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Transitive reductions and minimal equivalent subgraphs have proven to be a powerful concept to simplify networks and to measure their redundancy. Here we consider a generalization of the minimal equivalent subgraph problem where a set of arcs is already given. For two digraphs D = (V,A), D′ = (V,A′) with A′ ⊆ A, we ask for the minimal set of edges of D that have to be added to D′ such that the transitive closure of D equals the transitive closure of D′.
We present a method to compute such an extension and show that if D is transitively closed, this problem can be solved in polynomial time.
Multi-valued network models can be described by their topology and a set of parameters capturing the effects of the regulators for each component. Dynamics can then be derived and represented as state transition systems.
Different network models may lead to the same transition system, meaning dynamics analysis of a representative model covers a larger class of models. While rather clear in the Boolean case, the properties contributing to this effect become more involved for multi-valued models. We analyse these properties and present a mathematical description of the resulting model equivalence classes.