  We study the growth of random networks under a constraint that the diameter,
defined as the average shortest path length between all nodes, remains
approximately constant. We show that if the graph maintains the form of its
degree distribution then that distribution must be approximately scale-free
with an exponent between 2 and 3. The diameter constraint can be interpreted as
an environmental selection pressure that may help explain the scale-free nature
of graphs for which data is available at different times in their growth. Two
examples include graphs representing evolved biological pathways in cells and
the topology of the Internet backbone. Our assumptions and explanation are
found to be consistent with these data.
