Most agent-based models include a social network that describes the structure of interactions within the artificial population. Because of the dramatic impact of this structure on the simulated dynamics, modelers create this network for it to match criteria of plausibility (e.g. the small-world property). Networks are actually created by one network generator compliant with these criteria, like the Watts-Strogatz algorithm in the case of small-world networks. However, this practice comes to study the model's dynamics over the specific networks generated by one algorithm instead of the dynamics over the class of networks of interest, possibly inducing a strong bias in results. We identify three problematics related to this bias: (i) representativity of a network generator to a class of networks, (ii) conclusiveness of simulations over a class of networks and (iii) the gain in conclusiveness when refining the criteria for network choice. We propose an experimental protocol and instantiate it on small-world networks for epidemics, opinion and culture dynamics. We show that (i) Watts-Strogatz networks are not representative of small-world networks (ii) simulation results over small-worlds are arguably inconclusive, and (iii) even small-world networks having the same size, density, transitivity and average path length do not lead to coherent results. Beyond questioning the relevance of simulation results obtained from artificial networks, this research also constitute one more argument for the exploration of other approaches that are not solely focused on networks' statistical properties.