Topics In Linear And Nonlinear Systems On Complex Networks

The study of complex networks has permeated from Mathematics and Physics to Biology, Information science, Engineering, and Sociology, and so on, with inter-disciplinary and fusion features. The study of dynamics on complex networks has become an interesting topic in complexity. In this paper, with Probability Theory, Stochastic Process and Differential Equation, we studied the biased random walks on the activity-driven time-varying network, and investigated the dynamics of nonlinear and linear quantum systems on complex networks with long-range steps. The main results are as follows.1. Since the degree of any given node in the instantaneous network is proportional to its activity (see the results in chapter two), we assumed the weight ωij of edge linking node i and node j as ωij=(kikj)θ (αiαj)θ, such a form of edge weights can be observed in various real networks, and investigated the biased random walk on time varying networks. In other words, the dynamics of the network evolved and the random walk proceed at the same time scale. By using the knowledge of homogeneous Markov chain, generating function method, we obtained the exact expressions of the stationary distribution, the mean first passage time between any two nodes, and the coverage function for three different edge-construction cases. We can tune the values of parameter θ to get more effective search strategy than the unbiased walk search strategy.2. we introduced the generalized random-walk rule by considering that the transi-tion probability Γij, not restricted to nearest neighbors, allowing transitions that follow a power law as a function of the shortest distance(integer number of steps) between nodes, this navigation strategy is more efficient than the normal random walk strategy. As far as we know, the long-range interaction in quantum mechanics declined as a power law function of the interaction nodes’distance, thus we studied the continuous time quantum walks on WS network with the generalized random-walk rule compared with the random walk which only allowed the motion to the nearest neighbors. If only considering the long-ranged interaction, it gives no influence to the transport efficiency; only considering rewiring probability p, localization phenomenon obviously occurred; If simultaneously considering these two factors, the relatively stronger localization have become.3. In order to analyze how the different network structure affected the nonlinear system (i.e. nonlinear schrodinger equation), using numerical method, we mainly considered the self-trapping transition of nonlinear system on one-dimensional regular ring with next-neighbor interaction. For the case of∈=-1.0, N=5, the results showed that the system with next nearest neighbor interactions is unstable, however it is stable under the nearest neighbor interactions; For the case of∈=-1.0, the same coupling strength of next-neighbor interaction hindered the self-trapping transition, while it almostly has no effect for∈=1.0case.Moreover, we also investigated two factors, the rewiring probability p which induced disorder structure and the long-range interaction, how they affect the self-trapping transition of nonlinear systems respectively, compared to the case of considering these two factors simultaneously.