Simplifying Boolean Networks

This paper explores the compressibility of complex systems by considering the simplification of Boolean networks. A method, which is similar to that reported by Bastolla and Parisi,4,5 is considered that is based on the removal of frozen nodes, or stable variables, and network “leaves,” i.e. those nodes with outdegree = 0. The method uses a random sampling approach to identify the minimum set of frozen nodes. This set contains the nodes that are frozen in all attractor schemes. Although the method can over-estimate the size of the minimum set of frozen nodes, it is found that the chances of finding this minimum set are considerably greater than finding the full attractor set using the same sampling rate. Given that the number of attractors not found for a particular Boolean network increases with the network size, for any given sampling rate, such a method provides an opportunity to either fully enumerate the attractor set for a particular network, or improve the accuracy of the random sampling approach. Indeed, the paper also shows that when it comes to the counting of attractors in an ensemble of Boolean networks, enhancing the random sample method with the simplification method presented results in a significant improvement in accuracy.