Scale-free growing networks imply linear preferential attachment -: art. no. 017102

It has been recognized for some time that a network grown by the addition of nodes with linear preferential attachment will possess a scale-free distribution of connectivities. Here we prove by some analytical arguments that the linearity is a necessary component to obtain this kind of distribution. However. the preferential linking rate does not necessarily apply to single nodes, but to groups of nodes of the same connectivity. We also point out that for a time-varying mean connectivity the linking rate A ill deviate from a linear expression by an extra asymptotically logarithmic term.