There is mounting evidence of the apparent ubiquity of scale-free networks among complex systems. Many natural and physical systems exhibit patterns of interconnection that conform, approximately, to the structure expected of a scale-free network. We propose an efficient algorithm to generate representative samples from the space of all networks defined by a particular (scale-free) degree distribution. Using this algorithm we are able to systematically explore that space with some surprising results: in particular, we find that preferential attachment growth models do not yield typical realizations and that there is a certain latent structure among such networks --- which we loosely term "hub-centric". We provide a method to generate or remove this latent hub-centric bias --- thereby demonstrating exactly which features of preferential attachment networks are atypical of the broader class of scale free networks. Based on these results we are also able to statistically determine whether experimentally observed networks are really typical realizations of a given degree distribution (scale-free degree being the example which we explore). In so doing we propose a surrogate generation method for complex networks, exactly analogous the the widely used surrogate tests of nonlinear time series analysis.