Ordinary percolation with discontinuous transitions

Percolation on a one-dimensional lattice and fractals, such as the Sierpinski gasket, is typically considered to be trivial, because they percolate only at full bond density. By dressing up such lattices with small-world bonds, a novel percolation transition with explosive cluster growth can emerge at a non-trivial critical point. There, the usual order parameter, describing the probability of any node to be part of the largest cluster, jumps instantly to a finite value. Here we provide a simple example in the form of a small-world network consisting of a one-dimensional lattice which, when combined with a hierarchy of long-range bonds, reveals many features of this transition in a mathematically rigorous manner. Percolation transitions indicate the threshold above which a network can operate. This work examines a general class of models known as hierarchical networks, and shows they can be made to percolate explosively, if they share features of so-called 'small-world' networks.