On limits of finite graphs

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We prove that for any weakly convergent sequence of finite graphs with bounded vertex degrees, there exists a topological limit graphing.

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Asymptotic behavior and distributional limits of preferential attachment graphs (2014, background): “Interestingly, it is not hard to show that the Benjamini– Schramm limit coincides with the limit defined via graph homomorphisms in the case of sparse graphs of bounded degree; see [16] for yet another equivalent notion of convergent sequences of graphs with bounded degrees.”
On exchangeable random variables and the statistics of large graphs and hypergraphs (2008, methodology): “In fact, Elek and Szegedy’s use of ultralimits has a predecessor in Hoover’s approach to the basic representation theorems for exchangeable arrays of random variables, as outlined in [42].”