Bootstrap Percolation on the Binomial Random k-uniform Hypergraph
arxiv(2024)
摘要
We investigate the behaviour of r-neighbourhood bootstrap percolation on
the binomial k-uniform random hypergraph H_k(n,p) for given integers k≥
2 and r≥ 2. In r-neighbourhood bootstrap percolation, infection spreads
through the hypergraph, starting from a set of initially infected vertices, and
in each subsequent step of the process every vertex with at least r infected
neighbours becomes infected. For our analysis the set of initially infected
vertices is chosen uniformly at random from all sets of given size. In the
regime n^-1≪ n^k-2p ≪ n^-1/r we establish a threshold such that if
the number of initially infected vertices remains below the threshold, then
with high probability only a few additional vertices become infected, while if
the number of initially infected vertices exceeds the threshold then with high
probability almost every vertex becomes infected. In fact we show that the
probability of failure decreases exponentially.
更多查看译文
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络
Chat Paper
正在生成论文摘要