  A variant of Kauffman's model of cellular metabolism is presented. It is a
randomly generated network of boolean gates, identical to Kauffman's except for
a small bias in favor of boolean gates that depend on at most one input. The
bias is asymptotic to 0 as the number of gates increases. Upper bounds on the
time until the network reaches a state cycle and the size of the state cycle,
as functions of the number of gates $n$, are derived. If the bias approaches 0
slowly enough, the state cycles will be smaller than $n^c$ for some $c<1$. This
lends support to Kauffman's claim that in his version of random network the
average size of the state cycles is approximately $n^{1/2}$.
