Evolution of high-order connected components in random hypergraphs

We consider high-order connectivity in k-uniform hypergraphs defined as follows: Two j-sets are j-connected if there is a walk of edges between them such that two consecutive edges intersect in at least j vertices. We describe the evolution of j-connected components in the k-uniform binomial random hypergraph Hk(n,p). In particular, we determine the asymptotic size of the giant component shortly after its emergence and establish the threshold at which Hk(n,p) becomes j-connected with high probability. We also obtain a hitting time result for the related random hypergraph process {Hk(n,M)}M – the hypergraph becomes j-connected exactly at the moment when the last isolated j-set disappears. This generalises well-known results for graphs and vertex-connectivity in hypergraphs.