A network forming four-site model associative fluid (with freely located sites) is investigated by means of associative Ornstein-Zernike integral equation theory supplemented by a Percus-Yevick-like closure relation. Since the model exhibits critical behavior, the structure relevant to the gaseous and to the liquid phases are discussed. The properties of network forming systems with different strengths of the site-site attraction are analyzed. This allows us to describe topologically asymmetrical network clusters and branching polymers. It is determined that the critical temperature as well as the critical density become lower with an increasing degree of asymmetry. NVT Monte Carlo simulations for the same model, but with a fixed location of sites, are presented. Theoretical predictions are compared to the simulation results. It is shown that the theory agrees well with the simulations, except for low densities and temperatures, where the simulations predict a well developed waterlike structure with a tetrahedral arrangement. This disagreement is shown to be caused by the difference in the site location imposed by the model potentials.