Giant component in the configuration model under geometric constraints

TL;DR

This paper investigates the emergence of giant components in a constrained configuration model on a torus, revealing how local geometric restrictions influence global connectivity and percolation properties.

Contribution

It introduces a new geometric constrained configuration model and analyzes the conditions for giant component emergence, bridging lattice and random graph behaviors.

Findings

Giant component emerges when compartments grow quickly enough.

Fixed compartment sizes may prevent giant component formation.

Model combines local lattice structure with global random graph properties.

Abstract

We study the emergence of a giant component in the configuration model subject to additional constraints on the edges. We partition a $d$-dimensional torus into a cubic lattice with a diverging number of compartments containing vertices and allow only local edges inside and between neighbouring compartments. We show that, when the number of vertices per compartment grows quickly enough, a giant component emerges under similar conditions as for the standard configuration model. Conversely, when the compartment sizes are fixed, our model might not feature a giant component even if the standard configuration model does have one. Locally, our model resembles the configuration model, while globally, it has properties more akin to a $d$-dimensional lattice. Nonetheless the model remains analytically tractable using multitype branching processes with infinite number of types and opens new potential ways to study percolation in graphs with geometric properties.