Consider a large Boolean network with a feed forward structure. Given a
probability distribution on the inputs, can one find, possibly small,
collections of input nodes that determine the states of most other nodes in the
network? To answer this question, a notion that quantifies the determinative
power of an input over the states of the nodes in the network is needed. We
argue that the mutual information (MI) between a given subset of the inputs X =
{X_1, ..., X_n} of some node i and its associated function f_i(X) quantifies
the determinative power of this set of inputs over node i. We compare the
determinative power of a set of inputs to the sensitivity to perturbations to
these inputs, and find that, maybe surprisingly, an input that has large
sensitivity to perturbations does not necessarily have large determinative
power. However, for unate functions, which play an important role in genetic
regulatory networks, we find a direct relation between MI and sensitivity to
perturbations. As an application of our results, we analyze the large-scale
regulatory network of Escherichia coli. We identify the most determinative
nodes and show that a small subset of those reduces the overall uncertainty of
the network state significantly. Furthermore, the network is found to be
tolerant to perturbations of its inputs