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Understanding what types of phenomena lead to discontinuous phase transitionsin the connectivity of random networks is an outstanding challenge. Here weshow that a simple stochastic model of graph evolution leads to a discontinuouspercolation 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 examinedone at a time while a cap, $k$, on the maximum allowed component size isenforced. Edges whose addition would exceed $k$ can be simply rejected providedthe accepted fraction of edges never becomes smaller than a function whichdecreases 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 largestcomponent is dominated by an overtaking mechanism leading to a discontinuoustransition. If $\beta > 1$, once $k \ge n^{1/\beta}$, there are situations whena sampled edge must be accepted leading to direct growth dominated bystochastic fluctuations and a "weakly" discontinuous transition. We also showthat the distribution of component sizes and the evolution of component sizesare distinct from those previously observed and show no finite size effects forthe range of $\beta$ studied.

Deriving an underlying mechanism for discontinuous percolation