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why use euclidean distance in the function for defining weights? Doesn't it mean cities that are farther in distance with fewer friend connections would get a higher weight than cities that are close by? Not sure how it helps visualize or how to interpret it.

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Presumably they're using euclidean distance to scale down the brightness of long-distance links -- to ensure that long-distance relationships don't overpower short ones in the visualization. Otherwise, a pair of friends thousands of miles apart would be contributing the same "energy" to the image as a thousand pairs a few miles apart... You'd never be able to see local detail in the areas heavily crossed by global connections (e.g. the Caribbean).

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