A Simple Person’s Approach to Understanding the Contagion Condition for Spreading Processes on Generalized Random Networks

  • Peter Sheridan DoddsEmail author
Part of the Computational Social Sciences book series (CSS)


We present derivations of the contagion condition for a range of spreading mechanisms on families of generalized random networks and bipartite random networks. We show how the contagion condition can be broken into three elements, two structural in nature, and the third a meshing of the contagion process and the network. The contagion conditions we obtain reflect the spreading dynamics in a clear, interpretable way. For threshold contagion, we discuss results for all-to-all and random network versions of the model, and draw connections between them.


  1. 1.
    Ahn YY, Ahnert SE, Bagrow JP, Barabási AL (2011) Flavor network and the principles of food pairing. Nat Sci Rep 1:196Google Scholar
  2. 2.
    Barabási AL, Albert R (1999) Emergence of scaling in random networks. Science 286:509–511Google Scholar
  3. 3.
    Boguñá M, Ángeles Serrano M (2005) Generalized percolation in random directed networks. Phys Rev E 72:016106Google Scholar
  4. 4.
    de Solla Price DJ (1965) Networks of scientific papers. Science 149:510–515Google Scholar
  5. 5.
    de Solla Price DJ (1976) A general theory of bibliometric and other cumulative advantage processes. J Am Soc Inform Sci 27:292–306Google Scholar
  6. 6.
    Dodds PS, Harris KD, Payne JL (2011) Direct, physically motivated derivation of the contagion condition for spreading processes on generalized random networks. Phys Rev E 83:056122Google Scholar
  7. 7.
    Eom YH, Jo HH (2014) Generalized friendship paradox in complex networks: the case of scientific collaboration. Nat Sci Rep 4:4603Google Scholar
  8. 8.
    Fforde J (2001) The Eyre affair: a thursday next novel. New English Library, LondonGoogle Scholar
  9. 9.
    García-Pérez LP, Serrano MA, Boguñá M (2014) The complex architecture of primes and natural numbers.
  10. 10.
    Gleeson JP (2008) Cascades on correlated and modular random networks. Phys Rev E 77:046117Google Scholar
  11. 11.
    Gleeson JP, Cahalane DJ (2007) Seed size strongly affects cascades on random networks. Phys Rev E 75:056103Google Scholar
  12. 12.
    Goh KI, Cusick ME, Valle D, Childs B, Vidal M, Barabási AL (2007) The human disease network. Proc Natl Acad Sci 104:8685–8690Google Scholar
  13. 13.
    Granovetter M (1978) Threshold models of collective behavior. Am J Sociol 83(6):1420–1443Google Scholar
  14. 14.
    Granovetter MS, Soong R (1983) Threshold models of diffusion and collective behavior. J Math Sociol 9:165–179Google Scholar
  15. 15.
    Granovetter MS, Soong R (1986) Threshold models of interpersonal effects in consumer demand. J Econ Behav Organ 7:83–99Google Scholar
  16. 16.
    Granovetter M, Soong R (1988) Threshold models of diversity: Chinese restaurants, residential segregation, and the spiral of silence. Sociol Methodol 18:69–104Google Scholar
  17. 17.
    Harris KD, Payne JL, Dodds PS (2014) Direct, physically-motivated derivation of triggering probabilities for contagion processes acting on correlated random networks.
  18. 18.
    Hidalgo CA, Klinger B, Barabási AL, Hausman R (2007) The product space conditions the development of nations. Science 317:482–487.
  19. 19.
    Kuran T (1991) Now out of never: the element of surprise in the east European revolution of 1989. World Polit 44:7–48Google Scholar
  20. 20.
    Kuran T (1997) Private truths, public lies: the social consequences of preference falsification, reprint edn. Harvard University Press, CambridgeGoogle Scholar
  21. 21.
    Lazarsfeld P, Merton R (1954) Friendship as social process: a substantive and methodological analysis. In: Berger M, Abel T, Page C (eds) Freedom and control in modern society. Van Nostrand, New York, pp 18–66Google Scholar
  22. 22.
    Molloy M, Reed B (1995) A critical point for random graphs on a fixed degree sequence. Random Struct Algorithms 6:161–180Google Scholar
  23. 23.
    Momeni N, Rabbat M (2016) Qualities and inequalities in online social networks through the lens of the generalized friendship paradox. PLoS One 11:e0143633Google Scholar
  24. 24.
    Munz P, Hudea I, Imad J, Smith? RJ (2009) When zombies attack!: mathematical modelling of an outbreak of zombie infection. In: Tchuenche JM, Chiyaka C (eds) Infectious disease modelling research progress. Nova Science, New York, pp 133–150Google Scholar
  25. 25.
    Newman MEJ (2003) The structure and function of complex networks. SIAM Rev 45(2):167–256Google Scholar
  26. 26.
    Newman MEJ, Strogatz SH, Watts DJ (2001) Random graphs with arbitrary degree distributions and their applications. Phys Rev E 64:026118Google Scholar
  27. 27.
    Oliver PE (1993) Formal models of collective action. Ann Rev Sociol 19:271–300Google Scholar
  28. 28.
    Oliver PE, Marwell G, Teixeira R (1985) A theory of the critical mass. I. Interdependence, group heterogeneity, and the production of collective action. Am J Sociol 91(3):522–556Google Scholar
  29. 29.
    Olson M (1971) The logic of collective action: public goods and the theory of groups, revised edn. Harvard Economic Studies, Harvard University Press, CambridgeGoogle Scholar
  30. 30.
    Schelling TC (1971) Dynamic models of segregation. J Math Sociol 1:143–186Google Scholar
  31. 31.
    Schelling TC (1973) Hockey helmets, concealed weapons, and daylight saving: a study of binary choices with externalities. J Confl Resolut 17:381–428Google Scholar
  32. 32.
    Schelling TC (1978) Micromotives and macrobehavior. Norton, New YorkGoogle Scholar
  33. 33.
    Stauffer D, Aharony A (1992) Introduction to percolation theory, 2nd edn. Taylor & Francis, WashingtonGoogle Scholar
  34. 34.
    Teng CY, Lin YR, Adamic LA (2012) Recipe recommendation using ingredient networks. In: Proceedings of the 3rd annual ACM web science conference, WebSci ’12. ACM, New York, pp 298–307Google Scholar
  35. 35.
    TV Tropes (2017).
  36. 36.
    Watts DJ (2002) A simple model of global cascades on random networks. Proc Natl Acad Sci 99(9):5766–5771Google Scholar
  37. 37.
    Watts DJ, Dodds PS (2007) Influentials, networks, and public opinion formation. J Consum Res 34:441–458Google Scholar
  38. 38.
    Watts DJ, Strogatz SJ (1998) Collective dynamics of ‘small-world’ networks. Nature 393:440–442Google Scholar
  39. 39.
    Watts DJ, Dodds PS, Newman MEJ (2002) Identity and search in social networks. Science 296:1302–1305Google Scholar
  40. 40.
    Wilf HS (2006) Generating functionology, 3rd edn. A K Peters, NatickGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Vermont Complex Systems Center, Computational Story Lab, the Vermont Advanced Computing Core, Department of Mathematics & StatisticsThe University of VermontBurlingtonUSA

Personalised recommendations