Network Structure and Activity in Boolean Networks

Abstract : In this paper we extend the notion of activity for Boolean networks introduced by Shmulevich and Kauffman (2004). Unlike the existing notion, we take into account the actual graph structure of the Boolean network. As illustrations, we determine the activity of all elementary cellular automata, and $d$-regular trees and square lattices where the vertex functions are bi-threshold and logical nor functions.The notion of activity measures the probability that a perturbation in an initial state produces a different successor state than that of the original unperturbed state. Apart from capturing sensitive dependence on initial conditions, activity provides a possible measure for the significance or influence of a variable. Identifying the most active variables may offer insight into design of, for example, biological experiments of systems modeled by Boolean networks. We conclude with some open questions and thoughts on directions for future research related to activity.