How is that complex network complex?

Evidence of complex networks in real world settings abounds. Many data sets for physical and social systems display characteristics consistent with various models of complex networks - the most typical examples being scale-free and small-world networks. However, theory does not always match reality. While we see a wide range of real complex networks, simulated data most usually comes from a limited range of generative models (the Barabási-Albert model for scale-free networks, Watt-Strogatz's model for small world networks, and Erdos-Renyi's model of a random graph are the three usual archetypes). We argue that there is much to be learnt by examining what real world data does that these algorithms do not. To do this we propose a variety of new network generation algorithms. These algorithms allow us to sample, in a statistically unbiased manner, from the family of all networks (of a given size N) consistent with a given degree distribution. Using this technique we are able to determine which distributions really are likely origins for various observed data and (equally importantly) observe when particular real world networks are atypical. Examples include the observation that many collaboration networks are not consistent with the Barabási-Albert (BA) model but are typical of the family of graphs that exhibit a power-law degree distribution, Biological networks (protein-protein interaction and cellular metabolic processes) are scale-free (but not BA) networks with atypically large diameter.