A recent example would be the generalization of graphs to continuous objects graphons, symmetric measurable functions on a square. Topological properties of spaces of these objects (e.g., compactness) have been shown to yield known properties in the discrete setting (existence of Szemeredi partitions). Also the continuous setting allows graphons to be interpreted as probability distributions which form a very general model for random graphs. Such things are detailed in the book of Lovasz, "Large networks and graph limits".