As part of our work on scaling and allometry in the UK Urban System, we have briefly explored the notion that centrality or accessibility, defined using the conventional measure of gravitational potential, scales with population itself. We consider this to be an index of diversity of a city that takes account not only of its size but its relationship to all other places/cities in the urban system. This is something that we hope to develop in the next few months as we wish to connect scaling as it has been developed for individual cities to the linkages between individual cities that interact in a system of cities.
We have computed an index of potential for each city i by summing the population of every city j, other than i, weighted by the inverse square of the distance d(i,j) from i to j: Sum over j not i: N(j)/d(i,j)^2. We have added the population of i, N(i) to this and this in fact implies that this population is also a potential with distance from itself to itself equal to 1, that is N(i)/(d(i,i)^2) = N(i). We have computed this measure from each of 473 cities to all 8,850 wards in England and Wales over the road network and we show these interconnections in the map below where we eight each link by its accessibility of the nearest town to that link. The scale is from highest to lowest accessibility on the spectrum from red to blue where the measures are based on the formulas above. If we were to scale these measures with the top accessibility 100, the lowest would be 0.01 and thus the range would be over four orders of magnitude.
In fact we have scaled only the towns over 10,000 population of which there are 282 against population but we get super-linear scaling with a coefficient of about 1.09 which suggests that accessibility does indeed scale positively with population. This also implies that we need to add some measure of connectivity to population to generate the sort of economies of scale that we see in the UK urban system. We show the scaling for these accessibilities in the figure below but we should also note that if we take all cities over 100,000 population, the next order of magnitude up, of which there are 48 cities in England and Wales, the scaling coefficient increases slightly to 1.10. Note that the graph below is on a log-log scale and that the version produced in printed version of the Science magazine has an incorrect captions on the x and y scales. This has been corrected in the online version which you can access clicking here. The original paper by Luis Bettencourt is here but you will need a subscription to Science to read the full versions.
Peter Ferguson, Elsa Arcaute, and Michael Batty, June 4th, 2013