Limits of dense graph sequences

Laszlo Lovasz, Balazs Szegedy

Published 2004-08-12, updated 2004-09-22Version 2

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 2-variable function on [0,1]. This limit object determines all the limits of subgraph densities. We also show that the graph parameters obtained as limits of subgraph densities can be characterized by ``reflection positivity'', semidefiniteness of an associated matrix. Conversely, every such function arises as a limit object. Along the lines we introduce a rather general model of random graphs, which seems to be interesting on its own right.