aijp(xj(k))(1− xj(k)) ;

i = 1; 2; : : : ; N ; (14)

where the parameter p is 8xed at the stable value p=2:5, and in this network N=1000and the coupling matrix A = (aij)∈RN×N represents its coupling con8guration as ascale-free network generated by the B–A model with m = m0 = 3. For comparison,an E–R random network of coupled Logistic maps with 1000 nodes and 3000 links(connections) is also simulated in this study. In the simulations, all the nodes areinitialized randomly around the stable 8xed point 0.6. By using Eq. (9), the thresholdvalues c1 of the two types of networks are calculated, resulting in csf1 =e−h0 −1=|�sfN |=0:0098 and crn1 = e−h0 − 1=|�rnN |= 0:0624, where �sfN =−102:5288 and �rnN =−16:0492,respectively.Fig. 4 presents the bifurcation diagrams of some nodes in a 1000-node coupled

scale-free network and in a 1000-node coupled random network. The selected nodesare the 1st and the 7th nodes, whose degrees are 1 and 101, and they are the nodeswith the lowest and the highest degrees in the scale-free network, respectively. Theircounterparts in the random network are the 93th and the 384th nodes, whosedegrees are 1 and 14, respectively. It can be observed that as the coupling strength c is

376 X. Li et al. / Physica A 338 (2004) 367–378

Fig. 4. Bifurcation diagrams of the selected nodes in two networks of coupled Logistic maps. (a),(b) The1000-node scale-free network; (c),(d) The 1000-node random network; (a),(c) are the nodes with the highestdegrees, and (b),(d) are the nodes with the lowest degrees, respectively, in the networks.

increased, those chosen nodes, which were originally stable at their 8xed points, startto have bifurcation behaviors and 8nally reach chaotic states. Figs. 5(a),(b) are thePoincare sections of those selected nodes in the scale-free network, with the couplingstrength c = 0:028, and Figs. 5(c),(d) are that of those selected nodes in the randomnetwork, with the coupling strength c = 0:146. It can be clearly observed that as theheterogeity of the network topology increases, the coupling strength c decreases inthe nonchaos–chaos transition, in both types of coupled networks. In other words, ifthe topology is more heterogenous, then the coupling strength required to achieve thetransition can be weaker.

5. Conclusions

In this work, we have studied the transition from nonchaotic to chaotic states in twotypical types of coupled networks: the B–A scale-free and the E–R random networks.In these networks, where all individual nodes are originally in non-chaotic states, pos-itive Lyapunov exponents of the coupled network can be generated with a properuniform coupling strength, leading to bifurcations and chaos. We have found that thethreshold of this coupling strength is determined by the complexity of the network

X. Li et al. / Physica A 338 (2004) 367–378 377

Fig. 5. Poincare sections of the selected nodes in the two networks of coupled Logistic maps. (a),(b) The1000-node scale-free network with coupling strength c= 0:028; (c),(d) the 1000-node random network withcoupling strength c = 0:146; (a),(c) are the nodes with the highest degrees and (b),(d) are the nodes withthe lowest degrees, respectively, in the networks.

topology. Moreover, we have found that the topological e6ects can be visualized fromthe network topological sensitivity to random failures and intentional attacks. Simula-tions on a 1000-node scale-free network and a 1000-node random network of Logisticmaps were carried out, showing that, during the transition from nonchaotic to chaoticstates in such networks, if the topology is more heterogenous then the coupling strengthrequired to achieve the transition can be weaker. The phenomenon discovered here mayindeed pertain to many other types of coupled complex networks, which will be furtherinvestigated in the near future.

Acknowledgements

This research was supported by the Hong Kong Research Grants Council under theCERG grant City U 1031/01E.

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