Network synchronization: Spectral versus statistical properties

We consider synchronization of weighted networks, possibly with asymmetrical connections. We show that the synchronizability of the networks cannot be directly inferred from their statistical properties. Small local changes in the network structure can sensitively affect the eigenvalues relevant for synchronization, while the gross statistical network properties remain essentially unchanged. Consequently, commonly used statistical properties, including the degree distribution, degree homogeneity, average degree, average distance, degree correlation, and clustering coefficient, can fail to characterize the synchronizability of networks

Synchronization in discrete-time networks with general pairwise coupling

We consider complete synchronization of identical maps coupled through a general interaction function and in a general network topology where the edges may be directed and may carry both positive and negative weights. We define mixed transverse exponents and derive sufficient conditions for local complete synchronization. The general non-diffusive coupling scheme can lead to new synchronous behavior, in networks of identical units, that cannot be produced by single units in isolation. In particular, we show that synchronous chaos can emerge in networks of simple units. Conversely, in networks of chaotic units simple synchronous dynamics can emerge; that is, chaos can be suppressed through synchrony

On global convergence of consensus with nonlinear feedback, the Lure problem, and some applications

We give a rigorous proof of convergence of a recently proposed consensus algorithm with output constraint. Examples are presented to illustrate the efficacy and utility of the algorithm

Achieving synchronization in arrays of coupled differential systems with time-varying couplings

In this paper, we study complete synchronization of the complex dynamical networks described by linearly coupled ordinary differential equation systems (LCODEs). The coupling considered here is time-varying in both the network structure and the reaction dynamics. Inspired by our previous paper [6], the extended Hajnal diameter is introduced and used to measure the synchronization in a general differential system. Then we find that the Hajnal diameter of the linear system induced by the time-varying coupling matrix and the largest Lyapunov exponent of the synchronized system play the key roles in synchronization analysis of LCODEs with the identity inner coupling matrix. As an application, we obtain a general sufficient condition guaranteeing directed time-varying graph to reach consensus. Example with numerical simulation is provided to show the effectiveness the theoretical results.Comment: 22 pages, 4 figure

Synchrony and Elementary Operations on Coupled Cell Networks

Given a finite graph (network), let every node (cell) represent an individual dynamics given by a system of ordinary differential equations, and every arrow (edge) encode the dynamical influence of the tail node on the head node. We have then defined a coupled cell system that is associated with the given network structure. Subspaces that are defined by equalities of cell coordinates and left invariant under every coupled cell system respecting the network structure are called synchrony subspaces. They are completely determined by the network structure and form a complete lattice under set inclusions. We analyze the transition of the lattice of synchrony subspaces of a network that is caused by structural changes in the network topology, such as deletion and addition of cells or edges, and rewirings of edges. We give sufficient, and in some cases both sufficient and necessary, conditions under which lattice elements persist or disappear

On Synchronization in Coupled Dynamical Systems on Hypergraphs

Here we study both global and local synchronization in coupled dynamical systems on hypergraph. Diffusion Matrices, representing the diffusive influence of internal connectivities of the hypergraph on the dynamics occurring in the vertices, are constructed. Models are proposed for discrete-time and continuous-time dynamical systems on (weighted and un-weighted) hypergraphs. The interplay between the diffusion matrices and the dynamical systems on the vertices are analyzed to study synchronization of the dynamics under the influence of hyperedge couplings. Sufficient conditions for synchronization in dynamical systems on hypergraphs are derived. Some of the derived sufficient conditions incorporate structural properties of the underlying hypergraphs. Numerical examples are given to illustrate the theoretical results. The results in this article can initiate and stimulate the generalization of the underlying structure considered in the study of dynamical networks from graphs to hypergraphs.Comment: Title is replaced, some figures are replaces,some typos are removed, some unnecessary steps are remove

SYNCHRONIZATION OF DISCRETE-TIME DYNAMICAL NETWORKS WITH TIME-VARYING COUPLINGS

networks with time-varying couplings b