Convergent Sequences of Dense Graphs II: Multiway Cuts and Statistical Physics

We consider sequences of graphs (Gn) and define various notions of convergence related to these sequences including “left convergence,” defined in terms of the densities of homomorphisms from small graphs into Gn, and “right convergence,” defined in terms of the densities of homomorphisms from Gn into small graphs. We show that right convergence is equivalent to left convergence, both for simple graphs Gn, and for graphs Gn with nontrivial nodeweights and edgeweights. Other equivalent conditions for convergence are given in terms of fundamental notions from combinatorics, such as maximum cuts and Szemer´edi partitions, and fundamental notions from statistical physics, like energies and free energies. We thereby relate local and global properties of graph sequences. Quantitative forms of these results express the relationships among different measures of similarity of large graphs.