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 r n i follows a k i ∑ j k j + ( 1 − a ) r n i − α ∑ j r n j − α , where k i 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 .