Some Mathematics behind Graph Property Testing

Abstract

Graph property testing is the third reincarnation of the same general question, after statistics and learning theory. In its simplest form, we have a huge graph (we don't even know its size), and we draw a sample of the node set of bounded size. What properties of the graph can be deduced from this sample? The graph property testing model was first introduced by Goldreich, Goldwasser and Ron (but related questions were considered before). In the context of dense graphs, a very general result is due to Alon and Shapira, who proved that every hereditary graph property is testable. Using the theory of graph limits, Lovasz and Szegedy defined an analytic version of the (dense) graph property testing problem, which can be formulated as studying an unknown 2-variable symmetric function through sampling from its domain and studying the random graph obtained when using the function values as edge probabilities. This analytic version allows for simpler formulation of the problems, and leads to various characterizations of testable properties. These results can be applied to the original graph-theoretic property testing. In particular, they lead to a new combinatorial characterization of testable graph properties. We survey these results, along with analogous results for graphs with bounded degree.