Article

# Cluster aggregation model for discontinuous percolation transitions.

Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea.

Physical Review E (Impact Factor: 2.31). 03/2010; 81(3 Pt 1):030103. DOI: 10.1103/PhysRevE.81.030103 Source: PubMed

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**ABSTRACT:**Complex networks are a highly useful tool for modeling a vast number of different real world structures. Percolation describes the transition to extensive connectedness upon the 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 intensely in the past three years. In a recent article [ O. Riordan and L. Warnke 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, and we resolve this apparent paradox. We identify and analyze a process that is continuous in the sense defined by Riordan and Warnke but still exhibits infinitely many discontinuous jumps in an arbitrary vicinity of the transition point: a Devil’s staircase. We demonstrate analytically that continuity at the first connectivity transition and discontinuity of the percolation process are compatible for certain competitive percolation systems.Physical Review X. 08/2012; 2(3):031009. -
##### Article: Impact of Single Links in Competitive Percolation -- How complex networks grow under competition

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**ABSTRACT:**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.Nature Physics 03/2011; 7. · 19.35 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The suitable interpolation between classical percolation and a special variant of explosive percolation enables the explicit realization of a tricritical percolation point. With high-precision simulations of the order parameter and the second moment of the cluster size distribution a fully consistent tricritical scaling scenario emerges yielding the tricritical crossover exponent 1/φ(t)=1.8 ± 0.1.Physical Review Letters 03/2011; 106(9):095703. · 7.73 Impact Factor

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