Small World Problem

Small World Problem

Christopher McCarty

Small World Phenomenon

• You meet someone, seemingly randomly, who has a connection to someone you know– Person you meet on a plane who went to school

with a relative– Killworth’s example– McCarty example

• Is this a bizarre coincidence or is there an underlying explanation?

Six Degrees of Kevin Bacon

• http://oracleofbacon.org/

• LinkedIn

John Barnes (1969) Networks and Political Process in Social Networks in Urban Situations

Small World StudyJeffrey Travers and Stanley Milgram (1969) An Experimental Study of the Small

World Problem, Sociometry 32(4): 425-443

• What is the probability that any two randomly selected people know each other?

Method• 296 respondents in Nebraska and Boston are asked to send a

packet to someone they knew who had the best chance of knowing a target person in Massachusetts

• The target was a Boston stockbroker

• 100 respondents owned stocks, the rest were randomly selected

• There were 453 intermediaries

• There were no incentives

Method (cont.)

• The profile contained name, address, occupation, place of employment, college, military service, his wife’s maiden name and home town

• Roster included to prevent looping

Results

• 217 of the 296 sent the packet on

• 64 chains were completed (29 percent) with an average length of 5.2

• 48 percent of the chains passed through 3 people before reaching the target

Characteristics of intermediaries• There was a division of

labor among the 3

• One person was a clothing merchant and handled chains using residence

• Others were scattered around Boston and used occupation (stockbrokers)

Future study

• Throughout the article Travers and Milgram suggest a different study where they manipulate the starter and target information

H. Russell Benard, Peter D. Killworth and Christopher McCarty (1982) Index: An Informant-Defined Experiment

in Social Structure, Social Forces 61(1): 99-133• 50 mythical targets

• Each started with set characteristics

• 50 respondents were instructed to select the person they knew with the best chance of knowing the target

• The respondent had to decide which questions to ask

• As new information was given it was recorded on the target’s profile

Preset Target Characteristics

Peter D. Killworth, H. Russell Bernard and Christopher McCarty (1984) Measuring Patterns of Acquaintanceship, Current

Anthropology 25(4): 381-397

• Reverse Small World

• 500 mythical targets, 400 around the world and 100 in the U.S.

• Distributions of 10 areas of the world, occupations, sex, age, education, marital status

• 40 respondents

Suggests line would asymptote around 250. This is another measure of network size

Duncan J. Watts and Steven H. Strogatz (1998) Collective Dynamics of ‘Small World’ Networks,

Nature 393(4): 440-442• Mathematical re-examination of the small

world problem

• Highly ordered networks are re-wired to introduce disorder

• These lead to highly clustered networks with small path lengths (small world)

Peter Sheridan Dodds, Roby Muhamad and Duncan J. Watts (2003) An Experimental Study of Search in Global

Networks, Science 301 (5634): 827-829• Internet based study

• 18 target people from 13 countries

• Different occupations

• 98,847 people registered at web site, about 25% provided personal information

• 61,168 participants from 166 countries

• 24,163 chains – 384 (1.6% completed)

• Average chain length was 4.05

Results

Peter D. Killworth, Christopher McCarty, H. Russell Bernard and Mark House (2006) The Accuracy of Smal World Chains in Social

Networks, Social Networks 28: 85-96

• 105 members of survey lab were shown a roster of all other names

• They indicated if they knew each one

• For those they did not know they nominated someone who would have the best chance of knowing them

Results

• Mean length of shortest actual path was 2.3 (S.D. 0.71)

• 21.7% of the conceptual paths terminated as they included people who did not participate

• Another 23.7% ended up in loops (i chooses j chooses i)

• The remaining 54.6% reach completion with mean 3.23 (S.D. 2.06), 40% longer than actual length