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.
The existence of a zero epidemic threshold for scale-free networks was first shown by Pastor-Satorrás and Vespignani (2001a).
- 2.
- 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.
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.
Pastor-Satorrás and Vespignani (2001a) used this specification.
- 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.
For example, a rule where agents adopt only if at least two sampled agents have adopted does not satisfy this assumption.
- 8.
- 9.
- 10.
- 11.
Notice that \(\Pi ^{t} = \Pi {\ast} \Pi {\ast}\ldots {\ast}\Pi \), t times.
- 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, \(n(i)\left \langle d\right \rangle _{i}\pi _{ij} = n(j)\left \langle d\right \rangle _{j}\pi _{ji}\).
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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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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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