Defining City Size

To the mind's eye, cities are dense agglomerations where, historically, people have come together to trade and engage in diverse social relationships. In abstract generalization, they are considered points in space where the friction of distance that restricts our ability to relate to one another is minimized. Ever since cities emerged from our nomadic past some 10 000 years ago they have been dominated by a relatively high density of living where people crowd together. Until about 200 years ago their size was limited by how far we could walk or at least ride a horse and cities rarely reached a population of more than 1 million. When they did, as in Rome, they fell apart at the limits of known technology. But the industrial revolution, which sparked the exploitation of mechanical energy, changed all that, just as the current seamless transition from energy to information is destined to do so again in the current period. It took fifty years for the internal combustion engine to make an impact on mass travel through the railways and nearly another 100 to miniaturize the technology to the point where individual travel using the automobile was possible. Cities took a long time to physically reflect these changes and only by the late 20th century had the dense concentrations that dominated the past begun to change through sprawl, globalization, and even greater polarization in their cores. This transition to an urban society, where we will all be living in cities of one kind or another, will be complete by the end of this century (Batty, 2011a), but the shadow of the historical city will be with us for a long time yet. The physical form of the city must thus be seen increasingly as a series of accidental or even purposive events frozen in time, the products of earlier technologies and social preferences. If people live in crowded conditions, cheek by jowl as in the medieval city for example, the area that is inhabited increases less slowly than population size. If the area is A r 2 and population increases as the volume P r , then we can express this positive allometry of the population as P A . Of course, this is an idealization, but it is likely that in compact cities which are dominated by very slow movement, the tendency to such a relationship holds. If people begin to spread out, as they are able to do with personal and individualized rapid transportation, then it is likely that this superlinearity or positive allometry might degrade to linearity or even sublinearity or negative allometry. The implication is that, as cities come to rely more and more on fast transport, people spread and sprawl. In this way the average density of the city at time D(t), given by D(t)  P(t)=A(t) A(t)a t, falls, with the parameter a(t) falling too. We can of course test this hypothesis by estimating the parameter a(t) from data at each cross section in time. The first known estimate of the allometric parameter was made by Stewart in his 1947 Science article that introduced social physics. In an almost throwaway line, he stated that the parameter 1 a(1940) for all US cities with populations greater than 2500 in 1940 was 4=3. His analysis would not satisfy our audience in this day and age, for there was no statistical scrutiny of the robustness of his conclusion, discussion of the error bounds, or interpretation of its significance. Yet Stewart's work heralded in a number of studies of the allometric parameter, which until about 1980 demonstrated positive allometry. For the last thirty years, allometric studies have been fewer, but the recent ones we have conducted with US and UK data indicate strong negative allometry and, in assessing all the studies available, there is an implication that the Editorial Environment and Planning B: Planning and Design 2011, volume 38, pages 753 ^ 756