We consider a family of long-range percolation models (G p )p>0 on ℤd that allow dependence between edges and have the following connectivity properties for p ∈ (1/d, ∞): (i) the degree distribution of vertices in G p has a power-law distribution; (ii) the graph distance between points x and y is bounded by a multiple of logpd logpd |x - y | with probability 1 - o(1); and (iii) an adversary can delete a relatively small number of nodes from G p (ℤd ∩ [0, n]d ), resulting in two large, disconnected subgraphs.
