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Hierarchical synchronization in complex networks with heterogeneous degrees
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10.1063/1.2150381
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1 Institute of Physics, University of Potsdam PF 601553, 14415 Potsdam, Germany
Chaos 16, 015104 (2006)
/content/aip/journal/chaos/16/1/10.1063/1.2150381
http://aip.metastore.ingenta.com/content/aip/journal/chaos/16/1/10.1063/1.2150381
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## Figures

FIG. 1.

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

FIG. 2.

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.

FIG. 3.

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 .

FIG. 4.

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 .

FIG. 5.

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.

FIG. 6.

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.

FIG. 7.

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).

FIG. 8.

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.

FIG. 9.

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. ] for globally coupled oscillators. The networks have the same mean degree and size .

FIG. 10.

(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 .

FIG. 11.

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

FIG. 12.

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

FIG. 13.

(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 .

FIG. 14.

(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 .

FIG. 15.

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. .

/content/aip/journal/chaos/16/1/10.1063/1.2150381
2006-03-31
2014-04-20

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