The sharp threshold for jigsaw percolation in random graphs
Oliver Cooley, Tobias Kapetanopoulos, Tamás Makai
TL;DR
This work determines the sharp threshold for jigsaw percolation when two random graphs on the same vertex set act jointly. By introducing and exploiting an absorption process and bounding the count of minimal percolating configurations, the authors prove a precise threshold at $p_1p_2=\frac{1}{4n\ln n}$, with subcritical non-percolation for $p_1p_2\le \frac{1-\varepsilon}{4n\ln n}$ and supercritical percolation (conditional on connectedness) for $p_1p_2\ge \frac{1+\varepsilon}{4n\ln n}$. The subcritical analysis bounds the number of potential percolating configurations via intricate combinatorial counting (captured by $M_{k,\ell,r}$ and $M'_{k,\ell}$), while the supercritical analysis uses a two-stage exposure and Poisson approximation to construct a large percolating set that then expands to cover the graph. The results generalise the Erdős–Rényi threshold paradigm to a two-graph setting, yielding a precise double-graph threshold with implications for joint connectivity in networks; the work also outlines the critical window and potential generalisations to more graphs and other random graph models.
Abstract
We analyse the jigsaw percolation process, which may be seen as a measure of whether two graphs on the same vertex set are `jointly connected'. Bollobás, Riordan, Slivken and Smith proved that when the two graphs are independent binomial random graphs, whether the jigsaw process percolates undergoes a phase transition when the product of the two probabilities is $Θ\left( \frac{1}{n\ln n} \right)$. We show that this threshold is sharp, and that it lies at $\frac{1}{4n\ln n}$.