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Okay so why can't we get any reporting with any numeric literacy? How many people does the average person make contact with in a 5 year time span? Let's call

Okay so why can't we get any reporting with any numeric literacy? How many people does the average person make contact with in a 5 year time span? Let's call this number X for hop 1. How many people does each of the people in X then make contact with in the same 5 years? If we assume that on the whole people know ABOUT as many people as others, we can also call that number X for hop 2. Now of course there's going to be overlap, out of X people somewhere between 0% and probably 70% might overlap between any one person of interest, hop 1 and hop 2. Let's say that on average there's a 40% overlap. And let's call the Sweep(X) the number of people that get caught up in surveillance as a result of a single suspected terrorist. Sweep(X) = 0.4 * X (first hop) x 0.4 * X (second hop) Sweep(X) = 0.16 * X ^ 2 That's quadratic and as anyone who knows math, and especially people who know about Big-Oh notation, it's bad because Sweep(X) grows much faster than X does. But how big is X? Well we can look to Dunbar's number to give us somewhere to start. Wikipedia says that it's between 100 and 250 but most people use 150 as a reasonable approximation. Of course that's STABLE relationships, not acquaintances, friends-of-friends, or anything like that. I think it's pretty reasonable to assume the 150 number and then assume that you probably make contact with one random person every week. Over 5 years, that's another 250. http://en.wikipedia.org/wiki/Dunbar's_number http://en.wikipedia.org/wiki/Dunbar's_number So if we assume that X is 150 + 250 = 400 what do we get? Sweep(400) = 0.16 * 400 * 400 = 25,600 let's just say 25k for short. Now there are around 300 million people in the US. 300mm / 25k = 12000. That means that the NSA need only find 12,000 people (on average) to get all the call records for everyone in the US for the last 5 years. It gets worse, though. What happens if you make contact with TWO random people every week? Then X = 150 + 500 = 650 Sweep(650) = 0.16 * 650 * 650 = 67,600. 300mm / 67k = 4477 people suspected of nefarious deeds in order to gain all call records. As you can see this gets really out of hand really fast. And that's why I'm a bit upset that nobody's doing the math around the "two hops" argument because I assure you that the folks at the NSA aren't rolling over and giving up. This is something that SOUNDS reasonable but in actuality means little/no reform whatsoever. EDIT: wikipedia link

k2enemy 13y ago
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.
- msandford 13y ago
  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." This two hops thing really is a scam.
  - sbirchall 13y ago
    "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? [EDIT: s/Twitter/Facebook/WhatsApp/SnapChat/etc.]
j2kun 13y ago
It could be because the only people who want to read an article that defines functions and expects them to be familiar with Big-O notation can reach these conclusions without reading the article in the first place. That being said, it's generally believed to be (or measured) that the social network of the world is a sparse graph. So the fact that the sweep function is quadratic in the degree of a vertex is irrelevant. Using your notation, X is bounded by a constant for all intents and purposes. In other words, there's nothing inherently bad about the quadratic behavior, and "knowing math" doesn't give you much except perhaps a false sense of intuition. The arithmetic (and your ad-hoc estimates) just works out in favor of your argument. Which is not all that surprising conditioned on the fact that you posted an argument :)
- msandford 13y ago
  Sure, I'm not suggesting that the author give it the kind of treatment I did. Simply that the notion "oh two hops is better right?!" isn't valid. Yes social networks are sparse. With 300 million Americans and the capacity to know perhaps only a few thousand, yes, things will be sparse. But that doesn't mean you cannot construct a spanning tree of that sparse graph. My PC isn't directly connected to Amazon or Netflix or really anyone of interest. But in a couple of dozen hops I'm connected to just about anyone in the world. Sparse doesn't mean unconnected. The "badness" of the quadratic is that the "two hops" restriction SEEMS like a real restriction when in reality, it's not. It's bad in that the "two" in "two hops" sounds linear but the effects are quadratic. The quadratic-ness isn't inherently bad, I would never argue that. But the deception is.
  - j2kun 12y ago
    I agree that your point is right, these policy makers are douchebags. I just think your analysis is not much more enlightening with the arithmetic. Change 40% to 10% and your final number goes up to over a million. It would be wonderful to have better estimates than "somewhere between 0% and 70%."
    - msandford 12y ago

k2enemy 13y ago

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.

msandford 13y ago
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." This two hops thing really is a scam.
- sbirchall 13y ago
  "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? [EDIT: s/Twitter/Facebook/WhatsApp/SnapChat/etc.]

j2kun 13y ago

It could be because the only people who want to read an article that defines functions and expects them to be familiar with Big-O notation can reach these conclusions without reading the article in the first place. That being said, it's generally believed to be (or measured) that the social network of the world is a sparse graph. So the fact that the sweep function is quadratic in the degree of a vertex is irrelevant. Using your notation, X is bounded by a constant for all intents and purposes. In other words, there's nothing inherently bad about the quadratic behavior, and "knowing math" doesn't give you much except perhaps a false sense of intuition. The arithmetic (and your ad-hoc estimates) just works out in favor of your argument. Which is not all that surprising conditioned on the fact that you posted an argument :)

msandford 13y ago
Sure, I'm not suggesting that the author give it the kind of treatment I did. Simply that the notion "oh two hops is better right?!" isn't valid. Yes social networks are sparse. With 300 million Americans and the capacity to know perhaps only a few thousand, yes, things will be sparse. But that doesn't mean you cannot construct a spanning tree of that sparse graph. My PC isn't directly connected to Amazon or Netflix or really anyone of interest. But in a couple of dozen hops I'm connected to just about anyone in the world. Sparse doesn't mean unconnected. The "badness" of the quadratic is that the "two hops" restriction SEEMS like a real restriction when in reality, it's not. It's bad in that the "two" in "two hops" sounds linear but the effects are quadratic. The quadratic-ness isn't inherently bad, I would never argue that. But the deception is.
- j2kun 12y ago
  I agree that your point is right, these policy makers are douchebags. I just think your analysis is not much more enlightening with the arithmetic. Change 40% to 10% and your final number goes up to over a million. It would be wonderful to have better estimates than "somewhere between 0% and 70%."
  - msandford 12y ago