The Small World Phenomenon

Abhijit Mahabal

The Kevin Bacon Game

• http://www.cs.virginia.edu/oracle

Bacon Number, Erdös Number

• Kevin Bacon has been in 56 movies so far– Any body who has acted in a film with Bacon has a

bacon number of 1.– Anybody who does not have a bacon number 1 but has

worked with somebody who does, they have bacon number 2, and so on

• Most people in American movies have a number 4 or less. Given that there are about 225000 such people, this is remarkable.

• Why?? What makes this happen?

Six Degrees

• Mails (Milgram, 60’s)

• The Weak Version:– There exists a short path from anybody to

anybody else

• The Strong Version:– There is a path that can be found using local

information only.

The Caveman World

• Many caves, and people know only others in their caves, and know all of them.

• Clearly, there is no way to get a letter across to somebody in another cave.

• If we change things so that the head-person of a cave is likely to know other head-people, letters might be got across, but still slowly.

• There is too much “acquaintance-overlap”

The world of Chatting

• People meet others over the net

• In these over-the-net-only interactions, I think that there are almost no common friends.

• Again, if a message needed to be sent across, it’d be hard to figure out how to route it

Small Worlds Are Between These Extremes

• When there is some, but not very high, overlap between acquaintances of two people who are acquainted, small worlds results.

• If somebody knows people in different groups (caves?), they can act as linchpins that connect the small world.

• For example, cognitive scientists are lynchpins that connect philosophers, linguists, computer scientists etc.

• Bruce Lee is a linchpin who connects Hollywood to its Chinese counterpart.

Why Bother?

• In many of our earlier discussions, we have talked about how space was considered or ignored (in spread of wealth, language etc).

• Spatial interactions are more like those in a small world, rather than on a 2 D grid. The small world graphs are a lot more complicated, and cannot be embedded in a small dimensional space.

• Kareiva (‘90) has studied spread of disease.• Kretzschamer & Morris studied the spread of disease as a

function of the network structure• Nowak & May (94, etc) have studied the evolution of

prisoners’-dilemma strategies over various networks

Modeling this Middle Ground(Jon Kleinberg)

• Agents are on a grid.– Everybody is

connected to their neighbors

– But they are also connected to k other agents randomly.

The Random Neighbors

• The distribution could be uniform, or biased towards closer agents.

• It could be inversely proportional to the distance from us to that agent, or inversely proportional to the square of the distance (sort of like gravitation?)

• These can be represented as inversely proportional to d to the power r, where r is 0, 1 or 2 above.

• If r is 0, the neighbor is chosen randomly, and the world is like the solarium world.

• If r is very high, you only know your immediate neighbors: and the world is like the caveman world.

• For intermediate values, we get more and more small-world-like behavior.

• There is always a findable path whose length is not too big (log square (size of the world)) only when– R is 2!!

• For any other R (smaller or bigger than 2), the expected length of a findable path is a polynomial in n.