Math 443/543 Graph Theory Notes 10: Small world phenomenon and decentralized search

The small world phenomenon is the principle that all people are linked by short chains of acquaintances. For instance, “six degrees of separation” is the idea that everyone is linked by chains of people at most 6 persons long. That is, given any persons A and B; there are a chain of people such that A knows C who knows D who knows E who knows F who knows G whoe knows H who knows B: Other examples are the Kevin Bacon game (that every actor can be linked to Kevin Bacon via actors they have acted with) and the Erdos number (related to collaborations on mathematics papers. The …rst real empirical evidence is from an experiment by the social psychologist Stanley Milgram during the 1960s. He tested it in the following way. He chose a target person, as stockbroker living near Boston. And he took a random person in, say, Nebraska and asked them to get a letter to the target. Each person was only allowed to forward the letter to a single acquaintance they know on a …rst name basis. The only things the person knew about the target was his name, occupation, address, and some personal information. One can think of the letter as tracing a path in the graph of all people who know each other (the acquaintance graph). Milgram found that sometimes the letters made it to the target, and among those that did make it to the target, the median length of the path was around 6, hence the idea of 6 degrees of separation. For our purposes, we want to understand why a graph like the acquaintance graph might have short paths, and then think about how to …nd short paths. Simplistic idea: Suppose each person knows 100 others on a …rst name basis. Then in 2 steps, one should be able to reach 10,000 people, in three steps, 100 x 100x100 = 1,000,000 people, and then by 5 steps 10 billion people.. The population of the US is estimated just above 300 million and the population of the world is estimated to be less than 7 billion. This idea is a bit too simplistic, since many people who you know on a …rst name basis also know each other. This means that, in reality, 5 steps will not reach nearly this many people. The implicit assumption made above is that everyone knows 100 random people of all possible people.