Degrees of separation: Erdös, Einstein, Bacon and more

A basic Buddhist insight is that everything is connected, nothing exists in isolation.  The technical term is “emptiness” and the Heart Sutra expresses it as “Form is emptiness, emptiness is form.” One pop culture expression of this insight is the Six Degrees of Separation idea. 

This is the idea that all people are six or fewer social connections away from each other. As early as 1967, Stanley Milgram tested this idea in his small world experiment, where the goal was to send a letter from a random person in Kansas to a random person in Boston via a chain of friends. The letters on average reached their destinations after five and half people. Incidentally Milgram never used the term “six degrees of separation”. This was popularized in a 1990 play Six Degrees of Separation and a 1993 film of the same name starring Stockard Channing, Will Smith, Donald Sutherland and Ian McKellan among others.

The concept has been popularized in a number of offline and online games. This Wikipedia article gives a nice overview of these, as well as more serious attempts to estimate average degrees of separation in various contexts. Wikipedia itself has become the focus of another popular six degrees of freedom game, which my sons have told me about playing, in which the player attempts to find the shortest link path between any two given Wikipedia articles.

I’ll focus below on the Erdös number (mathematicians), the Bacon number (actors), various extensions of the Bacon number, plus a couple I’ve made up myself because I score well on them (what better reason could there be). Disclaimer: none of these numbers say anything about a person’s career success, or social influence, they just shine a light on the wild and wacky ways that we can be connected and that connections are often closer than we would think.

The Erdös Number

Just for fun, mathematicians like to quote their Erdös number, the degrees of collaboration distance from Paul Erdös, one of the most prolific modern writers of mathematical papers. Someone who has co-authored a paper with Erdös has an Erdös number of 1.  Another author’s Erdös number  is one greater than the lowest Erdös number of any of their collaborators.  Some years ago, when bored with work, I decided to try to work out my Erdös number. I found paths through two different co-authors that gave me an Erdös number of 6.

A couple of years I published with a co-author who reduced my number 5. Chatting to colleagues, I boasted that my Erdös number had dropped to 5, the medium number among professional mathematicians, and one of them asked what an Erdös number was. I explained, and he obviously went back to his office to see if he could work out his, because he returned a few minutes later to say that his was 3, and I was 4 (one less than the median for mathematicians. 

Here is my Erdös pathway:

  1. Erdös, P.; Babu, G. Jogesh; Ramachandra, K. An asymptotic formula in additive number theory. Acta Arith. 28 (1976) no. 4, 405-412.
  2. Mukherjee, S., Feigelson, E.D., Babu, G.J., Murtagh, F., Fraley, C. and Raftery, A.E. Three types of gamma ray bursts. Astrophysical Journal 1998; 508, 314-327.
  3. Le Bao, Josh A Salomon, Tim Brown, Adrian E Raftery, Daniel R Hogan. Modelling national HIV/AIDS epidemics: revised approach in the U Estimation and Projection Package 2011. Sexually transmitted infections 2012, 88 (Suppl 2), i3-i10.
  4. Colin D Mathers, Ritu Sadana, Josh Salomon, Christopher JL Murray, Lopez AD. Healthy life expectancy in 191 countries, 1999. The Lancet 2001, Vol 357: 1685-1691.

The Erdös number has been criticized as having a temporal bias. The further a person is in time from Erdös (who died in ) the higher his Erdös number will be on average. However, this irrelevant, the Erdös number is nothing other than a measure of the degrees of separation from Erdös and of course it must increase with time. Measures of career impact are available that are based on citations, not degrees of separation. The sciences have widely adopted the H-index which is a measure of the number of papers published and widely cited.  The H-index has its limitations also as it varies by discipline and length of career.

The Einstein Number

Since my original research field was physics, I was curious to see whether physicists have an analogous index for collaboration distance from a famous physicist, such as Einstein. I have only found two web pages (here and here) that mention degrees of separation from Einstein, one in terms of personal encounters and the other in terms of papers cited by a paper (not co-authorship as for Erdos). Einstein has an Erdos number 2, since he wrote papers with Ernst Straus, and Straus co-authored 20 papers with Erdos. Here is one example:

  1. Albert Einstein and Ernst G. Straus. The Influence of the Expansion of Space on the Gravitation Fields Surrounding the Individual Stars. Annals of Mathematics 1946; 47(4): pp 731-741.
  2. Paul Erdős and Ernst G. Straus. On products of consecutive integers, Number theory and algebra, pp. 63–70, Academic Press, New York, 1977.

So I automatically have an Einstein number of 6 via Erdos. I also found a separate path, avoiding Erdos, which also gives me an Einstein number of 6.

And a path consisting only of physics papers, giving me an Einstein number of 7, as follows:

  1. Albert Einstein and Wolfgang Pauli, Non-existence of regular stationary solutions of relativistic field equations, Annals of Mathematics, ser. 2, 44 (1943) 131-137.
  2. Wolfgang Pauli, L Rosenfeld and Victor F. Weisskopff . Niels Bohr and the development of physics. New York: Pergamon Press (1955) – 195 pp.
  3. John M Blatt, Victor F Weisskopf. . Theoretical nuclear physics. New York: Chapman and Hall (1952) – 896pp.
  4. JM Blatt, Stuart T Butler. Superfluidity of an ideal Bose-Einstein gas. Physics Review (1955) 100: 476-480.
  5. Stuart T Butler, Robert M May. Production of Highly Excited Neutral Atoms for Injection into Plasma Devices. Physics Review (1965) 137: A10–A16
  6. Robert M May, Neil F Cramer. Energy loss of fast test ions in a plasma in a weak magnetic field. Physics Letters A (1969), 30: 10-11.
  7. Colin D Mathers, Neil F Cramer.  The Effect of Ionization and Recombination on the Resistivity of a Partially Ionized Plasma in a Magnetic Field.  Australian Journal of Physics  (1978) 31: 171-9.

Kevin Bacon and the Bacon Number

In a 1994 interview, Kevin Bacon mentioned that he’d either been in a movie with everyone in Hollywood or someone who had worked with them. The comment morphed into a popular game for movie buffs connecting actors to Bacon, via a chain of the movies they have made together.

A valid Bacon number is assessed through co-starring roles in mutual films, television programs, and documentaries verifiable by the Internet Movie Database. Kevin Bacon has a Bacon number of 0, anyone who has appeared in a movie or television program with the actor has a Bacon number of 1, individuals who have appeared with one of Bacon’s co-stars (but not directly with Bacon) have a Bacon number of 2, and so on. Sometimes, this is relaxed to appearances in a movie, rather than co-starring role.

Two random examples of politicians with low Bacon numbers: Donald Trump has a Bacon number of two from his cameo in Home Alone 2, and Vladimir Putin has a three.

According to the Oracle of Bacon, based on actors appearing in movies and TV shows (excluding news, reality and talk shows), the distribution of Bacon numbers is as follows:

An astonishing 2,759 actors have appeared in movies with Kevin Bacon. The average Bacon number 3.03 and only 888 of the 1,375,157 actors had a Bacon number of 7 or greater, that’s less than 0.1%.

Erdös-Bacon Number

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. The Bacon number used here is often the broader form counting onscreen filmmaking collaborations rather than only actors in starring roles.

Natalie Portman has a Erdős-Bacon number of six, second-lowest among professional actors, as does Danica McKellar. The actor with the lowest number is apparently Albert M. Chan, who appeared with Bacon in Patriots Day. His number is four. Colin Firth’s number is seven. Coming from the other direction, Carl Sagan’s number was four. Stephen Hawking’s is seven.

The lowest number that anyone is known to have is three, held by mathematicians Daniel Kleitman and Buce Reznick. Daniel Kleitman, for example, was a math advisor for Good Will Hunting (which is two steps from Kevin Bacon via Minnie Driver’s appearance in Sleepers) and appeared in the film as an extra.

Elon Musk, who is neither a scientist nor an actor, has an Erdős–Bacon number of 6. In 2010 Musk had a cameo in the film Iron Man 2. Since actor Mickey Rourke played a role in both Iron Man 2 and in Diner where Kevin Bacon aso played a role, Musk has a Bacon number of 2. In 2021 Musk coauthored a peer-reviewed scientific paper on COVID-19 together with Pardis Sabeti, among others. Since Sabeti has an Erdős number of 3, Musk has an Erdős number of 4 (same as me) and consequently an Erdős–Bacon number of 6.

I don’t have a Bacon number, and the only way I could get one is if my friend James, a professional documentary producer, could be persuaded to put me in one of his documentaries. He has a Bacon number of 3, and he just needs to offer me a part in his next film.

Erdos-Bacon-Sabbath number

Legendary heavy metal band Black Sabbath is famous for having more members (35 touring and session players) than albums (19). So of course there is a Black Sabbath number for people who have connected with Black Sabbath through musical performances.

To have an Erdos-Bacon-Sabbath number, you must have: co-written a scientific paper with someone who eventually connects to Erdos; appeared in a film with someone who eventually connects to Kevin Bacon; and performed musically with someone who eventually connects to Black Sabbath. A perfect EBS number would be three. No-one has that number. The lowest known number is 8, held by Stephen Hawking, futurist Ray Kurzweil; and neuroscientist Daniel Levitin.

Natalie Portman (actress) and Brian May ( of Queen fame) both have EBS numbers of eleven. Lower EBS numbers have been claimed for both (solid 10 for Portman and 8 for Brian May, the latter through a dubious Erdos connection).

Handshake separation from Adolf Hitler

And one I won’t be adding to my CV is my 2 degrees of (handshake) separation from Adolf Hitler. Back in 2003 when I was visiting Germany, I met and shook hands with an elderly woman who had shaken hands with Hitler when she was young.