### Defining Cities

We recently showed that urban measures can vary greatly depending on the city boundary considered ^{[1]} (see section on Scaling). In addition, statistical properties of roads within cities are also dependant on city boundaries ^{[2]} (see section on Networks). Defining city limits beyond administrative boundaries is however, not a trivial problem. On the one hand, cities can be defined in terms of their urbanized extent, and on the other, metropolitan areas considering commuters, are also important for the functioning of cities as integrated entities. Using density thresholds on demographic data, we are able to address both aspects.

The following subsections present this methodology, and the application of percolation theory to the problem of defining cities.

*Density Thresholds*

We have developed a method to define the extent of the morphological city using census data at a high geographical resolution. The details of the methodology applied to the UK can be found in ^{[1]}. Census data provides population size for the whole cover area of the UK. We used the data at the resolution of wards. Then we aggregated neighbouring areas whose density reached at least a specific threshold ρ0. We obtained different realisations of systems of cities with different boundaries by varying the threshold (from 40 to 1 pers/ha in units of steps). At very high densities, 40 pers/ha, only the cores of cities are obtained, while at very small densities, cities merge to form big clusters, see **Fig. 1**. The extremes values do not reproduce realistic systems of cities, nevertheless these are important to keep in order to test any hypotheses related with the urban space, and their dependencies on agglomerations effects.

This procedure produces realistic morphologies of cities for some density thresholds. Details can be found in ^{[1]}. There we show that we are able to recover a system of cities that not only conforms to a Zipf distribution of city sizes, but that is also in high correspondence to the area defined as “urbanized” from the land cover classification of satellite images.

*Commuting thresholds*

The formalism is extended to metropolitan areas, by including wards from which people commute to work. This data is also obtained from the census. The methodology is as follows. Boundaries are initially constructed using population density thresholds. Then, for each of the 40 realisations obtained, wards from which people commute to work are added to these original clusters, if and only if a minimum proportion of the population commutes to work there. The commuting threshold takes the form of a percentage varying from 100% (everybody needs to commute from a ward to a cluster to be added) to 0% (a single individual needs to commute for the ward to be added to a cluster) in steps of 1%.

In order to allow for small settlements initially obtained at a specific density threshold to be added to bigger clusters (e.g. satellite towns around London), we impose a minimum population size P0 on the original clusters, so that the remaining clusters smaller than P0 can be part of the wards considered from which people commute to work. Different population cutoffs are considered: P0={0;104;5×104;105;1.5×105}.

**Fig. 2** shows some of the maps for different commuting thresholds for an initial system of cities given at a density of 14 pers/ha.

### Percolation theory applied to cities

Even though the formalism described above gives rise to a good morphological description of cities, it is subjected to the biases and availability of census data. This can be overcome by using data that is more readily available, defined in a consistent way and intrinsic to cities, such as the street network. Following the ideas by Gallos et al ^{[3]}, we apply percolation theory ^{[4]} to the road network in Britain, and show how cities emerge in the same way as modules emerge in the brain through multiple percolation transitions. A hierarchical process takes place, and through recursive percolations on each of the emerging clusters, cities are devised. Modelling Britain as the brain, we use road intersections as the occupied sites in space, connected between each other through proximity only. At this first stage, the roads themselves are removed, and the algorithm is very similar to the CCA (City Clustering Algorithm) defined by Rozenfeld et al ^{[5,6]} based on population in space, and the natural cities definition given also in terms of intersection roads by Jiang et al ^{[7]}. These algorithms differ from models of urban growth based on correlated percolation, see Makse et al ^{[8,9]} and Murcio et al ^{[10]}. Our algorithm is defined in terms of a distance parameter that determines clusters of intersection points in which every point has a neighbour at a distance equal or smaller to the given threshold. The process starts from the maximum distance at which all the points in Britain are connected. Lowering the distance threshold, leads to a series of percolation transitions that divide the space into clusters. The first transition separates Scotland from England and Wales. The following transitions give rise to regions that are defined by the natural geographical barriers, such as National Parks. The process is continued for each of these emerging clusters, until cities are obtained, see **Fig. 3**. Such a method allows us to define cities in a more accurate morphological way, since the threshold can be tuned locally, contrary to the method defined in ^{[1]}, in which a global population density was applied throughout the space. For an animation of the percolation process see Percolation Clustering of the UK Road Network.

### Fractals vs multi-fractals

The set of clusters obtained through the percolation approach, can be represented in terms of their road networks. We analyse these using different approaches to compute their fractal dimension, and our results show that for the largest clusters, the fractal dimension is very sensitive to data and methods. Investigating further, we find that these objects are multi-fractal. Based on these results, we propose a classification of cities based on their multi-fractal spectra.

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