Groups, Graphs, and Erdös Numbers

Mathematical research is generally thought to be a solitary pursuit. And some mathematicians do indeed spend their professional lives in lone contemplation of a single problem.

The dramatic announcement in 1993 by Andrew Wiles of Princeton University that he had proved Fermat’s last theorem appeared to belong to this category of discovery. Wiles had isolated himself from the rest of the mathematical community for nearly 8 years to work on the problem. Only a select few were aware of what he was trying to accomplish.

Nonetheless, Wiles relied heavily on the work of mathematicians who had previously tackled the same problem. He occasionally tested his ideas on a handful of trusted experts in areas of mathematics relevant to his approach. When reviewers later discovered a flaw in his original chain of logic, Wiles obtained help from one of his former graduate students, Richard Taylor, to fill in the gap and complete the proof.

In general, mathematical research is a remarkably social process. Colleagues meet constantly to compare notes, discuss problems, look for hints, and work on proofs together. The abundance of conferences, symposia, workshops, colloquia, seminars, and other gatherings devoted to mathematical topics attests to a strong desire for interaction. Electronic communication speeds and facilitates such interaction worldwide.

Perhaps more than any other mathematician in modern times, Paul Erdös (1913–1996) epitomized the strength and breadth of mathematical collaboration. Because he had no permanent home and no particular job, Erdös simply traveled from one mathematical center to another, sometimes seeking new collaborators and sometimes continuing a work in progress. His well-being was the collective responsibility of mathematicians throughout the world.

At the time of his death of a heart attack in 1996, Erdös had more than 1,500 published papers to his credit. His interests were mainly in number theory and combinatorics, though they ranged into topology and other areas of mathematics. He was fascinated by relationships among numbers, and numbers served as raw materials for many of his conjectures, questions, and proofs.

What’s astonishing, however, is the extent to which Erdös worked with other mathematicians to produce joint papers. Collaboration on such a scale had never been seen before in mathematics, and it has now entered the folklore of the mathematical community.

Of course, there’s a characteristically mathematical way to describe this webbiness—a quantity called the Erdös number.

Mathematicians assign Erdös the number 0. Anyone who has coauthored a paper with him has the cherished Erdös number 1. By the end of 2003, there were 509 such coauthors. Another 6,984 mathematicians had the Erdös number 2, because they wrote a paper not with Erdös himself but with someone who wrote a paper with Erdös.

The Erdös number 3 goes to anyone who has collaborated with someone who has collaborated with someone who coauthored a paper with Erdös.

Thus, any person not yet assigned an Erdös number who has written a joint mathematical paper with a person having an Erdös number n earns the Erdös number n + 1. Anyone left out of this assignment process has the Erdös number infinity. People belonging to the first three categories (n = 1 to 3) already encompass a significant portion of all mathematicians in academia today.

Keeping track of these mathematical links has become a kind of game, punctuated by published, tongue-in-cheek explorations of the properties of Erdös numbers. Albert Einstein, for instance, has an Erdös number of 2. Andrew Wiles has an Erdös number no greater than 3.

One can look at mathematical collaborations as a graph—an array of points connected by lines. Each point represents a mathematician, and lines join mathematicians who have collaborated with each other on at least one published paper.

The resulting tangle is one of the largest, most elaborate graphs available to mathematicians. Some people have conjectured that this monstrous graph snares nearly every present-day mathematician and has threads into the physical, life, and social sciences (and even into movies and baseball). Its breadth is astonishing and vividly demonstrates the vital role of collaboration in math and science.

Some time ago, mathematician Jerrold W. Grossman of Oakland University in Rochester, Mich., took on the task of maintaining a comprehensive, up-to-date listing of mathematicians who have earned an Erdös number of 1 or 2. Working with Patrick D.F. Ion of Mathematical Reviews and Rodrigo De Castro of the Universidad Nacional de Colombia, Bogota, he has continued to update, correct, and expand his files on Erdös numbers (see http://www.oakland.edu/enp/).

Grossman’s collaboration lists provide fascinating glimpses of mathematicians and how they choose to do mathematical research.

For example, the average number of authors per research article in the mathematical sciences increased dramatically during Erdös’s lifetime. About 60 years ago, more than 90 percent of all papers published were solo works, according to Grossman. Nowadays, barely half of the papers are individual efforts.

In the same period, the fraction of two-author papers has risen from less then one-tenth to roughly one-third. In 1940, very few papers had three authors. Now, about 10 percent of all papers have three or more authors. Erdös himself may have played a role in this relatively recent explosion of collaboration.

Grossman’s lists and statistics are fun to explore. A mathematician can try to work out his or her own Erdös number, just to see where he or she fits into the increasingly networked world of mathematics. Nonmathematicians can see the tangle of human relationships that play a key role in advancing mathematics.

The same way of tracking collaboration has also been applied to movie actors, basketball players, and other groups.

For example, the Oracle of Bacon at Virginia (http://www.cs.virginia.edu/oracle/) puts movie actor Kevin Bacon at the center of a collaboration graph that links actors who appeared with one another in the same movie. Going to the online “oracle” allows you to submit the name of an actor to find out his or her Bacon number. Elvis Presley has a Bacon number of 2, for example.

The Oracle of Baseball (http://www.baseball-reference.com/oracle/) lets you find a link between any two major league baseball players via the shortest possible list of teammates. The NBA Basketball Player Link Finder at http://www.math.uaa.alaska.edu/~afkjm/nbawelcome.html does the same thing for professional basketball. The Black Sabbath Game at http://www.imjustabill.com/blacksabbathgame links rock musicians.

Interestingly, there are mathematicians who have links to the movie actor, baseball, or rock musician groups.

It’s a highly connected world out there!

Originally posted: 6/15/96

Updated: 6/12/04