This paper addresses the question of whether one can generate networks with a given global structure (defined by selected blockmodels, i.e., cohesive, core-periphery, hierarchical, and transitivity), considering only different types of triads. Two methods are used to generate networks: (i) the newly proposed method of relocating links; and (ii) the Monte Carlo Multi Chain algorithm implemented in the ergm package in R. Most of the selected blockmodel types can be generated by considering all types of triads. The selection of only a subset of triads can improve the generated networks' blockmodel structure. Yet, in the case of a hierarchical blockmodel without complete blocks on the diagonal, additional local structures are needed to achieve the desired global structure of generated networks. This shows that blockmodels can emerge based only on local processes that do not take attributes into account.