Bootstrap percolation in random k -uniform hypergraphs

Abstract

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 ≪ n k−2 p ≪ 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.

High-order bootstrap percolation in hypergraphs

“…The answers to these questions are in general heavily dependent on the choice of the initially infected set A, whose vertices may be selected independently at random or, in a more recent approach, one can try to find the smallest contagious set A in G [3,15]. A closely related process is graph bootstrap percolation [3,4,8], originally proposed by Bollobás in 1968. Bootstrap percolation has also been studied in (random) k-uniform hypergraphs in [19], where an infection process on the vertices of a random hypergraph was studied; by contrast, inspired by recent work on high-order connectedness and percolation processes in hypergraphs (e.g., [9,12,13]), which evolve on sets of vertices rather than the vertices themselves, for any 1 ≤ j ≤ k − 1 we introduce a process on the j-sets (of vertices) of the k-uniform hypergraph, which has not been considered previously in the literature.…”

“…Note also that if k = 2 and j = 1, this is identical to the standard bootstrap percolation process for graphs. The only case which has been previously studied for hypergraphs is the case j = 1 [19].…”

High-dimensional bootstrap processes in evolving simplicial complexes

“…There are other models of high-dimensional bootstrap processes on hypergraphs -see for example [22,25].…”

A large deviation approach to super-critical bootstrap percolation on the random graph Gn,p

“…Instead, above the critical number a (n) c , the process percolates through the entire random graph, reaching, as n → ∞, a final size of active nodes which is of the same order as n (super-critical case), i.e., in mathematical terms, A * n /n converges to 1 in probability. In [27] the results of [26] were extended to k-uniform random hypergraphs. Bootstrap percolation on random graphs obtained by combining G n,pn with a regular lattice was investigated in [35].…”