Cut-norm and entropy minimization over weak* limits

We prove that the accumulation points of a sequence of graphs $G_1,G_2,G_3,\dots$ with respect to the cut-distance are exactly the weak${}^{\ast }$ limit points of subsequences of the adjacency matrices (when all possible orders of the vertices are considered) that minimize the entropy over all weak${}^{\ast }$ limit points of the corresponding subsequence. In fact, the entropy can be replaced by any map $W\mapsto \int \int f(W(x,y))$, where $f$ is a continuous and strictly concave function. Our proofs are elementary, and do not use the regularity lemma.