Evolution of locally dependent random graphs

In this paper we study $d$-dependent random graphs – introduced by Brody and Sanchez – which are the family of random graph distributions where each edge is present with probability $p$, and each edge is independent of all but at most $d$ other edges. For this random graph model, we analyze degree sequences, jumbledness, connectivity, and subgraph containment. Our results mirror those of the classical Erdős–Rényi random graph, which are recovered by specializing our problem to $d=0$, although we show that in many regards our setting is appreciably more nuanced. We survey what is known for this model and conclude with a variety of open questions.