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Abstract:
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Graphons and graphexes are limits of graphs which allow us to model and estimate properties of large-scale networks. This pair of talks will review the theory of dense graph limits, and give two alternative theories for limits of sparse graphs (those that occur most often in nature) -- one leading to unbounded graphons over probability spaces, and the other leading to bounded graphons and graphexes over sigma-finite measure spaces. Lecture I will review the general theory, highlighting unbounded graphons, and show how they can be used to consistently estimate properties of large sparse networks. This talk will also give an application of these sparse graphons to collaborative filtering on sparse bipartite networks. Lecture II will recast limits of dense graphs in terms of exchangeability and the Aldous-Hoover Theorem, and generalize this to obtain sparse graphons and graphexes as limits of subgraph samples from sparse graph sequences. This will provide a dual view of sparse graph limits as processes and random measures, an approach which allows a generalization of many of the well-known results and techniques for dense graph sequences.
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