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On the topology of synchrony optimized networks of a Kuramoto-model with non-identical oscillators

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10.1063/1.3590855

### Abstract

We study synchrony optimized networks. In particular, we focus on the Kuramoto model with non-identical native frequencies on a random graph. In a first step, we generate synchrony optimized networks using a dynamic breeding algorithm, whereby an initial network is successively rewired toward increased synchronization. These networks are characterized by a large anti-correlation between neighbouring frequencies. In a second step, the central part of our paper, we show that synchrony optimized networks can be generated much more cost efficiently by minimization of an energy-like quantity *E* and subsequent random rewires to control the average path length. We demonstrate that synchrony optimized networks are characterized by a balance between two opposing structural properties: A large number of links between positive and negative frequencies of equal magnitude and a small average path length. Remarkably, these networks show the same synchronization behaviour as those networks generated by the dynamic rewiring process. Interestingly, synchrony-optimized network also exhibit significantly enhanced synchronization behaviour for weak coupling, below the onset of global synchronization, with linear growth of the order parameter with increasing coupling strength. We identify the underlying dynamical and topological structures, which give rise to this atypical local synchronization, and provide a simple analytical argument for its explanation.

© 2011 American Institute of Physics

Received 31 January 2011
Accepted 25 April 2011
Published online 28 June 2011

Lead Paragraph:

Synchronization of networks is a ubiquitous phenomenon. The Kuramoto model describing the interaction of a network of identical oscillators has attracted mathematical interest because of its analytical tractability. In this paper, we will study the Kuramoto model with non-identical oscillators. An important question is how a network should be organized to allow for optimal synchronization behaviour of the oscillators. Several computational methods have been recently introduced to establish synchrony optimized networks, based on dynamic rewiring^{5,6} and simulated annealing.^{9,8} However, these methods, although successful, are computationally extremely costly and prohibit the study of large networks. In this paper, we will propose a very simple and cheap algorithm to create synchronization optimized networks.

Acknowledgments: We would like to thank Takashi Nishikawa for a critical reading of an earlier version of our manuscript. The research of GAG was supported in part by the Australian Research Council.

Article outline:

I. INTRODUCTION

II. THE MODEL

III. SYNCHRONY OPTIMIZED NETWORKS

A. DRA

B. Topological structure of DRA-optimized networks

C. *E*-networks

D. -networks

IV. LOCAL SYNCHRONIZATION AT WEAK COUPLING

A. Linear scaling of the local synchronization regime for weak coupling

V. DISCUSSION

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2011-06-28

2014-04-23

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