Incremental Update for Graph Rewriting

Graph rewriting formalisms are well-established models for the representation of biological systems such as protein-protein interaction networks. The combinatorial complexity of these models usually prevents any explicit representation of the variables of the system, and one has to rely on stochastic simulations in order to sample the possible trajectories of the underlying Markov chain. The bottleneck of stochastic simulation algorithms is the update of the propensity function that describes the probability that a given rule is to be applied next. In this paper we present an algorithm based on a data structure, called extension basis, that can be used to update the counts of predefined graph observables after a rule of the model has been applied. Extension basis are obtained by static analysis of the graph rewriting rule set. It is derived from the construction of a qualitative domain for graphs and the correctness of the procedure is proven using a purely domain theoretic argument.