Noise stability of synchronization and optimal network structures

Abstract

We provide a theoretical framework for quantifying the expected level of synchronization in a network of noisy oscillators. Through linearization around the synchronized state, we derive the following quantities as functions of the eigenvalues and eigenfunctions of the network Laplacian using a standard technique for dealing with multivariate Ornstein-Uhlenbeck processes: the magnitude of the fluctuations around a synchronized state and the disturbance coefficients α i that represent how strongly node i disturbs the synchronization. With this approach, we can quantify the effect of individual nodes and links on synchronization. Our theory can thus be utilized to find the optimal network structure for accomplishing the best synchronization. Furthermore, when the noise levels of the oscillators are heterogeneous, we can also find optimal oscillator configurations, i.e., where to place oscillators in a given network depending on their noise levels. We apply our theory to several example networks to elucidate optimal network structures and oscillator configurations.PACS numbers: 05.45.Xt, 82.40.Bj, 64.60.aq Synchronization of rhythmic elements is essential in many systems. To function properly and well, rhythmic elements are required to maintain an appropriate synchronization pattern precisely. What is the best network structure for accomplishing the best synchronization? In other word, which elements should each element have look at? Here, we develop a measure to quantify the precision of synchronization for a given network. Using this measure, we can quantitatively compare the stability of different networks and find the optimal network structure. We can also determine where reliable or unreliable elements should be placed in a given network.