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

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.

Optimal synchronization of Kuramoto oscillators: A dimensional reduction approach

“…Our condition allowed us to rederive analytically some recent results [10][11][12][13] obtained numerically. Our approach was also extended to the Kuramoto model with attractive and repulsive interactions, leading also to simple analytical condition to optimize the networks, complementing in this way some recent works [27][28][29] which considered the optimization of synchronization in these models numerically.…”

“…However, there are some potential issues, as for instance, the algorithm might get stuck in a local minimal state far from the global minimum. Nevertheless, our numerical simulations show that this is very rare and we can indeed get networks with pronounced enhancement of the synchronization capacity from this simple algorithm, which is indeed the most used one in the literature of optimization of synchronization in complex networks (see, for instance, [10][11][12]). …”

“…A very natural question to set forth here is: given N vertices with natural frequencies {ω i } N i=1 and m edges, what is the best way to connect the vertices (forming a connected network) in order to optimize synchronization (maximize r)? In the last years, it has become clear [10][11][12][13] that optimal (in the synchronization sense) networks of Kuramoto oscillators typically have a strong negative correlation between the natural frequencies of adjacent oscillators, and a positive correlation between the frequency magnitude |ω i | and the degree k i = N j=1 A i j of the vertex where the i oscillator lies. This situation favors some heterogeneity in the degree distribution: vertices with large positive (negative) natural frequencies tend to have higher degrees and to be surrounded by vertices with negative (positive) natural frequencies.…”

Frustration tuning and perfect phase synchronization in the Kuramoto-Sakaguchi model

“…Inspired by Watts and Strogatz' introduction of small world networks [12], the literature has strongly focussed on analysing the model on sparse complex networks [4,5]. A particular interest in this line of thinking has been to understand conditions under which synchronisation can be improved, e.g., determining the types of correlations of native frequencies along the links of a network that can strongly influence overall synchronisation behaviour [13][14][15][16][17][18].…”

Introduction to Focus Issue: Synchronization and Cascading Processes in Complex Networks

“…In contrast, Stout et al 31 closely examine how synchronized clusters form and merge as the coupling strength increases further toward global synchronization, using different network models and varying the natural frequency distribution and the network size. Taking a different perspective for the same class of systems, Kelly and Gottwald 32 study the structures of the networks optimized for synchronization. Other papers focus more on analytical methods.…”