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arXiv · 1704.07144

Bootstrap percolation in random $k$-uniform hypergraphs

Mihyun Kang·Christoph Koch·Tamás Makai
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Abstract

We investigate bootstrap percolation with infection threshold $r> 1$ on the binomial $k$-uniform random hypergraph $H_k(n,p)$ in the regime $n^{-1}\ll n^{k-2}p \ll n^{-1/r}$, when the initial set of infected vertices is chosen uniformly at random from all sets of given size. We establish a threshold such that if there are less vertices in the initial set of infected vertices, then whp only a few additional vertices become infected, while if the initial set of infected vertices exceeds the threshold then whp almost every vertex becomes infected. In addition, for $k=2$, we show that the probability of failure decreases exponentially.

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Mihyun Kang, Christoph Koch, Tamás Makai. 2017-04-24. Bootstrap percolation in random $k$-uniform hypergraphs. https://doi.org/10.1016/j.endm.2015.06.081

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