It is actually much worse than that. You're assuming a uniform distribution over the numbers that people call, but there are some numbers that a significant fraction of the population call. For example, how many people are three hops away from a "target" through Comcast, Time Warner, Dell, etc?
Of course the NSA would probably ignore these spurious connections to generate leads for an actual investigation. But I assume that they would use them to create the largest possible list of people that they are "allowed" to collect and store information on.
Yes I was being conservative so that few reasonable people could challenge the assumptions I made.
But you're completely correct. I bet Amazon gets you the second hop to at least 20% of the whole country. Once you add in all the 800, 866, etc phone numbers I'd imagine that two hops gets you to very nearly every person in the US with a starting pool of perhaps only a few dozen people.
I could easily see someone's job being to figure out which (presumably innocent) people to "target" for the sake of ensuring a full or nearly full data collection. You know, to make things more efficient. "So we don't always have to go crawling to judges and the phone companies every single time we think something bad might be going down."
"I could easily see someone's job being to figure out which (presumably innocent) people to "target" for the sake of ensuring a full or nearly full data collection."
Twitter makes that job damn easy, don't you think?
Of course the NSA would probably ignore these spurious connections to generate leads for an actual investigation. But I assume that they would use them to create the largest possible list of people that they are "allowed" to collect and store information on.