Liberalizing Erdős-Bacon Numbers

The normal definition of an Erdős-Bacon number (henceforth EBN, or EB number) is that is the sum of an individual's Erdős number (EN) and that same person's Bacon number (BN), where the EN is the number of links of coauthorship between that person and Paul Erdős and the BN is the number of links of costarring between that person and Kevin Bacon.  The purpose of this post is to present another way to compute the EBN.  While, at present, this procedure's results would be consistent with the current method of computation, it is not equivalent, and could lead to different results in some circumstances, which will be described.  In fact, the change is so radical that you might well think that I've actually introduced a new concept entirely—which may be so, but I think it is equally well described by the title "Erdős-Bacon number".

It is well-known that one can make graphs illustrating ENs and BNs.  Each node corresponds to a person, and the graph is centered on either Erdős or Bacon.  Edges between two nodes establish either coauthorship, in the case of an EN graph, or costardom, in the case of a BN graph.  Suppose we have an arrangement of nodes and draw both a BN graph and an EN graph.  Then we will see that there are some people included in both: Natalie Portman, Carl Sagan, Danica McKellar, and more.

Now call a person's Erdős-Bacon path (EBP) the shortest path including Erdős, Bacon, and that person.  For each person who is connected to Bacon by costardom, and to Erdős by coauthorship, the length of that person's path is equal to his or her EBN as traditionally construed.  However, the EBP is defined for any node in either graph.  So the first step in our revision will be: the EBN for a person is simply the length of that person's EBP, should one exist.

The second step is prompted by the question: why should we only allow edges of one kind in one direction, and only edges of another kind in the other direction?  If we start from Bacon, then we must continue, at present, only with costardom-edges, and at some point we must switch to coathorship-edges, and cannot switch back.  (This means, incidentally, that the shortest possible EBP would be a cycle between Erdős and Bacon—we would have to go from one and then back; a single edge would not suffice.)  But why should we submit to this rule?  We aren't simply talking about Erdős numbers and Bacon numbers; we're talking about Erdős-Bacon numbers.  So our second revision: we allow coauthorship edges and costardom edges to intermingle freely.

This means, of course, that one only needs to have a defined EN or a defined BN to have a defined EBN.  For instance, a mathematician M with an EN, who has coauthored a paper with a mathematician with a BN, does not himself have a BN, but does have an EBN.  This is a consequence of our first revision.  As a consequence of our second revision, however, a second mathematician with an EN N could shorten his or her EBN, not by coauthoring a paper with M, but by costarring in a movie with him or her; indeed, someone with neither an EN nor a BN could gain an EBN in that way—by coauthoring a math paper with a film star or by costarring in a movie with a mathematician.

Comments

[standpipe]

This is a good post.

[Christine]

This means, of course, that one only needs to have a defined EN or a defined BN to have a defined EBN.

Taking a union of those sets is retarded.

[Matt Weiner]

This is indeed a good post. But may I suggest a modification, for reasons of more-funness? Redefine x's EBP as the shortest nonrepetitive path from Erdos to x to Bacon (WLoG). Otherwise the existence of a three-edge Erdos-Bacon path (see) will mean that for the vast majority of people the w-EBN will simply be min[EN, BN]+4. For instance, my EN is 3, I have no BN, and my w-EBN is surely 7. (I had to count on my fingers to calculate it.) Even Danica McKellar has a w-EBN of min[EN, BN]+4 (probably). But the way I have defined it, EBNs are more entertaining to calculate.

[ben wolfson]

That won't work, though, for the case of the X who coauthored a paper with Erdős, never starred in a movie, and never coauthored any other papers. There is no path from Erdős to X to Bacon (or the other way 'round), but there is a path containing all three. I think we still want to say that this person still has an EBN.

[ben wolfson]

There is no noncyclical path, rather, and allowing the cycle would change the length.

[Matt Weiner]

What's the cyclical path? I guess what I meant to exclude by talking about nonrepetitive.

I was going to slang off such people as dead ends on the Erdos-Bacon tree, but I realized that I am such a person (if we change "with Erdos" to "with someone with an EN of 2") and the whole point of this was to mention my EN, so I concede.

[ben wolfson]

In the example above, it would be E -> X -> E [cycle!] -> ... -> B. Whereas if you do it my way, you can just do X -> E -> ... -> B.