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

As the only brand-new basic model that can study networks with scale change characteristics at present, the Barabási–Albert scale-free network evolution model (BA model) can well describe the entry phenomenon of a single subject. However, for many evolutionary networks in the real world, the entry of the subject is more realized in a team way. Therefore, the BA model’s assumption concerning one new node entering alone per unit of time can only be used as a particular case of a team entry. This deficiency makes it difficult for the BA model to analyze how the team entry mechanism affects the network’s performance and thus leads to the BA model’s certain limitations. To make up for this shortcoming, with the help of the motif’s concept, an extended BA model with a star-like coupling motif embedding (SCME-BA model) is constructed, where the motif’s scale obeys an arbitrary discrete probability distribution. The exact explicit expression of the network’s steady-state degree distribution of the SCME-BA model is first obtained by the Markov chain analytic method, and its numerical characteristics are explored. Then, the correctness of the analytic expression is verified by numerical simulation. The study results show that, compared with the BA model, the left tail of the SCME-BA model’s steady-state degree distribution can reflect the distribution law of the embedded motif scale well while the right one still has power-law attenuation characteristics whose power-law exponent is adjustable. Specifically, the power-law exponent is positively and negatively correlated with the expectation of the motif’s scale and the initial number of existing nodes connected to the new embedded motif, respectively. Moreover, the SCME-BA model can degenerate into the BA model when the motif’s scale is subject to the one-point distribution with parameter 1. Finally, the rationality of the SCME-BA model is verified through a case study.