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Impact of single links in competitive percolation


How a complex network is connected crucially impacts its dynamics and function. Percolation, the transition to extensive connectedness on gradual addition of links, was long believed to be continuous, but recent numerical evidence of ‘explosive percolation’ suggests that it might also be discontinuous if links compete for addition. Here we analyse 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 showing 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 substantial gaps, because gap sizes scale weakly algebraically with system size. Several essentially macroscopic clusters coexist immediately before the transition, announcing discontinuous percolation. These results explain how single links may drastically change macroscopic connectivity in networks where links add competitively.

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Figure 1: Competitive versus non-competitive percolation processes.
Figure 2: Dynamics of largest cluster size in competitive percolation processes.
Figure 3: Gap sizes decay algebraically with system size N for weakly discontinuous transitions.
Figure 4: Weakly discontinuous transition in stochastic mixture of largest-cluster growth (with probability p) and suppressed growth.
Figure 5: Coexistence of several large clusters.


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We thank N. Goldenfeld and I. Kanter for discussions. Supported by the Federal Ministry of Education and Research (BMBF) Germany under grant number 01GQ1005B (A.L. and M.T.) and by a grant of the Max Planck Society to M.T.

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All authors conceived and designed the research, contributed analysis tools and analysed the data. J.N. carried out the numerical experiments. All authors worked out the theory and wrote the manuscript.

Corresponding author

Correspondence to Jan Nagler.

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The authors declare no competing financial interests.

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Nagler, J., Levina, A. & Timme, M. Impact of single links in competitive percolation. Nature Phys 7, 265–270 (2011).

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