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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Strogatz, S. H. Exploring complex networks. Nature 410, 268–276 (2001).
Newman, M. E. J., Watts, D. J. & Strogatz, S. H. Random graph models of social networks. Proc. Natl Acad. Sci. 99, 2566–2572 (2002).
Song, C., Havlin, S. & Makse, H. A. Origins of fractality in the growth of complex networks. Nature Phys. 2, 275–281 (2006).
Parshani, R., Bulyrev, S. V. & Havlin, S. Interdependent networks: Reducing the coupling strength leads to a change from a first to second order percolation transition. Phys. Rev. Lett. 105, 048701 (2010).
Parshani, R., Bulyrev, S. V., Stanley, H. E. & Havlin, S. Catastrophic cascade of failures in interdependent networks. Nature 464, 1025–1028 (2010).
Ben-Avraham, D. & Havlin, S. Diffusion and Reactions in Fractals and Disordered Systems (Cambridge Univ. Press, 2001).
Solomon, S. G., Weisbucha, L. d. A., Janc, N. & Stauffer, D. Social percolation model. Physica A 277, 239–247 (2000).
Goldstone, R. L. & Janssena, M. A. Computational models of collective behavior. Trends Cognitive Sci. 9, 424–430 (2005).
Dorogovtsev, S. N., Goltsev, A. V. & Mendes, J. F. F. Critical phenomena in complex networks. Rev. Mod. Phys. 80, 1275–1335 (2008).
Rozenfeld, H. D., Gallos, L. K. & Makse, H. A. Explosive percolation in the human protein homology networks. Eur. Phys. J. E 75, 305–310 (2010).
Hufnagel, L., Brockmann, D. & Geisel, T. Forecast and control of epidemics in a globalized world. Proc. Natl Acad. Sci. USA 101, 15124–15129 (2004).
D’Souza, R. M. Complex networks: Structure comes to random graphs. Nature Phys. 5, 627–628 (2009).
Bunde, A. & Havlin, S. Fractals and Disordered Systems (Springer, 1996).
Bollobás, B. & Riordan, O. Percolation (Cambridge Univ. Press, 2006).
Stauffer, D. & Aharony, A. Introduction to Percolation Theory (Taylor and Francis, 1993).
Grimmett, G. Percolation (Springer, 1999).
Achlioptas, D., D’Souza, R. M. & Spencer, J. Explosive percolation in random networks. Science 323, 1453–1455 (2009).
Bohmann, T. Emergence of connectivity in networks. Science 323, 1438–1439 (2009).
Cho, Y. S., Kim, J. S., Park, J., Kahng, B. & Kim, D. Percolation transitions in scale-free networks under the Achlioptas process. Phys. Rev. Lett. 103, 135702 (2009).
Radicchi, F. & Fortunato, S. Explosive percolation in scale-free networks. Phys. Rev. Lett. 103, 168701 (2009).
Ziff, R. M. Explosive growth in biased dynamic percolation on two-dimensional regular lattice networks. Phys. Rev. Lett. 103, 045701 (2009).
Cho, Y. S., Kahng, B. & Kim, D. Cluster aggregation model for discontinuous percolation transitions. Phys. Rev. E 81, 030103 (2010).
Friedman, E. J. & Landsberg, A. S. Construction and analysis of random networks with explosive percolation. Phys. Rev. Lett. 103, 255701 (2009).
Moreira, A. A., Oliveira, E. A., Reis, S. D. S., Herrmann, H. J. & Andrade, J. S. Hamiltonian approach for explosive percolation. Phys. Rev. E 81, 040101 (2010).
Radicchi, F. & Fortunato, S. Explosive percolation: A numerical analysis. Phys. Rev. E 81, 036110 (2010).
D’Souza, R. M. & Mitzenmacher, M. Local cluster aggregation models of explosive percolation. Phys. Rev. Lett. 104, 195702 (2010).
Manna, S. S. & Chatterjee, A. A new route to explosive percolation. Physica A 390, 177–182 (2011).
Breskin, I., Soriano, J., Moses, E. & Tlusty, T. Percolation in living neural networks. Phys. Rev. Lett. 97, 188102 (2006).
Soriano, J., Martinez, M. R., Tlusty, T. & Moses, E. Development of input connections in neural cultures. Proc. Natl Acad. Sci. USA 105, 13758–13763 (2008).
Beveridge, A., Bohman, T., Frieze, A. & Pikhurko, O. Product rule wins a competitive game. Proc. Am. Math. Soc. 135, 3061–3071 (2007).
Binder, K. Theory of first-order phase transitions. Rep. Prog. Phys. 50, 783–859 (1987).
Goldenfeld, N. D. Lectures on Phase Transitions and the Renormalisation Group (Addison-Wesley, 1992).
Ziff, R. M. Scaling behavior of explosive percolation on the square lattice. Phys. Rev. E 82, 051105 (2010).
da Costa, R. A., Dorogovtsev, S. N., Goltsev, A. V. & Mendes, J. F. F. Explosive percolation transition is actually continuous. Preprint at http://arxiv.org/abs/1009.2534v2 (2010).
Spencer, J. & Wormald, N. Birth control for giants. Combinatorica 27, 587–628 (2008).
Spencer, J. The giant component: The golden anniversary. Not. Am. Math. Soc. 57, 720–724 (2010).
Araújo, N. A. M. & Herrmann, H. J. Explosive percolation via control of the largest cluster. Phys. Rev. Lett. 105, 035701 (2010).
Timme, M. Does dynamics reflect topology in directed networks? Europhys. Lett. 76, 367–373 (2006).
Memmesheimer, R. M. & Timme, M. Designing complex networks. Physica D 224, 182–201 (2006).
Urbanc, B. et al. Aggregation and disaggregation of senile plaques in Alzheimer disease. Proc. Natl Acad. Sci. USA 94, 7612–7616 (1996).
Dinsmore, A. D., Prasad, V., Wong, I. Y. & Weitz, D. A. Microscopic structure and elasticity of weakly aggregated colloidal gels. Phys. Rev. Lett. 96, 185502 (2006).
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.
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). https://doi.org/10.1038/nphys1860
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