Small world phenomenon

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The small-world-phenomenon ( English small-world experiment ) is a socio - psychological term coined by Stanley Milgram in 1967 , which describes the high degree of shortening paths through personal relationships within the social network in modern society . The hypothesis to every human being (social actor) connected to the world with each other over a surprisingly short chain of acquaintance relationships. This is possible even though the "density" of the social network of "all" actors - measured as the ratio of the real to the mathematically possible contacts of "the contact persons" of any actor - is close to zero.

The phenomenon is often referred to as Six Degrees of Separation . The underlying idea was presented in a short story by the Hungarian Frigyes Karinthy , published in 1929 - there, however, over five levels.

Milgram's Small World Experiment

experiment

The first small world experiment was carried out in 1967 by the American psychologist Stanley Milgram , then at Harvard University . Milgram created a kind of information package that 60 randomly selected participants had to send to a predetermined person in Boston . As starting points, he selected people from the cities of Omaha and Wichita, which are socially and geographically distant from the target city . The task of the participants was not to send the package directly to the target person, unless they knew them personally (addressed by their first name), but to someone they knew personally and who was more likely to receive the Target person knew. At the same time, the participants were asked to note down basic data about themselves in a table and to send a postcard to the scientists to make the chain understandable.

Results

A total of three packages reached the target persons with an average path length of 5.5 or six rounded up. The scientists concluded that every person in the US population is separated from every other person in the US by an average of six people, or, to put it the other way around, can be reached by an average of six people.

In an experiment carried out two years later with 296 possible chains, 217 parcels were dispatched, 64 of which reached their destination. Another attempt followed in 1970, which, in addition to the distance between people, was also intended to examine possible boundaries between ethnically different groups. Of 270 parcels that were started with African-American people as their destination, 13% reached this person, while 33% of 270 parcels addressed to "white" people reached their destination.

criticism

Both the experiment and the conclusions drawn from the results are controversial and are not considered to be conclusive. In a study published in 2002, the American psychologist Judith Kleinfeld criticized in particular the insufficient data situation (Kleinfeld speaks of a “5% chain completion rate”), which leads to the conclusion that it is a “small world” per se , do not allow. In their opinion, the investigations following the first experiment are also based on too few successful connections in the chain. The 1970 experiment in particular shows that “we do not live in a small, interwoven world, but in a world separated by racial barriers” (“The results suggest again that, far from living in a small, inter-connected world, we live in a world with racial barriers. ").

However, Kleinfeld recognizes the fascination of Milgram's enthusiasm for the small world phenomenon and cites a Canadian study from 1976 that, in contrast to Milgram's studies, had a high success rate of 85% and was carried out over the phone. She therefore advocates continuing the investigations and recommends more comprehensible methods such as contacting us by phone or email. Kleinfeld argues that the empirical evidence points to some very well-connected people, but also to less well-connected evidence, i.e. overall the reality of social relationship systems does not follow the elegance of mathematical models.

Small world networks

The small world phenomenon can also be transferred to other networks and graphs , as mathematical network research has been trying to show, especially since the late 1990s. The basic principle is that individual objects, e.g. B. People, are represented as nodes between which there is an edge if a certain relationship (e.g. acquaintance) exists between them. The Erdős number and the Bacon number are defined according to this pattern . Co-authorship chains, for example in psychology, can also be presented and researched in this way.

In 2003 the phenomenon was confirmed for the Internet in an experiment in which the e-mail traffic of 60,000 test persons from 166 countries was evaluated. Critics, however, argued against applying the results to the world's population.

In 2008, Microsoft scientists Jure Leskovec and Eric Horvitz were able to empirically confirm the thesis of the small world based on a network among instant messenger users (180 million nodes with 9.1 billion edges).

Two phenomena can be observed in small world networks:

Transitivity

First, the probability is very high that two nodes, each with an edge to a third node, are also connected to each other ( transitivity ). Transferred to social networks, this means that a person's friends are usually also known to one another because they got to know each other through their mutual friend (principle of transitivity). Mathematically, this fact is described using the clustering coefficient , which is very high on average for small world networks. This claim is of course controversial, because it assumes that the actors (nodes) have no head-rich (e.g. urban) networks and themselves have few social roles .

Small diameter

Second, the diameter of these networks is relatively small. This means that a message that is passed on from one node via an edge to all of its neighboring nodes has reached all nodes in the network in the shortest possible time. So-called short chains as connections to individual, distant nodes are of particular importance . This, too, is controversial because “closeness” by virtue of activatable acquaintances does not have to mean that certain messages spread as quickly as the experimental letter described.

Examples of scale-free networks

In the course of dealing with small world networks, mathematical network research has established a plurality of structural patterns and has placed particular emphasis on so-called scale - free networks . These are networks in which a few nodes ( english hubs ) potentially infinite have many connections, while much of the rest of the nodes relatively few relationships with other nodes has ( power law ).

Well-known "small worlds" are, for example, the American power grid, almost all subsets of social networks, a subset of the pages of the WWW, other articles, for example in an encyclopedia, which are linked to one another by references, and also the routers of the Internet. To assess the susceptibility of these networks to faults, this is an important approach, because a fault can also be viewed as a message. However, it is currently still disputed to what extent the networks mentioned really all have a scale-free structure. The system theory not treated such grids as "Small World", but as - tight or loose - coupled systems.

The special networking of a scale-free network makes it robust against the accidental failure of a few nodes or edges. However, if "important" nodes ( english hubs are) selectively removed, the network breaks down quickly into subnets. This is the reason why the failure of just a few routers on the Internet can have far-reaching effects. Conversely, the scale-free structure of the Internet also results in the rapid spread of computer viruses once they have reached the nodes. Research suggests that the same applies to the spread of HIV in sexual networks.

Modeling

First approaches

The first modeling approaches to describe the small world phenomenon were on the one hand a strongly connected grid model and on the other hand the Erdős-Rényi random graph . They were created shortly after Milgram's letter chain experiment was published, but were not yet able to model the social network satisfactorily.

  • In the strongly connected grid model, all integer points of the plane are taken and not only direct neighbors are connected by edges, but all points at which the coordinates differ by at most a fixed amount . For a point is connected to all points within a diamond, a total of 960 points.
  • Erdős-Rényi random graphs also assume grid points on a plane; here, however, the edges between "all" points of the (finite) plane are set according to a given probability .

Both models can, however, only represent "one" aspect of small world networks: the grid model represents the local connections of an individual, while the random graph models the global connections.

Further development of the model

The decisive further development was presented in 1998 by Duncan Watts and Steven Strogatz . The main approach is to link the two models presented in order to map the various relationships in the “real world”.

The model starts with an existing, regularly connected network. A small proportion of the connections are then broken and placed with random new neighbors. The result is a so-called "egalitarian" network, that is, because every node has roughly the same number of edges to other nodes. This idea was later developed by Jon Kleinberg . While the Watts and Strogatz model can describe the short observed paths, it fails to explain how the people in Milgram's experiment actually found these paths. Kleinberg was able to show that such paths can be found efficiently if connections are used not purely randomly, but randomly, but taking into account a certain length distribution.

A further model is the Barabási-Albert model published by Albert-László Barabási and Réka Albert in 1999 . Here you start with a fully connected network of three nodes and add new nodes one by one to the network. These each form a certain number of new connections to the existing network. The probability that an existing node will be chosen as a partner is proportional to the number of connections it already has: The rich are getting richer . Networks of this structure are also referred to as "aristocratic" or "hierarchical".

Both simulations generate networks with a small world effect. Barabási Albert networks are also scale-free .

Computer simulation

The possibilities of computer physics make it possible to empirically check models that are supposed to explain the emergence of networks with properties such as the small world phenomenon.

Application of the model

Spanish researchers at the University of Barcelona want to use the small world phenomenon to optimize the routing tables of Internet routers, reduce their complexity and thus significantly reduce them.

Online networks

This phenomenon can be observed in reality in online networks such as XING , StudiVZ or localists . You can access this network after registering yourself or at the invitation of an existing member, i. That is, often everyone is connected to at least one other person. However, if one takes a person out of this network at random, the direct path from oneself to this person is always shown, which rarely comprises more than five members. Anyone who is logged in without a connection does not appear in connection paths.

The small world phenomenon can only be transferred directly to social network sites to a limited extent , since all possible connections between all people are not stored in any online service and, on the other hand, connections can also be stored that do not exist in reality.

literature

Web links

Individual evidence

  1. ^ Six Degrees of Separation. (PDF; 95 kB) Center for Complex Network Research at Northeastern University , Boston, March 12, 2002, archived from the original on June 4, 2013 ; accessed on April 8, 2014 .
  2. ↑ Zelkadis Elvi, Lauren Lyster: Just Explain It: Six Degrees of Separation. yahoo .com, May 8, 2013, accessed April 8, 2014 .
  3. ^ Travers, J., & Milgram, S .: An experimental study of the small world problem. 1969, Sociometry 32, pp. 425-443.
  4. ^ Korte, C., & Milgram, S .: Acquaintance links between White and Negro populations: Application of the small world method. 1970, Journal of Personality and Social Psychology 15 (2), pp. 101-108
  5. ^ Judith S. Kleinfeld: Could It Be A Big World After All? The "Six Degrees of Separation" Myth. Working paper ; Six Degrees: Urban Myth? , Psychology Today , 2001 psychologytoday.com ( Memento from May 1, 2007 in the web archive archive.today )
  6. ↑ Musch, J., & Winter, D. The small world of psychology. ( Memento of May 9, 2009 in the Internet Archive ) Search engine for co-authorship relationships in psychology.
  7. ^ Anja Ebersbach, Markus Glaser, Richard Heigl: Social Web (=  UTB for science / university paperbacks . Volume 3065 ). 3. Edition. UVK, Konstanz / Munich 2016, ISBN 978-3-8252-3933-6 , p. 96 ( limited preview in Google Book Search - revised edition).
  8. ↑ Jure Leskovec, Eric Horvitz: Planetary-Scale Views on a Large Instant Messaging Network. In: erichorvitz.com . April 2008, accessed on April 13, 2020 (English; full text available as PDF).
  9. ↑ Florian Rötzer: Microsoft scientists confirm the thesis of the small world. In: heise.de. August 4, 2008. Retrieved February 26, 2019 .
  10. ↑ Erica Naone: The social life of the router. In: heise.de - Technology Review . December 19, 2008, accessed October 10, 2019 .