Optimal Synchronization of Complex Networks

Abstract

We study optimal synchronization in networks of heterogeneous phase oscillators. Our main result is the derivation of a synchrony alignment function that encodes the interplay between network structure and oscillators' frequencies and can be readily optimized. We highlight its utility in two general problems: constrained frequency allocation and network design. In general, we find that synchronization is promoted by strong alignments between frequencies and the dominant Laplacian eigenvectors, as well as a matching between the heterogeneity of frequencies and network structure. PACS numbers: 05.45.Xt, 89.75.Hc A central goal of complexity theory is to understand the emergence of collective behavior in large ensembles of interacting dynamical systems. Synchronization of networkcoupled oscillators has served as a paradigm for understanding emergence [1][2][3][4], where examples arise in nature (e.g., flashing of fireflies [5] and cardiac pacemaker cells [6]), engineering (e.g., power grid [7] and bridge oscillations [8]), and at their intersection (e.g., synthetic cell engineering [9]). We consider the dynamics of N network-coupled phase oscillators θ i for i = 1, . . . , N , whose evolution is governed bẏHere ω i is the natural frequency of oscillator i, K > 0 is the coupling strength, [A ij ] is a symmetric network adjacency matrix, and H is a 2π-periodic coupling function [10]. We treat H(θ) with full generality so long as H ′ (0) > 0. The choices H(θ) = sin(θ) and H(θ) = sin(θ − α) with the phase-lag parameter α ∈ (−π/2, π/2) yield the classical Kuramoto [10] and Sakaguchi-Kuramoto models [11].Considerable research has shown that the underlying structure of a network plays a crucial role in determining synchronization [12][13][14][15][16][17][18][19][20][21], yet the precise relationship between the dynamical and structural properties of a network and its synchronization remains not fully understood. One unanswered question is, given an objective measure of synchronization, how can synchronization be optimized? One application lies in synchronizing the power grid [22], where sources and loads can be modeled as oscillators with different frequencies. To this end, we ask: what structural and/or dynamical properties should be present to optimize synchronization?We measure the degree of synchronization of an ensemble of oscillators using the Kuramoto order parameterHere re iψ denotes the phases' centroid on the complex unit circle, with the magnitude r ranging from 0 (incoherence) to 1 (perfect synchronization) [10]. In general, the question of optimization (maximizing r) is challenging due to the fact that the macroscopic dynamics depend on both the natural frequencies and the network structure. To quantify the interplay between node dynamics and network structure, we derive directly from Eqs. (1) and (2) a synchrony alignment function which is an objective measure of synchronization and can be used to systematically optimize a network's synchronization. We highlight this result by addressing two classes of op...

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

“…where L ij = δ ij k i − A ij is the combinatorial Laplacian whose presence in such fluctuation problems is wellknown [17]. Multiplying by the cos λ i gives a weighted version of the Laplacian.…”

“…To understand the basic properties of the local Kuramoto-Sakaguchi model we repeat the steady state solution considered for the global case, such as in [17]. We consider solutions of the form…”

“…We indeed find this to be the case, to a degree even more than an equilibrium analysis reveals, where enhancement occurs in two ways: macroscopic synchronisation is found at lower values of critical coupling than for the normal Kuramoto system of Eq. (1); and this synchronisation occurs at collective frequencies Ω which may be different from (both less and greater than)ω, something already observed for the global Kuramoto-Sakaguchi model [17,20]. Synchronisation in this setting requires specifically tuned frustration parameters and one might wonder how appropriate configurations of such parameters may be reached in real world systems.…”

“…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].…”

Network Dynamics as an Inverse Problem

“…Recent studies coming from different fields suggest that, apart from structure, there may be other important factors defining and shaping network dynamics [28,29,57,85]. For example, it has been shown that models of spiking neural networks [25-27, 34,35,40] may exhibit specific spike patterns which may be generated by any network from a high-dimensional family of networks.…”