# 6 degrees of weak statistics

February 3, 2008 8:36 AM Subscribe

The

Kevin Bacon may be lonelier than we thought.

**other**Milgram experiment had less than shocking results. In fact, the famous six degrees of separation appear to be more folklore than science.Kevin Bacon may be lonelier than we thought.

Yeah, the Discovery article mentions the Watts experiment. However, I'm surprised that the Fast Company article left out this important tidbit concerning it:

posted by tkolar at 8:53 AM on February 3, 2008

*in the end, 61,168 starters signed on, and 24,163 chains were begun. Of those, only 384 were completed.*posted by tkolar at 8:53 AM on February 3, 2008

Facebook already has the data to confirm Milgram

posted by niccolo at 9:02 AM on February 3, 2008

*and*determine the viral nature of trends. I would love to see studies on the Facebook database. Indeed, the Facebook application, Six Degrees is currently reporting the average separation s 6.38 degreesposted by niccolo at 9:02 AM on February 3, 2008

The small world phenomenon is a mathematical property of certain types of networks, as shown by Watts and Strogatz. The Milgram letter experiment is not "bad science", it obtained a very interesting result: of the letters that were received, the number of people the letter went through was much smaller than anticipated.

posted by demiurge at 9:05 AM on February 3, 2008

posted by demiurge at 9:05 AM on February 3, 2008

I think this may be social scientists massively overthinking a relatively simple mathematical subject.

My impression was always that the six degrees thing came from Paul Erdős' work in mathematical network topology. Academic mathematicians compute their Erdős number, which like the six degrees thing indicates how many steps away you are (in terms of academic collaboration) from someone who has published a paper with Erdős.

Contrary to what it says in that Milgram article this sort of math has been very thoroughly analyzed. I think the sense that the Six Degrees of Kevin Bacon thing is true more broadly is if you include acquaintances - people you've only met once and people you see regularly but don't know their names.

posted by XMLicious at 9:06 AM on February 3, 2008 [1 favorite]

My impression was always that the six degrees thing came from Paul Erdős' work in mathematical network topology. Academic mathematicians compute their Erdős number, which like the six degrees thing indicates how many steps away you are (in terms of academic collaboration) from someone who has published a paper with Erdős.

Contrary to what it says in that Milgram article this sort of math has been very thoroughly analyzed. I think the sense that the Six Degrees of Kevin Bacon thing is true more broadly is if you include acquaintances - people you've only met once and people you see regularly but don't know their names.

posted by XMLicious at 9:06 AM on February 3, 2008 [1 favorite]

*Kevin Bacon may be lonelier than we thought.*

Well, one of my horses has a Bacon number of four, so it cannot be that bad.

posted by effbot at 9:07 AM on February 3, 2008

All that Kleinfeld has proven is that we don't have an easy time locating random people at the distant edges of their social networks.

posted by bunnytricks at 9:09 AM on February 3, 2008

posted by bunnytricks at 9:09 AM on February 3, 2008

*Facebook already has the data to confirm Milgram and determine the viral nature of trends*

I disagree. Facebook has a limited cross-section of the real-world social network. The tendencies of Facebook users to connect with one another may differ from the rest of the population. This isn't even just me antagonistically generating alternative hypotheses; Facebook is specifically a site that specifically promises social benefits, so it's very reasonable to suspect that the connectivity information provided by Facebook is not necessarily representative of the real-world social network as a whole.

*The Milgram letter experiment is not "bad science"*

Whether or not Milgram's work was bad science, it's completely irresponsible to publish in the popular press before going through peer-reviewed journals.

The small world phenomenon is a mathematical property of certain types of networks

The small world phenomenon is a mathematical property of certain types of networks

I am still waiting on confirmation that the real-world social network is one of those "certain types of networks". We can do math all day, but that won't say anything about the real-world social network unless the real-world social network is an instance of the classes of networks for which the small-world property is proven.

We don't even know whether the real-world social network is a connected graph. If we try to get around that by saying "individuals or communities isolated from my connected component of the social network don't matter; the six degrees hypothesis works for my connected component" then we are making the six degrees hypothesis profoundly weaker.

It's also not clear what constitutes a connection. If I consider myself isolated, to only have enemies, about to shoot up my workplace and kill myself, am I connected to anyone simply by virtue of knowing their names? If someone else, despite my antipathy towards them, deeply cares for me, are they connected to me? (which is to ask: "does the analysis still hold if the graph is directed?")

posted by Jpfed at 10:06 AM on February 3, 2008 [1 favorite]

The Facebook "six degrees" application stats are meaningless. The 6-point-whatever it reports is only valid for people who have added the application (and of course, those are people on facebook, which is hardly a representative sample of the world population) People find the application in the first place because they see a notice that a friend has in turn added that application. Also you'll note it's an average - there are many people with more degrees. The six degrees theory says: you are no more than six degrees separated from anyone else. By having an average above 6, the Facebook thing actually shows that even in a select subpopulation most people are only related by more than six degrees.

For the 99.999 percent of the world's population who aren't on facebook, the application has nothing to say.

posted by drmarcj at 10:09 AM on February 3, 2008 [1 favorite]

For the 99.999 percent of the world's population who aren't on facebook, the application has nothing to say.

posted by drmarcj at 10:09 AM on February 3, 2008 [1 favorite]

I have a Kibo Number of 1...

posted by benzo8 at 10:11 AM on February 3, 2008 [1 favorite]

posted by benzo8 at 10:11 AM on February 3, 2008 [1 favorite]

*I am still waiting on confirmation that the real-world social network is one of those "certain types of networks". We can do math all day, but that won't say anything about the real-world social network unless the real-world social network is an instance of the classes of networks for which the small-world property is proven.*

The "certain type" only has to do with the ratio of vertices to nodes in the network. It's a pretty simple property to check, but considering that the author of this article managed to do all the research on this and write three pages without mentioning the related math, so presumably without encountering any of the relevant math in social science literature, you may be waiting a long time.

This isn't saying that any social network you could come up with would be interconnected enough to work. The network of people who know each other well enough to be able to persuade each other to cooperate in the Milgram experiment is quite possibly too sparse.

I agree that the Facebook figures would be useless, not only because of the large number of non-human nodes (like schools and bands) but the fact that people who want high numbers will friend others they don't know. One of the things Erdős showed was that adding even a small number of vertices to a sparse network, at a certain point, can result in an explosion of complexity and interconnectedness.

posted by XMLicious at 10:41 AM on February 3, 2008

Yeah, well, I have a zero degree of separation from AWESOME. So there.

posted by miss lynnster at 11:26 AM on February 3, 2008 [2 favorites]

posted by miss lynnster at 11:26 AM on February 3, 2008 [2 favorites]

Plus one degree of separation from Wilt Chamberlain's ass. :)

posted by miss lynnster at 11:27 AM on February 3, 2008

posted by miss lynnster at 11:27 AM on February 3, 2008

*It's also not clear what constitutes a connection.*

It all depends on the application. For computer networks, connections are physical wires or perhaps wireless connections that can be itemized. For epidemiological studies, it might be intimate contact or just being in the area of someone else. It may be that for many sociological applications the idea of a "connection" is something that is not well defined, but that doesn't mean that the small world effect doesn't exist.

I think that as global communication gets easier and easier, connections to places outside of your geographically local social networks will become more common. There will undoubtedly be people in isolated networks that are very hard to reach, but I think the average number of hops neccessary to find someone who can help out has gone down in the last thirty years. Something like Ask Metafilter was impossible back then.

posted by demiurge at 11:33 AM on February 3, 2008

(Actually, it's only 99.99% of the world's population who aren't on Facebook...)

posted by hoverboards don't work on water at 11:38 AM on February 3, 2008

posted by hoverboards don't work on water at 11:38 AM on February 3, 2008

XMLicious- it

1. For any N, an N-degrees rule requires that the graph be connected. (Proof: select a point in one connected subgraph, and another point in a different connected subgraph. By definition there is no path between those two points, let alone a path of length N or less.)

2. For any vertex-to-edge ratio R, it is possible to construct a graph that is not connected. (proof: consider the class of graphs with a complete graph of size n, plus one unconnected vertex. The vertex-to-edge ratio for such graphs is 2(n+1)/(n^2-n+1). By selecting a large enough n, this can be made to approach zero arbitrarily closely- so for any R, there is a member of this class of graphs that has a vertex-to-edge ratio of R or less.)

3. Therefore, for any edge-to-vertex ratio, it is possible to construct a graph that does not obey

posted by Jpfed at 12:02 PM on February 3, 2008

*cannot*be true that "any graph that has at most this vertex-to-edge ratio exhibits an N-degrees rule for some N". Proof:1. For any N, an N-degrees rule requires that the graph be connected. (Proof: select a point in one connected subgraph, and another point in a different connected subgraph. By definition there is no path between those two points, let alone a path of length N or less.)

2. For any vertex-to-edge ratio R, it is possible to construct a graph that is not connected. (proof: consider the class of graphs with a complete graph of size n, plus one unconnected vertex. The vertex-to-edge ratio for such graphs is 2(n+1)/(n^2-n+1). By selecting a large enough n, this can be made to approach zero arbitrarily closely- so for any R, there is a member of this class of graphs that has a vertex-to-edge ratio of R or less.)

3. Therefore, for any edge-to-vertex ratio, it is possible to construct a graph that does not obey

*any*N-degrees rule.posted by Jpfed at 12:02 PM on February 3, 2008

(sorry- that was a little sloppy. The proof for step 1 should have read "

posted by Jpfed at 12:06 PM on February 3, 2008

*In a non-connected graph*, select a point...")posted by Jpfed at 12:06 PM on February 3, 2008

Interesting post, thanks!

Huh? How can this possibly be purely a mathematical subject, whether simple or not? If humans were antisocial and basically lived alone, meeting other humans only rarely and reluctantly, are you saying the numbers would be the same? That makes no sense.

posted by languagehat at 12:29 PM on February 3, 2008

*I think this may be social scientists massively overthinking a relatively simple mathematical subject.*Huh? How can this possibly be purely a mathematical subject, whether simple or not? If humans were antisocial and basically lived alone, meeting other humans only rarely and reluctantly, are you saying the numbers would be the same? That makes no sense.

posted by languagehat at 12:29 PM on February 3, 2008

Uh, I didn't say that, Jpfed. I didn't think you were expecting a mathematical proof, to demonstrate that something like the set of all actors related via the movies they were in together is equivalent to a digraph. I was just mentioning one salient way that the graph theory / network topology¹ stuff I'm familiar with would be used to analyze things like this, which I don't think is so difficult despite the impression the article gives that despite de Sola Pool and Kochen's best efforts the problem doesn't yield to mathematical analysis (when in fact there's an entire field of math devoted to exactly this kind of thing.)

Obviously, if you think that it's pertinent to mathematically disprove a statement about graph theory you believed I made, doesn't that mean you think that graph theory is pretty significant to this six degrees thing?

Maybe I'm simply misunderstanding you. What is the "six degrees hypothesis" you mention hypothesizing?

1. I'm aware that they're different things, I just said "network" because "graph" has more of a tendency to confuse people.

posted by XMLicious at 12:33 PM on February 3, 2008

Obviously, if you think that it's pertinent to mathematically disprove a statement about graph theory you believed I made, doesn't that mean you think that graph theory is pretty significant to this six degrees thing?

Maybe I'm simply misunderstanding you. What is the "six degrees hypothesis" you mention hypothesizing?

1. I'm aware that they're different things, I just said "network" because "graph" has more of a tendency to confuse people.

posted by XMLicious at 12:33 PM on February 3, 2008

*Huh? How can this possibly be purely a mathematical subject, whether simple or not? If humans were antisocial and basically lived alone, meeting other humans only rarely and reluctantly, are you saying the numbers would be the same? That makes no sense.*

I'm not saying that it's a purely mathematical subject, I'm saying that trying to analyze the properties of networks - like how the number of connections between people relates to the number of hops between two given people - by doing social experiments, without delving into the substantial amount of analysis that has already been brought to bear on those topics in the last hundred or so years, would be silly.

Every time I've seen this topic discussed in a social science context there's always omission of the fact that there's an entire branch of mathematics devoted to the study of this stuff, that has been cranking away at it for almost a century, and that at least the success of the "Six Degrees of Kevin Bacon" analysis of movie casts is unsurprising.

And I get the sense that consequently, lots of people view this as some edgy, mist-shrouded, unexplored field instead of something that has been thoroughly traversed in at least one discipline. The way this article goes through without mentioning even the existence of graph theory or network topology is like covering all of U.S. history without ever mentioning the U.K.

posted by XMLicious at 12:50 PM on February 3, 2008

*I didn't think you were expecting a mathematical proof, to demonstrate that something like the set of all actors related via the movies they were in together is equivalent to a digraph.*

No proof necessary for that.

*Obviously, if you think that it's pertinent to mathematically disprove a statement about graph theory you believed I made, doesn't that mean you think that graph theory is pretty significant to this six degrees thing?*

Of course it's significant (more in a later reply). I presented the proof because I had misinterpreted what you were saying (I thought you were saying something much more specific than you what you probably intended to say). Further, I ended up proving something that may not have been relevant in the first place. I had mentally pieced together the following snippets:

demiurge said "The small world phenomenon is a mathematical property of certain types of networks, as shown by Watts and Strogatz." The only small-world phenomenon mentioned up to this point was the "six degrees hypothesis".

I then said "I am still waiting on confirmation that the real-world social network is one of those 'certain types of networks'." You then quoted me, saying "The 'certain type' only has to do with the ratio of vertices to nodes in the network."

The resulting impression was that you were saying that a network with the proper ratio of vertices to nodes would exhibit a small-world property. Since vertices and nodes are the same thing, I mentally corrected this to "vertices-to-edges ratio". Since the only small-world property under discussion was the six-degrees hypothesis, but it is obvious that some very large graphs must require more than 6 edges to connect their vertices, I thought you must be referring to a generalized form of the 6-degrees rule- an N-degree rule.

Maybe I'm simply misunderstanding you. What is the "six degrees hypothesis" you mention hypothesizing?

Maybe I'm simply misunderstanding you. What is the "six degrees hypothesis" you mention hypothesizing?

No, it was me misunderstanding the topic as a whole. I thought that the "6-degrees hypothesis" was "any two people can be linked by at most 6 connections" (and that's why I proved what I did) whereas I now think the hypothesis is "the expected length of the shortest path between two randomly selected people is around 6".

posted by Jpfed at 12:58 PM on February 3, 2008

To express it in a hopefully clearer way, if the question is posed

That's what I meant by "relatively simple", in that a whole lot of really heavy lifting has already been done and much of the remaining job is applying existing tools.

On preview - Jpfed, I totally clumsified up the statement I tried to make. Yeah, vertices and nodes are synonyms, I always mess that up. I was actually trying to whip off a simple expression along the lines of how full the adjacency matrix is and failed miserably. (I handle this stuff from the software engineering end of things more than from the proofs end most of the time.)

posted by XMLicious at 1:22 PM on February 3, 2008

*Is it possible that in the real world outside of movie casts, everyone is actually within six steps of each other?*I think part of the immediate answer ought to be "Yes, definitely it's possible. And we know many sorts of hypothetical networks where it would definitely be true, and we've got all kind of mathematical tools for analyzing the degree of connectivity in a given network, and one thing that's needed is to examine social networks with these existing tools."That's what I meant by "relatively simple", in that a whole lot of really heavy lifting has already been done and much of the remaining job is applying existing tools.

On preview - Jpfed, I totally clumsified up the statement I tried to make. Yeah, vertices and nodes are synonyms, I always mess that up. I was actually trying to whip off a simple expression along the lines of how full the adjacency matrix is and failed miserably. (I handle this stuff from the software engineering end of things more than from the proofs end most of the time.)

posted by XMLicious at 1:22 PM on February 3, 2008

Is it distance-regular graphs we're looking for, that social networks would need to be equivalent to? I didn't feel like going to the effort of getting it all straight in my head.

posted by XMLicious at 1:32 PM on February 3, 2008

posted by XMLicious at 1:32 PM on February 3, 2008

On most sites, this post would have 500 "D00D my BFF's brother's cousin was a dancer on Britney's last tour thats only 4 degrees" comments.

Here, we have 20 comments featuring a thoroughly impenetrable debate regarding the relative merits of qualitative sociology and quantitative network analysis.

posted by googly at 1:58 PM on February 3, 2008

Here, we have 20 comments featuring a thoroughly impenetrable debate regarding the relative merits of qualitative sociology and quantitative network analysis.

*Ain't Metafilter grand?*posted by googly at 1:58 PM on February 3, 2008

(thoroughly impenetrable but highly informative, I should add)

posted by googly at 1:59 PM on February 3, 2008

posted by googly at 1:59 PM on February 3, 2008

Thanks for the informative post and ensuing discussion, tkolar. Incidentally, I feel compelled to link to the Erdős–Bacon number.

posted by Monochrome at 5:14 PM on February 3, 2008

posted by Monochrome at 5:14 PM on February 3, 2008

I'm disappointed in Carl Sagan there, trailing the pack with an Erdős–Bacon of 9. But on the other hand Natalie Portman is extra-hawt for having co-authored a paper on Frontal Lobe Activation.

(But I must admit, co-authoring a paper on Riemannian manifolds is even hotter. Kiralee Hayashi for president.)

1. That potentially sounds quite lascivious, but all it involves is miss lynnster completely ignoring her at a dance party.

2. I love having footnotes in comments. Some day I'll figure out how to get a TOC and index in.

posted by XMLicious at 6:13 PM on February 3, 2008

**miss lynnster**, you need to have an ignored-at-a-dance-party¹ story with her. She sure activates*my*frontal lobe.(But I must admit, co-authoring a paper on Riemannian manifolds is even hotter. Kiralee Hayashi for president.)

1. That potentially sounds quite lascivious, but all it involves is miss lynnster completely ignoring her at a dance party.

2. I love having footnotes in comments. Some day I'll figure out how to get a TOC and index in.

posted by XMLicious at 6:13 PM on February 3, 2008

It was sort of odd to use bin Laden as an example. George W. Bush has a lower bin Laden number than almost anyone in the United States (certainly 3, but very possibly 2 -- that is, assuming he's met one of Osama's siblings, which seems likely). Which has of course sparked any number of implied dramatic drum-rolls.

I don't think that anything has been proven about human networks except that the folklore about the magic number 6 has been overstated. I haven't led an exceptionally gregarious life to date, but I start out with 3-degree separations from people like Enrico Fermi and Saul Alinsky (my parents were schoolmates of their children) and through work I've been 2-degree separations from people like Christie Brinkley. And technically I have a Feingold number of one (I was present at a very private pre-Senate fundraiser), though two is probably more honest (his brother is my attorney). That puts me in low numbers for a lot of people. Now, the proverbial low-caste rural farmer in India might be another matter, but I still wonder how high the number could possibly be -- I might even be indirectly related to one through my second cousin's Indian-American wife. And so forth.

As another example, you take a random proposed-for-deletion Wikipedia article like Asa Bird Gardiner, and even though I recognized the name had some importance, once I began researching his life story it made the Reconstruction-era US Army and turn-of-the-century New York City both seem like small towns where everybody knew each other. Or the case of the closely related Presidents, which is worth considering if you're a potential Hillary voter. But as Mark Humphrys has found, the mathematical separation of everyone is not as great as we think, and almost everyone in the US is probably "descended from royalty", or even Muhammad. At the same time, we in the US continue to choose presidents who all seem to come out of this smallish group of descendants of the same people that can't be much larger than 100,000 or so.

It's an attractive little myth to say "six" but I think debunking "six" gets in the way of the larger point, which is that there's an astonishing amount of interconnectedness even for a world of six billion people.

posted by dhartung at 9:25 PM on February 3, 2008

I don't think that anything has been proven about human networks except that the folklore about the magic number 6 has been overstated. I haven't led an exceptionally gregarious life to date, but I start out with 3-degree separations from people like Enrico Fermi and Saul Alinsky (my parents were schoolmates of their children) and through work I've been 2-degree separations from people like Christie Brinkley. And technically I have a Feingold number of one (I was present at a very private pre-Senate fundraiser), though two is probably more honest (his brother is my attorney). That puts me in low numbers for a lot of people. Now, the proverbial low-caste rural farmer in India might be another matter, but I still wonder how high the number could possibly be -- I might even be indirectly related to one through my second cousin's Indian-American wife. And so forth.

As another example, you take a random proposed-for-deletion Wikipedia article like Asa Bird Gardiner, and even though I recognized the name had some importance, once I began researching his life story it made the Reconstruction-era US Army and turn-of-the-century New York City both seem like small towns where everybody knew each other. Or the case of the closely related Presidents, which is worth considering if you're a potential Hillary voter. But as Mark Humphrys has found, the mathematical separation of everyone is not as great as we think, and almost everyone in the US is probably "descended from royalty", or even Muhammad. At the same time, we in the US continue to choose presidents who all seem to come out of this smallish group of descendants of the same people that can't be much larger than 100,000 or so.

It's an attractive little myth to say "six" but I think debunking "six" gets in the way of the larger point, which is that there's an astonishing amount of interconnectedness even for a world of six billion people.

posted by dhartung at 9:25 PM on February 3, 2008

It appears as if all the major players in the neocon revolution have less than 6 degrees of separation from Leon Trotsky

posted by kigpig at 1:13 PM on February 4, 2008

posted by kigpig at 1:13 PM on February 4, 2008

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