  Many networks exhibit scale free behavior where their degree distribution
obeys a power law for large vertex degrees. Models constructed to explain this
phenomena have relied on preferential attachment where the networks grow by the
addition of both vertices and edges, and the edges attach themselves to a
vertex with a probability proportional to its degree. Simulations hint, though
not conclusively, that both growth and preferential attachment are necessary
for scale free behavior. We derive analytic expressions for degree
distributions for networks that grow by the addition of edges to a fixed number
of vertices, based on both linear and non-linear preferential attachment, and
show that they fall off exponentially as would be expected for purely random
networks. From this we conclude that preferential attachment alone might be
necessary but is certainly not a sufficient condition for generating scale free
networks.
