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Towards real-world complexity: an introduction to multiplex networks

  • Colloquium
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Abstract

Many real-world complex systems are best modeled by multiplex networks of interacting network layers. The multiplex network study is one of the newest and hottest themes in the statistical physics of complex networks. Pioneering studies have proven that the multiplexity has broad impact on the system’s structure and function. In this Colloquium paper, we present an organized review of the growing body of current literature on multiplex networks by categorizing existing studies broadly according to the type of layer coupling in the problem. Major recent advances in the field are surveyed and some outstanding open challenges and future perspectives will be proposed.

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References

  1. Networks of networks: The last frontier of complexity, edited by G. D’Agostino, A. Scala (Springer, 2014)

  2. M. Kivelä, A. Arenas, M. Barthélemy, J.P. Gleeson, Y. Moreno, M.A. Porter, J. Compl. Netw. 2, 203 (2014)

    Google Scholar 

  3. S. Boccaletti, G. Bianconi, R. Criado, C.I. del Genio, J. Gómez-Gardeñes, M. Romance, I. Sendiña-Nadal, Z. Wang, M. Zanin, Phys. Rep. 544, 1 (2014)

    ADS  MathSciNet  Google Scholar 

  4. P.J. Mucha, T. Richardson, K. Macon, M.A. Porter, J.-P. Onnela, Science 328, 876 (2010)

    ADS  MATH  MathSciNet  Google Scholar 

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

    ADS  Google Scholar 

  6. C.D. Brummitt, K.-M. Lee, K.-I. Goh, Phys. Rev. E 85, 045102(R) (2012)

    ADS  Google Scholar 

  7. S. Gómez, A. Díaz-Guilera, J. Gómez-Gardeñes, C.J. Pérez-Vicente, Y. Moreno, A. Arenas, Phys. Rev. Lett. 110, 028701 (2013)

    ADS  Google Scholar 

  8. G. Bianconi, Phys. Rev. E 87, 062806 (2013)

    ADS  Google Scholar 

  9. J.Y. Kim, K.-I. Goh, Phys. Rev. Lett. 111, 058702 (2013)

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  12. S. Wasserman, K. Faust, Social Network Analysis: Methods and Applications (Cambridge University Press, 1994)

  13. A. Nagurney, Q. Qiang, Fragile Networks: Identifying Vulnerabilities and Synergies in an Uncertain World (Wiley, 2009)

  14. E.A. Leicht, R.M. D’Souza, arXiv:0907.0894 (2009)

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

    ADS  Google Scholar 

  16. F. Radicchi, A. Arenas, Nat. Phys. 9, 717 (2013)

    Google Scholar 

  17. M.E.J. Newman, Networks: An Introduction (Oxford University Press, 2010)

  18. M. Szell, R. Lambiotte, S. Thurner, Proc. Natl. Acad. Sci. USA 107, 13636 (2010)

    ADS  Google Scholar 

  19. A. Vespignani, Nature 464, 984 (2010)

    ADS  Google Scholar 

  20. J. Gao, S.V. Buldyrev, H.E. Stanley, S. Havlin, Nat. Phys. 8, 40 (2012)

    Google Scholar 

  21. P. Holme, J. Saramäki, Phys. Rep. 519, 97 (2012)

    ADS  Google Scholar 

  22. Temporal Networks, edited by P. Holme, J. Saramäki (Springer, Heidelberg, 2013)

  23. B. Kapferer, in Social networks in urban situations, edited by J.C. Mitchell (Manchester University Press, 1969)

  24. L.M. Verbrugge, Social Forces 57, 1286 (1979)

    Google Scholar 

  25. M.J. Minor, in Applied Network Analysis: A Methodological Introduction, edited by R.S. Burt, M.J. Minor (Sage, 1983)

  26. J.F. Padgett, C.K. Ansell, Am. J. Sociol. 98, 1259 (1993)

    Google Scholar 

  27. M. Szell, S. Thurner, Soc. Netw. 32, 313 (2010)

    Google Scholar 

  28. S. Son, A.R. Kang, H.-C. Kim, T. Kwon, J. Park, H.K. Kim, PLoS One 7, e33918 (2012)

    ADS  Google Scholar 

  29. K.-M. Lee, J.Y. Kim, S. Lee, K.-I. Goh, Multiplex networks, in Networks of networks: The last frontier of complexity, edited by G. D’Agostino, A. Scala (Springer, 2014)

  30. G. Menichetti, D. Remondini, P. Panzarasa, R.J. Mondragon, G. Bianconi, PLoS One 9, e97857 (2014)

    Google Scholar 

  31. M. De Domenico, A. Lima, P. Mougel, M. Musolesi, Sci. Rep. 3, 2980 (2013)

    ADS  Google Scholar 

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

    ADS  MathSciNet  Google Scholar 

  33. C.-G. Gu, S.-R. Zou, X.-L. Xu, Y.-Q. Qu, Y.-M. Jiang, D.R. He, H.-K. Liu, T. Zhou, Phys. Rev. E 84, 026101 (2011)

    ADS  Google Scholar 

  34. A. Cardillo, M. Zanin, J. Gómez-Gardeñes, M. Romance, A.J. García del Amo, S. Boccaletti, Eur. Phys. J. Special Topics 215, 23 (2013)

    ADS  Google Scholar 

  35. R. Parshani, C. Rozenblat, D. Ietri, C. Ducruet, S. Havlin, Europhys. Lett. 92, 68002 (2010)

    ADS  Google Scholar 

  36. R.G. Little, J. Urban Technol. 9, 109 (2002)

    Google Scholar 

  37. V. Rosato, L. Issacharoff, F. Tiriticco, S. Meloni, S. Porcellinis, R. Setola, J. Critical Infrast. 4, 63 (2008)

    Google Scholar 

  38. M. Barigozzi, G. Fagiolo, D. Garlaschelli, Phys. Rev. E 81, 046104 (2010)

    ADS  Google Scholar 

  39. R. Foschi, M. Riccaboni, S. Schiavo, Phys. Rev. E 90, 022817 (2014)

    ADS  Google Scholar 

  40. S. Kühner et al., Science 326, 1235 (2009)

    ADS  Google Scholar 

  41. E. Yus et al., Science 326, 1263 (2009)

    ADS  Google Scholar 

  42. M. Güell et al., Science 326, 1268 (2009)

    ADS  Google Scholar 

  43. J.-F. Rual et al., Nature 437, 1173 (2005)

    ADS  Google Scholar 

  44. N.C. Duarte, S.A. Becker, N. Jamshidi, I. Thiele, M.L. Mo, T.D. Vo, R. Srivas, B.Ø. Palsson, Proc. Natl. Acad. Sci. USA 104, 1777 (2007)

    ADS  Google Scholar 

  45. J.R. Karr, J.C. Sanghvi, D.N. Macklin, M.V. Gutschow, J.M. Jacobs, B. Bolival Jr., N. Assad-Garcia, J.I. Glass, M.W. Covert, Cell 150, 389 (2012)

    Google Scholar 

  46. J.G. White, E. Southgate, J.N. Thomson, S. Brenner, Philos. Trans. R. Soc. London B 314, 1 (1986)

    ADS  Google Scholar 

  47. V. Nicosia, V. Latora, arXiv:1403.1546 (2014)

  48. M.J.O. Pocock, D.M. Evans, J. Memmott, Science 335, 973 (2012)

    ADS  Google Scholar 

  49. F. Battiston, V. Nicosia, V. Latora, Phys. Rev. E 89, 032804 (2014)

    ADS  Google Scholar 

  50. M. De Domenico, A. Solé-Ribalta, E. Cozzo, M. Kivelä, Y. Moreno, M.A. Porter, S. Gómez, A. Arenas, Phys. Rev. X 3, 041022 (2013)

    Google Scholar 

  51. A. Solé-Ribalta, M. De Domenico, N.E. Kouvaris, A. Díaz-Guilera, S. Gómez, A. Arenas, Phys. Rev. E 88, 032807 (2013)

    ADS  Google Scholar 

  52. E. Cozzo, R.A. Baños, S. Meloni, Y. Moreno, Phys. Rev. E 88, 050801(R) (2013)

    ADS  Google Scholar 

  53. L. Solá, M. Romance, R. Criado, J. Flores, A.G. del Amo, S. Boccaletti, Chaos 23, 033131 (2013)

    ADS  Google Scholar 

  54. A. Halu, R.J. Mondragón, P. Panzarasa, G. Bianconi, PLoS One 8, e78293 (2013)

    ADS  Google Scholar 

  55. M.E.J. Newman, Phys. Rev. Lett. 89, 208701 (2002)

    ADS  Google Scholar 

  56. V. Nicosia, G. Bianconi, V. Latora, M. Barthélemy, Phys. Rev. Lett. 111, 058701 (2013)

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  58. F. Tan, Y. Xia, W. Zhang, X. Jin, Europhys. Lett. 102, 28009 (2013)

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  61. B. Min, S. Lee, K.-M. Lee, K.-I. Goh, Chaos Soliton Fractals, in press (2015)

  62. J.F. Donges, H.C.H. Schultz, N. Marwan, Y. Zou, J. Kurths, Eur. Phys. J. B 84, 635 (2011)

    ADS  Google Scholar 

  63. P. Bródka, K. Skibicki, P. Kazienko, K. Musial, in 2011 International Conference on Computational Aspects of Social Networks (CASoN) (IEEE, 2011)

  64. P. Bródka, P. Kazienko, arXiv:1212.2425 (2012)

  65. A. Cardillo, M. Zanin, J. Gómez-Gardeñes, M. Romance, A.J. García del Amo, S. Boccaletti, Eur. Phys. J. Special Topics 215, 23 (2013)

    ADS  Google Scholar 

  66. E. Cozzo, M. Kivelä, M. De Domenico, A. Solé, A. Arenas, S. Gómez, M.A. Porter, Y. Moreno, arXiv:1307.6780 (2013)

  67. R.G. Morris, M. Barthélemy, Phys. Rev. Lett. 109, 128703 (2012)

    ADS  Google Scholar 

  68. S.V. Buldyrev, N.W. Shere, G.A. Cwilich, Phys. Rev. E 83, 016112 (2011)

    ADS  MathSciNet  Google Scholar 

  69. B. Söderberg, Phys. Rev. E 68, 015102(R) (2003)

    ADS  Google Scholar 

  70. B. Söderberg, Phys. Rev. E 68, 026107 (2003)

    ADS  MathSciNet  Google Scholar 

  71. D.-H. Kim, B. Kahng, D. Kim, Eur. Phys. J. B 38, 193 (2004)

    ADS  Google Scholar 

  72. A. Halu, S. Mukherjee, G. Bianconi, Phys. Rev. E 89, 012806 (2014)

    ADS  Google Scholar 

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

    ADS  MathSciNet  Google Scholar 

  74. B. Podobnik, D. Horvatić, M. Dickson, H.E. Stanley, Europhys. Lett. 100, 50004 (2012)

    ADS  Google Scholar 

  75. V. Nicosia, G. Bianconi, V. Latora, M. Barthélemy, Phys. Rev. E 90, 042807 (2014)

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  77. S.-W. Son, G. Bizhani, C. Christensen, P. Grassberger, M. Paczuski, Europhys. Lett. 97, 16006 (2012)

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  79. C.M. Schneider, N.A.M. Araújo, H.J. Herrmann, Phys. Rev. E 87, 043302 (2013)

    ADS  Google Scholar 

  80. S. Hwang, S. Choi, D. Lee, B. Kahng, arXiv:1409.1147 (2014)

  81. G.J. Baxter, S.N. Drogovtsev, A.V. Goltsev, J.F.F. Mendes, Phys. Rev. E 82, 011103 (2010)

    ADS  Google Scholar 

  82. G.J. Baxter, S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, Phys. Rev. E 83, 051134 (2011)

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  84. R. Parshani, S.V. Buldyrev, S. Havlin, Proc. Natl. Acad. Sci. USA 108, 1007 (2011)

    ADS  Google Scholar 

  85. Y. Hu, B. Ksherim, R. Cohen, S. Havlin, Phys. Rev. E 84, 066116 (2011)

    ADS  Google Scholar 

  86. J. Gao, S.V. Buldyrev, S. Havlin, H.E. Stanley, Phys. Rev. Lett. 107, 195701 (2011)

    ADS  Google Scholar 

  87. J. Gao, S.V. Buldyrev, S. Havlin, H.E. Stanley, Phys. Rev. E 85, 066134 (2012)

    ADS  Google Scholar 

  88. J. Gao, S.V. Buldyrev, H.E. Stanley, X. Xu, S. Havlin, Phys. Rev. E 88, 062816 (2013)

    ADS  Google Scholar 

  89. G. Bianconi, S.N. Dorogovtsev, J.F.F. Mendes, Phys. Rev. E 91, 012804 (2015)

    ADS  Google Scholar 

  90. G. Bianconi, S.N. Dorogovtsev, Phys. Rev. E 89, 062814 (2014)

    ADS  Google Scholar 

  91. M. Barthélemy, Phys. Rep. 499, 1 (2011)

    ADS  MathSciNet  Google Scholar 

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

    ADS  Google Scholar 

  93. Y. Berezin, A. Bashan, S. Havlin, Phys. Rev. Lett. 111, 189601 (2013)

    ADS  Google Scholar 

  94. W. Li, A. Bashan, S.V. Buldyrev, H.E. Stanley, S. Havlin, Phys. Rev. Lett. 108, 228702 (2012)

    ADS  Google Scholar 

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

    Google Scholar 

  96. L.D. Valdez, P.A. Macri, H.E. Stanley, L.A. Braunstein, Phys. Rev. E 88, 050803(R) (2013)

    ADS  Google Scholar 

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

    Google Scholar 

  98. J. Chalupa, P.L. Leath, G.R. Reich, J. Phys. C 12, L31 (1979)

    ADS  Google Scholar 

  99. J. Adler, A. Aharony, J. Phys. A 21, 1387 (1988)

    ADS  MathSciNet  Google Scholar 

  100. N. Azimi-Tafreshi, J. Gómez-Gardneñes, S.N. Dorogovtsev, Phys. Rev. E 90, 032816 (2014)

    ADS  Google Scholar 

  101. S. Havlin, R. Cohen, Complex networks: Structure, robustness and function (Cambridge University Press, 2010)

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

    ADS  Google Scholar 

  103. R. Albert, H. Jeong, A.-L. Barabási, Nature 406, 378 (2000)

    ADS  Google Scholar 

  104. R. Cohen, K. Erez, D. ben-Avraham, S. Havlin, Phys. Rev. Lett. 86, 3682 (2001)

    ADS  Google Scholar 

  105. D.S. Callaway, M.E.J. Newman, S.H. Strogatz, D.J. Watts, Phys. Rev. Lett. 85, 5469 (2000)

    ADS  Google Scholar 

  106. X. Huang, J. Gao, S.V. Buldyrev, S. Havlin, H.E. Stanley, Phys. Rev. E 83, 065101(R) (2011)

    ADS  Google Scholar 

  107. G. Dong, J. Gao, L. Tian, R. Du, Y. He, Phys. Rev. E 85, 016112 (2012)

    ADS  Google Scholar 

  108. G. Dong, J. Gao, R. Du, L. Tian, H.E. Stanley, S. Havlin, Phys. Rev. E 87, 052804 (2013)

    ADS  Google Scholar 

  109. Y. Berezin, A. Bashan, M.M. Danziger, D. Li, S. Havlin, arXiv:1310.0996 (2013)

  110. O. Yağan, J. Zhang, IEEE Trans. Parallel Distribut. Syst. 23, 1708 (2012)

    Google Scholar 

  111. C.M. Schneider, N. Yazdani, N.A.M. Araújo, S. Havlin, H.J. Herrmann, Sci. Rep. 3, 1969 (2013)

    ADS  Google Scholar 

  112. A. Barrat, M. Barthélemy, A. Vespignani, Dynamical Processes on Complex Networks (Cambridge University Press, 2008)

  113. O. Yağan, V. Gligor, Phys. Rev. E 86, 036103 (2012)

    ADS  Google Scholar 

  114. K.-M. Lee, K.-I. Goh, I.-M. Kim, J. Korean Phys. Soc. 60, 641 (2012)

    ADS  Google Scholar 

  115. C.D. Brummitt, R.M. D’Souza, Proc. Natl. Acad. Sci. USA 109, E680 (2012)

  116. T.C. Schelling, J. Conflict Resolution 17, 381 (1973)

    Google Scholar 

  117. M. Granovetter, Am. J. Sociol. 83, 1420 (1978)

    Google Scholar 

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

    ADS  MATH  MathSciNet  Google Scholar 

  119. J.P. Gleeson, Phys. Rev. E 77, 046117 (2008)

    ADS  Google Scholar 

  120. S. Melnik, M.A. Porter, P.J. Mucha, J.P. Gleeson, Chaos 24, 023106 (2014)

    ADS  Google Scholar 

  121. A.E. Motter, Y.-C. Lai, Phys. Rev. E 66, 065102(R) (2002)

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  123. M.E.J. Newman, Phys. Rev. Lett. 103, 058701 (2009)

    ADS  Google Scholar 

  124. M.E.J. Newman, G. Ghoshal, Phys. Rev. Lett. 100, 138701 (2008)

    ADS  Google Scholar 

  125. J.D. Noh, H. Rieger, Phys. Rev. Lett. 92, 118701 (2004)

    ADS  Google Scholar 

  126. S. Hwang, D.-S. Lee, B. Kahng, Phys. Rev. Lett. 109, 088701 (2012)

    ADS  Google Scholar 

  127. J.P. Garrahan, I. Lesanovsky, arXiv:1406.4706 (2014)

  128. M. De Domenico, V. Nicosia, A. Arenas, V. Latora, arXiv:1405.0425 (2014)

  129. L.M. Pecora, T.L. Carroll, Phys. Rev. Lett. 80, 2109 (1998)

    ADS  Google Scholar 

  130. T. Nishikawa, A.E. Motter, Y.-C. Lai, F.C. Hoppensteadt, Phys. Rev. Lett. 91, 014101 (2003)

    ADS  Google Scholar 

  131. Q. Xuan, F. Du, L. Yu, G. Chen, Phys. Rev. E 87, 032809 (2013)

    ADS  Google Scholar 

  132. M. Asllani, D.M. Busiello, T. Carletti, D. Fanelli, G. Planchon, Phys. Rev. E 90, 042814 (2014)

    ADS  Google Scholar 

  133. M. Salehi, R. Sharma, M. Marzolla, D. Montesi, P. Siyari, M. Magnani, arXiv:1405.4329 (2014)

  134. R.K. Ahuja, T.L. Magnanti, J.B. Orlin, Network Flows: Theory, Algorithms, and Applications (Prentice Hall, 1993)

  135. M. Kurant, P. Thiran, Phys. Rev. Lett. 96, 138701 (2006)

    ADS  Google Scholar 

  136. M. Kurant, P. Thiran, Phys. Rev. E 74, 036114 (2006)

    ADS  Google Scholar 

  137. R. Pastor-Satorras, C. Castellano, P. Van Mieghem, A. Vespignani, arXiv:1408.2701 (2014)

  138. M. Dickison, S. Havlin, H.E. Stanley, Phys. Rev. E 85, 066109 (2012)

    ADS  Google Scholar 

  139. A. Saumell-Mendiola, M.Á. Serrano, M. Boguañ´, Phys. Rev. E 86, 026106 (2012)

  140. O. Yağan, D. Qian, J. Zhang, D. Cochran, IEEE JSAC Special Issue on Network Science 31, 1038 (2013)

    Google Scholar 

  141. H. Wang, Q. Li, G. D’Agostino, S. Havlin, H.E. Stanley, P. Van Mieghem, Phys. Rev. E 88, 022801 (2013)

    ADS  Google Scholar 

  142. R. Vida, J. Galeano, S. Cuenda, Physica A 421, 134 (2015)

    MathSciNet  Google Scholar 

  143. C. Buono, L.G. Albarez-Zuzek, P.A. Macri, L.A. Braunstein, PLoS One 9, e92200 (2014)

    ADS  Google Scholar 

  144. B. Min, K.-I. Goh, arXiv:1307.2967 (2013)

  145. C. Granell, S. Gómez, A. Arenas, Phys. Rev. Lett. 111, 128701 (2013)

    ADS  Google Scholar 

  146. C. Granell, S. Gómez, A. Arenas, Phys. Rev. E 90, 012808 (2014)

    ADS  Google Scholar 

  147. W. Wang, M. Tang, H. Yang, Y. Do, Y.-C. Lai, G. Lee, Sci. Rep. 4, 5097 (2014)

    ADS  Google Scholar 

  148. E. Massaro, F. Bagnoli, Phys. Rev. E 90, 052817 (2014)

    ADS  Google Scholar 

  149. J. Sanz, C.-Y. Xia, S. Meloni, Y. Moreno, Phys. Rev. X 4, 041005 (2014)

    Google Scholar 

  150. F.D. Sahneh, C. Scoglio, Phys. Rev. E 89, 062817 (2014)

    ADS  Google Scholar 

  151. L. Chen, F. Ghanbarnejad, W. Cai, P. Grassberger, Europhys. Lett. 104, 50001 (2013)

    ADS  Google Scholar 

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

    MathSciNet  Google Scholar 

  153. J. Aguirre, D. Papo, J.M. Buldú, Nat. Phys. 9, 230 (2013)

    Google Scholar 

  154. K. Zhao, G. Bianconi, J. Stat. Phys. 152, 1069 (2013)

    ADS  MATH  MathSciNet  Google Scholar 

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

    ADS  MATH  MathSciNet  Google Scholar 

  156. S. Jang, J.S. Lee, S. Hwang, B. Kahng, arXiv:1410.0769 (2014)

  157. E. Cozzo, A. Arenas, Y. Moreno, Phys. Rev. E 86, 036115 (2012)

    ADS  Google Scholar 

  158. J. Um, P. Minnhagen, B.J. Kim, Chaos 21, 025106 (2011)

    ADS  MathSciNet  Google Scholar 

  159. V.H.P. Louzada, N.A.M. Araújo, J.S. Andrade Jr., H.J. Herrmann, Sci. Rep. 3, 3289 (2013)

    ADS  Google Scholar 

  160. J. Aguirre, R. Sevilla-Escoboza, R. Gutiérrez, D. Papo, J.M. Buldú, Phys. Rev. Lett. 112, 248701 (2014)

    ADS  Google Scholar 

  161. V. Nicosia, P.S. Skardal, V. Latora, A. Arenas, arXiv:1405.5855 (2014)

  162. J. Gómez-Gardeñes, I. Reinares, A. Arenas, L.M. Floría, Sci. Rep. 2, 620 (2012)

    Google Scholar 

  163. J. Gómez-Gardeñes, C. Gracia-Lázaro, L.M. Floría, Y. Moreno, Phys. Rev. E 86, 056113 (2012)

    ADS  Google Scholar 

  164. M. Diakonova, M.S. Miguel, V.M. Eguíluz, Phys. Rev. E 89, 062818 (2014)

    ADS  Google Scholar 

  165. J. Ashkin, E. Teller, Phys. Rev. 64, 178 (1943)

    ADS  Google Scholar 

  166. W. Selke, Phys. Rep. 70, 213 (1988)

    ADS  MathSciNet  Google Scholar 

  167. S. Galam, S.R. Salinas, Y. Shapir, Phys. Rev. B 51, 2864 (1995)

    ADS  Google Scholar 

  168. P.W. Anderson, The Theory of Superconductivity in the High- T c Cuprate Superconductors (Princeton University Press, 1997)

  169. S.-W. Cheong, M. Mostovoy, Nat. Mater. 6, 13 (2007)

    ADS  Google Scholar 

  170. A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, Rev. Mod. Phys. 81, 109 (2009)

    ADS  Google Scholar 

  171. Y.-Y. Liu, J.J. Slotine, A.-L. Barabási, Nature 473, 167 (2011)

    ADS  Google Scholar 

  172. T. Nepusz, T. Vicsek, Nat. Phys. 8, 568 (2012)

    Google Scholar 

  173. C. Luo, X. Wang, H. Liu, Chaos 24, 033108 (2014)

    ADS  Google Scholar 

  174. Z. Yuan, C. Zhao, W.-X. Wang, Z. Di, Y.-C. Lai, New. J. Phys. 16, 103036 (2014)

    ADS  MathSciNet  Google Scholar 

  175. S.N. Goltsev, A.V. Dorogovtsev, J.F.F. Mendes, Rev. Mod. Phys. 80, 1275 (2003)

    Google Scholar 

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Author notes

  1. Byungjoon Min

    Present address: Department of Physics and Levich Institute, City College of New York, 136-713, New York, NY, USA

Authors and Affiliations

  1. Department of Physics and Institute of Basic Science, Korea University, 136-713, Seoul, Korea

    Kyu-Min Lee, Byungjoon Min & Kwang-Il Goh

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  1. Kyu-Min Lee
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  2. Byungjoon Min
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  3. Kwang-Il Goh
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Correspondence to Kwang-Il Goh.

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Lee, KM., Min, B. & Goh, KI. Towards real-world complexity: an introduction to multiplex networks. Eur. Phys. J. B 88, 48 (2015). https://doi.org/10.1140/epjb/e2015-50742-1

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