WHAT has Hollywood actor Kevin Bacon got in common with worms and the US power grid? All have been linked by a mathematical explanation of the so-called small-world effect, in which apparently unrelated people turn out to have friends in common. This effect could also help our understanding of a wide range of problems, from the way disease spreads to brain seizures.
It turns out that the effect turns up in any type of network, from the film industry to the nervous system of living organisms. All that鈥檚 needed is for a tiny proportion of the connections to link up with distant parts of the network.
Duncan Watts and Steven Strogatz at Cornell University in Ithaca, New York, made their discovery while they were studying networks whose connections could be altered at random. Until now, most research into networks has focused on two broad types. So-called regular networks have every point connected in the same way to the same number of neighbours: for example, just four connections to the four nearest neighbours. These networks are also highly clustered: nearby points are connected by a high density of links.
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At the other extreme, a random network has points connected haphazardly 鈥� some to their nearest neighbours, but many more to distant parts of the network. As a result, the network has a relatively small 鈥渟tep-length鈥�, a measure of the typical number of steps between any two points. However, such graphs also have little clustering of nearby points. But what about networks that are in between 鈥� neither random nor regular? In computer simulations of these, Watts and Strogatz have shown that introducing just a few distant connections in a regular graph is enough to drastically cut the step-length 鈥� and yet still retain high levels of clustering.
In other words, such 鈥渟emi-random鈥� networks are very likely to show the small-world effect, with apparently unrelated clusters joined to others via few steps. Watts and Strogatz found that having just 1 per cent of the connections linked to distant parts can trigger the effect.
To find out if this was more than just a curiosity, Watts and Strogatz calculated the step-lengths and clustering levels of three real-life networks: film actors, the electric grid of the western US and the nervous system of a nematode worm. In each case, the step-lengths were small 鈥� suggesting that they were random networks. But all three also had the high levels of clustering expected of regular networks, confirming that they exhibit the small-world effect.
According to Strogatz, film buffs have long recognised its presence in the movie world. 鈥淭here鈥檚 even a game people play where you have to connect any given actor to Kevin Bacon,鈥� says Strogatz. 鈥淚t turns out he can be connected to virtually any other in no more than four steps.鈥� It鈥檚 now clear that the Bacon game works because a small number of actors have played opposite some very unusual partners. 鈥淏acon can be linked to Chaplin in just three steps, through films starring Laurence Fishburne and Marlon Brando,鈥� says Strogatz.
As Watts and Strogatz report in Nature (vol 393, p 440), the small-world effect also has some serious implications. A small-world model of infections reveals that a few people can dramatically speed the spread of disease by crossing social or geographical boundaries. 鈥淵ou don鈥檛 need to know any of these people yourself to be connected to a high-risk group,鈥� says Watts.
鈥淭his work is exceptionally significant,鈥� says William Ditto of the Georgia Institute of Technology, Atlanta. 鈥淚t鈥檚 amazing that one principle connects so many phenomena. We鈥檙e already busy looking at its effects in electronic circuits and duration of seizures in the brain.鈥�
