Here we consider the anecdotes and lore surrounding human connections known as social networks. Many of us have met strangers, only to discover that we share with them common acquaintances. Such connections have been publicized in dramatical productions, such as the well-known play Six Degrees of Separation by John Guare. In that play a character, named Ouisa, claims that any two individuals on our planet are separated by only six other people and, thus everyone is bound to everyone else by a trail of six people.

The Kevin Bacon Game - Movie Star Networks

The Kevin Bacon game concerns the actor, Kevin Bacon, who has, in reality. not been in many films. It seeks to identify the Bacon Number of actors in American films such that if an actor or actress has even been in a film with Kevin Bacon, then he/she has a Bacon Number of one; if they have never been in a film with Bacon, but  have been in a film with somebody that has, then they have a Bacon  Number of two, and so on. The claim is that few actors/actresses, who have ever appeared in an American film, have a Bacon Number greater than four. Woody Allen has, for example, a Bacon number of 2. While Allen never costarred with Bacon, Woody Allen was in The Curse of the Jade Scorpion (2001) with Charlize Theron, who costarred with Kevin Bacon in 24 Hours (2001).

The University of Virginia Computer Science Department maintains a site  http://www.cs.virginia.edu/oracle providing the Bacon Number and shortest path from any actor/actress to Kevin Bacon. Data for the site is from the Internet Movie Database http://www.imdb.com which references almost all movies ever made globally, with approximately 150,000 films  and about 300,000 actors. The underlying Kevin Bacon graph or network can also be viewed as a collaborative graph.

Paul Erdos and the Erdos Number - Networks of Mathematicians

Paul Erdos, the renowned twentieth-century mathematician, who authored or co-authored over 1400 papers in his lifetime and was a known eccentric, is the first person to have his name associated with small-world phenomena. A person's Erdos Number is a measure of how close one is to Erdos via their co-authors. The 507 people who have co-authored a paper with Erdos hold an Erdos Number of one. An Erdos Number of two means that the person has not published a paper with Erdos but has published with someone who has, and so on. For more information on Erdos Numbers see http://www.oakland.edu/enp/

Corporate Directorships - The Old Boy Network

The old boy network consisting of the directors of the largest companies in the United States can be described as a network/graph in which the nodes or vertices represent the directors/boards and the pairs of nodes are joined by edges (undirected links) if and only if the corresponding directors sit together on one or more boards. In 1999, Vernon Jordan sat on no fewer than nine distinct boards of Fortune 1000 companies, a group that consists of the largest US companies according to revenue. Mr. Jordan is one of a group who sit on many corporate boards, creating an overlap that joins virtually all major US companies into a huge network or web of corporate governance. The data on interlocking directorates can be represented by a bipartite (two sets of nodes) graph or network, with the top tier or set of nodes corresponding to the distinct boards and the bottom or second tier of nodes to the individual members. The edges (undirected links) from a top tier node to a bottom tiered node represents the connection of the board with a particular director. For example, the figure below represents the case in which there are 4 boards and 16 members but only 11 of the members are distinct.