We study a system of perturbed graph-directed iterated functions. In our situation, the unperturbed system has several Gibbs measures associated with the dimension of the limit set and on the other hand, the perturbed system possesses a unique Gibbs measure for   Namely, our perturbed system forms a metastable state. In this paper, under suitable conditions, we give a necessary and sufficient condition for convergence  weakly for some probability vector  in terms of solutions of Bowen’s equations and measure-theoretic entropies of Gibbs measures.

graph directed function systems, metastable system, perturbation.