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Recent Advances of Percolation Theory in Complex Networks

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Abstract

During the past two decades, percolation has long served as a basic paradigm for network resilience, community formation and so on in complex systems. While the percolation transition is known as one of the most robust continuous transitions, the percolation transitions occurring in complex systems are often of different types such as discontinuous, hybrid, and infinite-order phase transitions. Thus, percolation has received considerable attention in network science community. Here we present a very brief review of percolation theory recently developed, which includes those types of phase transitions, critical phenomena, and finite-size scaling theory. Moreover, we discuss potential applications of theoretical results and several open questions including universal behaviors.

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References

  1. K. Ziemelis and L. Allen, Nature (London) 410, 241 (2001).

    Google Scholar 

  2. R. Gallagher and T. Appenzeller, Science 284, 87 (1999).

    ADS  Google Scholar 

  3. G. Parisi, Physica A (Amsterdam) 263, 557 (1999).

    ADS  Google Scholar 

  4. J. P. Sethna, K. A. Dahmen and C. R. Myers, Nature (London) 410, 242 (2001).

    ADS  Google Scholar 

  5. H. E. Stanley et al., Proc. Natl. Acad. Sci. U.S.A. 99, 2561 (2002).

    ADS  Google Scholar 

  6. M. Buchanan, Nature (London) 419, 787 (2002).

    ADS  Google Scholar 

  7. D. J. Watts and S. H. Strogatz, Nature (London) 393, 440 (1998).

    ADS  Google Scholar 

  8. A-L. Barabási and R. Albert, Science 286, 509 (1999).

    ADS  MathSciNet  Google Scholar 

  9. R. Albert and A-L. Barabási, Rev. Mod. Phys. 74, 47 (2002).

    ADS  MathSciNet  Google Scholar 

  10. S. N. Dorogovtsev and J. F. F. Mendes, Adv. Phys. 51, 1079 (2002).

    ADS  Google Scholar 

  11. M. E. J. Newman, SIAM Rev. 45, 167 (2003).

    ADS  MathSciNet  Google Scholar 

  12. P. Erdős and A. Rényi, Publ. Math. Inst. Hung. Acad. Sci. 5, 17 (1960).

    Google Scholar 

  13. R. Albert, H. Jeong and A-L. Barabási, Nature 401, 130 (1999).

    ADS  Google Scholar 

  14. M. Faloutsos, P. Faloutsos and C. Faloutsos, Comput. Commun. Rev. 29, 251 (1999).

    Google Scholar 

  15. J-H. Kim, K-I. Goh, B. Kahng and D. Kim, Phys. Rev. Lett. 91, 058701 (2003).

    ADS  Google Scholar 

  16. R. Albert, H. Jeong and A. L. Barabsi, Nature 406, 378 (2000).

    ADS  Google Scholar 

  17. R. Cohen, K. Erez, D. ben-Avraham and S. Havlin, Phys. Rev. Lett. 85, 4626 (2000).

    ADS  Google Scholar 

  18. P. L. Krapivsky, S. Redner and F. Leyvraz, Phys. Rev. Lett. 85, 4629 (2000).

    ADS  Google Scholar 

  19. S. N. Dorogovtsev, J. F. F. Mendes and A. N. Samukhin, Phys. Rev. Lett. 85, 4633 (2000).

    ADS  Google Scholar 

  20. D-S. Lee, K-I. Goh, B. Kahng and D. Kim, Nucl. Phys. B. 696, 351 (2004).

    ADS  Google Scholar 

  21. S. Boccalettia, V. Latora, Y. Moreno, M. Chavez and D-U. Hwang, Phys. Rep. 424, 175 (2006).

    ADS  MathSciNet  Google Scholar 

  22. R. Cohen and S. Havlin, Phys. Rev. Lett. 90, 058701 (2003).

    ADS  Google Scholar 

  23. M. Molloy and B. Reed, Random Struct. Algorithms 6, 161 (1995).

    Google Scholar 

  24. M. Molloy and B. Reed, Combinatorics, Probab. Comput. 7, 295 (1998).

    MathSciNet  Google Scholar 

  25. K-I. Goh, B. Kahng and D. Kim, Phys. Rev. Lett. 87, 278701 (2001).

    Google Scholar 

  26. F. Chung and L. Lu, Annals Combinatorics 6, 125 (2002).

    MathSciNet  Google Scholar 

  27. S. R. Broadbent and J. M. Hammersley, Cambridge Philos. Soc. 53, 629 (1957).

    ADS  Google Scholar 

  28. P. J. Flory, J. Am. Chem. Soc. 63, 3083 (1941).

    Google Scholar 

  29. P. J. Flory, J. Am. Chem. Soc. 63, 3091 (1941).

    Google Scholar 

  30. P. J. Flory, J. Am. Chem. Soc. 63, 3096 (1941).

    Google Scholar 

  31. J. D. Murray, Mathematical Biology, 3rd edn. (Springer, Berlin, 2005).

    MATH  Google Scholar 

  32. D. S. McLachlan, M. Blaszkiewicz and R. E. Newnham, J. Am. Ceram. Soc. 73, 2187 (1990).

    Google Scholar 

  33. L. S. Schulman and P. E. Seiden, Science 233, 425 (1986).

    ADS  Google Scholar 

  34. L. Bergqvist, O. Eriksson, J. Kudrnovský, V. Drchal, P. Korzhavyi and I. Turek Phys. Rev. Lett. 93, 137202 (2004).

    ADS  Google Scholar 

  35. R. Albert, H. Jeong and A. L. Barabási, Nature 406, 378 (2000).

    ADS  Google Scholar 

  36. F. Morone and H. A. Makse, Nature 524, 65 (2015).

    ADS  Google Scholar 

  37. D. J. Watts, Proc. Natl. Acad. Sci. 99, 5766 (2002).

    ADS  Google Scholar 

  38. J. Shao, S. Havlin and H. E. Stanley, Phys. Rev. Lett. 103, 018701 (2009).

    ADS  Google Scholar 

  39. R. Degraeve, G. Groeseneken, R. Bellens, M. Depas and H. E. Maes Tech. Dig. 1995 Inter. Electron Dev. Meeting 863 (1995).

    Google Scholar 

  40. R. Degraeve et al., IEEE Trans. Electron Dev. 51, 1392 (2004).

    ADS  Google Scholar 

  41. D. Stauffer and A. Aharony, Introduction to Percolation Theory, 2nd edn. (Taylor, and Francis, London, 1994).

    MATH  Google Scholar 

  42. M. E. Fisher and J. W. Essam, J. Math. Phys. 2, 609 (1961).

    ADS  Google Scholar 

  43. M. E. Fisher, J. Math. Phys. 2, 620 (1961).

    ADS  Google Scholar 

  44. P. W. Kasteleyn and C. M. Fortuin J. Phys. Soc. Jpn. (Suppl.) 26, 11 (1969).

    ADS  Google Scholar 

  45. J. Berg and M. Lässig, Phys. Rev. Lett. 89, 228701 (2002).

    ADS  Google Scholar 

  46. M. Baiesi and S. S. Manna, Phys. Rev. E 68, 047103 (2003).

    ADS  Google Scholar 

  47. Z. Burda, J. D. Correia and A. Krzywicki, Phys. Rev. E 64, 046118 (2001).

    ADS  Google Scholar 

  48. S. N. Dorogovtsev, J. F. F. Mendes and A. N. Samukhin, Nucl. Phys. B 666, 396 (2003).

    ADS  Google Scholar 

  49. I. Farkas, I. Derenyi, G. Palla and T. Viscek, Lecture notes in Physics: Networks: structure, dynamics, and function, edited by E. Ben-Naim, H. Frauenfelder and Z. Toroczkai (Springer, 2004).

  50. M. E. J. Newman, S. H. Strogatz and D. J. Watts, Phys. Rev. E 64, 026118 (2001).

    ADS  Google Scholar 

  51. G. Caldarelli, A. Capocci, P. De Los Rios and M. A. Mu˜noz, Phys. Rev. Lett. 89, 258702 (2002).

    ADS  Google Scholar 

  52. B. Söderberg, Phys. Rev. E 66, 066121 (2002).

    ADS  MathSciNet  Google Scholar 

  53. W. Aiello, F. Chung and L. Lu, Exp. Math. 10, 53 (2001).

    Google Scholar 

  54. F. Y. Wu, Rev. Mod. Phys. 54, 235 (1982).

    ADS  Google Scholar 

  55. R. Cohen, D. ben-Avraham and S. Havlin, Phys. Rev. E 66, 036113 (2002).

    ADS  Google Scholar 

  56. D. Achlioptas, R. M. D’Souza and J. Spencer, Science 323, 1453 (2009).

    ADS  MathSciNet  Google Scholar 

  57. J. S. Andrade, H. J. Herrmann, A. A. Moreira and C. L. N. Oliveira, Phys. Rev. E 83, 031133 (2011).

    ADS  Google Scholar 

  58. N. A. M. Araújo, J. S. Andrade, R. M. Ziff and H. J. Herrmann, Phys. Rev. Lett. 106, 095703 (2011).

    ADS  Google Scholar 

  59. E. J. Friedman and A. S. Landsberg, Phys. Rev. Lett. 103, 255701 (2009).

    ADS  Google Scholar 

  60. H. Hooyberghs and B. V. Schaeybroeck, Phys. Rev. E 83, 032101 (2011).

    ADS  Google Scholar 

  61. Y. S. Cho and B. Kahng, Sci. Rep. 5, 11905 (2015).

    ADS  Google Scholar 

  62. R. M. Ziff, Phys. Rev. Lett. 103, 045701 (2009).

    ADS  Google Scholar 

  63. R. M. Ziff, Phys. Rev. E 82, 051105 (2010).

    ADS  Google Scholar 

  64. Y. S. Cho, J. S. Kim, J. Park, B. Kahng and D. Kim Phys. Rev. Lett. 103, 135702 (2009).

    ADS  Google Scholar 

  65. R. Filippo and F. Santo, Phys. Rev. Lett. 103, 168701 (2009).

    Google Scholar 

  66. S. Boccaletti et al., Phys. Rep. 424, 175 (2006).

    ADS  MathSciNet  Google Scholar 

  67. Y. S. Cho and B. Kahng, Phys. Rev. Lett. 107, 275703 (2011).

    Google Scholar 

  68. K. J. Schrenk, N. A. M. Araujo and H. J. Herrmann, Phys. Rev. E 84, 041136 (2011).

    ADS  Google Scholar 

  69. K. J. Schrenk, A. Felder and S. Deflorin, Phys. Rev. E 85, 031103 (2012).

    ADS  Google Scholar 

  70. R. A. da Costa, S. N. Dorogovtsev, A. V. Goltsev and J. F. F. Mendes Phys. Rev. Lett. 105, 255701 (2010).

    ADS  Google Scholar 

  71. O. Riordan and L. Warnke Science 333, 322 (2011).

    ADS  Google Scholar 

  72. P. Grassberger, C. Christensen and G. Bizhani, Phys. Rev. Lett. 106, 225701 (2011).

    ADS  Google Scholar 

  73. H. K. Lee, B. J. Kim and H. Park, Phys. Rev. E 84, 020101 (2011).

    ADS  Google Scholar 

  74. Y. S. Cho, S. Hwang, H. J. Herrmann and B. Kahng Science 339, 1185 (2013).

    ADS  Google Scholar 

  75. K. J. Schrenk, N. A. M. Araújo, J. S. Andrade Jr. and H. J. Herrmann, Sci. Rep. 2, 348 (2012).

    ADS  Google Scholar 

  76. A. Bar and D. Mukamel, Phys. Rev. Lett. 112, 015701 (2014).

    ADS  Google Scholar 

  77. S. Jang, J. S. Lee, S. Hwang and B. Kahng, Phys. Rev. E 92, 022110 (2015).

    ADS  MathSciNet  Google Scholar 

  78. K. Panagiotou, R. Sphöel, A. Steger and H. Thomas, Elec. Notes Discret. Math. 38, 699 (2011).

    Google Scholar 

  79. Y. S. Cho, J. S. Lee, H. J. Herrmann and B. Kahng, Phys. Rev. Lett. 116, 025701 (2016).

    ADS  Google Scholar 

  80. K Choi, D. Lee, Y. S. Cho, J. C. Thiele, H. J. Herrmann and B. Kahng, Phys. Rev. E 96, 042148 (2017).

    ADS  Google Scholar 

  81. J. Chalupa, P. L. Leath and G. R. Reich, J. Phys. C 12, L31 (1981).

    Google Scholar 

  82. S. N. Dorogovtsev, A. V. Goltsev and J. F. F. Mendes, Phys. Rev. Lett. 96, 040601 (2006).

    ADS  Google Scholar 

  83. A. V. Goltsev, S. N. Dorogovtsev and J. F. F. Mendes Phys. Rev. E 73, 056101 (2006).

    ADS  MathSciNet  Google Scholar 

  84. G. J. Baxter, S. N. Dorogovtsev, K. E. Lee, J. F. F. Mendes and A. V. Goltsev, Phys. Rev. X. 5, 031017 (2015).

    Google Scholar 

  85. P. S. Dodds and D. J. Watts, Phys. Rev. Lett. 92, 218701 (2004).

    ADS  Google Scholar 

  86. H-K Janssen, M. Müller and O. Stenull, Phys. Rev. E 70, 026114 (2004).

    ADS  MathSciNet  Google Scholar 

  87. W. Cai, L. Chen, F. Ghanbarnejad and P. Grassberger, Nat. Phys. 11, 936 (2015).

    Google Scholar 

  88. N. A. M. Araújo and H. J. Herrmann, Phys. Rev. Lett. 105, 035701 (2010).

    ADS  Google Scholar 

  89. S. V. Buldyrev, R. Parshani, G. Paul, H. E. Stanley and S. Havlin, Nature 464, 1025 (2010).

    ADS  Google Scholar 

  90. S-W Son, P. Grassberger and M. Paczuski, Phys. Rev. Lett. 107, 195702 (2011).

    ADS  Google Scholar 

  91. G. J. Baxter, S. N. Dorogovtsev, A. V. Goltsev and J. F. F. Mendes, Phys. Rev. Lett. 109, 248701 (2012).

    ADS  Google Scholar 

  92. A. Bashan, Y. Berezin, S. V. Buldyrev and S. Havlin, Nat. Phys. 9, 667 (2013).

    Google Scholar 

  93. D. Cellai, E. López, J. Zhou, J. P. Gleeson and G. Bianconi, Phys. Rev. E 88, 052811 (2013).

    ADS  Google Scholar 

  94. D. Zhou, A. Bashan, R. Cohen, Y. Berezin, N. Shnerb and S. Havlin, Phys. Rev. E 90, 012803 (2014).

    ADS  Google Scholar 

  95. S. D. S. Reis, Y. Hu, A. Babino, J. S. S. Andrade Jr., S. Canals, M. Sigman and H. A. Makse, Nat. Phys. 10, 762 (2014).

    Google Scholar 

  96. S. Boccaletti, G. Bianconi, R. Criado, C. I. del Genio, J. Gómez-Gardenes, M. Romance, I. Sendi˜na-Nadal, Z. Wang and M. Zanin, Phys. Rep. 544, 1 (2014).

    ADS  MathSciNet  Google Scholar 

  97. M. Kivelä, A. Arenas, M. Barthelemy, J. P. Gleeson, Y. Moreno and M. A. Porter, J. Complex Netw. 2, 203 (2014).

    Google Scholar 

  98. J. M. Schwarz, A. J. Liu and L. Q. Chayes, EPL 73, 560 (2006).

    ADS  Google Scholar 

  99. A. V. Goltsev, S. N. Dorogovtsev and J. F. F. Mendes, Phys. Rev. E 73, 056101 (2006).

    ADS  MathSciNet  Google Scholar 

  100. D. Lee, S. Choi, M. Stippinger, J. Kertész and B. Kahng, Phys. Rev. E 93, 042109 (2016).

    ADS  Google Scholar 

  101. D. Lee, M. Jo and B. Kahng, Phys. Rev. E 94, 062307 (2016).

    ADS  Google Scholar 

  102. D. Lee, W. Choi, J. Kertész and B. Kahng, Sci. Rep. 7, 5723 (2017).

    ADS  Google Scholar 

  103. W. Choi, D. Lee and B. Kahng, Phys. Rev. E 95, 022304 (2017).

    ADS  Google Scholar 

  104. W. Choi, D. Lee and B. Kahng, Phys. Rev. E 95, 062115 (2017).

    ADS  MathSciNet  Google Scholar 

  105. E. Ben-Naim and P. L. Krapivsky, Phys. Rev. E 71, 026129 (2005).

    ADS  Google Scholar 

  106. S-W. Son, G. Bizhani, C. Christensen, P. Grassberger and M. Paczuski, EPL 97, 16006 (2012).

    ADS  Google Scholar 

  107. K-M. Lee, B. Min and K-I. Goh, Eur. Phys. J. B. 88, 28 (2015).

    ADS  Google Scholar 

  108. R. Parshani, S.V. Buldyrev and S. Havlin, Phys. Rev. Lett. 105, 048701 (2010).

    ADS  Google Scholar 

  109. K. Zhao and G. Bianconi, J. Stat. Mech. 2013, P05005 (2013).

    Google Scholar 

  110. F. Radicchi and G. Bianconi, Phys. Rev. X 7, 011013 (2017).

    Google Scholar 

  111. B. Min, S. D. Yi, K-M. Lee and K-I. Goh, Phys. Rev. E 89, 042811 (2014).

    ADS  Google Scholar 

  112. Y. Hu, D. Zhou, R. Zhang, Z. Han, C. Rozenblat and S. Havlin, Phys. Rev. E 88, 052805 (2013).

    ADS  Google Scholar 

  113. B. Min, S. Lee, K-M. Lee and K-I. Goh, Chaos Soliton Fractals 72, 49 (2015).

    ADS  Google Scholar 

  114. B. Min and K-I. Goh, Phys. Rev. E 89, 040802(R) (2014).

    ADS  Google Scholar 

  115. G. J. Baxter, S. N. Dorogovtsev, J. F. F. Mendes and D. Cellai, Phys. Rev. E 89, 042801 (2014).

    ADS  Google Scholar 

  116. C. D. Brummit, K-M. Lee and K-I. Goh, Phys. Rev. E 85, 045102(R) (2012).

    ADS  Google Scholar 

  117. K-M. Lee, C. D. Brummitt and K-I. Goh, Phys. Rev. E 90, 062816 (2014).

    ADS  Google Scholar 

  118. E. A. Leicht and R. M. D’Souza, arXiv:0907.0894v1.

  119. K-M. Lee, J. Y. Kim, W-k. Cho, K-I. Goh and I-M. Kim, New. J. Phys. 14, 033027 (2012).

    ADS  Google Scholar 

  120. A. Hackett, D. Cellai, S. Gómez, A. Arenas and J. P. Gleeson, Phys. Rev. X 6, 021002 (2016).

    Google Scholar 

  121. M. De Domenico, A. Solé-Ribalta, S. Gómez and A. Arenas, Proc. Natl. Acad. Sci. USA 111, 8351 (2014).

    ADS  MathSciNet  Google Scholar 

  122. B. Min, S-H. Gwak, N. Lee and K-I. Goh, Sci. Rep. 6, 21392 (2016).

    ADS  Google Scholar 

  123. A-L. Barabási, Network science (Cambridge University Press, Cambridge, 2016).

    MATH  Google Scholar 

  124. D. Lee, K-I. Goh, B. Kahng and D. Kim, Phys. Rev. E 82, 026112 (2010).

    ADS  Google Scholar 

  125. D. S. Callaway, J. E. Hopcroft, J. M. Kleinberg, M. E. J. Newman and S. H. Strogatz, Phys. Rev. E 64, 041902 (2001).

    ADS  Google Scholar 

  126. R. V. Solé, R. Pastor-Satorras, E. D. Smith and T. Kepler, Adv. Complex Syst. 05, 43 (2002).

    Google Scholar 

  127. A. Vázquez, A. Flammini, A. Maritan and A. Vespignani, ComPlexUs 1, 38 (2003).

    Google Scholar 

  128. J. Kim, P. L. Krapivsky, B. Kahng and S. Redner, Phys. Rev. E 66, 055101 (2002).

    ADS  Google Scholar 

  129. J. M. Kosteritz, Rev. Mod. Phys. 89, 040501 (2017).

    ADS  Google Scholar 

  130. J. M. Kosteritz and D. J. Thouless, J. Phys. C. 5, L124 (1972).

    ADS  Google Scholar 

  131. F. Duncan and M. Haldane, Rev. Mod. Phys. 89, 040502 (2017).

    MathSciNet  Google Scholar 

  132. S. M. Oh, S-W. Son and B. Kahng, Phys. Rev. E 93, 032316 (2016).

    ADS  MathSciNet  Google Scholar 

  133. S. M. Oh, S-W. Son and B. Kahng (unpublished).

  134. R. M. D’Souza and J. Nagler, Nat. Phys. 11, 531 (2015).

    Google Scholar 

  135. N. Araújo, P. Grassberger, B. Kahng, K. J. Schrenk and R. M. Ziff, Eur. Phys. J.: Spec. Top. 223, 2307 (2014).

    Google Scholar 

  136. D. Lee, Y. S. Cho and B. Kahng, J. Stat. Mech. P124002 (2016).

    Google Scholar 

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Authors and Affiliations

  1. CCSS, CTP and Department of Physics and Astronomy, Seoul National University, Seoul, 08826, Korea

    Deokjae Lee & B. Kahng

  2. Department of Physics, Chonbuk National University, Jeonju, 54896, Korea

    Y. S. Cho

  3. Department of Physics, Korea University, Seoul, 02841, Korea

    K.-I. Goh

  4. Department of Physics, Inha University, Incheon, 22212, Korea

    D.-S. Lee

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  1. Deokjae Lee
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  2. B. Kahng
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Lee, D., Kahng, B., Cho, Y.S. et al. Recent Advances of Percolation Theory in Complex Networks. J. Korean Phys. Soc. 73, 152–164 (2018). https://doi.org/10.3938/jkps.73.152

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