  Despite their apparent simplicity, random Boolean networks display a rich
variety of dynamical behaviors. Much work has been focused on the properties
and abundance of attractors. The topologies of random Boolean networks with one
input per node can be seen as graphs of random maps. We introduce an approach
to investigating random maps and finding analytical results for attractors in
random Boolean networks with the corresponding topology. Approximating some
other non-chaotic networks to be of this class, we apply the analytic results
to them. For this approximation, we observe a strikingly good agreement on the
numbers of attractors of various lengths. We also investigate observables
related to the average number of attractors in relation to the typical number
of attractors. Here, we find strong differences that highlight the difficulties
in making direct comparisons between random Boolean networks and real systems.
Furthermore, we demonstrate the power of our approach by deriving some results
for random maps. These results include the distribution of the number of
components in random maps, along with asymptotic expansions for cumulants up to
the 4th order.
