Cut-norm and entropy minimization over weak⁎ limits

Abstract We prove that the accumulation points of a sequence of graphs G 1 , G 2 , G 3 , … with respect to the cut-distance are exactly the weak ⁎ limit points of subsequences of the adjacency matrices (when all possible orders of the vertices are considered) that minimize the entropy over all weak ⁎ limit points of the corresponding subsequence. In fact, the entropy can be replaced by any map W ↦ ∬ f ( W ( x , y ) ) , where f is a continuous and strictly concave function. As a corollary, we obtain a new proof of compactness of the cut-distance topology.

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