There are roughly 300 million people in the US. A given person has, let's say, ~10,000 tenuous connections.

How often would a random selection of 1000 people from the 300 million be expected to intersect with the 10,000?

Napkin math: For just 1 person to intersect: Each of those 10000 people have a (10000/30000000)/100 % chance to get selected, or .00000033%. Multiply that by a thousand selections = 0.00033% chance. Those selections are refreshed maybe 100 times a year so 0.033% chance per year.

There are billions of facebook users, so this should be happening all the time by pure chance. But of course it's not pure chance. Facebook will be selecting from a pool much much smaller than 300 million, and the selections won't be random.