The Erdös Number Project

Just as the idea of the small-world phenomenon famously inspired “Six Degrees of Kevin Bacon” in the movie industry, so too do mathematicians have their own version of the game. This one centers around the late Hungarian mathematician Paul Erdös, who had a reputation for, among other things, publishing an incredibly vast body of academic work. After his death, his friends invented the Erdös number, a tongue-in-cheek measure of how closely all other mathematicians were “related” to Erdös. In fact, the Erdös number is much older than the Kevin Bacon game — the former was created in the late 1960s, the latter in the early 1990s.

In the Erdös-number sense, if two people have co-authored an academic paper, they are said to be connected. The Erdös number itself is a measure of the shortest path of connections from any person to Erdös. Of course, Erdös himself has a number of 0. All those who have collaborated with him have an Erdös number of 1, all those who have collaborated with his collaborators have a number of 2, and so on. The description given by the website strengthens this definition with some formalities:

In graph-theoretic terms, the mathematics research collaboration graph C has all mathematicians as its vertices; the vertex p is Paul Erdös. There is an edge between vertices u and v if u and v have published at least one mathematics article together. (There is no reason to restrict this to the field of mathematics, of course.) We will usually adopt the most liberal interpretation here, and allow any number of other coauthors to be involved; for example, a six-author paper is responsible for 15 edges in this graph, one for each pair of authors. Other approaches would include using only two-author papers (we do consider this as well), or dealing with hypergraphs or multigraphs or multihypergraphs. The Erdös number of v, then, is the distance (length, in edges, of the shortest path) in C from v to p. The set of all mathematicians with a finite Erdös number is called the Erdös component of C. It has been conjectured that the Erdös component contains almost all present-day publishing mathematicians (and has a not very large diameter), but perhaps not some famous names from the past, such as Gauss. (We have some information about the conjecture on this site.) Clearly, any two people with a finite Erdös number can be connected by a string of coauthorships, of length at most the sum of their Erdös numbers.

It’s intriguing to see how some famous people stack up using this metric. The page “Famous paths to Paul Erdös” lists some distinguished professionals — Nobel Prize winners, Fields Medalists, etc. — and their respective numbers. As expected, mathematicians generally have the lowest numbers. Andrew Wiles, who solved Fermat’s Last Theorem, has a number of 3. Einstein’s is 2. Bill Gates has a number of 4. And the only mathematician said to be as prolific as Erdös is Euler, whose number is…well, undefined. Of course, like the Kevin Bacon game, the Erdös number was never meant to be taken seriously, and interestingly, because Euler lived so long before the 20th century, no definitive link has been found between his works and those of Erdös.

As a side note, an Erdös-Bacon number also exists, which is defined as the sum of a person’s Erdös and Bacon numbers. As given by Wikipedia, Paul Erdös has a Bacon number of 3, although it would seem that Kevin Bacon doesn’t have an Erdös number.