A few columns back I posed the problem of computing how many people it takes to have even odds that at least two of them have the same birth date. This and a related problem involving the number of coincidences often surprise people who have not been exposed to them. In this case, the required size is only 23 people. That assumes all the usual simplifying things like taking equal probabilities for every day of the year and non-correlation between the 23 individuals.

It turns out that much work has been done on networking theory along the lines of six degrees of Kevin Bacon. This is a particular example of the six degrees of connectivity hypothesis which says that you are connected to every other human being on the Earth by no more than six mutual acquaintances. That is, you know someone who knows someone, who knows someone, etc. for six times and the last someone can be anyone.

The Kevin Bacon subset of that observation (or Erdos number for those of you more into math) started as a game relating actors who have acted in movies with Kevin Bacon or acted with actors who have acted with Kevin Bacon, etc.

One result that comes out of serious investigations into this type of network is that the overall connectivity can be changed greatly by only a few highly connected individuals. That is, the results are really different if everyone knows the same number of other people - for example, assume everyone knows 100 people - and if you have a population in which a small minority of celebrities or politicians know many more than the average. Only a few such vectors can dominate the overall connectivity.

This result is much more important than simply playing games with Kevin Bacon. It can relate to the spread of AIDS, growth of terrorist cells, and data collisions on the Internet. A few hyperactive individuals with many sexual partners accelerate the spread of AIDS. A similar thing can happen with a few extremely popular Web sites.

When I select puzzles and problems for this column, I try to get ones that have an underlying universality. It might not be obvious that the parlor puzzle of coincident birthdays might lead to an epidemiological discussion on the transmission of AIDS, but it is that unexpected connectivity (that word again - maybe we have connectivity about connectivity and we should call it meta-connectivity) between types of problems that I find interesting.

However, in a short column, only a small subset of the truly fascinating paradoxes and puzzles can be explored and those must be intermixed with purely entertaining puzzles. If you have a favorite paradox of puzzle that I have not hit upon, please let me know and I will try to include it in a future article.

For those who wish to delve further into decision theory without wading through a lot of equations, I have posted a tutorial on elementary decision theory. It shows examples of faulty physicians’ diagnoses (important for those considering surgery) and how to evaluate anti-terrorist activities (important for everyone). That tutorial can be found here.