Relating the cut distance and the weak* topology for graphons
The theory of graphons is ultimately connected with the so-called cut norm. In this paper, we approach the cut norm topology via the weak* topology (when considering a predual of $L^{1}$-functions). We prove that a sequence $W_1,W_2,W_3,\ldots$ of graphons converges in the cut distance if and only if we have equality of the sets of weak* accumulation points and of weak* limit points of all sequences of graphons $W_1',W_2',W_3',\ldots$ that are weakly isomorphic to $W_1,W_2,W_3,\ldots$. We further give a short descriptive set theoretic argument that each sequence of graphons contains a subsequence with the property above. This in particular provides an alternative proof of the theorem of Lovász and Szegedy about compactness of the space of graphons. We connect these results to "multiway cut" characterization of cut distance convergence from [Ann. of Math. (2) 176 (2012), no. 1, 151-219]. These results are more naturally phrased in the Vietoris hyperspace $K$ over graphons with the weak* topology. We show that graphons with the cut distance topology are homeomorphic to a closed subset of $K$, and deduce several consequences of this fact. From these concepts a new order on the space of graphons emerges. This order allows to compare how structured two graphons are. We establish basic properties of this "structurdness order".