Centroid

The centroid of a country (or area) can de defined in multiple ways. On the earth surface it is more useful to speak of the geographical center. It is possible to find lists of these centers for area's on earth.

One quickly realises that these centers are not (always) good enough. Either they are in a location far away from population centers, or they fall in between a set of islands, etc. So we need to come up with a better estimate.

Any better estimate must reflect what we intend to achieve in the first place. We are talking about transportation and its impact. So we talk about distances, mode of transportation, the amount transported and the impact of it all. Any centroid should take this into account.

If we know only the country a product is bought in, we can either assumes the worst or the best. The worst is the maximum impact and the best the minimum impact. The minimum impact will imply transportation from the most centered location in a country.

What is now the most centered location? The distances to all other locations should be small as possible. And the usage of these distances should be as small as possible. This will take into account the population sizes of all the locations.

This seems to be a Weber problem. This problem is not solvable exactly, but must be solved iteratively. The locations are then the towns and cities in a country. And the weights are defined by the population sizes of each towns. And the distances are defined by the distances by road.

In practice it will be quite difficult to solve this for all towns in an area. But can we do it for the largest towns? And how many towns should we incorporate?

Recipe

Using the above considerations we can setup a recipe:

1. Largest cities - find the largest cities (by population size) for a country. Star with the 5 largest cities;
2. Population - list the population size for each of the largest cities, and calculate the fraction of the total population of all largest cities;
3. distances - calculate the distances between all the largest cities. We can use the directions of OpenStreetMap for this;
4. weighted distance - calculate the weighted distances, i.e. the distance times the city fraction;
5. weighted distance sum - add all the weighted fractions for a specific city (the centroid city);
6. minimum weighted distance sum - find the weighted distance sum that is the lowest. The corresponding city is the centroid city;
7. add cities - more cities can be added to see whether the found centroid city is correct. For this recommence on step 1;

Result

This recipe has been tried on the following countries:

Country	Centroid city	spreadsheet
Belgium	Brussels	link
Denmark	Copenhagen	link
France	Paris	link
Germany	Hannover	link
Italy	Rome	link
Netherlands	Utrecht	link
Norway	Skien	link
Portugal	Lisboa	link
Spain	Madrid	link
Sweden	Nörrköping	link
Switzerland	Olten	link

This approach of finding a centroid seems to give good results. For Belgium there is not a large difference with the geographic centroid, which is expected for a densely populated country. For Norway it is much closer to the population area's.

Observations

The city population centroids have been found a step by step process. This allows to see how the centroid changes after a city has been added. In general the largest city dominates the position of the centroid. And if that city is also close to the geographic centroid, it will be the centroid in the end (Belgium, Spain). If a large city is surrounded by other cities, then one of those can become the centroid (Norway, Switzerland). The layout of motorways can influence the selection between two cities (France).

Some individual country observations:

• Belgium - the runnerup is Mechelen, so if more cities are added, the centroid might move towards that city along the motorway.
• Denmark - no surprise that this is Copenhagen. This is by far the largest city of Denmark. And thus much different from the geographic centroid.
• France - the centroid lies along the motorway between Paris and Lyon. There are no cities that are large enough in the top 20 to push the centroid away from Paris towards Lyon. Adding Dijon did not help. Probably due to a lack of motorways to the west from Dijon.
• Germany - the country is dominated by Berlin. Only thanks to the existence of a large town to the west, the centroid could be moved there (Hannover).
• Italy - the centroid is dominated by Rome, but might move upwards towards Florence, when more cities in the populous north are added.
• Netherlands - as expected in Utrecht. This is in the center of the country and also a center of motorways;
• Norway - the centroid is in Skien, but could have been in Oslo, or in between these two cities. The centroid is much lower than the actual geographic centroid.
• Spain - Madrid is very well placed centrally in the country. Madrid and Barcelona are the greatest influencers on the centroid. The cities on the Canaries have been removed, but the Baleares have been kept.
• Sweden - the country is dominated by the Oslo and surrounded cities. Due to the cities close to Denmark, the centroid is moved to the south.
• Switzerland - Olten is one of the smaller towns in the valleys around Zurich, which dominates the centroid. Note that for transport a route through Italy is sometimes more advantageous.