On Small World Statistics

Milgram [1967] formulates the basic dimensions of the small-world problem in terms of a series of questions: (a) Given any two randomly chosen people in the world, what is the probability that they know each other? (b) If they don’t know each other, do they share a mutual acquaintance? (c) If they don’t share a mutual acquaintance, how many intermediate acquaintances, linked in a chain, are needed in order to connect them? To explore these dimensions, Milgram and various coauthors [see Milgram, 1967] conducted a series of observational experiments designed to trace the line of acquaintances linking a pair of individuals in distant cities. Each experiment began with the selection of a random sample of starting persons in one city (e.g. Omaha, Nebraska) and a target person in a remote city (e.g. Sharon, Massachusetts). Each starting person was given a packet, containing background information on the target, person, with instructions to send the packet to the target person only if he or she knew the target person on a first-name basis; otherwise, he should forward the packet to the acquaintance whom he thinks would be most likely to know the target. Each recipient of the packet was instructed to send a postcard informing the experimenter to whom the packet had been forwarded. This device allowed the experimenter to keep track of all the intermediate steps of each chain. Therefore, each chain could be completed in only two ways: either the packet was received by the target person or it was discarded at an intermediate step. All of the small-world experiments carried out by Milgram share two features: a large percentage of dropouts and a small median number of