The Weak Core and the Structure of Elites in Social Multiplex Networks

  • Bernat Corominas-MurtraEmail author
  • Stefan Thurner
Part of the Understanding Complex Systems book series (UCS)


Recent approaches on elite identification highlighted the important role of intermediaries, by means of a new definition of the core of a multiplex network, the generalised K-core. This newly introduced core subgraph crucially incorporates those individuals who, in spite of not being very connected, maintain the cohesiveness and plasticity of the core. Interestingly, it has been shown that the performance on elite identification of the generalised K-core is sensibly better that the standard K-core. Here we go further: Over a multiplex social system, we isolate the community structure of the generalised K-core and we identify the weakly connected regions acting as bridges between core communities, ensuring the cohesiveness and connectivity of the core region. This gluing region is the Weak core of the multiplex system. We test the suitability of our method on data from the society of 420,000 players of the Massive Multiplayer Online Game Pardus. Results show that the generalised K-core displays a clearly identifiable community structure and that the weak core gluing the core communities shows very low connectivity and clustering. Nonetheless, despite its low connectivity, the weak core forms a unique, cohesive structure. In addition, we find that members populating the weak core have the best scores on social performance, when compared to the other elements of the generalised K-core. The weak core provides a new angle on understanding the social structure of elites, highlighting those subgroups of individuals whose role is to glue different communities in the core.


Weak Core Multiplex Networks (MPN) Multiplex Social System Coral Communities Core Subgraph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work was supported by the Austrian Science Fund FWF under KPP23378FW, the EU LASAGNE project, no. 318132 and the EU MULTIPLEX project, no. 318132.


  1. 1.
    Mills, C.W.: The Power Elite. Oxford University Press, Oxford (1956)Google Scholar
  2. 2.
    Mills, C.W.: The structure of power in american society. Br. J. Sociol. 9(1), 29–41 (1958)CrossRefGoogle Scholar
  3. 3.
    Keller, S.: Beyond the Ruling Class. Strategic Elites in Modern Society. Random House, New York (1963)Google Scholar
  4. 4.
    William, F.G.: Who Rules America? McGraw-Hill, New York (1967)Google Scholar
  5. 5.
    Bottomore, T.: Elites and Society, 2nd edn. Routledge, London (1993)Google Scholar
  6. 6.
    Friedkin, N.E.: Structural cohesion and equivalence explanations of social homogeneity. Sociol. Methods Res. 12, 235–261 (1984)CrossRefGoogle Scholar
  7. 7.
    Corominas-Murtra, B., Fuchs, B., Thurner, S.: Detection of the elite structure in a virtual multiplex social system by means of a generalised K-core. PLoS ONE 9(12), e112606 (2014)CrossRefADSGoogle Scholar
  8. 8.
    Mucha, P.J., Richardson, T., Macon, K., Porter, M.A., Onnela, J.P.: Community structure in time-dependent, multiscale, and multiplex networks. Science 328, 876–878 (2010)CrossRefADSMathSciNetzbMATHGoogle Scholar
  9. 9.
    Szell, M., Thurner, S.: Measuring social dynamics in a massive multiplayer online game. Soc. Netw. 39, 313–329 (2010)CrossRefGoogle Scholar
  10. 10.
    Nicosia, V., Bianconi, G., Latora, V., Barthelemy, M.: Growing multiplex networks. Phys. Rev. Lett. 111, 058701 (2013)CrossRefADSGoogle Scholar
  11. 11.
    Wasserman, S., Faust, K.: Social Network Analysis. Cambridge University Press, Cambridge/New York (1994)CrossRefzbMATHGoogle Scholar
  12. 12.
    Seidman, S.B.: Network structure and minimum degree. Soc. Netw. 5, 269–287 (1983)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Bollobás, B.: The evolution of sparse graphs. In: Graph Theory and Combinatorics, Proc Cambridge Combinatorial Conf in Honor to Paul Erdös, pp. 35–57. Academic, London (1984)Google Scholar
  14. 14.
    Dorogovtsev, S.N., Goltsev, A.V., Mendes, J.F.F.: k-core organization of complex networks. Phys. Rev. Lett. 96, 040601 (2006)Google Scholar
  15. 15.
    Szell, M., Lambiotte, R., Thurner, S.: Multirelational organization of large-scale social networks in an online world. Proc. Natl. Acad. Sci. 107, 13636–13641 (2010)CrossRefADSGoogle Scholar
  16. 16.
    Castronova, E.: Synthetic Worlds: The Business and Culture of Online Games. University of Chicago Press, Chicago (2005)Google Scholar
  17. 17.
    Szell, M., Thurner, S.: Social dynamics in a large-scale online game. Adv. Complex Syst. 15, 1250064 (2012)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Szell, M., Sinatra, R., Petri, G., Thurner, S., Latora, V.: Understanding mobility in a social Petri dish. Sci. Rep. 2, 457 (2012)CrossRefADSGoogle Scholar
  19. 19.
    Thurner, S., Szell, M., Sinatra, R.: Emergence of good conduct, scaling and zipf laws in human behavioral sequences in an online world. PLoS ONE 7, e29796 (2012)CrossRefADSGoogle Scholar
  20. 20.
    Szell, M., Thurner, S.: How women organise social networks different from men: gender-specific behavior in large-scale social networks. Sci. Rep. 3, 1214 (2013)CrossRefADSGoogle Scholar
  21. 21.
    Fuchs, B., Thurner, S.: Behavioral and network origins of wealth inequality: insights from a virtual world. PLoS ONE 9(8), e103503 (2014). doi:10.1371/journal.pone.0103503CrossRefADSGoogle Scholar
  22. 22.
    Bianconi, G.: Statistical mechanics of multiplex networks: entropy and overlap. Phys. Rev. E 87, 062806 (2013)CrossRefADSGoogle Scholar
  23. 23.
    Colomer-de-Simón, P., Serrano, M.Á., Beiró, M.G., Alvarez-Hamelin, J.I., Boguñá, M.: Deciphering the global organization of clustering in real complex networks. Sci. Rep. 3, 2517 (2013). doi:10.1038/srep02517CrossRefADSGoogle Scholar
  24. 24.
    Rapoport, A.: Spread of information through a population with socio-structural bias: I. Assumption of transitivity. Bull. Math. Biol. 15, 523–533 (1953)MathSciNetGoogle Scholar
  25. 25.
    Granovetter, M.: The strength of weak ties. Am. J. Sociol. 78, 1360–1380 (1973)CrossRefGoogle Scholar
  26. 26.
    Davidsen, J., Ebel, H., Bornholdt, S.: Emergence of a small world from local interactions: modeling acquaintance networks. Phys. Rev. Lett. 88, 128701 (2002)CrossRefADSGoogle Scholar
  27. 27.
    Klimek, P., Thurner, S.: Triadic closure dynamics drives scaling laws in social multiplex networks. New J. Phys. 15, 063008 (2013)CrossRefADSMathSciNetGoogle Scholar
  28. 28.
    Harary, F., Ross, I.C.: A procedure for clique detection using the group matrix. Sociometry 20, 205–215 (1957)CrossRefMathSciNetGoogle Scholar
  29. 29.
    Colizza, V., Flammini, A., Serrano, M.A., Vespignani, A.: Detecting rich-club ordering in complex networks. Nat. Phys. 2, 110–115 (2006)CrossRefGoogle Scholar
  30. 30.
    Corominas-Murtra, B., Valverde, S., Rodríguez-Caso, C., Solé, R.V.: K-scaffold subgraphs of complex networks. EPL (Europhys. Lett.) 77, 18004 (2007)Google Scholar
  31. 31.
    Corominas-Murtra, B., Mendes, J.F.F., Solé, R.V.: Nested subgraphs of complex networks. J. Phys. A: Math. Theor. 41, 385003 (2008)CrossRefADSMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Section for Science of Complex SystemsMedical University of ViennaViennaAustria
  2. 2.Santa Fe InstituteSanta FeUSA
  3. 3.IIASALaxenburgAustria

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