# joel’s proof

#### eon 2012-03-08

On Fri, Apr 8, 2011 at 2:28 PM, Joel Tenenbaum wrote:

Would it matter if I could show that, statistically, I’ve probably only distributed 1 copy of each of the mentioned songs to the KaZaA network?

Cheers,

Joel

On Sat, Apr 9, 2011 at 11:06 AM, Charles Nesson wrote:

how would you do that?

Date: Sat, Apr 9, 2011 at 12:06 PM

Every download is someone else’s distribution. I’ve presumably downloaded every song in my collection at most once. 1000 songs on my computer is at most 1000 uploads from someone else. But in the entire KaZaA network, every download is a single upload so the total number of uploads and downloads in the network is equal. Therefore, on average, each person only uploads about as many copies as he downloads. There are exceptional cases – “philanthropist” sharers and “leech” downloaders – but he who pays little mind to his status on that spectrum winds up sharing about what he takes.

There are also many connections to this problem in percolation theory. You ask the question: given a region with some kind of specified spatial structure (e.g. like a grid) and a probably p of making each connection within that structure (e.g. 0.6), will you be able to get from one side to the other? What is the critical value of p (denoted p_c) that makes this happen?

In a scale-free network which is the structure of most social connections, p_c=0. E.g. If the probability of passing a rumor on to the next person is anything except 0, a rumor will reach the majority of the population. I’m not sure of the specifics of KaZaA, but if all traffic is directed through supernodes, you wind up with this structure. (If it’s actually the case that anyone can download through anyone, you have a mean field situation and it takes even less effort from each person to get a song spread around.)

Hence, a song will get distributed completely if each person only passes it along once. Interestingly, this also predicts that virtually everyone who has had sex, is connected to everyone else in the world through that sexual network.

Hence, is highly unlikely that I distributed more than a few copies of any given song. Their claim that it could have happened thousands of times is statistical nonsense.

Cheers,

Joel