US Counties and power laws

Yesterday I heard that the county I live in, Harris County, is the 3rd largest is the United States. (In population. It's nowhere near the largest in area.) Somehow I've lived here a couple decades without knowing that. Houston is the 4th largest city in the US, so it's no shock that Harris County the 3rd largest county, but I hadn't thought about it.

I knew that city populations followed a power law, so I wanted to see if county populations do too. I thought they would, but counties are a little different than cities. For example, cities grow and shrink over time but counties typically do not.

To cut to the chase, county populations do indeed follow a power law. They have the telltale straight line graph on a log-log plot. That is for the largest counties. The line starts to curve down at some point and later drops precipitously. That's typical. When you hear that something "follows a power law" that means it approximately has a power law distribution over some range. Nothing has exactly a power law distribution, or any other ideal distribution for that matter. But some things follow a power law distribution more closely and over a longer range than others.

Even though Los Angeles County (10.1 million) is the largest by far, it doesn't stick out on a log scale. It's population compared to Cook County (5.2 million) and Harris County (4.6 million) is unremarkable for a power law.