Did Zipf Anticipate Socio-Economic Spatial Networks?

An avalanche of empirical studies has addressed the validity of the rank-size rule (or Zipf’s law) in a multi-city context in many countries. City size in most countries seems to obey Zipf’s law, but the question under which conditions (e.g. sample size, spatial scale) this ‘law’ holds remained largely underinvestigated. Another complementary question is whether socio-economic networks in space also show a similar hierarchical pattern. Against this background, the present paper investigates – from a methodological viewpoint – the relationship between network connectivity and the rank-size rule (or Zipf’s law) in an urban-economic network constellation. After a review of the literature, we address in particular the following methodological issues: (i) the (aggregate) behavioral foundation underlying the rank-size rule/Zipf’s law in the light of spatial-economic network theories (e.g. entropy maximization, spatial interaction theory, etc.); (ii) the nature of the analytical relationship between social-spatial network analysis and the rank-size rule/Zipf’s law. We argue that the rank size rule is compatible with conventional economic foundations of spatial network models. Consequently, a spatial-economic interpretation – as well as a network connectivity interpretation – of the rank-size rule coefficient is provided. Our methodological contribution forms the foundation for the subsequent empirical analysis applied to spatial networks in a socio-economic context. The aim here is to test the sensitivity of empirical findings for changes in scale, functional forms, time periods, and network structures. Our application is concerned with an extensive spatio-temporal panel database related to the evolution of urban population in Germany. We test the relevance of the rank-size rule/Zipf’s law, and its evolution over the years, and – in parallel – the related ‘socio-economic’ connectivity in these urban networks. In particular, we will show that Zipf’s law (i.e., with the rank-size coefficient equal to 1) is only valid under particular conditions of the sample size. The paper concludes with some retrospective and prospective remarks.