Time-cumulative scale-free networks without both growth and preferential attachment

A model of nongrowing networks with time-cumulative scale-free (SF) property based on the apply–reply process is proposed in this paper. This model is not only without growth but also without a direct expression of preferential attachment. The network structure is evolved by the change of nodes’ active probability, which is mathematically equivalent to a special diffusion process. This model only allows zero or one edge connecting on a node at any moment, and the power-law degree distributions can be exhibited from the statistic of the network after a long time accumulation. Our results imply that both growth and direct preferential attachments are not necessary in the generation of the SF property.