The Small World Phenomenon Abhijit Mahabal

The Kevin Bacon Game

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 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.