| Complex networks are a highly useful tool to model a vast number of different real world structures. Percolation describes the transition to extensive connectedness upon gradual addition of links. Whether single links may ''explosively'' change macroscopic connectivity in networks where, according to certain rules, links are added competitively has been debated controversially in the past three years. In the recent article [Science 333, 322 (2011)], O. Riordan and L. Warnke conclude that (i) ''any rule based on picking a fixed number of random vertices gives a continuous transition'', and (ii) that ''explosive percolation is continuous''. In contrast we show that it is equally true that certain percolation processes based on picking a fixed number of random vertices are discontinuous. Here we resolve this apparent paradox. We identify and analyze this by studying a process that is continuous in the sense of Riordan and Warnke but still exhibits infinitely many discontinuous jumps in an arbitrary vicinity of the transition point: a Devil's staircase. |
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