  Understanding what types of phenomena lead to discontinuous phase transitions
in the connectivity of random networks is an outstanding challenge. Here we
show that a simple stochastic model of graph evolution leads to a discontinuous
percolation transition and we derive the underlying mechanism responsible:
growth by overtaking. Starting from a collection of $n$ isolated nodes,
potential edges chosen uniformly at random from the complete graph are examined
one at a time while a cap, $k$, on the maximum allowed component size is
enforced. Edges whose addition would exceed $k$ can be simply rejected provided
the accepted fraction of edges never becomes smaller than a function which
decreases with $k$ as $g(k) = 1/2 + (2k)^{-\beta}$. We show that if $\beta < 1$
it is always possible to reject a sampled edge and the growth in the largest
component is dominated by an overtaking mechanism leading to a discontinuous
transition. If $\beta > 1$, once $k \ge n^{1/\beta}$, there are situations when
a sampled edge must be accepted leading to direct growth dominated by
stochastic fluctuations and a "weakly" discontinuous transition. We also show
that the distribution of component sizes and the evolution of component sizes
are distinct from those previously observed and show no finite size effects for
the range of $\beta$ studied.
