Lipschitz property for systems of linear mappings and bilinear forms

Let G be a graph with undirected and directed edges. Its representation is given by assigning a vector space to each vertex, a bilinear form on the corresponding vector spaces to each directed edge, and a linear map to each directed edge. Two representations A and A′ of G are called isomorphic if there is a system of linear bijections between the vector spaces corresponding to the same vertices that transforms A to A′. We prove that if two representations are isomorphic and close to each other, then their isomorphism can be chosen close to the identity.