Percolation threshold is not a decreasing function of the average coordination number.

Abstract

It is commonly believed that the percolation critical probability is a monotonically decreasing function of the average coordination number for periodic lattice graphs in the same dimension. This paper provides counterexamples-a pair of planar lattices for which the bond percolation critical probabilities and average coordination numbers are in the same order, and a pair for which the site percolation critical probabilities and average coordination numbers are in the same order. These counterexamples confirm the existence of this counterintuitive phenomenon, which was observed in one case in numerical estimates by van der Marck.