2011-3-3  六度空间理论(Six Degrees of Separation) 一个数学领域的猜想，名为Six Degrees of Separation，中文翻译包括以下几种： 六度分割理论或小世界理论等。 理论指出：你和任何一个陌生人之间所间隔的人不会超过六个，也就是说，最多通过6个中间人你. Six Degrees of Separation is one in a long line of 90s film adaptations of a stage play, centering around a young man, played by Will Smith, who enters the lives of two socialites, both confounding and informing the people he crosses paths with. It's a very unique film, though not entirely successful.

Problem statement

You want to determine the separation between two people in a population (LinkedIn, Facebook, etc).

Assume that there's a getFriends function that will return the list of friends for a given Person.

You can then get the friends-of-friends by calling getFriends on each of the friends in the original result set. Then you can get friends-of-friends-of-friends and so on.

So, given Person A and Person B how can you determine if they are connected, and if so, how many 'links' or 'degrees' separate the two people?

Data structure analysis

You have two trees, with Person A as the root node of 1st tree, and Person B as the root node of the 2nd tree.

Search strategy analysis

DFS vs BFS

When traversing a tree, you have two basic options -- depth-first traversal (DFS) and breadth-first traversal (BFS).

Friend graphs in social networks tend to be shallow but broad (remember you are only a handful of degrees removed from anyone on earth!).

So for this use case, BFS is clearly superior to DFS, because DFS will take a lot longer to discover that you are merely 1 degree removed away from a friend.

Single-direction BFS

So you can start a Person A and traverse the friend-tree looking for Person B, counting the 'levels' as you go.

To illustrate, let's assume naively that each person anywhere in the population has 100 friends and in this particular case all the friends are distinct.

So Person A has 100 friends, who then have 10,000 friends, who then have 1,000,000 friends. Person B is somewhere in the A's third-degree group of 1,000,000 friends.

If you search directly for Person B, it looks like this:

Launch from both directions

Instead of looking directly for Person B, it's a lot more efficient on average to launch the BFS from both directions and look for a common friend.

Same scenario as above -- Person A has 100 friends, who then have 10,000 friends, who then have 1,000,000 friends. And the same holds for Person B.

Notice how the 'common friends' approach can dramatically cut down the processing.

Side note

This is assuming no further optimizations. In particular, we are not checking for an intersection while fetching friends -- the supplied diagrams assume that the entire set of friends at a given degree is assembled first, then intersection is computed, which of course need not be the case. However the diagrams give the basic idea -- meeting in the middle is going to be a lot less work on average.

Reference implementation

You may or may not need to provide an actual implementation to answer this problem. Often, you don't -- it suffices to provide an algorithmic description.

However for reference here is a simple implementation (gist with rough tests).

How to scale this

When you have huge user populations as large as LinkedIn or Facebook, and then track the ginormous number of relationships between those users, you generally can't fit all that data into memory on one machine.

Adn even if you theoretically could, you probably still want to parallelize the processing for speed's sake, at least across different CPU cores on the same machine, or much more likely, distributed across different machines.

So, how can you break up this problem for distribution?

Analyze the processing phases for distribution / parallelization

To answer that, what are the different aspects of processing?

• the collection of node A friends into a friend set
• the collection of node B friends into a friend set
• the intersection of the friend set
• the orchestration of the above, including tracking current degrees of separation

You can envision each of the above pieces living in a different process and/or a different server, and tying the processing together with network calls.

The orchestrator would begin the processing for each degree of separation, break up the work into chunks, farm out the chunks as messages/jobs with a message-bus/job-queuing part of your system architecture. External workers would do the fetching of friends and build up the friend set in a common transient data store. This would automatically start happening in parallel once the number of friends at given degree became larger than the per-worker batch size, thus dramatically speeding up the fetch-and-store operation.

When the jobs are all done for a given phase (note: a 'phase' could correspond to the first half or the second half of the while loop in the calculateDegreesOfSeparation above), then the orchestrator could either directly compute the intersection between sets or ask the common transient data store to do so.

redis as an analogy

For example, Redis has a native Set data type, as well as facilities for detecting and fully-computing intersections between sets. LinkedIn and Facebook don't use Redis that way, because that tool doesn't scale well enough for their tremendous data sets, but you can speak to what you know, and could offer that as an analogy for how this might work.

map/reduce as a real potential solution

You can look at map/reduce as an abstraction for organizing this type of processing. Here's a random article describing that.

Six Degrees of Kevin Bacon or 'Bacon's Law' is a parlour game based on the 'six degrees of separation' concept, which posits that any two people on Earth are six or fewer acquaintance links apart. Movie buffs challenge each other to find the shortest path between an arbitrary actor and prolific actor Kevin Bacon. It rests on the assumption that anyone involved in the Hollywood film industry can be linked through their film roles to Bacon within six steps. In 2007, Bacon started a charitable organization called SixDegrees.org.

History[edit]

In a January 1994 interview with Premiere magazine Kevin Bacon mentioned while discussing the film The River Wild that 'he had worked with everybody in Hollywood or someone who's worked with them.'^[1] Following this, a lengthy newsgroup thread which was headed 'Kevin Bacon is the Center of the Universe' appeared.^[2] Four Albright College students, including Brian Turtle, claim to have invented the game that became known as 'Six Degrees of Kevin Bacon'^[1] after watching two movies featuring Bacon back to back, Footloose and The Air Up There. During the second they began to speculate on how many movies Bacon had been in and the number of people with whom he had worked.^[3] In the interview, Brian Turtle explained how 'it became one of our stupid party tricks I guess. People would throw names at us and we'd connect them to Kevin Bacon.'

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They wrote a letter to talk show host Jon Stewart, telling him that 'Kevin Bacon was the center of the entertainment universe' and explaining the game.^[4] They appeared on The Jon Stewart Show and The Howard Stern Show with Bacon to explain the game. Bacon admitted that he initially disliked the game because he believed it was ridiculing him, but he eventually came to enjoy it. The three inventors released a book, Six Degrees of Kevin Bacon (ISBN9780452278448), with an introduction written by Bacon.^[4] A board game based on the concept was released by Endless Games.

Bacon also appeared in a commercial for the Visacheck card that parodied the game. In the commercial, Bacon wants to write a check to buy a book, but the clerk asks for his ID, which he does not have. He leaves and returns with a group of people, then says to the clerk, 'Okay, I was in a movie with an extra, Eunice, whose hairdresser, Wayne, attended Sunday school with Father O'Neill, who plays racquetball with Dr. Sanjay, who recently removed the appendix of Kim, who dumped you sophomore year. So you see, we're practically brothers.'^[5] In a similar vein, Dave Barry, in a column describing the unexpected complications that emerged when he attempted to find out the precise wording of the Lone Ranger's catchphrase, connected the Lone Ranger to Kevin Bacon in the following way: the Lone Ranger was the Green Hornet's great-uncle; the Green Hornet and O. J. Simpson both hung out with people named Kato; Simpson and Robert Wagner co-starred in The Towering Inferno; Wagner and Bacon co-starred in Wild Things.^[6]

The concept was also presented in an episode of the TV show Mad About You dated November 19, 1996, in which a character expressed the opinion that every actor is only three degrees of separation from Kevin Bacon. Bacon spoofed the concept himself in a cameo he performed for the independent film We Married Margo.^[7] Playing himself in a 2003 episode of Will and Grace, Bacon connects himself to Val Kilmer through Tom Cruise and jokes 'Hey, that was a short one!'.^[8] The headline of The Onion, a satirical newspaper, on October 30, 2002, was 'Kevin Bacon Linked To Al-Qaeda'.^[9] Bacon provides the voice-over commentary for the NY Skyride attraction at the Empire State Building in New York City. At several points throughout the commentary, Bacon alludes to his connections to Hollywood stars via other actors with whom he has worked.

Turner Classic Movies’ annual 31 Days of Oscar festival sometimes utilizes a promotional tactic called “360 Degrees of Oscar” (based on Bacon’s theoretical game) where TCM takes a significant actor, actress, director, or producer, and evolves its entire festival around him/her (the first and last films in the festival feature his/her works). It has been used as recently as 2020.

In 2009, Bacon narrated a National Geographic Channel show 'The Human Family Tree'^[10] – a program which describes the efforts of that organization's Genographic Project to establish the genetic interconnectedness of all humans. In 2011, James Franco made reference to Six Degrees of Kevin Bacon while hosting the 83rd Academy Awards. In the summer of 2012, Google began to offer the ability to find an actor's Bacon number on its main page, by searching for the actor's name preceded by the phrase 'bacon number'.EE began a UK television advertising campaign on November 3, 2012, based on the Six Degrees concept, where Kevin Bacon illustrates his connections and draws attention to how the EE 4G network allows similar connectivity.^[11]

The most highly connected nodes of the Internet have been referred to as 'the 'Kevin Bacons' of the Web,' inasmuch as they enable most users to navigate to most sites in 19 clicks or less.^[12][13] In 'Weird Al' Yankovic's song 'Lame Claim to Fame,' one of the lines is, 'I know a guy who knows a guy who knows a guy who knows a guy who knows a guy who knows Kevin Bacon.'

Bacon numbers[edit]

The Bacon number of an actor is the number of degrees of separation he or she has from Bacon, as defined by the game. This is an application of the Erdős number concept to the Hollywood movie industry. The higher the Bacon number, the greater the separation from Kevin Bacon the actor is.

The computation of a Bacon number for actor X is a 'shortest path' algorithm, applied to the co-stardom network:

• Kevin Bacon himself has a Bacon number of 0.
• Those actors who have worked directly with Kevin Bacon have a Bacon number of 1.
• If the lowest Bacon number of any actor with whom X has appeared in any movie is N, X's Bacon number is N+1.

Examples[edit]

Elvis Presley:

• Elvis Presley was in Change of Habit (1969) with Edward Asner
• Edward Asner was in JFK (1991) with Kevin Bacon

Therefore, Asner has a Bacon number of 1, and Presley (who never appeared in a film with Bacon) has a Bacon number of 2.

Ian McKellen:

• Ian McKellen was in X-Men: Days of Future Past (2014) with Michael Fassbender and James McAvoy
• McAvoy and Fassbender were in X-Men: First Class (2011) with Kevin Bacon

Therefore, McAvoy and Fassbender have Bacon numbers of 1, and McKellen has a Bacon number of 2.

Because some people have both a finite Bacon and a finite Erdős number because of acting and publications, there are a rare few who have a finite Erdős–Bacon number, which is defined as the sum of a person's independent Erdős and Bacon numbers.

Center of the Hollywood Universe[edit]

While at the University of Virginia, Brett Tjaden created the Oracle of Bacon.^[14] A previous version of this computer program used information on some 800,000 people from the Internet Movie Database (IMDb), while the current implementation uses data drawn from Wikipedia. The algorithm calculates 'how good a center'^[15] an individual IMDb personality is, i.e. a weighted average of the degree of separation of all the people that link to that particular person. The site returns an average personality number, e.g. for Clint Eastwood, it returns an average 'Clint Eastwood Number.' From there the Oracle site posits 'The Center of the Hollywood Universe' as being the person with the lowest average personality number. Kevin Bacon, as it turns out, is not the 'Center of the Hollywood Universe' (i.e. the most linkable actor). In fact, Bacon does not even make the top 100 list of average personality numbers.^[16] While he is not the most linkable actor, this still signifies being a better center than more than 99% of the people who have ever appeared in a film. Since each actor's average personality number can change with each new film made, the center can and does shift. 'Centers' have included Rod Steiger, Donald Sutherland, Eric Roberts, Dennis Hopper, Christopher Lee and Harvey Keitel.

A 2020 study from Cardiff University also used further measures to determine the center of the movie universe.^[17] These included degree centrality, closeness centrality and betweenness centrality, which were implemented using the NetworkX Python Library. The study also looked at movies released in different decades and found that, between 2000 and 2020, the most central actors included Angelina Jolie, Brahmanandam, Samuel L. Jackson, and Ben Kingsley.

Photography book[edit]

Inspired by the game, the British photographer Andy Gotts tried to reach Kevin Bacon through photographic links instead of film links.

Gotts wrote to 300 actors asking to take their pictures, and received permission only from Joss Ackland. Ackland then suggested that Gotts photograph Greta Scacchi, with whom he had appeared in the film White Mischief. Gotts proceeded from there, asking each actor to refer him to one or more friends or colleagues. Eventually, Christian Slater referred him to Bacon. Gotts' photograph of Bacon completed the project, eight years after it began. Gotts published the photos in a book, Degrees (ISBN0-9546843-6-2), with text by Alan Bates, Pierce Brosnan, and Bacon.^[18]

See also[edit]

• Morphy Number, connections via chess games to Paul Morphy
• Shusaku number, equivalent in the Go world with Honinbo Shusaku

References[edit]

1. ^ ^abTeotonio, Isabel (September 13, 2012). 'Google adds Six Degrees of Kevin Bacon to search engine'. Toronto Star. Retrieved January 31, 2018.
2. ^Ruthven, Alexander (April 7, 1994). 'Kevin Bacon is the Center of the Universe'. rec.arts.movies. Google groups. Retrieved July 19, 2009.
3. ^Interview with inventors in the college's magazine, The Albright Reporter, Spring 1999
4. ^ ^abFass, Craig; Turtle, Brian; Ginelli, Mike (1996). Six Degrees of Kevin Bacon. New York City: Plume. ISBN978-0-452-27844-8.
5. ^'VISA CHECK CARD The Six Degrees Of Kevin Bacon'. YouTube. Retrieved September 4, 2017.
6. ^Dave Barry, Boogers Are My Beat (New York: Three Rivers Press, 2003), 101.
7. ^'We Married Margo'. J.D. Shapiro's Official Website.
8. ^Will and Grace, Season 5 episode 2 'Bacon and Eggs'
9. ^'The Onion, Volume 38, Issue 40'. theonion.com. Retrieved May 26, 2013.
10. ^'The Human Family Tree'. National Geographic Channel.
11. ^'EE debuts multi-million pound marketing launch campaign starring Kevin Bacon'. The Lovemarks Company, Saatchi & Saatchi London. Retrieved November 5, 2012.
12. ^Stromberg, Joseph (February 18, 2013). 'Any Two Pages on the Web Are Connected By 19 Clicks or Less'. Smithsonian. Retrieved March 22, 2013.
13. ^Albert-László Barabási (February 18, 2013). 'Discussion: Network science'. Philosophical Transactions of the Royal Society. 371 (1987): 20120375. doi:10.1098/rsta.2012.0375. PMID23419844.
14. ^oracleofbacon.org
15. ^Reynolds, Patrick. 'How good a center an actor is?'. The Oracle of Bacon.
16. ^Reynolds, Patrick. '1000 best centers'. The Oracle of Bacon.
17. ^Lewis, R. 'Who is the Centre of the Movie Universe? Using Python and NetworkX to Analyse the Social Network of Movie Stars', arXiv:2002.11103, Feb 2020
18. ^'Andy Gotts' Degrees Exhibition'. Clooney Studio. Archived from the original on October 13, 2007.

External links[edit]

• The Oracle of Bacon computes the Bacon number of any actor or actress from Wikipedia data. A previous implementation used IMDB data.
• Six Degrees of James A. Conrad A how-to demonstration for those wishing to compile their own 'degrees of' list by a Hollywood author who is three degrees of Bacon.
• Cinema FreeNet Movie Connector finds links between stars, but can also use directors and producers.
• Filmlovr.com browse the extensive film library to find films to connect via their actors aka determine the Bacon number.
• Six Degrees of Lois Weisberg suggests that Bacon connects to many actors because he acts in many different kinds of roles and films.