Network skeleton for synchronization: Identifying redundant connections

Synchronization is an important dynamical process on complex networks with wide applications. In this paper, we design a greedy link removal algorithm and find that many links in networks are actually redundant for synchronization, i.e. the synchronizability of the network is hardly affected if these links are removed. Our analysis shows that homogeneous networks generally have more redundant links than heterogeneous networks. We denote the reduced network with the minimum number of links to preserve synchronizability (eigenratio of the Laplacian matrix) of the original network as the synchronization backbone. Simulating the Kuramoto model, we confirm that the network synchronizability is effectively preserved in the backbone. Moreover, the topological properties of the original network and backbone are compared in detail

Autonomous Synchronization of Chemically Coupled Synthetic Oscillators

Synthetic biology has recently provided functional single-cell oscillators. With a few exceptions, however, synchronization across a population has not been achieved yet. In particular, designing a cell coupling mechanism to achieve autonomous synchronization is not straightforward since there are usually several different design alternatives. Here, we propose a method to mathematically predict autonomous synchronization properties, and to identify the network structure with the best performance, thus increasing the feasibility for a successful implementation invivo. Our method relies on the reduction of ODE-based models for synthetic oscillators to a phase description, and the subsequent analysis of the phase model either in the spatially homogeneous or heterogeneous case. This analysis identifies three major factors determining if and when autonomous synchronization can be achieved, namely cell density, cell to cell variability, and structural design decisions. Moreover, when considering a spatially heterogeneous medium, we observe phase waves. These waves may hinder synchronization substantially, and their suppression should be considered in the design process. In contrast to previous work, we analyze the synchronization process of models of experimentally validated synthetic oscillators in mammalian cells. Alternative designs for cell-to-cell communication via a quorum sensing mechanism differ in few mechanistic details, but these differences have important implications for autonomous synchronization. Our analysis suggests that not only the periodical transcription of the protein producing the signaling molecule, but also of the receptor protein is necessary to achieve good performanc

The spark of synchronization in heterogeneous networks of chaotic maps

We investigate the emergence of synchronization in heterogeneous networks of chaotic maps. Our findings reveal that a small cluster of highly connected maps is responsible for triggering the spark of synchronization. After the spark, the synchronized cluster grows in size and progressively moves to less connected maps, eventually reaching a cluster that may remain synchronized over time. We explore how the shape of the network's degree distribution affects the onset of synchronization and derive an expression based on the network's construction that determines the expected time for a network to synchronize. Understanding how the network structure affects the spark of synchronization is particularly important for the control and design of more robust systems that require some level of coherence between a subset of units for better functioning. Numerical simulations in finite-sized networks are consistent with this analysis

Multiplex PI-Control for Consensus in Networks of Heterogeneous Linear Agents

In this paper, we propose a multiplex proportional-integral approach, for solving consensus problems in networks of heterogeneous nodes dynamics affected by constant disturbances. The proportional and integral actions are deployed on two different layers across the network, each with its own topology. Sufficient conditions for convergence are derived that depend upon the structure of the network, the parameters characterizing the control layers and the node dynamics. The effectiveness of the theoretical results is illustrated using a power network model as a representative example.Comment: 13 pages, 6 Figures, Preprint submitted to Automatic

Optimal synchronization of complex networks

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.Comment: 5 pages, 4 figure

Synchronization of heterogeneous oscillators under network modifications: Perturbation and optimization of the synchrony alignment function

Synchronization is central to many complex systems in engineering physics (e.g., the power-grid, Josephson junction circuits, and electro-chemical oscillators) and biology (e.g., neuronal, circadian, and cardiac rhythms). Despite these widespread applications---for which proper functionality depends sensitively on the extent of synchronization---there remains a lack of understanding for how systems evolve and adapt to enhance or inhibit synchronization. We study how network modifications affect the synchronization properties of network-coupled dynamical systems that have heterogeneous node dynamics (e.g., phase oscillators with non-identical frequencies), which is often the case for real-world systems. Our approach relies on a synchrony alignment function (SAF) that quantifies the interplay between heterogeneity of the network and of the oscillators and provides an objective measure for a system's ability to synchronize. We conduct a spectral perturbation analysis of the SAF for structural network modifications including the addition and removal of edges, which subsequently ranks the edges according to their importance to synchronization. Based on this analysis, we develop gradient-descent algorithms to efficiently solve optimization problems that aim to maximize phase synchronization via network modifications. We support these and other results with numerical experiments.Comment: 25 pages, 6 figure