Fig. 2: Spatial distribution of nodes according to the model described below with parameters r = 0.5 (where 1 is the side of the unit square), and p-bias = 0.2 (i.e. probability of random placement is 0.2. There are about 1000 nodes shown here.
Fig. 1: Spatial distribution of FON hot spots in a portion of England. THere is correlation between the population density and FON hotspots. Have we modeled wireless access points or do we actually have a more general parsimonious population distribution model?
Fig. 3: Probability distribution of city areas in the US. l is a data collection methodology parameter (see paper) My only concern with the authors' interpretation of the data is that it follows exactly Zipf's Law, which apparently it does not.
Fig. 4: Probability distribution of cluster areas as generated by our model for 10,000 people/nodes; radius of 0.25 (on a unit square; and p-bias = 0.2 (i.e. probability of random placement is 0.2). Notice that although the distribtuion is long-tailed, the generated curve does not quite follow Zipf's law, matching the real world data for the US city sizes.
How does the population of cities grow?
People have observed that the population distribution of urban areas follows closely a Zipfian distribution.
A number of complex socio-economic factors have been suggested to explain and model this population distribution shape. However, the extent to which these models have been successful is not entirely clear (see for instance the good work by H. D. Rozenfeld et al. "The Area and Population of Cities: New Insights from a Different Perspective on Cities").
The following story carries a moral: when in doubt submit for publication :-).
Back in 2012, I was interested in modeling the distribution of connecitons and spatial positions across nodes in wireless networks grwoing over time. For instance, community based WiFi networks such as FON have very intriguing spatial patterns, similar to the ones shown in Fig. 1. At the time, I wanted to find out what organic process could generate such patterns.
I stumbled on the following generative process. Suppose a node corresponds to a FON network wireless device.
Suppose the FON network grows in the following pattern:
1: place a node, uniformly at random, on the unit square
2: toss a p-biased coin
3: if Heads, select a node uniformly at random from the already placed nodes and pick a point, uniformly at random, from the disk with radius r centered at the selected node; if Tails, pick a random point in the unit square
4: place a node at the point selected in 3
5: go to 2
The above process generates spatial distributions similar to the ones shown in Fig. 2; it is easily tunable since it depends on two parameters: the coin bias p and the radius of a node. Notice the similarity between the process outcome on Fig. 2 and the one on Fig. 1.
I then wondered wouldn't a similar process account for the population and area growth of cities? After all, if nodes correspond to people, the probability that an individual or a small group of individuals move to start new settlement may be modeled as the outcome of a biased coin. The bias corresponds to the propensity of people to 1) move to areas where they would be connected with their acquaintances; and 2) be born in areas where there are already people.
Sure enough, when plotted the distribution of cluster areas (corresponding to cities area) generated by the process, matched very closely the real world data on city sizes Fig. 3 and 4. Notice that city sizes distribution is a long-tail one but do not completely follow a Zipf law. This detail is also matched by our model, where we have a long-tail distribution but an abberation leading to not a "complete" power-law distribution of cluster sizes.
Back in 2012, we had a preliminary document (short excerpt available here; notice that the content of this document has not been peer-reviewed; it is now several years old and as such may be even more error-prone). However, we decided that these results and observations would not be interesting and are not worth pursuing further. I have recently stumbled on an article published in Phys. Rev. X titled "Spatially Distributed Social Complex Networks" published in 2014 and submitted only in 2013. Guess what their generative model was... yes, apparently our results were of some interest.
Noticeably, noone has pursued the community WiFi distribution model. I currently think that is not of sufficient interest. But a later work by someone else could prove me wrong :).