  Models based on preferential attachment have had much success in reproducing
the power law degree distributions which seem ubiquitous in both natural and
engineered systems. Here, rather than assuming preferential attachment, we give
an explanation of how it can arise from a more basic underlying mechanism of
competition between opposing forces.
  We introduce a family of one-dimensional geometric growth models, constructed
iteratively by locally optimizing the tradeoffs between two competing metrics.
This family admits an equivalent description as a graph process with no
reference to the underlying geometry. Moreover, the resulting graph process is
shown to be preferential attachment with an upper cutoff. We rigorously
determine the degree distribution for the family of random graph models,
showing that it obeys a power law up to a finite threshold and decays
exponentially above this threshold.
  We also introduce and rigorously analyze a generalized version of our graph
process, with two natural parameters, one corresponding to the cutoff and the
other a ``fertility'' parameter. Limiting cases of this process include the
standard Barabasi-Albert preferential attachment model and the uniform
attachment model. In the general case, we prove that the process has a power
law degree distribution up to a cutoff, and establish monotonicity of the power
as a function of the two parameters.
