Structure properties of evolutionary spatially embedded networks

This work is a modeling of evolutionary networks embedded in one or two dimensional configuration space. The evolution is based on two attachments depending on degree and spatial distance. The probability for a new node n to connect with a previous node i at distance rni follows aki∑jkj+(1−a)rni−α∑jrnj−α, where ki is the degree of node i, α and a are tunable parameters. In spatial driven model (a=0), the spatial distance distribution follows the power-law feature. The mean topological distance l and the clustering coefficient C exhibit phase transitions at same critical values of α which change with the dimensionality d of the embedding space. When a≠0, the degree distribution follows the “shifted power law” (SPL) which interpolates between exponential and scale-free distributions depending on the value of a. ► Spatially constrained network is a promising type of complex network. ► The model with two kinds of preferential attachment is closer to real network. ► The network structure exhibits phase transitions at two critical values. ► The degree distribution interpolates between exponential and scale-free distribution.