In this paper, subgraphs and complementary graphs are used to analyze the network synchronizability. Some sharp and attainable bounds are provided for the eigenratio of the network structural matrix, which characterizes the network synchronizability, especially when the network’s corresponding graph has cycles, chains, bipartite graphs or product graphs as its subgraphs.

In this paper, a new susceptible-infected-susceptible model with infective medium is proposed, which describes epidemics (e.g. malaria) transmitted by infective media (e.g. mosquitoes) on various complex networks. The dynamic behaviours of the model on a homogeneous network and a heterogenous scale-free network are considered, respectively, where the absence of the threshold on the scale-free network is demonstrated. Then, analytical and simulated results are given to show that the proportional immunization strategy to the new model is very effective on scale-free networks. Furthermore, it is shown that the immune density of nodes depends not only on the infectivity between individual persons, but also on the infectivity between persons and mosquitoes. This reveals that the absence of a critical immunization threshold not only is due to the unbounded connectivity fluctuation of the underlying scale-free network, but also is determined by the path of the epidemic spreading.

In this paper, stability analysis and decentralized control problems are addressed for linear and sector-nonlinear complex dynamical networks. Necessary and sufficient conditions for stability and stabilizability under a special decentralized control strategy are given for linear networks. Especially, two types of linear regular networks, star-shaped networks and globally coupled networks, are studied in detail. A dynamical network is viewed as a large-scale system composing of some subsystems, based on which the relationship between the stability of a network and the stability of its corresponding subsystems is investigated. It is pointed out that some subsystems must be unstable for the whole network to be stable in some special cases. Moreover, a controller design method based on a parameter-dependent Lyapunov function is provided. Furthermore, interconnected Lur’e systems and symmetrical networks of Lur’e systems are similarly studied. The test of absolute stability of a network of Lur’e systems is separated into the test of absolute stability of several independent Lur’e systems. Finally, several numerical examples are given to illustrate the theoretical results.

In this paper, the commonly concerned issue of synchronization regions of complex dynamical networks is investigated, for the case when the synchronous state is an equilibrium point. Some simple sufficient conditions for a network to have or have no unbounded synchronization regions of the form ð 1; a1Þ are established, where a1 is a constant. In addition, a sufficient condition for the existence of a bounded synchronization region of the form ða2; a3Þ is derived, where a2 and a3 are constants, by using the parameter-dependent Lyapunov function method. Furthermore, some effective controller design methods are presented that can change the synchronization regions, thereby managing the synchronizability of the network. Finally, some numerical examples are given to show that a dynamical network may have disconnected synchronization regions, particularly it may have the coexistence of unbounded and bounded synchronization regions in the form of ð 1; a1Þ [ ða2; a3Þ].

This paper considers global properties of a class of nonlinear systems with infinite equilibria. Time-domain and frequency-domain criteria for dichotomy and gradient-like behavior are established. Similar to the sector condition in absolute stability problems, the bounded derivative condition of nonlinear functions can be used to reduce the conservativeness of the given criteria. Compared with the frequency-domain conditions, the time-domain inequalities can be solved easily by linear matrix inequality (LMI) toolbox and used to discuss controller design problems. Several examples are given to illustrate the effectiveness of the results.