Evolution of locally dependent random graphs

TL;DR

This paper investigates $d$-dependent random graphs, analyzing their properties like degree sequences and connectivity, revealing nuanced differences from classical Erdős–Rényi graphs and highlighting open research questions.

Contribution

It provides a comprehensive analysis of $d$-dependent random graphs, extending classical results and exploring their unique properties and complexities.

Findings

Mirror results of Erdős–Rényi graphs when $d=0$

Showcase nuanced differences for $d>0$

Survey of existing knowledge and open questions

Abstract

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 Erd\H{o}s--R\'enyi 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.