This paper addresses the fundamental problem of complex network synchronizability from a graphtheoretic approach. First, the existing results are briefly reviewed. Then, the relationships between the network synchronizability and network structural parameters e.g., average distance, degree distribution, and node betweenness centrality are discussed. The effects of the complementary graph of a given network and some graph operations on the network synchronizability are discussed. A basic theory based on subgraphs and complementary graphs for estimating the network synchronizability is established. Several examples are given to show that adding new edges to a network can either increase or decrease the network synchronizability. To that end, some new results on the estimations of the synchronizability of coalescences are reported. Moreover, a necessary and sufficient condition for a network and its complementary network to have the same synchronizability is derived. Finally, some examples on Chua circuit networks are presented for illustration.

Real neurons can exhibit various types of firings including tonic spiking, bursting as well as silent state, which are frequently observed in neuronal electrophysiological experiments. More interestingly, it is found that neurons can demonstrate the co-existing mode of stable tonic spiking and bursting, which depends on initial conditions. In this paper, synchronization in gap-junctioncoupled neurons with co-existing attractors of spiking and bursting firings is investigated as the coupling strength gets increased. Synchronization transitions can be identified by means of the bifurcation diagram and the correlation coefficient. It is illustrated that the coupled neurons can exhibit different types of synchronization transitions between spiking and bursting when the coupling strength increases. In the course of synchronization transitions, an intermittent synchronization can be observed. These results may be instructive to understand synchronization transitions in neuronal systems.

We study the spatial dynamics of spiral waves in noisy Hodgkin–Huxley neuronal ensembles evoked by different information transmission delays and network topologies. In classical settings of coherence resonance the intensity of noise is fine-tuned so as to optimize the system’s response. Here, we keep the noise intensity constant, and instead, vary the length of information transmission delay amongst coupled neurons. We show that there exists an intermediate transmission delay by which the spiral waves are optimally ordered, hence indicating the existence of delay-enhanced coherence of spatial dynamics in the examined system. Additionally, we examine the robustness of this phenomenon as the diffusive interaction topology changes towards the small-world type, and discover that shortcut links amongst distant neurons hinder the emergence of coherent spiral waves irrespective of transmission delay length. Presented results thus provide insights that could facilitate the understanding of information transmission delay on realistic neuronal networks.

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, 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].