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Limits of compact decorated graphs

Following a general program of studying limits of discrete structures, and motivated by the theory of limit objects of converge sequences of dense simple graphs, we study the limit of graph sequences such that every edge is labeled by an element of a compact second-countable Hausdorff space K. The "local structure" of these objects can be explored by a sampling process, which is shown to be equivalent to knowing homomorphism numbers from graphs whose edges are decorated by continuous functions on K. The model includes multigraphs with bounded edge multiplicities, graphs whose edges are weighted with real numbers from a finite interval, edge-colored graphs, and other models. In all these cases, a limit object can be defined in terms of 2-variable functions whose values are probability distributions on K.