The source for cities and their populations was: Tertius Chandler, Four Thousand Years of Urban Growth, 2nd ed. (The Edwin Mellen Press, 1987), "Tables of the World's Largest Cities." I picked the top ten in each century. I supplemented his with a list of calculated populations for 2012 found in world-gazetteer.com, acceessed February 6, 2013.
For each of these cities I chose an area in which it was found whose area was between one hundred and one million square kilometers. If a nation fell in this range, I used it. If a nation was too large I split it into its second order political units, and so on iteratively until the unit was smaller than one million square kilometers. If the nation, or a unit of one, was smaller than one hundred thousand kilometers, I aggregated it with those nations or units next to it that were also under-sized, completing a full 'circle,' but skipping over larger adjacent nations/ units. If necessary, this was repeated, picking (arbitrarily) one of the bounding units and adding its under-sized bounding units that were not already included, doing this iteratively until at the end of a cycle the lower threshhold was met or exceeded. Where an under-sized unit was completed surrounded by this process, it was also included in the aggregate. (I applied a similar rule where a small sea-bordering unit was left by itself.) Where the left-over unit was surrounded by two or more already defined areas, it was allocated to the smaller one, even if that one was not an aggregate. Where a small aggregate was left over I treated it similarly. Which was the smaller aggregate at this decision point depended arbitrarily on what had already been aggregated with it. In one case, Danmark was left over and was deemed to border both Deutschland (and several nations already aggregated with it) and Sverige (by straits), and was added to Sverige. The United Kingdom was aggregated with Eire based on its land border on the island of Ireland.
For each of these areas, the like-chosen areas that form their periphery were also chosen. Directly bordering places were chosen, and to this added those separated only by straits. Nippon was deemed to border the aggregate that included Sakhalin Island, and the aggregate the included the Korean Peninsula.
For each century, three groups of cities were included: those in the top ten that century (n); those in the top ten in century n-1 for which there was data in century n; and those in the top ten in century n+1 (where n isn't 2012) for which there was data in century n. I then looked at the range of magnitudes, where the bottom of the range for each magnitude was in the series 40 thousand, 100 thousand, 200 thousand, 400 thousand, one million, two million, four million, ten million, 20 million. Starting from the largest, a maximum of four magnitudes were chosen. Only for 1900 was there a consequent reduction in the included cities. Then I chose the areas in which these cities were found and the areas defined as peripheral to the initial set. All the cities that fell in the allowed range in century n and were within the chosen areas were included for use in the maps.
This methodology was for some centuries limited by the lower bound used as an inclusion criterion by Chandler.
The maps cover change over a century, n to n + 1. The areas for n (defined using n and n-1 criteria) and n+1=m (using m criteron) are used in the dynamic maps.
In general there is no text outside of the maps, and the legend explains the symbols used for the cities. Where a city's population crossed two magnitude bounds (as defined above), I added a brief explanation of why it changed. This is meant to be a proximate cause, not a profound examination. The magnitude bounds included Chandler's table's lowest bound.
Other aspects of my methodology are either explicit in map legends, or easily deduced from them.
Minor points: the source for national areas was generally the World Almanac 2016 and for smaller areas, a Wikipedia article. For example, areas of prefectural-level units within Nei Mongol Province of Zhonghua were found in the Nei Mongol Wikipedia article. These were accessed February and March, 2017. Where separate figures for area were included, one with water and one without, the former was used. Ontario's sub units were its two regions rather than its census divisions. Where Chandler's lower bound changed between the centuries, the higher bound was used as the data cutoff for both centuries; however, in choosing whether or not to include text, if it is known that the lower bound would not have been crossed except in consequence of raising the bound, the lower bound is considered not crossed.
The color coding does not indicate a value judgment about growth.