Is there a measure for the "dominance" of a city's population?
Exciting question... it's something I've wondered as well, both for interest purposes and for utility in programs like you mention (I posed a fairly related question in the cartography SE site proposal).  I've searched a long time but could not find any term/calculation for this... perhaps I too just did not know the correct term to use?

As a result, for a very basic calculation, I created an Excel spreadsheet and worked with US locations to formulate a simplistic prominence... calculating just the distance each city is from the nearest larger city.  The outcome was this Sporcle quiz.

My results are likely not of great use for within a mapping program, as it takes no account of the degree of relative population size... but certainly would be interested in knowing more about that as well, as I've looked for a way to better prioritize the cities shown in a weather mapping solution I use.  Not sure you'd be able to use contouring methods given the discretized nature of population.  Instead I'd imagine perhaps iterating until you find the radius from the city border centroid where the surrounding population matches the population of the city.   That sound like a more complex GIS problem, one I've considered asking before but haven't even been sure how to narrow down into a basic enough question for SE!  Still, if there's no luck here of existing formulas, and you work out what way you want to calculate such a prominence (and can then boil it down to a basic question) GIS StackExchange may be the place to try.  Hopefully you'll give us the link to any questions you propose there so users here can follow any resolution!

If the basic prominence I calculated is somehow useful, I could dig up the full spreadsheet.  I believe I also might have found a potential global dataset as well, but never followed it up.

I cannot exaggerate how exciting it is to see someone else thinking about this topic, and hope your question brings some further better answers from those in the know!