How does the shape of a network change as its size increases? Although random graph models provide some expectations for such "scaling behaviors" in the structure of networks, relatively little is known about how empirical network structure scales with network size or how well random graphs explain those empirical patterns. Using a large, structurally diverse corpus of networks from four scientific domains, we first characterize the empirical scaling laws of real-world networks, considering how mean degree, transitivity, mean geodesic distance, and degree assortativity vary with network size. We show that networks from all four scientific domains exhibit a consistent set of scaling laws on these measures of network structure, but with differing scaling rates. We then assess the extent to which these empirical scaling laws are explained by three random graph models with different structural assumptions, showing that configuration model random graphs are a remarkably good model of network scaling behavior, although null models with modular structure are slightly better. These findings identify a new set of common patterns in the network structure of complex systems, provide new validation targets for models of network structure, and shed new light on the role of randomness in shaping the large-scale structure of networks.