  Despite original claims of a first-order transition in the product rule model
proposed by Achlioptas et al. [Science 323, 1453 (2009)], recent studies
indicate that this percolation model, in fact, displays a continuous
transition. The distinctive scaling properties of the model at criticality,
however, strongly suggest that it should belong to a different universality
class than ordinary percolation. Here we introduce a generalization of the
product rule that reveals the effect of non-locality on the critical behavior
of the percolation process. Precisely, pairs of unoccupied bonds are chosen
according to a probability that decays as a power-law of their Manhattan
distance, and only that bond connecting clusters whose product of their sizes
is the smallest, becomes occupied. Interestingly, our results for
two-dimensional lattices at criticality shows that the power-law exponent of
the product rule has a significant influence on the finite-size scaling
exponents for the spanning cluster, the conducting backbone, and the cutting
bonds of the system. In all three cases, we observe a continuous variation from
ordinary to (non-local) explosive percolation exponents.
