Structure and behavior in Boolean monotonic model pools

One of the main theoretical questions in the field of discrete regulatory networks is the question what aspects of the dynamics - the structure of the state transition graph - are already imposed by structural descriptions of the network such as the interaction graph. For Boolean networks, prior work has concentrated on different versions of the Thomas conjectures that link feedback cycles in the network structure to attractor properties. Other approaches check algorithmically whether certain properties hold true for all models sharing specific structural constraints, e.g. by using model checking techniques. In this work we investigate the behavior of the pool of Boolean networks in agreement with a given interaction graph using a different approach. Grouping together states that are updated consistently across the pool we derive an equivalence relation and analyze a corresponding quotient graph on the state space. By construction this graph yields information about the dynamics of all functions in the pool. Our main result is that this graph can be computed efficiently without enumerating and analyzing all individual functions. This opens up new possibilities for applications, where such model pools arise when modeling under uncertainty.