Product Measure Approximation of Symmetric Graph Properties

Abstract

In the study of random structures we often face a trade-off between realism and tractability, the latter typically enabled by assuming some form of independence. 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 Gn be the set of all graphs on n vertices and let S be an arbitrary subset of Gn, e.g., the set of 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 ⊆ Gn to be approximable by a product measure?

Spatial Gibbs random graphs

“…(We are not aware of spatially embedded versions of these models.) Yet another direction is explored in [1], where the authors give conditions ensuring that the uniform measure on a set of graphs satisfying some constraints can be well-approximated by a product measure on the edges.…”

“…Hence, part (2) of the proposition is a consequence of Corollary 2.10. We now turn to part (1). Throughout the argument, we denote by c > 0 a generic constant whose value may change from place to place to be as small as necessary, and is not allowed to depend on N .…”