  It has previously been shown that the network of connected minima on a
potential energy landscape is scale-free, and that this reflects a power-law
distribution for the areas of the basins of attraction surrounding the minima.
Here, we set out to understand more about the physical origins of these
puzzling properties by examining how the potential energy landscape of a
13-atom cluster evolves with the range of the potential. In particular, on
decreasing the range of the potential the number of stationary points increases
and thus the landscape becomes rougher and the network gets larger. Thus, we
are able to follow the evolution of the potential energy landscape from one
with just a single minimum to a complex landscape with many minima and a
scale-free pattern of connections. We find that during this growth process, new
edges in the network of connected minima preferentially attach to more
highly-connected minima, thus leading to the scale-free character. Furthermore,
minima that appear when the range of the potential is shorter and the network
is larger have smaller basins of attraction. As there are many of these smaller
basins because the network grows exponentially, the observed growth process
thus also gives rise to a power-law distribution for the hyperareas of the
basins.
