# Comments on The Tipping Point

by Allen Downey

On August 31, 2006, I participated in a panel discussion of Malcolm Gladwell's book The Tipping Point as part of the first-year book program at Olin College.

Here is an approximation of what I said, based on the notes I wrote ahead of time:

I used Tipping point in Computational Modeling as an appetizer. We read it (and 15 other books) at the beginning of the semester and then looked in more detail at some of the topics that it raised.

There are some interesting ideas lurking between Gladwell's anecdotes, so I want to call one of them out and, maybe, whet your appetite to look a little deeper.

The notion I want to explore is the small-world graph. In Gladwell's book this comes up in the discussion of Milgram's small world experiment, where he asked his subjects to deliver a letter to a particular person, with the restriction that they have to pass the letter along to someone they know personally.

Milgram's goal was to characterize the properties of the social network, which is a graph that represents relationships among people. What I mean by a graph is a set of nodes and edges, where each node represents a person and an edge between two nodes means that those two people know each other personally.

So if I know you, there is an edge connecting us. If I don't know you, there wouldn't be an edge between us, but there might be a path connecting us; that is, a sequence of edges and nodes from me to you through one or more intermediaries.

The characteristic of this graph that Milgram wanted to measure was the average path length between any two people in the world, and it turns out to be much smaller than you would have guessed, something like six. People have questioned whether or not it is really six, and whether the global network has this property, but it is pretty well established that there are many social networks (and other kinds of networks as well) that have this property.

Part of the reason this result was surprising is that we expect social networks to be locally connected -- you are more likely to know people who live near you. As a result, we expect each hop in a path to span a relatively short distance, so we expect it to take many hops to cross the country.

So Milgram's result was surprising. Milgram proposed possible explanations for it, and there are still a number of possibilities, but one of the more interesting explanations didn't come out until 1998 (about 30 years after Milgram's paper), when Watts and Strogatz wrote a paper called "Collective dynamics of 'small-world' networks", which was published in Nature.

They started by studying a regular lattice -- one example is a graph where the nodes are laid out on a plane and each node is connected to its nearest neighbors.

A regular lattice does what we expected social networks to do; as the size of the graph increases, the average path length grows linearly.

But now what happens when someone in a regular lattice moves? They probably make new friends in the new neighborhood, but they keep some of their friends from the old neighborhood. The old friends create "long links" that make it possible to jump across the graph in a single hop.

In the Watts and Strogatz paper, they model long links by adding a small number of random edges to a regular lattice. The first thing they see is that the average path length drops off very fast as you add just a small number of random edges.

Also, once you have this structure, which they call a "small world graph", you can scale up the graph and the path length stays small. More specifically, the path length grows logarithmically as the number of nodes increases.

This result is interesting for a number of reasons, but part of the reason I talked about it in my class is that it is an unusual kind of explanation for the small world phenomenon.

In classical physics, we are used to explanations that come in the form of continuous models. For example, we explain planetary motion in terms of forces and differential equations, and if someone asks why plantary orbits are ellipses, we answer that orbits are governed by such and such an equation, and the solutions of that equation are ellipses. And that's very satisfying. As a people, we are quite proud of that kind of explanation.

The small world graph is a different kind of model. First of all, it is discrete; nodes and edges are countable; path lengths are integers; there are no continuous quantities. It is also a highly abstract model, which is to say not very realistic. Social scientists, broadly speaking, hate this kind of model. A node in a graph is a pretty coarse summary of a human being.

So if someone asks why path lengths in social networks are surprisingly short, and you say it's because in a regular lattice with a few random edges, the path length grows logarithmically, it's less clear that that's going to be a satisfying explanation.

So I think there is something interesting happening here, let's say in the late 20th early 21st century, that people are starting to propose different kinds of explanations for physical and social phenomena, and to some degree people are finding those explanations more acceptable than they did in the past. This is what Wolfram calls a New Kind of Science, which is the title of his great big book.

And that leads to a natural question, if there is a new kind of science, where is the new kind of engineering? I think that's a fascinating question and I've been thinking about it for a while. I won't even try to answer it here, but I will give you one example, which is that peer-to-peer file-sharing systems, which some of you might be familiar with, are often based on nodes that assemble themselves into small-world networks; that is, they start with a regular topology and deliberately sprinkle in a small number of long links in order to keep the average path length short.

Well, that's enough of that for now. I will wrap up by saying that I think Gladwell's book is a great appetizer. There is a lot of interesting material that he hints at without going into much detail, but I think the detail is interesting, too, maybe more so. So I hope your appetite has been whetted, and I hope you will take the opportunity to explore these ideas more during your time at Olin, and after.

Other essays by Allen Downey are available here.