MOTIVATION: Given a large-scale biological network represented as an influence graph, in this article we investigate possible decompositions of the network aimed at highlighting specific dynamical properties. RESULTS: The first decomposition we study consists in finding a maximal directed acyclic subgraph of the network, which dynamically corresponds to searching for a maximal open-loop subsystem of the given system. Another dynamical property investigated is strong monotonicity. We propose two methods to deal with this property, both aimed at decomposing the system into strongly monotone subsystems, but with different structural characteristics: one method tends to produce a single large strongly monotone component, while the other typically generates a set of smaller disjoint strongly monotone subsystems. AVAILABILITY: Original heuristics for the methods investigated are described in the article.