Convex Random Graph Models

We propose a principled framework for designing random graph models from minimal assumptions. Our central principle is to study the uniform measure over symmetric subsets of graphs. The symmetries we require are far less limiting than those in existing graph models and flexible enough to encompass both geometric features and properties of graph limits. Our main contribution is to derive natural sufficient conditions under which the uniform measure over a set of graphs can be approximated by a product measure over its edges. In particular, we prove that often the uniform measure over the set of graphs with a highly complex, global graph property collapses asymptotically to a distribution in which edges appear independently with different probabilities, the probability of each computable from the property. ∗A longer version of this work will be made available on the Arxiv. †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.