On certain perturbations of the Erdos-Renyi random graph

TL;DR

This paper investigates how specific geometric pattern-based perturbations affect the Erdős-Rényi random graph model, revealing a percolation transition and deriving related thresholds through perturbative methods.

Contribution

It introduces a novel perturbation approach based on geometric patterns and establishes the existence of a percolation transition with an explicit threshold expression.

Findings

Percolation transition exists under certain perturbations.

Derived explicit expression for the percolation threshold.

Compared the percolation criterion with Molloy-Reed criterion.

Abstract

We study perturbations of the Erdos-Renyi model for which the statistical weight of a graph depends on the abundance of certain geometrical patterns. Using the formal correspondance with an exactly solvable effective model, we show the existence of a percolation transition in the thermodynamical limit and derive perturbatively the expression of the threshold. The free energy and the moments of the degree distribution are also computed perturbatively in that limit and the percolation criterion is compared with the Molloy-Reed criterion.