# Largest Components in Random Hypergraphs

@article{Cooley2018LargestCI, title={Largest Components in Random Hypergraphs}, author={Oliver Cooley and Mihyun Kang and Yury Person}, journal={Combinatorics, Probability and Computing}, year={2018}, volume={27}, pages={741 - 762} }

In this paper we consider j-tuple-connected components in random k-uniform hypergraphs (the j-tuple-connectedness relation can be defined by letting two j-sets be connected if they lie in a common edge and considering the transitive closure; the case j = 1 corresponds to the common notion of vertex-connectedness). We show that the existence of a j-tuple-connected component containing Θ(nj) j-sets undergoes a phase transition and show that the threshold occurs at edge probability
$$\frac{(k-j…

## 15 Citations

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The `Pathfinder' algorithm is introduced, a depth-first search algorithm which discovers j-tight paths in a k-uniform hypergraph and it is proved that, in the supercritical case, with high probability this algorithm will find a long $j$-tight path.

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It is shown that the model of random simplicial complexes generated from the binomial random (k + 1)-uniform hypergraph has a single sharp threshold, and a hitting time result is proved, relating the vanishing of these cohomology groups to the disappearance of the last minimal obstruction.

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This study determines the structure, order, and size of the largest $j-connected components, with the help of a certain class of `hypertrees' and related objects, and establishes a symmetry between the subcritical random hypergraph and the hypergraph obtained from the supercritical randomhypergraph by deleting its giant $j$-connected component.

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We consider simplicial complexes that are generated from the binomial random 3-uniform hypergraph by taking the downward-closure. We determine when this simplicial complex is homologically connected,…

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A hitting time result is proved, relating the vanishing of these cohomology groups to the disappearance of the last minimal obstruction, and the asymptotic distribution of the dimension of the $j$-th cohomological group inside the critical window is studied.

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