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When you say that you randomly generated graphs, how did you sample from graph-space? If you used something like an Erdős–Rényi generating function (which appears to be the case to me) then your graphs are "random," but represent only a tiny subset of the structures seen in many other graphs. For example, the degrees of vertices in your ER graphs should follow a normal distribution with a mean governed by your 'probability' parameter, while many other graphs follow exponential or power-law distributions where a small number of vertices hoard many more edges than their peers.