The Traffic Of Particles On The Complex Network

Recent years, condensation on the complex network has been studied. In 2005, Noh has studied the Zero Range Process on the scale-free network with the grand canonical ensemble. Then M. Tang's mean-field approach gave the same results. In Zero Range Process, the particles in the same node have reactions with each other, and no reactions between the particles in different nodes. The jumping rate of each particle is related with the number of particles occupied on the node. If the jumping rate is smaller than the threshold, a large part of particles will condense on the hub node. Our work consists of two parts. Firstly, we consider the jumping rate of individuals in the phenomenon of condensations of the real social life is varying, For example, during the moring rush hour, people go to the work place from home, then in the same period the number of travellers is much more than that of the other time, thus the jumping rate of individual must be variational. So, we change the invariable jumping rate into a time varying quantity. Concretely, we make the jumping rate change according to the sine function and analyze the time series of the number of particles assembling on the hub node to explore the influence of the varying jumping rate on ZRP condensation. We find that the time series of the number of particles changes with the same period of the varying jumping rate. And the fluctuation of the series becomes larger and larger when the fluctuation of the jumping rate.Bianconi and Barab'asi have studied the condensation on the complex networks. By treating each link as a quantum particle, they found that at low temperature, the hub will take most of the particles, i.e., BEC. BEC on the complex network is not a condensation of real particles, but a competition and condensation of the edges. Our second work is to investigate the influences of the temperature on the particle condensation. We build a trap model to exploe the competition between the reaction of particles and the temperature which leads to the phenomenon of condensation. We consider the reaction among the particles in the same node as a trap, that is to say a node is a trap. The trap keeps particles staying there for a longer time or to prevent the escaping of all the particles at the same time. The stronger the attractive force is, the deeper the trap.The particles jump between the traps under the principle of ZRP. The jumping rate is depend on the depth of the trap and the temperature of the network. Suppose the potential energy of trap is determined by the degree of node.,(1) Ei=ki, (2) Ei=(?) and (3) Ei=lnki. We use the mean theory to classify the nodes by the degree to investigate the influence of the network topology on the dynamics of the system. We find that there is a optimal temperature Tc. When temperature is too low, the particles will be trapped in the traps and result in the lack of moving particles. On the other hand, when temperature is too high, the particles are very easy to move out of the traps and do not make them stay at the hub at the same time. Thus, only when the temperature is near, the particles will move out and stay at the hub for a relatively long time. This makes the phenomenon of condensation easy to show up at the hub. What's more, the optimal value Tc is not only depend on the depth of the trap, but also the topology of the network. Compared to the random network, the condensation is more likely to happen on the scale-free network and the phenomenon is more remarkable.