Among the many (mostly randomized) models proposed in the last decades to study how opinions of a set of individuals interconnected by pairwise relations evolve, a novel deterministic model is introduced in this paper that is able to encompass individual choices, strength and sign of relations, and asynchronism. In particular, asynchronism has been considered until now only in randomized settings. It is here studied in which cases the behavior of the resulting dynamical network is predictable, that is, in which cases the number of opinion configurations encountered by the set of individuals before the dynamical network enters a loop is polynomially bounded by the network size.