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Treebeard's Stumper Answer
21 September 2001

Six Degrees of Connection

I've been thinking about how close we all are in the world today. It's not that the world is any smaller of course, but our personal web of connections is larger and easier than ever before. It's been said that any two people in the world are linked by just six or fewer people. I know someone (that's 1) who knows someone (now 2) who... was there, and I've got the email forwards to prove it. Is it possible? What assumptions would justify this claim? Can you design a practical experiment so we can test this claim with mail or email around the world? We could do it!

I'm sure telephones and email have reduced the number of personal links that connect us all around the world. As a result we're exposed more personally to disaster and terror around the world, "contributing to our ... numb response," as mathematician John Allen Paulos adds. I feel that.

For a fun example of our connectedness, play with the Oracle of Bacon which finds links between film actors and Kevin Bacon! A more esoteric example is the Erdös Number Project linking mathematicians to publications co-authored by the prolific Paul Erdös.

The Exploding Dictionary is about words instead of people. You get a definition with every word cross-linked to its own definition. It's a fun game to start with two wildly different words and find the shortest route from one to the other just by clicking through definitions.

Our web of connections begins close and
personal, and spreads across the world.

Can there really be just "six degrees of connection" between any two people in our shrinking world? If I know 100 people well, and they each know 100 more, then I'm already connected to 100x100 = 1002 or 10,000 people. With just three more steps, I'm connected to 1005 = 10 billion people, far more than the world's 6+ billion. Of course we have many friends in common, but this shows it's not a crazy idea. Our web of connections explains how gossip and jokes and viruses can spread so fast, it makes coincidence the norm, and it brings tragedy home.


The math behind our connectedness across the world is simple and compelling. Suppose I have 100 unique friends, and so do they. That's probably not true, but it's an easy and reasonable place to start. Each step brings two more zeros:

1001001100 immediate friends
100 x 100100210,000 with one connection
100 x 100 x 10010031,000,000 with two connections
100 x 100 x 100 x 1001004100,000,000 with three connections
100 x 100 x 100 x 100 x 100100510,000,000,000 with four connections
100 x 100 x 100 x 100 x 100 x 10010061,000,000,000,000 with five connections

Whether that's five or six degrees of separation, it's more than enough to cover the world. With these exponential functions, it hardly matters where we start.

Graybear made it personal, and did the math with different assumptions:

When I read in the new stumper, "I know someone (that's 1) who knows someone (now 2) who... was there, and I've got the email forwards to prove it.", I thought, "Wow, that's pretty interesting that someone like you, in California, would be connected that quickly.", forgetting that I could be that in-between person. I have personally spoken with three members of a family that witnessed the second plane and watched the towers come down. I also know personally several people who were in the Pentagon at the time.

I have heard that it only takes three connections (two in-between people) to connect with almost anyone on the earth. I find that hard to believe, but if you know at least 500 people (say 100 from work/school, 100 from the extended neighborhood, 100 from church/social group, 100 relatives, and 100 miscellaneous), and each of them know 500 people, so on, then with one connection you have a web of 500. With two connections, assuming minimal overlap, your web has grown to 250,000 people. With three connections, you now have 125,000,000. Four connections, (three in-betweens) puts you in touch with 62,500,000,000 which is ten times the world population.

If you believe that the connection can be made through six or fewer people, that allows seven connections. If you can name only 25 people, who can each name 25 additional people, who can each name 25 additional people, and so on, seven connections will create a web of 6,103,515,626 people, approximately the population of the earth.

Graybear and I are working on an experiment to measure the connectedness of our respective schools. I'll report when I can. Further experiments may be called for.

I first thought about this stumper reading a piece in the Washington Post by William Raspberry (Sep. 15), written soon after the attacks on New York and Washington:

It will get worse when we discover by what few degrees we are separated from the direct victims of the catastrophe -- when it turns out that virtually all of us know someone who knows someone who lost someone.

My first thought was that it can't be true. I don't know anyone in New York or Washington, but I soon received email reports from friends and friends-of-friends. I really don't know anyone in Tibet, so how can there be any connection there? But wait. My son Ewan did a trek in Tibet with the group Where There Be Dragons a few years ago. If Ewan remembers just one friend from Tibet, or even a trip leader who remembers a guide, then I'm just a step or two away from another part of the world and millions more connections. One of those links [to links [...]] no doubt goes to Afganistan. It's really not very far away.

This shows the importance of the "bridge people" who connect different social groups around the world. Typhoid Mary and Patient Zero, the airline steward said to have spread AIDS around the world, come to mind. Representatives of more than 60 nations died in the World Trade Center on September 11. Just think how many links that creates around the world.

It's worth remembering. If a person has 10 sexual partners, and so do they, then that person is just six degrees of connection away from 1,000,000 sexual partners and too close to dire consequences:

Thatís a lot of people to be involved with, and a lot of sexually transmitted diseases (STDs) to be concerned about. Of the million people in this scenario, you might estimate that: Suddenly Six Degrees doesnít seem like such an innocent game. The ex-lover of your last date may have fooled around with a guy who tricked with a person with all sorts of unpleasant STDs. There are literally thousands of ways those STDs can be traced back through the chain to you.

Stories of tragedy and heroism, dumb jokes, urban legends, gossip, computer viruses, and human viruses quickly spread in this exponential way around the world. Coincidence is normal.

Here are some links for your own research:

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