Activities and sensitivities in boolean network models

Abstract

We study how the notions of importance of variables in Boolean functions as well as the sensitivities of the functions to changes in these variables impact the dynamical behavior of Boolean networks. The activity of a variable captures its influence on the output of the function and is a measure of that variable's importance. The average sensitivity of a Boolean function captures the smoothness of the function and is related to its internal homogeneity. In a random Boolean network, we show that the expected average sensitivity determines the well-known critical transition curve. We also discuss canalizing functions and the fact that the canalizing variables enjoy higher importance, as measured by their activities, than the noncanalizing variables. Finally, we demonstrate the important role of the average sensitivity in determining the dynamical behavior of a Boolean network.

FIG. 1

(color online). A 2-dimensional normalized histogram constructed by generating all Boolean functions of K 4 variables and computing the number of functions with a given normalized Hamming weight (which is the Hamming weight divided by 16) and with a given average sensitivity. For each fixed Hamming weight, the histogram of average sensitivities is normalized such that the sum is equal to 1, so that we can interpret each bar as a probability.

FIG. 2

(color online). Derrida curves corresponding to two different random Boolean networks, both with N = 100 elements and K = 8 inputs in each Boolean function. The solid curve corresponds to the case where one activity is much higher than all other activities, in each Boolean function. The dashed curve corresponds to the case where all activities are equal. The internal homogeneity for both networks is the same.