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A communicating system is $k$-synchronizable if all of the message sequence charts representing the executions can be divided into slices of $k$ sends followed by $k$ receptions. It was previously shown that, for a fixed given $k$, one could decide whether a communicating system is $k$-synchronizable. This result is interesting because the reachability problem can be solved for $k$-synchronizable systems. However, the decision procedure assumes that the bound $k$ is fixed. In this paper we improve this result and show, for mailbox communication, that it is possible to decide if such a bound $k$ exists.

In this paper, we numerically investigate the structural characteristics that affect the synchronizability of coupled identical oscillators on scale-free networks. By using the edge-exchange method, we can change the network structure with degree sequence fixed. An optimal algorithm, namely Tabu Search, is applied, respectively, to enhance and weaken the synchronizability. The numerical results indicate that the synchronizability is most sensitive to the maximal betweenness. By tracking the optimization process, most previous works suggest that the synchonizability can be enhanced by reducing assortative coefficient and clustering coefficient; however, interestingly, our numerical simulations show that the depression of synchronizability also corresponds to the reduction process of those two coefficients.

Most previous studies on the synchronization of complex networks were based on that each node managed to adjust its neighbors coupling strength to enhance synchronizability, i.e. each node tried to adjust its total input coupling strength in a proper way and the neighbor nodes were passively adjusted. From practical and engineering viewpoints, each node should manage to adjust its total output coupling strength to realize synchronization. Moreover, each node's total output coupling strength can be distributed to its neighbors with different proportions. In view of the above reasons, in this paper, we study the synchronization of complex networks under the assumptions that the total output coupling strength of each node is voluntarily/directly distributed to its neighbors with different proportions. What is more, we assume that the total output coupling strength of each node can be nonlinear to its degree. Our analysis and numerical simulations show that the synchronizability can be enhanced dramatically when the parameters are properly selected.

Many real-world networks show community structure characterized by dense intra-community connections and sparse inter-community links. In this paper we investigated the synchronization properties of such networks. In this work we constructed such networks in a way that they consist of a number of communities with scale-free or small-world structure. Furthermore, with a probability, the intra-community connections are rewired to inter-community links. Two synchronizability measures were considered as the eigenratio of the Laplacian matrix and the phase order parameter obtained for coupled nonidentical Kuramoto oscillators. We found a power-law relation between the eigenratio and the inter-community rewiring probability in which as the rewiring probability increased, the eigenratio decreased, and hence, the synchronizability enhanced. The phase order parameter also increased by increasing the rewiring probability. Also, small-world networks with community structure showed better synchronization properties as compared to scale-free networks with community structure.

For networks with fixed network topology, when the total coupling strength between nodes is limited and the coupling strength between nodes is saturated, the global optimization algorithms including genetic algorithm (GA) and particle swarm optimization (PSO) algorithm are used to adjust the coupling strength between nodes to improve the synchronizability of the network, respectively. Simulation results show that in WS small-world network, when the edge betweenness centrality of the edge is large, the coupling strength of the edge after optimization is greater. Furthermore, compared with GA, PSO has better performance.

This paper investigates the synchronizability of multi-layer small-world dynamical networks and discusses the significant factors affecting the synchronizability. First, we introduce the network model of multi-layer small-world dynamical networks with one-to-one connections in the inter-layer connections and each layer has the same topology. Second, we consider the topological parameters including adding probability, intra-layer coupling strength, inter-layer coupling strength, number of nodes per layer, initial node degree, and number of network layers which will influence the network’s synchronizability. Third, we adopt the master stability function method and numerical simulations to analyze the synchronizability of the network. We also examine how the topological parameters influence the synchronizability of multi-layer small-world dynamical networks and the relationships among these parameters. Finally, synchronization control experiments are conducted to verify our results. As a result, we find that only increasing the number of nodes per layer will weaken the synchronizability of multi-layer small-world dynamical networks, while increasing other topological parameters will enhance the synchronizability. The current findings enable us to gain a deeper understanding of the synchronization behavior and characteristics of multi-layer small-world dynamical networks.

Synchronization phenomena in dynamical systems with local-world topologies are investigated. The effect of local-world size M on the synchronizability is studied. A larger M makes networks more synchronizable. Then the ability of dynamical networks to resist random errors and attacks is analyzed and compared with those of scale-free and exponential networks. Two attacking strategies are adopted, and a better quantity is used to measure the changes in synchronizability after node removal.

A network is referred to have locally optimal structure for synchronizability if its synchronizability is always decreased, no matter an arbitrary edge is added to it or an arbitrarily existed edge is deleted from it or an arbitrary edge is rewired in it. Although it is difficult to generally determine which networks have locally optimal structure, this paper through various examples demonstrates that symmetrical networks indeed have this property. Actually, it is found that any structural perturbations which destruct the symmetry of an originally symmetrical network would decrease the network's synchronizability, no matter how the perturbations change the network's average path length or clustering coefficient.

This paper studies the synchronizability of power network with community structure. Second-order Kuramoto-like oscillators with dissimilar natural frequencies are used as a coarse-scale model for an electrical power network that contains generators and consumers. The impact of community structure on frequency synchronization of power network is investigated, focusing on the parameters such as community strength, community number and connection strategy between communities. Numerical simulations show that increasing the community strength above a certain critical threshold or adding new communities to the network will be beneficial for the synchronization. Of course, connecting high-degree nodes among communities will be a best strategy to enhance synchronization. Furthermore, it is observed that the synchronizability of the network is significantly influenced by adding new links with different characteristics.

In this paper, the synchronizability characterized by the Laplacian spectrum is applied to windmill networks where three types of parameters are introduced to control the number of deleted nodes. Using the network’s structures, exact solutions of the Laplacian eigenvalues are obtained and metrics of the synchronizability are correspondingly shown. Relationships between the synchronizability and the introduced parameters are presented. Then, the synchronizability of models with different settings of node failures is compared. The obtained results reveal distinct synchronizabilities originating from intrinsic structures of models and the setting forms of node failures. Finally, numerical examples are provided to verify the effectiveness of theoretical analysis.

Currently, synchronizability of networks is mainly studied in terms of the eigen ratio of the coupling matrix, which is a pure property of network topology. In this work, we clarify that although the eigen ratio is relevant to the possible range of coupling strength for achieving synchronization, it cannot fully determine the latter. The magnitude of the eigenvalues also plays a decisive role. We emphasize that synchronizability of networks is inherently related to the local dynamics on networks. It is not appropriate to discuss synchronizability of networks without considering the specific dynamics on them. For three typical types of local dynamics, we discuss the implication of synchronizability of networks.

Networks with high propensity to synchronization are desired in many applications ranging from biology to engineering. In general, there are two ways to enhance the synchronizability of a network: link rewiring and/or link weighting. In this paper, we propose a new link weighting strategy based on the concept of the neighborhood subgroup. The neighborhood subgroup of a node i through node j in a network, i.e. $G^{i\to j}$, means that node u belongs to $G^{i\to j}$ if node u belongs to the first-order neighbors of j (not include i). Our proposed weighting schema used the local and global structural properties of the networks such as the node degree, betweenness centrality and closeness centrality measures. We applied the method on scale-free and Watts–Strogatz networks of different structural properties and show the good performance of the proposed weighting scheme. Furthermore, as model networks cannot capture all essential features of real-world complex networks, we considered a number of undirected and unweighted real-world networks. To the best of our knowledge, the proposed weighting strategy outperformed the previously published weighting methods by enhancing the synchronizability of these real-world networks.

In this study, we concentrate on the synchronizability of multilayer totally homogeneous networks. The eigenvalue spectrum of four typical multilayer networks, including multilayer nearest-neighbor regular networks, multilayer small-world networks, multilayer random networks and multilayer totally homogeneous networks, has been rigorously derived and the synchronizability of four typical multilayer networks has been compared due to the master stability function. Furthermore, numerical simulations are performed to validate the synchronizability of these multilayer networks. The results indicate that multilayer totally homogeneous networks exhibit optimal synchronizability. The Euler characteristic numbers and Betti numbers of these networks are derived, and their relationships with the synchronizability of multilayer networks are analyzed. We can observe that a negative correlation is exhibited between the Euler characteristic number and the synchronizability of multilayer networks, while a positive correlation is exhibited between the Betti number and the synchronizability of multilayer networks.