Percolation on a one-dimensional lattice and fractals, such as the sierpinski gasket, is typicallyconsidered to be trivial, because they percolate only at full bond density. By dressing up suchlattices with small-world bonds, a novel percolation transition with explosive cluster growth canemerge at a non-trivial critical point. There, the usual order parameter, describing the probabilityof any node to be part of the largest cluster, jumps instantly to a finite value. Here we providea simple example in the form of a small-world network consisting of a one-dimensional latticewhich, when combined with a hierarchy of long-range bonds, reveals many features of thistransition in a mathematically rigorous manner.
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