Product Measure Approximation of Symmetric Graph Properties

Random structures often present a trade-off between realism and tractability, the latter predominantly enabled by independence. In pioneering random graph theory, Erdős and Rényi originally studied the set of all graphs with a given number of edges, seeking to identify its typical properties under the uniform measure, i.e., the G(n,m) model of random graphs. However, this approach quickly presents major challenges, most notably the lack of independence, leading to the approximation of the G(n,m) model by the G(n, p) model, wherein edges appear independently with probability p. Let Gn be the set of all graphs on n vertices. In this work we pursue the following question: What are general sufficient conditions for the uniform measure on a set of graphs S ⊆ Gn to be approximable by a product measure? ∗Research supported by a European Research Council (ERC) Starting Grant (StG-210743) and an Alfred P. Sloan Fellowship. †Supported in part by an Onassis Foundation Scholarship.