Limits of dense graph sequences

We show that if a sequence of dense graphs has the property that for every fixed graph F, the density of copies of F in these graphs tends to a limit, then there is a natural “limit object”, namely a symmetric measurable function 2-variable function on [0,1]. This limit object determines all the limits of subgraph densities. Conversely, every such function arises as a limit object. We also give a characterization of the graph parameters arising as limits in terms of “reflection positivity”. Along the lines we introduce a rather general model of random graphs, which seems to be interesting on its own right.