As mentioned in Barabasi's "Linked", Judith Kleinfeld's paper has been
published. See a related article in Discover magazine.
http://www.discover.com/june_02/featworks.htmlSix Degrees of
Speculation
Even in a small world, there's room for disagreement
By Karen Wright
You probably don't know Judith Kleinfeld. She's a psychologist at
the University of Alaska in
Fairbanks, and you can contact her by calling the university
switchboard or finding her e-mail
address online. But could you get to her through some extension of
your own social
network-by mailing a letter for her to a friend who might know
someone who knows someone
who knows her?
Who cares?
Judith Kleinfeld does, for starters. She's part of a
growing cadre of scientists reviving the so-called
small-world problem, a social-cum-mathematical
conundrum formulated in the last century to
characterize the interwoven webs of acquaintance
among friends, neighbors, colleagues, and kin. In the
mid-1960s the legendary social psychologist Stanley
Milgram asked randomly selected citizens of Kansas
and Nebraska to try to connect with social "targets" in
Massachusetts by mailing letters to likely
intermediaries. The average number of links between
strangers turned out to be surprisingly small. Milgram
claimed we're all connected, on average, by half a
dozen interpersonal avenues- a numinous network
popularized by the phrase "six degrees of separation."
Now some mathematicians say the dynamics of such
networks can describe a host of natural and
technological systems. But Kleinfeld, who has
reviewed the Milgram archive at Yale University, says the small
world may have no basis in
fact. "It's not a robust social-science finding," she asserts. "It's
a very odd one that has
never been replicated."
The original small-world question was posed in the 1960s by
political scientist Ithiel de Sola
Pool of the Massachusetts Institute of Technology and IBM
mathematician Manfred Kochen.
How many acquaintances, they wondered, connect any two people chosen
at random? Kochen
and Pool's calculations put the average number at three. Milgram
devised an experiment to
test that surprising prediction. He recruited volunteer "starters"
in Kansas and Nebraska
from mailing lists and newspaper ads and told them the name,
address, and occupation of a
target in Massachusetts. He asked the starters to forward a document
to someone they knew
on a first-name basis who might be able to reach the target. In a
1967 report of his results,
Milgram described how one folder had found its way to the wife of a
divinity student in just
four days and three steps: from a Kansas wheat farmer to an
Episcopal minister, from the
minister to a colleague in Cambridge, and thence to the divinity
student's wife. In that case,
Milgram wrote, "The number of intermediate links between starting
person and target person
amounted to two!" The average number of intermediates was five- six
steps, in other words,
connected Milgram's distant strangers.
As a graduate student in the late 1960s at Harvard, where Milgram
worked from 1963 to
1967, Kleinfeld had known of Milgram's studies. But it wasn't until
the 1990s that she
decided to try to repeat his study as an exercise for her graduate
students. When she went to
the Yale archives to find out more about his methods, Kleinfeld saw
that the rates of
completion in Milgram's studies were lower than she'd realized. In
the main study, using
almost 300 starters, only 29 percent of the documents reached a
Boston stockbroker- and
100 of the Nebraska starters owned blue-chip stocks. In a pilot
study- the one that yielded
the anecdote of the divinity student's wife- just 5 percent of 60
documents reached the
target, and they passed through an average of eight people. And
Kleinfeld couldn't find any
exact replications of Milgram's experiments in the literature.
"I was really upset and discouraged," she says. "I thought, 'Oh,
Stanley, how could you do
this?'"
Milgram, whose controversial studies of obedience to authority
earned him celebrity, died in
1984. But his small-world concept still has its defenders. "It's a
solid phenomenon,"
maintains Thomas Blass, a Milgram scholar at the University of
Maryland, Baltimore County,
who says Kleinfeld's criticism is overblown. Many investigators
don't report the results of
pilot studies, Blass says. And Milgram's low completion rates don't
necessarily mean that
his participants hit dead ends. "It's as likely that the low
completion rates are because people
just lost interest. If that's the case, it doesn't damage the
small-world phenomenon."
Support for Milgram's idea has come from another camp as well. In
the mid-1990s,
mathematician Steve Strogatz of Cornell University and his graduate
student Duncan Watts
began constructing theoretical models that explained how millions of
people could be linked by
relatively short social routes. Watts and Strogatz's models show
that the members of a big
network can be connected by short paths if the network comprises
clusters of close
associates joined by occasional, far-reaching links- "random"
elements in otherwise
structured groups.
To test whether such networks actually exist, Watts consulted one of
the few impeccably
documented social circles known to science: the parlor game Six
Degrees of Kevin Bacon. The
game, housed online by the Oracle of Bacon at Virginia Web site and
supported by the vast
Internet Movie Database, involves finding the shortest route between
any given thespian and
actor Kevin Bacon. Links are forged by appearing in a movie with
Bacon, or with someone
who has appeared with Bacon, or with someone who has appeared with
someone who has
appeared with Bacon, and so on. Among the hundreds of thousands of
actors living and dead,
the average "Bacon number" is 2.918, and no actor has been found to
be more than 10 degrees
of separation from Bacon.
Watts learned that the Bacon network conforms to his models of a
small world: Groups of
closely associated actors are joined by random "linchpins" that link
far-flung clusters.
Linchpin actors may have long careers, like Green Acres's Eddie
Albert, thus joining clusters
in time. They may link the films of different cultures, as Bruce Lee
does. Or they may span
genres, like Nicolas Cage (comedy and suspense). There's reason to
believe that ordinary
social networks operate according to similar principles, says
Watts. And he's found the same
kind of connectivity in systems ranging from the neural wiring of a
nematode to the electrical
power grid of the western United States. Small-world networks may
also describe the spread
of disease, fads, and social movements.
"Even just a little bit of randomness in any network is going to
generate short paths between
different nodes," says Watts. "Milgram showed that people can
actually find those paths-
and that's a different sort of problem. The models we developed to
show that these short
paths actually exist don't explain how you can find them."
How people recognize the routes to their distant affiliates is the
next big small-world
question. Computer scientist Jon Kleinberg at Cornell has proposed
an algorithm to explain it
(although Milgram himself conceded that "there is nothing more alien
to mathematically
untutored intuition than this form of thinking"). Watts is trying to
trace both the reasoning and
the routes through an e-mail replication of Milgram's
experiment. He's got 50,000
participants so far, with targets in India, Europe, Indonesia, and
Australia as well as the
United States. "We've already had some globe-crossing chains- from
Sydney to Siberia in
four links," he says. "It can be done."
Kleinfeld thinks an e-mail experiment misses the point, because
computers link people who
already have a lot in common- such as income and education. "I think
scientists live in a
small world," says Kleinfeld. "People who are low-income or
minorities aren't connected."
Watts concedes that the e-mail network is a practical
compromise. "In terms of getting very
large numbers of people [to participate], nothing can beat the Web,"
he says.
Blass gives Milgram the final word. "Milgram would be very happy
about it," he claims,
"because he was one of the first people to appreciate the
communicative value of PCs."
RELATED WEB SITES:
Kleinfeld, Judith S. "Could It Be a Big World After All?"
Forthcoming in Society (2002). See
www.uaf.edu/northern/big_world.html.
The Oracle of Bacon at Virginia Web site:
www.cs.virginia.edu/oracle. Sign up for Duncan
Watts's e-mail experiment: http://smallworld
.sociology.columbia.edu/target.html. Also see
Watts, Duncan J. Small Worlds: The Dynamics of Networks Between
Order and Randomness.
New Jersey: Princeton University Press, 1999.
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