Our results are shown in figures 2-4. In figure 2,
the convergence time in 10th-order mean field theory is
studied for rules of radius 2-4. Increasing line width
indicates increasing radius. Many rules were generated
at each possible value of 
, and the convergence-time
averaged over the set of rules with a given 
value.
For each radius a broad peak in convergence time is found.
The location and width of these peaks is in close correspondence
with previous empirical results using the 
parameter,
e.g. those Li et al. [LPL90]. Thus, our convergence-time
criterion provides an analytic measure of complexity in
quantitative agreement with empirical measures.
Figure 3 shows an example of how the mean field theory can
be used to refine the 
parameter analysis. Here
two mean field parameters, 
, and 
for radius-3 rules are varied,
while the other parameters
are held constant. To generate this figure,
all rules from figure 2 which happened to
fall in this hyperplane in mean-field parameter space were
selected. The height of the surface is this figure gives
the average convergence time for rules with specified
mean-field parameter values. Note that by restricting
consideration to this hyperplane, we greatly sharpen
the region in which rules of high convergence time
are found.
This point is emphasized in figure 4. In this figure, the
hyperplane is projected into the 
parameter space.
For this hyperplane, 
=
. It is seen that
the sharp mean-field peak is in the same location as the
parameterized peak. By increasing the order of the mean
field theory, we should be able to locate with greater
precision the region in parameter space in which high-convergence-
time reside.