A whole spectrum of networks is generated with a quasistatic growth model. N nodes are located in a 2D lattice and a fixed number of links are added between them. Links are created either preferentially with strenght γ or with a nearest neighbor, thus obtaining from regular lattices to random or scale free networks. Throw simulations, we study the efficiency of a search and congestion algorithm on such networks and relate the results to their topology. Some new figures and techniques are presented that clarify results in previous works. Finally, a certain change in the average load of networks for γ>l, which had no previous explanation, is found to be related with a transition in the average degree of the most connected node.