Like @ggruccio - a whole pile of trial and error - and finally came out with 275 airports.
Took me way too long to realise that there was no need to filter the "Node" input since that would just come out with degree of 0 if there were no connections! You can see these on the chart as the circle of disconnected items around the edge!
Great challenge @JoeM and team - thank you for giving us one that forces learning of a new tool!
Here's my solution, using Iceland to speed up the rendering time.
I found the sample workflow for the Network Analysis tool very helpful in understanding how my inputs needed to be configured.
Help > Sample Workflows > Predictive tool samples > Predictive Analysis > 25 Network Analysis Tool