Dynamic Processes On Scale-free Network And Weighted Network

Many social, biological, and communication systems can be properly described by complex networks. One important characteristic features in manyof these real-world networks is scale-free degree distribution. This distributionimplies that each vertex has a statistically significant probability of having avery large number of connections compared to the average degree of the network. Another feature is the weighted network whose edges having differentvalue of weight to characterize the interaction strength in real networks. Here,we inspect the effect of these complex connection features on the dynamicalprocesses taking place in networks.We first present the main results and analytical method for the behavior ofepidemic spread model with a feedback mechanism on scale-free networks. Weobtained that the infection prevalence in steady states has the self-consistencyformρ(?) 2e^{-1/mλ(1-αρ)} and the linear relationship between the reciprocal of thedensity of infected node 1/ρand lethality constantα,while the spreading rateλis fixed. These findings are extremely different from that obtained in exponential networks and give an important understanding of the epidemic dynamicswith feedback mechanics on complex networks. We studied the SIS model withtime delay and found out that the relationship between the spread thresholdλ_cand delay time T isλ_c～(T+1)^-1.Secondly, we study a simple population model with diffusion and birth/deathaccording to 2A→3A and A→φfor an individual A on scale-free networks.In the fully random diffusion case, the network topology can not affect the critical death rate, but the heterogenous connectivity will cause smaller steady population density and critical population density. In the modified diffusion strategycase, we can obtain larger critical death rate and steady population density, atthe meanwhile, lower critical population density, which is good for the species survival. All these mean-field analytical results are confirmed by computer simulations.Thirdly, we investigate the behavior of the excitable Greenberg-Hastingscellular automaton (GHCA) model on scale-free network and other kinds ofcomplex networks. We find that a maximal range precisely occuring at thecritical point, this phenomena represents a universal behavior of the excitableGHCA model on complex networks. Due to strong fluctuations in the degreedistribution on scale-free network, we calculated the average activity F^k(r) andthe dynamic range△^k (p) for nodes with given connectivity k. The two quantities F^k(r) and△^k(p) have the similar behavior as that of F(r) and△(p) inwhole network, respectively. It is interesting that nodes with larger degree havelarger optimal range. This property could be applied in biological experimentrevealing the network topology.Finally, we investigate the behavior of walks on weighted networks. Theexact expressions for the stationary distribution and the average return timeare derived for the two walk processes: weight-dependent walk and strengthdependent walk; then, we study the mean first passage time (MFPT) to edgein random walks on weighted networks and obtained the expressions for theMFPT from a node to edge along one direction,the MFPT from a node to anedge aong both directions, and the average MFPT to edge. All these expressionsare in excellent agreement with numerical results of random walks on weightednetworks.