Under the Microscope Last year when attending a conference in Baltimore, Maryland, I fell into conversation with a local man in a supermarket. When he found out that I was Irish, he told me he knew an Irish man who regularly visited his company. It turned out that the Irish visitor is a near neighbour of mine.
We all have occasional experiences like this that make us exclaim: "It's a small world." Well, it really is a small world and the reason is now understood mathematically. A special branch of mathematics, Small World Theory, is devoted to the study of the phenomenon. The theory has important implications for many practical things ranging from globalisation to the spread of infectious disease. Expanded details of Small World Theory can be found in 25 Big Ideas, by Robert Matthews (One World, 2005).
Basically the phenomenon hinges on the existence of a relatively small number of long-range random connections between people on a global basis. Imagine two extreme situations. In the first scenario each person in the world knows 10 other people, each a close neighbour. In the second scenario, each person in the world knows 10 other people, but these 10 are randomly distributed around the world. Let us say the world population is one million people.
Now imagine that 10 people are each told a story and asked to pass it on. In the first scenario, the story will take a long time to spread around the world, having to pass ponderously from one group of 10 to the next group of 10, and so on. In the second scenario, the message will quickly fill the globe, because each retelling of the story will inform 10 times more people than previously knew. Starting with 10, next 100 know the story, next 1,000, then 10,000, next 100,000 and then one million, ie six re-tellings and everyone in the world knows.
Neither of these scenarios corresponds to the real world, which exists in an intermediate situation. In other words, the ties between people are neither exclusively local nor entirely random, they are a mixture of both. But the random long-range connectors are sufficient to ensure that messages will pass through the entire network relatively quickly.
The mathematics underlying Small World Theory was given significant early development in the 1950s by two Hungarian mathematicians Paul Erdös and Alfred Renyi. Its most sophisticated modern treatment was published by Steven Strogatz and Duncan Watts in Nature, Volume 393 (1998).
Stanley Milgram provided a striking demonstration of the small world in action in 1967. Milgram, a professor of social psychology at Harvard University, devised an experiment to measure the size of our social networks. He posted packets to approximately 300 people in Boston and Nebraska and asked each to forward them on to a named person in Massachusetts. He told them the named person's occupation and a few other details, but did not supply the address. So, how were they to proceed not knowing the address? Milgram asked them to post the packets to someone they knew on a first-name basis who might be in a better position to know the address, who, in turn, would follow the same procedure. Amazingly, the packets arrived at their correct final destinations after five re-postings.
The implication of Milgram's experiment is that on average each person knew 50 others well enough to post the packet to them. The population of America is about 200 million. Therefore one further re-posting should be enough to find anyone on earth (200 million x 50 = 10 billion).
Small World Theory has important implications for many areas, including disease, business and communications. The world wide web is an example of the small world. At any given time 8 per cent of the web's crucial "router" computers are down, but some other fast route can usually be found between any two computers because of the random links that make up the web.
Small World Theory has obvious insights to offer those charged with preventing or minimising the spread of infectious disease. The key links here will be random long-range links and resources should be targeted at minimising the number of these links. The power of such links is dramatically illustrated by the statistic that 40 of the first 248 men diagnosed with Aids were linked by sexual activity with the same homosexual Canadian flight attendant.
Mathematics predicts that if 20 per cent of people at the site of an outbreak of infectious disease have random links beyond the site, a minor localised outbreak can change into an epidemic. This analysis explains how the pandemic of Aids developed in Africa. Large population movements, accelerated by wars, followed the departure of colonial powers. This greatly increased the formation of random long-range links and the consequent rapid spread of disease.
We are told that we are long overdue a pandemic of deadly influenza. The last one happened in 1918 and killed between 20 and 40 million people worldwide. Please God, Small World Theory will guide us in minimising the effects of the next one. This theory, by the way, is a wonderful example of how small, basic research can unexpectedly produce results of great practical value. Paul Erdös was a completely unworldly character who lived mainly on the charity of friends. He worked either alone or with a collaborator. We must continue to fund and encourage the research of brilliant individuals who work in small groups.
William Reville is associate professor of biochemistry and public awareness of science officer at UCC - http://understandingscience.ucc.ie