Graph theory is one of the most currently active branches of mathematics and interest in it is due mostly to the ever-increasing power of computers and the widespread applicability of graphs for modeling important realistic problems.

In an increasingly connected, wired, networked world (not to mention small) the nature of physical, social, and conceptual relationships is a hot topic in a wide variety of academic disciplines spanning mathematics, physical sciences, social sciences, and computer science.

Small-World Phenomenon

Remember the Kevin Bacon Game where an actor is selected at random and we see how few links are needed to connect that actor to Kevin Bacon? This is a perfect example of graph theory modeling the "small-world" phenomenon.

And this doesn't just apply to celebrities. Let's say you're sitting next to a stranger on an airplane, at a large party, or at a ball game and after talking a bit you realize that one of your relatives knows the sister of the stranger's best friend — in fact they work for the same company. Such a connection leads you to remark, "Guess it's a small world, huh?"

Read more about graph theory and an introduction to the small world phenomenon in this previously published article in our Consortium newsletter.

Exploring Six Degrees of Separation

Although the Kevin Bacon Game appeared on the scene in 1994, the six degrees of separation idea dates back to an experiment conducted in 1967.  Then in 1998, Professor Steven Strogatz and then-student Duncan Watts, Ph.D. published a mathematical model that launched the field of network science. The model they proposed not only provided a way to see and understand a wide range of networks, but it would mobilize work in scientific communities across many disciplines.

A key finding in the paper was that a large network could be made small very quickly with just a few random connections, connections joining already tightly clustered nodes.  Read more from Cornell here.

More Resources on Small-World Phenomenon

In addition to the links included above, here are some additional resources for exploring this phenomenon:

• Sync: The Emerging Science of Spontaneous Order by Steven Strogatz
• Small-World Networks of College Classes and the Spread of COVID
• Small-World Models, Artificial Neural Networks, and Diabetes
• Computer Science Text Case Study: Small World Example from Princeton