  How a complex network is connected crucially impacts its dynamics and
function. Percolation, the transition to extensive connectedness upon gradual
addition of links, was long believed to be continuous but recent numerical
evidence on "explosive percolation" suggests that it might as well be
discontinuous if links compete for addition. Here we analyze the microscopic
mechanisms underlying discontinuous percolation processes and reveal a strong
impact of single link additions. We show that in generic competitive
percolation processes, including those displaying explosive percolation, single
links do not induce a discontinuous gap in the largest cluster size in the
thermodynamic limit. Nevertheless, our results highlight that for large finite
systems single links may still induce observable gaps because gap sizes scale
weakly algebraically with system size. Several essentially macroscopic clusters
coexist immediately before the transition, thus announcing discontinuous
percolation. These results explain how single links may drastically change
macroscopic connectivity in networks where links add competitively.
