  A network is scale-free if its connectivity density function is proportional
to a power-law distribution. Scale-free networks may provide an explanation for
the robustness observed in certain physical and biological phenomena, since the
presence of a few highly connected hub nodes and a large number of small-
degree nodes may provide alternate paths between any two nodes on average --
such robustness has been suggested in studies of metabolic networks, gene
interaction networks and protein folding. A theoretical justification for why
biological networks are often found to be scale-free may lie in the well-known
fact that expanding networks in which new nodes are preferentially attached to
highly connected nodes tend to be scale-free. In this paper, we provide the
first efficient algorithm to compute the connectivity density function for the
ensemble of all secondary structures of a user-specified length, and show both
by computational and theoretical arguments that preferential attachment holds
when expanding the network from length n to length n + 1 structures. Since
existent power-law fitting software, such as powerlaw, cannot be used to
determine a power-law fit for our exponentially large RNA connectivity data, we
also implement efficient code to compute the maximum likelihood estimate for
the power-law scaling factor and associated Kolmogorov-Smirnov p-value.
Statistical goodness-of-fit tests indicate that one must reject the hypothesis
that RNA connectivity data follows a power-law distribution. Nevertheless, the
power-law fit is visually a good approximation for the tail of connectivity
data, and provides a rationale for investigation of preferential attachment in
the context of macromolecular folding.
