^{1,a)}and Jürgen Kurths

^{1}

### Abstract

We study synchronization behavior in networks of coupled chaotic oscillators with heterogeneous connection degrees. Our focus is on regimes away from the complete synchronization state, when the coupling is not strong enough, when the oscillators are under the influence of noise or when the oscillators are nonidentical. We have found a hierarchical organization of the synchronization behavior with respect to the collective dynamics of the network. Oscillators with more connections (hubs) are synchronized more closely by the collective dynamics and constitute the dynamical core of the network. The numerical observation of this hierarchical synchronization is supported with an analysis based on a mean field approximation and the master stability function.

Complex networks are playing an increasing role in the understanding of complex systems. The analysis of various real-world complex systems using the approach of complex networks has uncovered general and important principles in the structure organization of realistic systems. In particular, many complex networks are scale-free, characterized by a heterogeneous power-law distribution of the degrees. A problem of fundamental importance is the impact of the network topology on the dynamics of the complex systems, which has been recently studied intensively in the context of synchronization of coupled oscillators. Many previous works have focused on the global synchronizability, i.e., the ability of the network to synchronize completely for fully identical oscillators. In this paper we consider more natural situations where the networks are not in the complete synchronization state, for example, when the coupling is not strong enough, when the oscillators are in the presence of noise or when the oscillators are nonidentical. We have shown that complex networks of chaotic oscillators display significant collective oscillations in such regimes. More interestingly, we have found that in networks with heterogeneous degrees, the individual oscillators have different levels of synchronization with respect to the collective oscillations and they exhibit a hierarchical dependence on the connection degrees. The behavior can be understood analytically based on a mean field approximation and the linear stability analysis. Our results demonstrate that, in the context of synchronization, hubs having large degrees play the leading role in the formation of the dynamical core, which is the main contributor to the collective dynamics of the network. In the future, it is interesting to study hierarchical synchronization in more realistic networks whose connection topology and connection strengths are time varying and the results could have meaningful applications in the dynamics of real-world complex systems, such as the human brain.

The authors thank M. Matias and B. Blasius for helpful discussions. This work was supported by Grant No. SFB 555 (DFG) and the European Union through the Network of Excellence BioSim, Contract No. LSHB-CT-2004-005137.

I. INTRODUCTION

II. MODELS

A. Dynamical equations

B. Network models

III. IDENTICAL OSCILLATORS

A. Linear stability

B. Transition to synchronization

C. Desynchronization by noise

D. Effective synchronization clusters

E. Analysis of hierarchical synchronization

IV. NONIDENTICAL OSCILLATORS

A. Weak coupling: Nonsynchronization regime

B. Intermediate coupling: Phase synchronization

C. Strong coupling: Almost complete synchronization

D. Analysis of the coherent regime of HN

V. CONCLUSION AND DISCUSSION

### Key Topics

- Oscillators
- 70.0
- Networks
- 49.0
- Synchronization
- 42.0
- Network topology
- 27.0
- Mean field theory
- 22.0

## Figures

Chaotic attractors of the Rössler oscillator in the phase coherent regime (a) and phase noncoherent regime (b).

Chaotic attractors of the Rössler oscillator in the phase coherent regime (a) and phase noncoherent regime (b).

Degree sequence of a SFN with nodes. The inset shows the power-law distribution , averaged over 50 realizations of the networks. The flat tail results from finite size effects.

Degree sequence of a SFN with nodes. The inset shows the power-law distribution , averaged over 50 realizations of the networks. The flat tail results from finite size effects.

Transition to CS in the SFN and the HN, indicated by the synchronization error (squares) and the amplitude of the mean field (circles). The filled symbols are for the SFNs and the open symbols for the HNs. (a) Phase coherent oscillations at . (b) Phase noncoherent oscillations at . In both networks, and .

Transition to CS in the SFN and the HN, indicated by the synchronization error (squares) and the amplitude of the mean field (circles). The filled symbols are for the SFNs and the open symbols for the HNs. (a) Phase coherent oscillations at . (b) Phase noncoherent oscillations at . In both networks, and .

Synchronization difference of the oscillators with respect to the global mean field in the SFN (solid line) and HN (dotted line). The symbol (엯) denotes the degree of the nodes. Note the log-log scales of the plots. (a) When the coupling strength is weak and (b) when the synchronized state is perturbed by noise . Here the oscillations are phase noncoherent at and the behavior is very similar for phase coherent oscillations at .

Synchronization difference of the oscillators with respect to the global mean field in the SFN (solid line) and HN (dotted line). The symbol (엯) denotes the degree of the nodes. Note the log-log scales of the plots. (a) When the coupling strength is weak and (b) when the synchronized state is perturbed by noise . Here the oscillations are phase noncoherent at and the behavior is very similar for phase coherent oscillations at .

The average values as a function of at various coupling strength in the SFN. (a) Phase coherent regime . (b) Phase noncoherent regime . The solid line with slop is plotted for reference.

The average values as a function of at various coupling strength in the SFN. (a) Phase coherent regime . (b) Phase noncoherent regime . The solid line with slop is plotted for reference.

The average values as a function of in the SFN at various noise levels . (a) Phase coherent regime with the coupling strength in the CS region. (b) Phase noncoherent regime with . The solid line with slop is plotted for reference.

The average values as a function of in the SFN at various noise levels . (a) Phase coherent regime with the coupling strength in the CS region. (b) Phase noncoherent regime with . The solid line with slop is plotted for reference.

The effective synchronization clusters in the synchronized network in the presence of noise ( , , and ), represented simultaneously in the index space [(a), (b)] and in the degree space [(c), (d)]. A dot is plotted when . (a) and (c) for the threshold value , and (b) and (d) for . The solid lines in (a) and (b) denote and are also plotted in (c) and (d) correspondingly. Note the different scales in (a) and (b) and the log-log scales in (c) and (d).

The effective synchronization clusters in the synchronized network in the presence of noise ( , , and ), represented simultaneously in the index space [(a), (b)] and in the degree space [(c), (d)]. A dot is plotted when . (a) and (c) for the threshold value , and (b) and (d) for . The solid lines in (a) and (b) denote and are also plotted in (c) and (d) correspondingly. Note the different scales in (a) and (b) and the log-log scales in (c) and (d).

as a function of for networks with different degree distributions (power law and exponential) due to different aging exponents . The results obtained in the synchronization regime perturbed by noise ( , , and ) in these different networks collapse into a single curve.

as a function of for networks with different degree distributions (power law and exponential) due to different aging exponents . The results obtained in the synchronization regime perturbed by noise ( , , and ) in these different networks collapse into a single curve.

The amplitude of the mean field as a function of the coupling strength in the SFN (엯) and HN (◻). The solid line is the analytically obtained results [Eq. (20) ] for globally coupled oscillators. The networks have the same mean degree and size .

The amplitude of the mean field as a function of the coupling strength in the SFN (엯) and HN (◻). The solid line is the analytically obtained results [Eq. (20) ] for globally coupled oscillators. The networks have the same mean degree and size .

(a) Synchronization difference of the oscillators with respect to the global mean field in the SFN (solid line) and HN (dotted line). (b) The average values as a function of in the SFN. (c) and (d), as in (a) and (b), but for the oscillation amplitude and the average value of the oscillators. The results are averaged over 50 realizations of the random time-scale parameters . The coupling strength is .

(a) Synchronization difference of the oscillators with respect to the global mean field in the SFN (solid line) and HN (dotted line). (b) The average values as a function of in the SFN. (c) and (d), as in (a) and (b), but for the oscillation amplitude and the average value of the oscillators. The results are averaged over 50 realizations of the random time-scale parameters . The coupling strength is .

The mean oscillation frequencies of the oscillators in the SFN (a) and the HN (b) at the coupling strength .

The mean oscillation frequencies of the oscillators in the SFN (a) and the HN (b) at the coupling strength .

Time series of the mean field in the SFN (a) and HN (b) at the coupling strength .

Time series of the mean field in the SFN (a) and HN (b) at the coupling strength .

(a) Phase synchronization order parameter of node with respect to the mean field in the SFN (solid line) and HN (dotted line). (b) Average value of nodes with degree as a function of in the heterogeneous network. (c),(d) as (a) and (b), but for the distance and its average value , respectively. The results are averaged over 50 realizations of random distribution of the time-scale parameter . The coupling strength is .

(a) Phase synchronization order parameter of node with respect to the mean field in the SFN (solid line) and HN (dotted line). (b) Average value of nodes with degree as a function of in the heterogeneous network. (c),(d) as (a) and (b), but for the distance and its average value , respectively. The results are averaged over 50 realizations of random distribution of the time-scale parameter . The coupling strength is .

(a) Averaged phase difference between a node and the mean field in the SFN (solid line) and HN (dotted line). (b) Average value of nodes with degree as a function of in the SFN. (c),(d) as (a) and (b), but for the absolute difference and its average value , respectively. The solid line in (d) with slope are plotted for reference. The results are averaged over 50 realizations of the random time scale parameters . The coupling strength is .

(a) Averaged phase difference between a node and the mean field in the SFN (solid line) and HN (dotted line). (b) Average value of nodes with degree as a function of in the SFN. (c),(d) as (a) and (b), but for the absolute difference and its average value , respectively. The solid line in (d) with slope are plotted for reference. The results are averaged over 50 realizations of the random time scale parameters . The coupling strength is .

Bifurcation diagram of the collective oscillations with respect to the coupling strength . The dots plotted are the local maxima in the time series of . (a) SFN, (b) HN, and (c) Macroscopic equations in Eq. (20) .

Bifurcation diagram of the collective oscillations with respect to the coupling strength . The dots plotted are the local maxima in the time series of . (a) SFN, (b) HN, and (c) Macroscopic equations in Eq. (20) .

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