Jigsaw Percolation on Erdos-Renyi Random Graphs

TL;DR

This paper analyzes the jigsaw percolation model on Erdős-Rényi graphs, establishing thresholds for the effective edge probability that ensure percolation, extending understanding of connectivity in random graph-based puzzles.

Contribution

It extends the jigsaw percolation model to Erdős-Rényi graphs, deriving bounds for the critical effective probability for percolation.

Findings

Critical effective probability bounds are established.

Percolation occurs when effective probability exceeds a threshold.

Results depend on minimum edge probability in the graphs.

Abstract

We extend the jigsaw percolation model to analyze graphs where both underlying people and puzzle graphs are Erd\H{o}s-R\'enyi random graphs. Let $p_{\text{ppl}}$ and $p_{\text{puz}}$ denote the probability that an edge exists in the respective people and puzzle graphs and define $p_{\text{eff}}= p_{\text{ppl}}p_{\text{puz}}$, the effective probability. We show for constants $c_1>1$ and $c_2> \pi^2/6$ and $c_3<e^{-5}$ if $min(p_{\text{ppl}},p_{\text{puz}}) > c_1 \log n /n$ the critical effective probability $p^c_{\text{eff}}$, satisfies $c_3 < p^c_{\text{eff}}n\log n < c_2.$