  We discuss basic features of emergent complexity in dynamical systems far
from equilibrium by focusing on the network structure of their state space. We
start by measuring the distributions of avalanche and transient times in Random
Boolean Networks (RBNs) and in the \emph{Drosophila} polarity network by exact
enumeration. A transient time is the duration of the transient from a starting
state to an attractor. An avalanche is a special transient which starts as
single Boolean element perturbation of an attractor state. Significant
differences at short times between the avalanche and the transient times for
RBNs with small connectivity $K$ -- compared to the number of elements $N$ --
indicate that attractors tend to cluster in configuration space. In addition,
one bit flip has a non-negligible chance to put an attractor state directly
onto another attractor. This clustering is also present in the segment polarity
gene network of \emph{Drosophila melanogaster}, suggesting that this may be a
robust feature of biological regulatory networks. We also define and measure a
branching ratio for the state space networks and find evidence for a new time
scale that diverges roughly linearly with $N$ for $2\leq K \ll N$. Analytic
arguments show that this time scale does not appear in the random map nor can
the random map exhibit clustering of attractors. We further show that for K=2
the branching ratio exhibits the largest variation with distance from the
attractor compared to other values of $K$ and that the avalanche durations
exhibit no characteristic scale within our statistical resolution. Hence, we
propose that the branching ratio and the avalanche duration are new indicators
for scale-free behavior that may or may not be found simultaneously with other
indicators of emergent complexity in extended, deterministic dynamical systems.
