A percolation system with extremely long range connections and node dilution

We study the very long-range bond-percolation problem on a linear chain with both node and bond dilution. Very long-range means that the probability pij for a connection between two nodes i,j at a distance rij decays as a power-law, i.e. pij=ρ/[rijαN1−α] when α∈[0,1), and pij=ρ/[rijln(N)] when α=1. Node dilution means that the probability that a node is present in a site is ps∈(0,1]. The behavior of this model results from the competition between long-range connectivity which enhances the percolation, and node dilution which weakens percolation. The case α=0 with ps=1 is well-known, being the exactly solvable mean-field model. The percolation order parameter P∞ is investigated numerically for different values of α, ps and ρ. We show that in all range α∈[0,1] the percolation order parameter P∞ depends only on the average connectivity γ of the nodes, which can be explicitly computed in terms of the three parameters α, ps and ρ.