Consider a large Boolean network with a feed forward structure. Given a probability distribution for the inputs, can one find-possibly small-collections of input nodes that determine the states of most other nodes in the network? To identify these nodes , a notion that quantifies the determinative power of an input over states in the network is needed. We argue that the mutual information (MI) between a  subset of the inputs X = {X_1, ..., X_n} of  node i and the function f_i(X) associated with node i quantifies the determinative power of this subset of inputs over node i. To study the relation of determinative power to sensitivity to perturbations , we relate the MI to measures of perturbations, such as the influence of a variable, in terms of inequalities. The result shows that, maybe surprisingly, an input that has large influence does not necessarily have large determinative power. The main tool for the analysis is Fourier analysis of Boolean functions. Whether a function is sensitive to perturbations or not, and which are the determinative inputs, depends on which coefficients the Fourier spectrum is concentrated on. We also consider unate functions which play an important role in genetic regulatory networks . For those, a particular relation between the influence and MI is found . As an application of our methods , we analyze the large-scale regulatory network of E. colinumerically: We identify the most determinative nodes and show that a small set of those reduces the overall uncertainty of network states significantly. The network is also found to be tolerant to perturbations of its inputs , which can be seen from the Fourier spectrum of its functions .
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 .
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