Spatial patterns and size distributions of cities

City size distributions are known to be well approximated by power laws across many countries. One popular explanation for such power-law regularities is in terms of random growth processes, where power laws arise asymptotically from the assumption of iid growth rates among all cities within a given country. But this assumption has additional consequences. Since all subsets of cities have the same statistical properties, each subset must exhibit essentially the same power law. Moreover, this common power law (CPL) property must hold regardless of the spatial relations among cities. Using data from the US, this paper shows first that spatial partitions of cities based on geographical proximity are significantly more consistent with the CPL property than are random partitions. It is then shown that this significance becomes even stronger when proximity among cities is measured in terms of trade linkages rather than simple geographical distance. These results provide compelling evidence that spatial relations between cities do indeed matter for city-size distributions. Further analysis shows that these results hinge on the natural "spacing out" property of city patterns in which larger cities tend to be widely spaced apart with smaller cities organized around them.