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ABSTRACT: How do we draw sound and defensible conclusions from big data, for example in comparing two sets of observations, or evaluating goodness of model fit? In this talk I will discuss the current state of the art in one area of particular interest: big network data. Progress in this area includes the development of new large-sample theory that helps us to view and interpret networks as statistical data objects, along with the transformation of this theory into new statistical methods to model and draw inferences from network data in the real world. The insights that result from connecting theory to practice also feed back into pure mathematics and theoretical computer science, prompting new questions at the interface of combinatorics, analysis, probability, and algorithms.