Symmetric Graph Properties Have Independent Edges

Abstract

In the study of random structures we often face a trade-off between realism and tractability, the latter typically enabled by independence assumptions. In this work we initiate an effort to bridge this gap by developing tools that allow us to work with independence without assuming it. Let $\mathcal {G}_{n}$ be the set of all graphs on n vertices and let S be an arbitrary subset of $\mathcal {G}_{n}$, e.g., the set of all graphs with m edges. The study of random networks can be seen as the study of properties that are true for most elements of S, i.e., that are true with high probability for a uniformly random element of S. With this in mind, we pursue the following question: What are general sufficient conditions for the uniform measure on a set of graphs $S \subseteq \mathcal {G}_{n}$ to be well-approximable by a product measure on the set of all possible edges?

D. Achlioptas—Research supported by ERC Starting Grant StG-210743 and an Alfred P. Sloan Fellowship. Research co-financed by the European Union (European Social Fund ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) - Research Funding Program: ARISTEIA II.

P. Siminelakis—Supported in part by an Onassis Foundation Scholarship.