Emergence of a Core-Periphery Structure of Regions

Over the course of a simulation run, there is, in general, an increase in the complexity of the artifacts that are produced in a region. Moreover, in general, the average, minimum, and maximum complexity of artifacts also increase if there is an increase in either the maximum distance m over which artifacts can be transported or in the maximum distance n over which research collaboration is possible. However, there may be strong regional differences in the artifact complexity depending on the positions of regions in the spatial configuration. Notably, a core-periphery structure may emerge in which core regions have high complexity, while peripheral regions have low complexity. Whether such a core-periphery structure emerges or not depends also on the spatial layout of regions. Moreover,

Fig. 6 Example of an artifact of complexity 5

we find that also the (strength of the) effect of distances m and n is moderated by the spatial layout.

Firstly, whenever the spatial layout of regions takes the form of a circle, there may be regional differences in individual cases (e.g. in a particular simulation instance, firms in one region may produce rather complex artifacts while, in other regions, firms produce rather primitive artifacts), but over 50 different seeds, there is no structural difference.

Secondly, in a cluster layout, there are regional differences in technological performance (i.e. in advancedness of transformations and complexity of artifacts) discovered, but only for low maximum artifact transportation distance m < 1. As soon as m is higher, the complexity in all regions is about the same. Table 1 contains plots of the spatial layout with in each region indicated the average complexity over 50 runs (and 5 % and 95 %-percentiles between square brackets) of the most advanced artifact feasible after T = 600 periods. In Fig. 7, we see that the core region has relatively high average artifact complexity. For m = 1, the central region has access to artifacts (used as inputs) in all regions, but the peripheral regions have access to artifacts provided in only a subset of regions. Whenever agents can acquire input at distance m = 2, the peripheral regions also have access to input artifacts in all regions and the average complexity is about the same as that of the core region. The artifacts in the periphery regions become substantially more complex and the core-periphery structure vanishes. This is explained from the fact that whenever firms fail to produce artifacts of a complexity higher than t due to absence of particular inputs, they will focus on innovations in transformations with an advancedness of t -1 and lower, but not higher. So, the rationale behind the effect of m is that by increasing the distance over which inputs can be acquired, artifact complexity goes up (in expectation). In fact, this effect on technological progress is amplified as firms subsequently also engage in search for more advanced transformations (in turn providing more advanced artifacts, potentially). In fact, for low m < 1, the maximum distance n over which firms collaborate in innovation has little effect on the artifact complexity. It is further observed that primarily the q = 0.2 case in which n does have (a moderate) effect on the artifact complexity, caused by the extensive progressive cross-linking in the

Table 1 Average artifact complexity after T = 600 periods in a cluster for various maximum production collaboration distances m=l, 2

Fig. 7 m = 1, n = 1	Fig. 8 m = 2, n = 1	Fig. 9 »)=1,h = 0	Fig. 10 m = 1, n = 2

Table 2 Average artifact complexity after T = 600 periods in a string for various distances m = n

Fig. 11 m = n =	Fig. 12 m = n = 2	Fig. 13 m = n = 3	Fig. 14 m = n = 5 (global)

transformation blueprint. In Table 1, we have plotted the cases for, p = 0.8 and q = 0.8, however, for all four combinations of p = 0.2, 0.8 and q = 0.2, 0.8, a similar core-periphery structure emerges (although slightly less polarized, particularly for lower p) and similarly vanishes with an increase in m.

Thirdly, there is a particularly strong core-periphery structure in case of a string layout (in which four regions have only two neighbors and the two regions on either ends of the string have only one neighbor). Table 2 contains plots of the spatial layout with in each region indicated the average complexity over 50 runs (and 5 % and 95 %-percentiles between square brackets) of the most advanced artifact feasible after T = 600 periods. In line with intuition, the middle two regions have the highest complexity (and hence advancedness) and the two region at either ends the lowest, in general. However, this core-periphery emerges nor for low, nor for high n, m, but is rather particularly strong when the distances n, m are somewhat in the middle. So, the discrepancy between the highest and the lowest level of artifact complexity in the various regions (a measure of how strong the core-periphery phenomenon is present) follows an inverted-U shape. The reason is that the maximum difference in the number of regions that can be accessed is then highest: when n, m = 2 (n, m = 5), a firm in the middle region can access four (five) other regions, while a firm in an end region can access two (five) other regions, with the difference two (zero).

Apart from the spatial layout of regions and distances n, m, also the technological structure defined by p and q affects the emerging levels of artifact complexity in the various regions. Whenever the transformation blueprint is conservative (q is high), i.e. transformations primarily extend transformations that are already ancestors, combining transformations within the regions will already unlock more advanced transformations and thereby allow production of relatively complex artifacts. So, whenever q is high, artifact complexity in the region is relatively high and collaboration in innovation across regional boundaries has relatively little impact. Simulation results confirm this robustly for each spatial layout of regions.

Moreover, whenever p is relatively high, many transformations in the blueprint split into two more advanced transformations, whereby each of these two transformations, say at tier t, takes two inputs that are the outputs of two uniformly drawn transformations at tier t -1. In case of a split, one primitive transformation gives rise to two options to create an artifact of higher advancedness. Consequently, whenever there is more splitting, artifact complexity is expected to be higher. Simulation results confirm this robustly for each spatial layout of regions. Access to more potential inputs (i.e. a higher m) increases artifact complexity, particularly when there is more splitting.