Generating Fibonacci-model as evolution of networks with vertex-velocity and time-memory

The problem of how to clearer account for the significant evolutional mechanism gradually driving seemingly random complex networks into ones that have a few typical characters is of great interest in various science communities. Such discussed characters include small-world property, scale-free feature, and self-similarity, etc. However, most of previously generated models have no time memory, that is, the probability for each old vertex i of degree ki to obtain a new link from young vertex does only depend on its current degree. To address this issue, in this paper, we propose a class of prototype of evolving complex network models based on the additional feature, i.e., time memory. In contrast with presented models lacking of time memory, we can find that our model still follows power-law form of vertex-degree distribution and has almost the smallest diameter. More interestingly, this diameter of our model does not increase continuously with the development of model over time, but has discontinuity growth. From the generalization point of view, this thought behind the construction of our model can be thought of as a more general fashion in which a large number of published models can be rebuilt. With the validity of our model, it is worth to note that in the growth process of deterministic network models the number m of each existing vertex i obtaining new edges may be proportional to its degree ki, not always an integer multiple of ki. So as to describe and generate more available deterministic models as complex networks, we here introduce another parameter, vertex-velocity, corresponding to dynamic function on networks, which will be carefully discussed in this paper.