Study On Models And Dynamics For Complex Networks

Network is a set composed of many nodes and edges which connect different nodes. Regular network and random network are two classic network forms. Regular network has a large cluster coefficient and average distance, while random network has a small cluster coefficient and small average distance. And great research progresses for these two kinds of networks have been made. Recently, many studies found that there exists some special network form between regular network and random network—complex network. This kind of network may have a large cluster coefficient and small average distance at the same time. A lot of the natural and social issues can be modeled using complex network. Thus the study of the characteristics of complex network, as well as the development and control laws has important practical significance.Complex network not only allows us to better understand and explain the complexity of real-world networks, but also can apply theoretical research findings to the real application and served for us.In this paper, we give a detailed survey for the model and dynamics of complex networks. This paper introduces the basic concepts of a complex network, such as degree of nodes, shortest path, clustering coefficient and the classification of complex networks. And we also discussed the model of complex network, such as random graph model, small world network model, and scale free network model. We hence discussed the differences between the three models and the complexity of the practical application of the network. We focus on the small-world network and scale-free network. Small-world network has a larger aggregation of factors and smaller average path length, the node distribution subject to the uniform distribution or exponential distribution. Scale-free network degree distribution of nodes subject to the rate of distribution of power, in such a network there are a lot of node degrees, node degrees but most are small. There is no specific scale. Small-world network and scale-free network have small world characteristics. They are widely applied to our practical life. We introduce the application of the small world network in Internet, biology, economics. At the same time, we analyze the relation between the scale free network and war, worm circulation, finance system.On the basis of the complex network dynamics and research methods, we study a delayed two-dimensional small-world network model, which is a complex system with time delay and non-linear item. The two-dimensional small-world network is modeled byWhere,ξ?0 is Newman Watts length scale, 0?μ??l is the non-linear metricin network. V(t) is tainted amount.τ≥0 is time lag.First, we used the eigenvalue methods to analyze the systematic stability of the trivial equilibrium and found that for any given time-delay the trivial solution of this system is unstable. Meanwhile, we examined the controlled model in which the linear time delay feedback control is added in this model. We then got the stable interval of this controlled system and derived the maximum controlled time delay for linear time-delay feedback models.In addition, the Hopf bifurcation theorem is used to get the conditions for Hopf bifurcation.Lemma5.1: The equilibrium point of the equation (?) = -kx(t) + Ax(t—T) is stable in the following regions.Theorem5.1: If parameters A and T satisfied the Lemma, and all roots of the equation F(ω)≡[ω~2+Acos(ωT)]~2 +[Asin(ωT)]~2 -k~2 = 0 are simple roots, thereexits aτ_c(A,T)∈[0,2π/(?)) , whenτ∈[0,τ_c(A,T)) , the trival solution of the equation (?) = -kx(t-τ)-bx~2(t -τ) + Ax(t-T) is asymptotically stable;τ_c(A,T) is the Hopf bifurcation point of the equation (?) = -kx(t -τ)-bx~2(t-τ) + Ax(t - T). These studies in this paper have got a good foundation for further study of complex networks.