小世界实验

Historical context of the small-world problem

Guglielmo Marconi's conjectures based on his radio work in the early 20th century, which were articulated in his 1909 Nobel Prize address, may have inspired Hungarian author Frigyes Karinthy to write a challenge to find another person to whom he could not be connected through at most five people. This is perhaps the earliest reference to the concept of six degrees of separation, and the search for an answer to the small world problem.

Mathematician Manfred Kochen and political scientist Ithiel de Sola Pool wrote a mathematical manuscript, "Contacts and Influences", while working at the University of Paris in the early 1950s, during a time when Milgram visited and collaborated in their research. Their unpublished manuscript circulated among academics for over 20 years before publication in 1978. It formally articulated the mechanics of social networks, and explored the mathematical consequences of these (including the degree of connectedness). The manuscript left many significant questions about networks unresolved, and one of these was the number of degrees of separation in actual social networks.

The experiment

Milgram's experiment developed out of a desire to learn more about the probability that two randomly selected people would know each other. This is one way of looking at the small world problem. An alternative view of the problem is to imagine the population as a social network and attempt to find the average path length between any two nodes. Milgram's experiment was designed to measure these path lengths by developing a procedure to count the number of ties between any two people.

Basic procedure

Though the experiment went through several variations, Milgram typically chose individuals in the U.S. cities of Omaha, Nebraska, and Wichita, Kansas, to be the starting points and Boston, Massachusetts, to be the end point of a chain of correspondence. These cities were selected because they were thought to represent a great distance in the United States, both socially and geographically.

Information packets were initially sent to "randomly" selected individuals in Omaha or Wichita. They included letters, which detailed the study's purpose, and basic information about a target contact person in Boston.

The Reversal Small-World Experiment

The Reversal Small-World Experiment is a 1978 study conducted by Peter D. Killworth and H. Russell Bernard, aiming to test and refine the understanding of the small-world phenomenon. This phenomenon suggests that individuals in a social network are connected by surprisingly short chains of acquaintances. The study builds upon the pioneering work of Stanley Milgram. Killworth and Bernard introduced a reversal approach to the experiment, addressing key limitations in Milgram’s methodology and testing the validity of his conclusions regarding the structure and reachability of social networks.

Motivation

Milgram’s original experiment relied on forward routing, where participants were tasked with passing messages to a target person by selecting acquaintances they believed were closest to the destination. However, Milgram’s findings were limited by:

High attrition rates: Many message chains never reached their destination, leading to incomplete data.

Cognitive biases: Participants might not have accurately assessed who among their acquaintances was closest to the target.

Structural biases: The small-world model assumes a connected network, but real-world networks contain isolated subgroups.

Influence

The social sciences

The Tipping Point by Canadian journalist Malcolm Gladwell, based on articles originally published in The New Yorker, elaborates on the "funneling" concept. Gladwell condenses sociological research, which argues that the six-degrees phenomenon is dependent on a few extraordinary people ("connectors") with large networks of contacts and friends: these hubs then mediate the connections between the vast majority of otherwise weakly connected individuals.

Recent work in the effects of the small world phenomenon on disease transmission, however, have indicated that due to the strongly connected nature of social networks as a whole, removing these hubs from a population usually has little effect on the average path length through the graph (Barrett et al., 2005).

A corollary of network structures is that if the edges that connect nodes in a network, even a randomly constructed one, are above a certain threshold, then the shortest path between nodes, averaged across the entire network, is short. Subsequent research following Milgram’s experiment, namely by Watts and Strogatz, have aimed to reflect the highly-connected and highly-clustered networks of reality.

Current research on the small-world problem

The small-world question is still a popular research topic today, with many experiments still being conducted. For instance, Peter Dodds, Roby Muhamad, and Duncan Watts conducted the first large-scale replication of Milgram's experiment, involving 24,163 e-mail chains and 18 targets around the world.

Dodds et al. also found that the mean chain length was roughly six, even after accounting for attrition. A similar experiment using popular social networking sites as a medium was carried out at Carnegie Mellon University. Results showed that very few messages actually reached their destination. However, the critiques that apply to Milgram's experiment largely apply also to this current research.

Recent research suggests that the small-world effect is a phenomenon that appeared rather recently in human history, leading to a drastic reduction in the average chain distance in social and physical networks. This can be justified by studying evolution patterns of infectious diseases throughout history, notably the Black Plague in Medieval Europe. Past epidemics have been noticed to spread in waves from well-defined central points, which can be explained through the localized nature of interactions of medieval populations.

In popular culture

Social networks pervade popular culture in the United States and elsewhere. In particular, the notion of six degrees has become part of the collective consciousness. Social networking services such as Facebook, Linkedin, and Instagram have greatly increased the connectivity of the online space through the application of social networking concepts.