Robust network structures for conserving total activity in Boolean networks

One of the typical properties of biological systems is the law of conservation of mass, that is, the property that the mass must remain constant over time in a closed chemical reaction system. However, it is known that Boolean networks, which are a promising model of biological networks, do not always represent the conservation law. This paper thus addresses a kind of conservation law as a generic property of Boolean networks. In particular, we consider the problem of finding network structures on which, for any Boolean operation on nodes, the number of active nodes, i.e., nodes whose state is one, is constant over time. As a solution to the problem, we focus on the strongly-connected network structures and present a necessary and sufficient condition.