^{1,2}, Xingang Wang

^{3,a)}, Ying Fan

^{4}, Zengru Di

^{4}and Choy-Heng Lai

^{2,5}

### Abstract

By numerical simulations, we investigate the onset of synchronization of networked phase oscillators under two different weighting schemes. In scheme-I, the link weights are correlated to the product of the degrees of the connected nodes, so this kind of networks is named as the weight-degree correlated (WDC) network. In scheme-II, the link weights are randomly assigned to each link regardless of the node degrees, so this kind of networks is named as the weight-degree uncorrelated (WDU) network. Interestingly, it is found that by increasing a parameter that governs the weight distribution, the onset of synchronization in WDC network is monotonically enhanced, while in WDU network there is a reverse in the synchronization performance. We investigate this phenomenon from the viewpoint of gradient network, and explain the contrary roles of coupling gradient on network synchronization: gradient promotes synchronization in WDC network, while deteriorates synchronization in WDU network. The findings highlight the fact that, besides the link weight, the correlation between the weight and the node degree is also important to the networkdynamics.

In the past decades, complex networks have become an established framework to understand the behavior of a large variety of complex systems, ranging from biology to social sciences.

^{1–3}In network studies, one important issue is to investigate the influence of the network topology on the dynamics, e.g., the synchronization behaviors of coupled oscillators.

^{4}For many realistic systems, besides the degree, the weight of the network links also presents the heterogeneous distribution, i.e., the weighted networks. Depending on the correlation between the node degree and the link weight, the weighted systems can be roughly divided into two types: weight-degree correlated (WDC) and weight-degree uncorrelated (WDU) networks.

^{3}While the collective dynamics taking place on weighted networks have been extensively studied,

^{4}but so far, to the best of our knowledge, the impact of weight-degree correlation on the collective dynamics has not been addressed in literature. In this paper, we consider the effect of weight-degree correlation on the onset of synchronization of a generalized Kuramoto model with fixed total coupling cost. Interestingly, it is found that, by increasing the weight parameter that characterizes the distribution of links weights, the synchronization is monotonically enhanced in WDC network, while the synchronization could be deteriorated first and then enhanced in WDU network. Moreover, we explain qualitatively the fundamental mechanism from the viewpoint of gradient network. Our findings may be helpful to the design and optimization of realistic dynamical networks.

X. Wang is supported by NSFC under Grant No. 10805038 and by Chinese Universities Scientific Fund. Y. Fan and Z. Di are supported by 985 Project and NSFC under Grant Nos. 70771011 and 60974084. This work is also supported by DSTA of Singapore under Project Agreement POD0613356.

I. INTRODUCTION

II. THE MODEL

III. ONSET OF SYNCHRONIZATION IN WEIGHTED COMPLEX NETWORKS

IV. THE INFLUENCE OF WEIGHT-DEGREE CORRELATION ON NETWORK SYNCHRONIZATION

A. WDC-type networks

B. WDU-type networks

C. Mechanism analysis

V. DISCUSSION AND CONCLUSION

### Key Topics

- Networks
- 38.0
- Oscillators
- 12.0
- Network topology
- 9.0
- Coupled oscillators
- 6.0
- Numerical modeling
- 2.0

## Figures

(Color online) Under the WDC scheme, the variation of the synchronization order parameter, , as a function of the coupling strength, ɛ, for different weight parameters, α. The network is generated by the standard BA model (scale-free network), which has size *N* = 6400 and average degree . Onset of synchronization is identified as the coupling strength where starts to increase from 2 × 10^{−2}. The error bars are estimated by the standard deviation. Each data is averaged over 20 network runs.

(Color online) Under the WDC scheme, the variation of the synchronization order parameter, , as a function of the coupling strength, ɛ, for different weight parameters, α. The network is generated by the standard BA model (scale-free network), which has size *N* = 6400 and average degree . Onset of synchronization is identified as the coupling strength where starts to increase from 2 × 10^{−2}. The error bars are estimated by the standard deviation. Each data is averaged over 20 network runs.

(Color online) Under the WDU scheme, the variations of the synchronization order parameter, , as a function of the coupling strength, ɛ, for scale-free networks (a) and (c) and small-world networks (b). As a reference, the synchronization of non-weighted network (Binary) is also plotted in each subplot. All the networks have size *N* = 6400 and average degree 6. In constructing the WDU network, each link is arranged a weight chosen randomly from the range [1,100] for (a) and (b), and from the range [1,100000] for (c). The rewiring probability in generating the small-world networks is . The error bars are estimated by the standard deviation, and each data is averaged over 20 network runs.

(Color online) Under the WDU scheme, the variations of the synchronization order parameter, , as a function of the coupling strength, ɛ, for scale-free networks (a) and (c) and small-world networks (b). As a reference, the synchronization of non-weighted network (Binary) is also plotted in each subplot. All the networks have size *N* = 6400 and average degree 6. In constructing the WDU network, each link is arranged a weight chosen randomly from the range [1,100] for (a) and (b), and from the range [1,100000] for (c). The rewiring probability in generating the small-world networks is . The error bars are estimated by the standard deviation, and each data is averaged over 20 network runs.

(Color online) For the same scale-free network as used in Fig. 1, under WDC weighting scheme, the variation of the critical coupling strength, , as a function of the weight parameter, α, in (a) and the average network gradient, , in (b). Inset: versus α. Each data is averaged over 20 network runs.

(Color online) For the same scale-free network as used in Fig. 1, under WDC weighting scheme, the variation of the critical coupling strength, , as a function of the weight parameter, α, in (a) and the average network gradient, , in (b). Inset: versus α. Each data is averaged over 20 network runs.

(Color online) For the same scale-free network as used in Fig. 2(a), under WDU scheme, the variation of the critical coupling strength, , as a function of the weight parameter, β, in (a) and the averaged network gradient, , in (b). Insets: versus β. Each data is averaged over 20 network runs.

(Color online) For the same scale-free network as used in Fig. 2(a), under WDU scheme, the variation of the critical coupling strength, , as a function of the weight parameter, β, in (a) and the averaged network gradient, , in (b). Insets: versus β. Each data is averaged over 20 network runs.

(Color online) For the same small-world network as used in Fig. 2(b), under WDU scheme, the variation of the critical coupling strength, , as a function of the weight parameter, β, in (a) and the average network gradient, , in (b). Insets: versus β. Each data is averaged over 20 network runs.

(Color online) For the same small-world network as used in Fig. 2(b), under WDU scheme, the variation of the critical coupling strength, , as a function of the weight parameter, β, in (a) and the average network gradient, , in (b). Insets: versus β. Each data is averaged over 20 network runs.

(Color online) A schematic plot on the distribution of the gradient couplings in WDC and WDU networks. (a) The simplified model of undirected, non-weighted network. (b) The distribution of the gradient couplings under the WDC-scheme. The network nodes and the gradient couplings are organized into a spanned gradient tree, in which each node is reachable from the rooting node (numbered 1). (c) The distribution of the gradient couplings under the WDU-scheme, in which the weight on the link between nodes 3 and 4 is increased by Δ, while the remaining gradients keep identical to that of (b). As Δ exceeds some critical value, the gradient between the nodes 2 and 3 will switch its direction (the red arrow-line), manifesting a breaking of the gradient network. Dashed lines are links dominated by symmetric couplings (with negligible gradient).

(Color online) A schematic plot on the distribution of the gradient couplings in WDC and WDU networks. (a) The simplified model of undirected, non-weighted network. (b) The distribution of the gradient couplings under the WDC-scheme. The network nodes and the gradient couplings are organized into a spanned gradient tree, in which each node is reachable from the rooting node (numbered 1). (c) The distribution of the gradient couplings under the WDU-scheme, in which the weight on the link between nodes 3 and 4 is increased by Δ, while the remaining gradients keep identical to that of (b). As Δ exceeds some critical value, the gradient between the nodes 2 and 3 will switch its direction (the red arrow-line), manifesting a breaking of the gradient network. Dashed lines are links dominated by symmetric couplings (with negligible gradient).

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