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Erschienen in: Quality & Quantity 4/2015

01.07.2015

Diffusion of innovations in dense and sparse networks

verfasst von: Mario Eboli

Erschienen in: Quality & Quantity | Ausgabe 4/2015

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Abstract

This paper puts forward a comparison of the performance of sparsely and densely connected social networks in promoting the diffusion of innovations of uncertain profitability. To this end, we use a threshold model of innovation diffusion, based on a classic model of adoption of innovations via imitation by Jensen (Int. J. Ind. Organ. 6:335–350, 1988), to evaluate the probability of diffusion of an innovation in three classes of networks: the circular, the star-shaped and the complete networks. We find that, if agents hold a low prior confidence in the profitability of an innovation, then complete networks and star networks with informed agents (i.e., with agents who are aware of the structure of the network and use this information rationally) perform better than circles and than stars with myopic agents. The converse is true for innovations accompanied by initial high expectations about their profitability.

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Fußnoten
1
In threshold models of diffusion in networks, the structure of the network is represented by its adjacency matrix, which is utilised to compute recursively the contagion process started by an initial set of adoptions.
 
2
See the works cited in Sect. 4.
 
3
Complete networks, where everybody observes everybody else, depict well situations where the actions of all members of a society are publicly observable (e.g., public reports on the number of consumers who switched to a new product). This class of networks is considered an acceptable approximation of highly connected networks. Star-shaped networks, and its multi-center version known as the ‘royal family’, represent situations where individuals observe local behaviours as well as some common/public source of information. These networks tend to arise when there are one or more highly visible agents (successful companies, research centers, pop stars, etc.) who attract the attention of less prominent agents. This is the case, for instance, of the agricultural sector, where individual farmers observe their neighbouring farmers and all the farmers observe a few large and successful farms. Circular networks depict situations where the interaction among agents is severely limited by proximity (in a geographical or social sense), and are used in social learning models to represent restricted local interaction. See the comments in Goyal (2005) on the ‘royal family’ and the bilateral circular network.
 
4
Jackson and Wolinsky (1996), for instance, show that, in networks with bilateral connections, the complete structure is both efficient and pairwise stable for low values of the cost of forming connections, while the star structure is both efficient and pairwise stable for intermediate values of such a cost. Other notable examples are Bala and Goyal (2000) and Hojman and Szeidl (2008), where the star emerges as the unique equilibrium network structure if the cost of linking is not too high. The circle, the star and the complete network also appear, as representative structures, in the literature on learning in networks. In Gale and Kariv (2003), the analysis focuses on these three structures because, as the authors argue, they span all possible networks in three agents games. Sanjeev Goyal (2005), inter alia, compares the performance of the circle and of the ‘royal family’ structure.
 
5
This feature of Jensen’s model, expressed in the above theorem 1, enables us to characterise the probability of diffusion in a network without the need of modelling a dynamic stochastic process of adoptions, as will be clearer below.
 
6
In this paper we take these network structures as a datum of the problem at hand and do not investigate the behaviour of an agent in choosing the sample of other agents to be monitored.
 
7
We content ourselves with the analysis of the unilateral circle because the performance of the bilateral circle, in the present setting, is highly sensitive to the distribution of the early adopters along the circle, as argued in the next footnote.
 
8
The result is less neat if the circular network is bilateral, with each agent monitoring her neighbours on both sides. It can be checked that, in this case, there are three relevant ranges of the prior belief \(p_{0}.\) For high values of \(p_{0},\)the bilateral circle performs exactly like the unilateral one: one early adoption suffices to induce a full diffusion. The same applies for low values of \(p_{0}\): diffusion never occurs. For intermediate values of \(p_{0},\) the result remains ambigous because it depends on both the number of early adoption and on the location of the early adopters along the circle. For instance, a complete diffusion is guaranteed if half of the agents are early adopters and early and late adopters are distributed in an alternate fashion: each late adopter is surrounded by two early adopters, and vice versa. The same number of early adoptions does not induce a complete diffusion with a different distribution of early adopters along the circle.
 
9
Morris (2000) presents a similar result for unilateral circular networks. The author, under some restrictive assumptions, obtains that diffusion occurs for a prior belief lager than or equal to \(1/2.\) See Morris (2000), Sect. 3, p. 61.
 
10
In this sense, such agents are endowed with a bounded rationality.
 
11
Consider that – in a circular network where, say, agent \(i\) observes agent \( i-1\) – to agent \(i\) it does not matter whether agent \(i-1\) has adopted because she is an early adopter or because there have been one or more early adoption upstream along the circle. Even if agent \(i\) can observe the date at which agent \(i-1\) adopts the innovation, and infer the distance between \( i-1\) and the closest early adopter, this signal does not reveal whether there have been one or more than one early adoptions in the circle.
 
12
See Rolfe (2004) and Acemoglu et al. (2010), and the literature cited therein.
 
13
Threshold models of diffusion are also frequently embedded in models of social learning. See, inter alia, Gale and Kariv (2003), Ellison and Fudenberg (1993), Ellison (1993) and Peyton Young (2006).
 
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Metadaten
Titel
Diffusion of innovations in dense and sparse networks
verfasst von
Mario Eboli
Publikationsdatum
01.07.2015
Verlag
Springer Netherlands
Erschienen in
Quality & Quantity / Ausgabe 4/2015
Print ISSN: 0033-5177
Elektronische ISSN: 1573-7845
DOI
https://doi.org/10.1007/s11135-014-0069-9

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