An extended scale-free network evolution model based on global coupling motif embedding

The BA scale-free network evolution model assumes that one node enters at every unit time, which does not adequately reflect team entries that usually occur during the evolution of many practical networks, i.e. the phenomenon of motif embedment. Unfortunately, there are no specific studies on how the motif embedment mechanism affects the degree distribution of networks. In order to solve this problem, an extended scale-free network evolution model with global coupling motif embedment and with the motif size obeying an arbitrary discrete probability distribution (called the MEEBA model for short) is formulated with the help of the 'motif' concept. Using the Markov chain method, the accurate analytical expression of the network degree distribution of the MEEBA model is obtained and the correctness of the equation is verified through comparisons with numerical simulation results. The study results show that the right tail of the degree distribution of the MEEBA model still has a power-law behavior, while its left tail reflects the horse-head-like shapes of the degree distribution of many real networks. Furthermore, the power-law exponent and the horse-head shapes are both related to the distribution of the motif size. When the motif size follows a one-point distribution, the network degree distribution of the MEEBA model degenerates into a two-parameter Waring-like distribution and its horse-head shapes disappear. In particular, if the expectation of the one-point distribution is 1, the MEEBA model further degenerates to the BA model. Finally, the practicality and effectiveness of the MEEBA model are verified through a case study. (paper)