The threshold for jigsaw percolation on random graphs

TL;DR

This paper determines the critical threshold for jigsaw percolation to occur in social network models where both the social and puzzle graphs are Erdős–Rényi random graphs, advancing understanding of puzzle-solving dynamics in random environments.

Contribution

It establishes the percolation threshold for jigsaw percolation on Erdős–Rényi graphs, providing a key probabilistic boundary for the process.

Findings

Identifies the percolation threshold up to a constant factor.

Provides probabilistic bounds for the emergence of a single connected component.

Enhances understanding of puzzle-solving in random social networks.

Abstract

Jigsaw percolation is a model for the process of solving puzzles within a social network, which was recently proposed by Brummitt, Chatterjee, Dey and Sivakoff. In the model there are two graphs on a single vertex set (the `people' graph and the `puzzle' graph), and vertices merge to form components if they are joined by an edge of each graph. These components then merge to form larger components if again there is an edge of each graph joining them, and so on. Percolation is said to occur if the process terminates with a single component containing every vertex. In this note we determine the threshold for percolation up to a constant factor, in the case where both graphs are Erd\H{o}s--R\'enyi random graphs.