Six Degrees, again

Thursday, 7 August 2008 by bbbeard.

There is new research into the Milgram hypothesis (”six degrees of separation”), which was last discussed in Compuserve SCIMATH about two years ago. This time, some folks at Microsoft investigated the connectivity of the Microsoft Messenger network. They looked at 30 billion records of messages sent among 180 million users and came up with an average (not median!) link length of 6.6. From the WaPo article:

“To me, it was pretty shocking. What we’re seeing suggests there may be a social connectivity constant for humanity,” said Eric Horvitz, a Microsoft researcher who conducted the study with colleague Jure Leskovec. “People have had this suspicion that we are really close. But we are showing on a very large scale that this idea goes beyond folklore.”

Funny, this triggered a memory. From the BBC news story on Judith Kleinfeld:

Judith Kleinfeld, a professor psychology at Alaska Fairbanks University, went back to Milgram’s original research notes and found something surprising.

It turned out, she told us, that 95% of the letters sent out had failed to reach the target.

Not only did they fail to get there in six steps, they failed to get there at all.

Milgram was a giant figure in his world of research, but here was evidence that the claim he was famously associated with was not supported by his experiments.

“I was shocked. I was horrified,” she said.

Wow. Two shocked researchers. But then I recalled the other thing Milgram was famous for.

In Milgram’s first set of experiments, 65 percent (26 of 40) of experiment participants administered the experiment’s final 450-volt shock, though many were very uncomfortable doing so; at some point, every participant paused and questioned the experiment, some said they would refund the money they were paid for participating in the experiment. No participant steadfastly refused to administer shocks before the 300-volt level.

Hmm, I’m beginning to see a pattern. Perhaps the Milgram hypothesis is part of a diabolically Byzantine scheme to see how far science journalists can be pushed into torturing innocent researchers.

But seriously. To my mind there is nothing ideological about the Milgram hypothesis, although I know several folks who have gotten quite exercised in their interpretation of the meaning of the six degrees of separation. The hypothesis seems to me to be a more-or-less obvious property of percolating networks, essentially a consequence of the logarithmic nature of shortest-path length.

To bring you up to date, here is what I wrote about six-degrees in 2006:

It seems to me that one first has to proffer a specific meaning for “link between two people”. Let us suppose for the sake of argument that we settle on something like “a link is said to exist between two people if neither would be surprised to receive a communication from the other”. I suppose the sociologists or psychologists could spend a lot of time arguing about more precise meanings for “link” — but I conjecture that, except for pathological meanings of “link”, the connectedness of humanity as defined by the Milgram hypothesis and the discussion below would not vary very much.

Okay, so once we agree on what we mean by “link”, I suggest that the six in “six degrees of separation” is a median number, not a maximum. By that I mean: pick two living persons. Determine their Milgram number, that is, the minimum number of links traversed to connect them. Add that number to a list. Continue with a different pairs until all O(36E18) pairs have been examined. Order the list and find the median Milgram number. Then the Milgram hypothesis corresponds to the median Milgram number being six.

Note this allows the possibility that some persons (or small groups) could be entirely isolated from the rest of humanity. Simply enter their Milgram numbers as infinity. If the number of these “hermits” is not overwhelming, their existence won’t shift the median. For example, if there were 10,000 hermits, then there would be roughly six billion times ten thousand (6E13) pairs that give infinite entries in the list, and these comprise less than the last 0.0002% of the list. Also, there could *conceivably* be a number of people who are really hard to get to, but who have Milgram numbers less than infinity. It seems to me that having a Milgram number of, say, 100, requires that one or both ends of the chain connecting a pair of people is populated by folks who are linked to *exactly* two people — neither of whom know each other! (Hypothetically, the very ends of the chain could know just one other person). Once you connect to “normal” people who know dozens or hundreds of other people, the jig is up and the logarithmic nature of Milgram connectivity takes over.

Your mileage may vary. It seems to me that the important observation is that the Milgram number is small, something smaller in fact than the common log of the number of nodes in the network. FWIW it seems to me that people’s intuitive rebellion against the Milgram hypothesis has nothing to do with the number actually being 6. I don’t think that a person uncomfortable with 6 would be comfortable with 7. I can’t imagine what a person who thinks the Milgram number should be 100 or 1000, or greater, is thinking .

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