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A Discontinuity Model of Technological Change: Catastrophe Theory and Network Structure

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Abstract

Discontinuities as a crucial aspect of economic systems have been discussed both verbally—particularly in institutionality theory—and formally, chiefly using catastrophe theory. Catastrophe theory has, however, been criticized heavily for lacking micro-foundations and has mainly fallen out of use in economics and social sciences. The present paper proposes a simple catastrophe theory model of technological change with network externalities and reevaluates the value of such a model by adding an agent-based micro layer. To this end an agent-based variant of the model is proposed and investigated specifically with regard to the network structure among the agents. While the macro level of the model produces a classical cusp catastrophe—a result that is preserved in the agent-based form—it is found that the behavior of the model changes locally depending on the network structure, especially if networks with features that resemble social networks (low diameter, high clustering, power law distributed node degree) are considered. While the present work investigates merely an aspect out of a large possibility space, it encourages further research using agent-based catastrophe theory models especially of economic aspects to which catastrophe theory has previously successfully been applied; aspects such as technological and institutional change, economic crises, or industry structure.

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Notes

  1. In game theory models, rational anticipation of the response of the second mover may play a role but it is questionable if that would be preserved under stochastic perturbation and other modifications.

  2. Note that this equation has a canonic functional for equations for technological change with network externalities. It corresponds to s-shape equations (or strong s-property) in Arthur et al. (1983[1982], 1987).

  3. By substituting

    $$\begin{aligned} a=-\frac{1}{3}z^2 \end{aligned}$$

    and

    $$\begin{aligned} b=-\frac{2z^3}{27}+z\beta . \end{aligned}$$

  4. Dunbar (1993) for instance proposed a number of around 300. There are some models that nevertheless propose using Barabási-Albert networks as a model of social networks, e.g. Barash et al. (2012)—the argument of networks among social groups (as opposed to within social groups) may be relevant to this discussion.

  5. This has been suggested directly in some models (Dahui et al. 2006; Stephen and Toubia 2009, but since both the size of firms (profits, capital, number of employees) and the degrees of the internet (but also the sizes of urban centers and many other quantities related to network technologies in one way or another) are known to be power law distributed (Newman 2005; Delli Gatti et al. 2005, this is generally a plausible assumption related to other stylized facts.

  6. The diameter of a network is the longest distance (shortest path) between two nodes in the network. A small diameter compared to the size of the network (number of nodes) this means that the network are relatively well-connected, more specifically having Watts and Strogatz’ small world property (Watts and Strogatz 1998) (or having a huge number of links compared to the number of nodes, thus being a complete or almost complete network as this would also lead to a small diameter).

  7. These two properties (and the conjunction of these two properties) in social networks has received particular attention in the small-world network literature (Watts and Strogatz 1998); for an extensive overview, see Schnettler (2009).

  8. The concept of triadic closure is, however, older. It was first applied in sociology and made popular in network theory by Granovetter (1973).

  9. The betweenness centrailty of a node i is the share of shortest path \(sp_{j,k}\) in the network between any two nodes (j and k) of which it is part (denoted \(sp_{j,k}(i)\))

    $$\begin{aligned} bc(i)=\sum _{j\ne k\ne i} \frac{sp_{j,k}(i)}{sp_{j,k}}. \end{aligned}$$

  10. In fact, the distributions of the node’s betweenness centrality seem to decay according to a power law for the preferential attachment network and the preferential attachment network with triadic closure, but with different tail slopes.

  11. Note that this module uses a breadth-first search algorithm to obtain a single shortest path between any two nodes. As in regular grid networks there are always many equally short paths, one is selected randomly, therefore the computed betweenness centralities differ slighly between the nodes; the correct result would be a constant and equal value for all nodes.

  12. Further, for the network 7, the number of links is such that the network is directly comparable to a Barabási-Albert preferential attachment network of the same size (Fig. 9).

  13. However, some promising approaches exist; these include urn schemes (Arthur et al. 1987; Buendía 2013) and network externality based models.

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Acknowledgments

The author would like to thank an anonymous reviewer as well as the discussants of the 2014 EAEPE conference for many helpful comments. The usual disclaimers apply.

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  1. Institute for Institutional and Innovation Economics (IINO), University of Bremen, Hochschulring 4, 28359, Bremen, Germany

    Torsten Heinrich

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Heinrich, T. A Discontinuity Model of Technological Change: Catastrophe Theory and Network Structure. Comput Econ 51, 407–425 (2018). https://doi.org/10.1007/s10614-016-9609-9

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