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Complex Networks and Dynamics pp 27–48Cite as

An Overview of Diffusion in Complex Networks

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Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 683))

Abstract

We survey a series of theoretical contributions on diffusion in random networks. We start with a benchmark contagion process, referred in the epidemiology literature as the Susceptible-Infected-Susceptible model, which describes the spread of an infectious disease in a population. To make this model tractable, the interaction structure is considered as a heterogeneous sampling process characterized by the degree distribution. Within this framework, we distinguish between the case of unbiased-degree networks and biased-degree networks. We focus on the characterization of the diffusion threshold; that is, a condition on the primitives of the model that guarantees the spreading of the product to a significant fraction of the population, and its persistence. We also extend the analysis introducing a general diffusion model with features that are more appropriate for describing the diffusion of a new product, idea, behavior, etc.

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Notes

  1. 1.

    The existence of a zero epidemic threshold for scale-free networks was first shown by Pastor-Satorrás and Vespignani (2001a).

  2. 2.

    The so-called SIS model has extensively been studied in the literature (see e.g., Pastor-Satorrás and Vespignani 2001a; Jackson and Rogers 2007, etc.).

  3. 3.

    Note that in the context of a disease, it is implicitly assumed that there is no full immunization and therefore a recovered person can catch the disease again. An obvious instance is the standard flu.

  4. 4.

    Benaïm and Weibull (2003) show that the continuous (deterministic) approximation is appropriate when dealing with large populations. In particular, they find that if the deterministic population flow remains forever in some subset of the state space, then the stochastic process will remain in the same subset space for a very long time with a probability arbitrarily close to one, provided the population is large enough.

  5. 5.

    Pastor-Satorrás and Vespignani (2001a) used this specification.

  6. 6.

    Jackson and Rogers (2007) were the first to analyze the diffusion proprieties of networks ordered through the stochastic dominance of their degree distributions.

  7. 7.

    For example, a rule where agents adopt only if at least two sampled agents have adopted does not satisfy this assumption.

  8. 8.

    These threshold models have been extensively analyzed in the literature (see Granovetter 1978, Watts 2002, López-Pintado 2006, and Jackson and Yariv 2007).

  9. 9.

    The study of how collective outcomes depend on the details of the contagion process has also been highlighted by Young (2009), Galeotti and Goyal (2009), López-Pintado and Watts (2008), etc.

  10. 10.

    There are some exceptions such as Currarini et al. (2009), Golub and Jackson (2012b), among others.

  11. 11.

    Notice that \(\Pi ^{t} = \Pi {\ast} \Pi {\ast}\ldots {\ast}\Pi \), t times.

  12. 12.

    In the biased-degree case, certain constraints on the parameters of the model would be required in order to approximate it to an undirected network. For example, the number of interactions from type i to type j should coincide with the number of interactions from type j to type i in a unit of time. That is, \(n(i)\left \langle d\right \rangle _{i}\pi _{ij} = n(j)\left \langle d\right \rangle _{j}\pi _{ji}\).

References

  • Bailey NTJ (1975) The mathematical theory of infectious diseases. Griffin, London

    Google Scholar 

  • Bala V, Goyal S (1998) Learning from neighbors. Rev Econ Stud 65:595–621

    Article  Google Scholar 

  • Benaïm M, Weibull JW (2003) Deterministic approximation of stochastic evolution in games. Econometrica 71:873–903

    Article  Google Scholar 

  • Bollobás B (2001) Random graphs. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Conley TG, Udry CR (2001) Social learning through networks: the adoption of new agricultural technologies in Ghana. Am J Agric Econ 83:668–673

    Article  Google Scholar 

  • Currarini S, Jackson MO, Pin P (2009) An economic model of friendship: homophyly, minorities, and segregation. Econometrica 77:1003–1045

    Article  Google Scholar 

  • Erdös P, Rényi A (1959) On random graphs. Publ Math Debr 6:290–297

    Google Scholar 

  • Galeotti A, Goyal S (2009) A theory of strategic diffusion. Rand J Econ 40:509–532

    Article  Google Scholar 

  • Galeotti A, Goyal S, Jackson MO, Vega-Redondo F, Yariv L (2010) Network games. Rev Econ Stud 77:218–244

    Article  Google Scholar 

  • Golub B, Jackson MO (2012a) How homophily affects the speed of learning and best response dynamics. Q J Econ 127:1287–1338

    Article  Google Scholar 

  • Golub B, Jackson MO (2012b) Does homophily predict consensus times? Testing a model of network structure via a dynamic process. Rev Netw Econ 11:Article 9

    Google Scholar 

  • Golub B, Jackson MO (2012c) Network structure and the speed of learning: measuring homophily based on its consequences. Ann Econ Stat 107/108:33–35

    Article  Google Scholar 

  • Granovetter MS (1978) Threshold models of collective behavior. Am J Sociol 83:1420–1443

    Article  Google Scholar 

  • Jackson MO, López-Pintado D (2013) Diffusion and contagion in networks with heterogeneous agents and homophily. Netw Sci 1:49–67

    Article  Google Scholar 

  • Jackson MO, Rogers B (2007) Relating network structure to diffusion properties through stochastic dominance. B.E. J Theor Econ (Advances) 7:1–13

    Google Scholar 

  • Jackson MO, Yariv L (2007) Diffusion of behavior and equilibrium properties in network games. Am Econ Rev (Papers and Proceedings) 97:92–98

    Google Scholar 

  • López-Pintado D (2006) Contagion and coordination in random networks. Int J Game Theory 34:371–381

    Article  Google Scholar 

  • López-Pintado D (2008) Diffusion in complex social networks. Games Econ Behav 62:573–90

    Article  Google Scholar 

  • López-Pintado D (2012) Influence networks. Games Econ Behav 75:776–787

    Article  Google Scholar 

  • López-Pintado D, Watts DJ (2008) Social influence, binary decisions and collective dynamics. Ration Soc 20:399–443

    Article  Google Scholar 

  • Morris S (2000) Contagion. Rev Econ Stud 67:57–78

    Article  Google Scholar 

  • Pastor-Satorrás R, Vespignani A ( 2001a) Epidemic spreading in scale-free networks. Phys Rev Lett 86:3200–3203

    Google Scholar 

  • Pastor-Satorrás R, Vespignani A (2001b) Epidemic dynamics and endemic states in complex networks. Phys Rev E 63:066117

    Article  Google Scholar 

  • Price DJS (1965) Networks of scientific papers. Science 149:510–515

    Article  Google Scholar 

  • Rogers EM (1995) Diffusion of innovations. Free Press, New York

    Google Scholar 

  • Watts DJ (2002) A simple model of information cascades on random networks. Proc Natl Acad Sci 99:5766–5771

    Article  Google Scholar 

  • Young HP (2009) Innovation diffusion in heterogeneous populations: contagion, social influence, and social learning. Am Econ Rev 99:1899–1924

    Article  Google Scholar 

Download references

Acknowledgements

I want to thank Juan D. Moreno-Ternero for his helpful comments. Financial support from the Spanish Ministry of Economy and Competitiveness (ECO2014-57413-P) is gratefully acknowledged.

This chapter is based upon work from COST Action ISCH COST Action IS1104 “The EU in the new complex geography of economic systems: models, tools and policy evaluation”, supported by COST (European Cooperation in Science and Technology), www.cost.eu.

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Authors and Affiliations

  1. CORE, Université catholique de Louvain, Louvain-la-Neuve, Belgium

    Dunia López-Pintado

  2. Department of Economics, Universidad Pablo de Olavide, Carretera de Utrera, Km. 1, 41013, Seville, Spain

    Dunia López-Pintado

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  1. Dunia López-Pintado
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Correspondence to Dunia López-Pintado .

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Editors and Affiliations

  1. Department of Law, University of Naples "Federico II", Naples, Italy

    Pasquale Commendatore

  2. Faculty of Economics, UNED, Madrid, Spain

    Mariano Matilla-García

  3. Faculty of Physics, University of Santiago de Compostela, Santiago de Compostela, Spain

    Luis M. Varela

  4. Department of Applied Mathematics, Technical University of Cartagena, Cartagena, Spain

    Jose S. Cánovas

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López-Pintado, D. (2016). An Overview of Diffusion in Complex Networks. In: Commendatore, P., Matilla-García, M., Varela, L., Cánovas, J. (eds) Complex Networks and Dynamics. Lecture Notes in Economics and Mathematical Systems, vol 683. Springer, Cham. https://doi.org/10.1007/978-3-319-40803-3_2

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