v 1 . 1 . 4 Generative modeling of city - size scaling laws , 250 BCE – 2005 : Embedded co - evolution in the theory of long - term geopolitical dynamics ∗

Scaling world city size distributions are analyzed in terms of q-exponentials, the distributions which naturally emerge within nonextensive statistical mechanics. These distributions allow to estimate hierarchy and scale parameters for 29 historical periods, thus conveniently replacing the Zipfian scaling law by an historically sensitive and nuanced model of how long-term oscillations in urban hierarchies co-evolve with urban industrial, commercial and financial innovations. Our present model emerges from attempts to understand our scaling results by validation tests using multiple political and economic variables. We investigate the model so as to explore the extent and by-play of endogenous dynamics and exogenous shocks as causes of synchronization. One such step consists of identifying a network macro-structural variable, the average urban hub adjacency ratio, to help explain interactive network dynamics of endogenous synchronization. We find, further, in examining our validation-check variables, that innovation clusters in space and time – the leading sectors of urban economies – are differentially embedded in periods of alternating hierarchical and heterarchical urban scaling regimes. Urban infrastructure and demography, as expected, change slowly compared to the pace of Schumpeterian K-cyclic tendencies in bursts of competitive economic innovation, but the synchrony of these changes in time and space, while a surprising discovery, is consistent with our network-economic theory. [DOUG: I do not understand the sentence above.] Sandwiched in an intermediate layer of change in terms of its spatiotemporal scalings we find the shifts in the leading cities and states, as studied by Modelski and Thompson, forming a synchronously-embedded three-layer model of economic, political, and urban structural oscillations. These findings have potential for unifying a new perspective on glacially slow and then quickly triggered structural changes in the city hierarchies and infrastructure with perspectives on long-cycles as studied through the world political and economic perspectives.