Phase Transitions in Partially Structured Random Graphs

TL;DR

This paper explores a continuum of random graph models bridging Erdős-Rényi graphs and geometric percolation, revealing that most models exhibit a phase transition for giant cluster formation, with one remaining case behaving like a classic random graph.

Contribution

It introduces a unified family of random graph models and analyzes their phase transition behavior, filling gaps in existing theories.

Findings

Most models exhibit a phase transition with a giant cluster.

One remaining case behaves like a standard random graph.

The phase transition occurs when the expected degree exceeds one.

Abstract

We study a one parameter family of random graph models that spans a continuum between traditional random graphs of the Erd\H{o}s-R\'enyi type, where there is no underlying structure, and percolation models, where the possible edges are dictated exactly by a geometry. We find that previously developed theories in the fields of random graphs and percolation have, starting from different directions, covered almost all the models described by our family. In particular, the existence or not of a phase transition where a giant cluster arises has been proved for all values of the parameter but one. We prove that the single remaining case behaves like a random graph and has a single linearly sized cluster when the expected vertex degree is greater than one.