杏吧原创

Global village

Question: I have heard it said that every person on Earth knows everybody
else through a chain of no more than seven people, so I could trace everybody
through a friend of a friend of a friend of a friend of a friend of a friend of
a friend. Is this true?

Answer: I have often dismissed this story as a myth, but this time decided to
investigate. I would estimate that I know or remember meeting approximately 100
people in my life so far (which hasn鈥檛 been very long). Say these 100 people
also know 100 people each and those 10 000 people know 100 people and so on,
after the seventh circle of people my list would have reached 100 脳 100⁷. That
is 1 脳 10¹⁶, which is approximately 2 000 000 times the world population. The
statement is obviously true even though it seems hardly credible at first
glance.

But we also have to take into account the probability that each person you
contact knows the same people you have already contacted. The second person you
contact will have a chance of knowing someone else you have already contacted,
and the last will almost certainly know many of the people you have contacted.
However, even when this is taken into account, you are still very likely to get
through the whole world by just knowing seven friends.

Tania Jacob

Lesmurdie, Western Australia

Answer: The recent report of the discovery of a previously unrecorded tribe,
apparently completely isolated in a South American jungle, and found only by
chance when their dwellings were spotted from the air, implies that even if the
chain of seven contacts is a theoretical possibility it cannot be true in
practice.

Barrie Watson

Shoreham-by-Sea, West Sussex

Answer: The answer is clearly false because some of us have no friends.

Chris Jack

London

Many people sent in a calculation similar to that performed by Tania Jacob,
but fewer noted the risk that as your circle of friends grew you might be
counting the same people over and over again.

That problem is plain if you consider the Amazon village that is mentioned by
Barrie Watson. If there are 50 people living in this totally isolated village
and they all know one another, then any one particular individual would know 49
other people.

Although all of those people would also know 49 people, the number of the
first individual鈥檚 friends鈥� friends would obviously not be 49 脳 49 people,
because you would just be recounting the same 49 people. The key issue is that
although few people live in such isolation (with the possible exception of Chris
Jack), we all tend to have restricted circles of friends.

What happens if you bear in mind this reservation? An answer is provided in a
recent paper published in Nature by Duncan Watts and Steven Strogatz
(鈥淐ollective dynamics of `small world鈥� networks鈥�, vol 393. p 440, 4 June 1998),
outlined in the diagram below. Think of yourself as one of the dots on the
circles in the diagram above and that each dot has links to four other dots.

If you live in a 鈥渃liquey world鈥�, your situation is like that in the circle
on the left where you are linked only to your near neighbours. From the dot that
represents you, you are linked to two friends to the left of you and two to the
right. And each one of those friends has two friends to the left and two to the
right.

You will find that to travel around this small circle to the opposite side
through the friends of friends of friends and so on, takes a surprisingly large
number of steps. Imagine how many steps it would take to be connected to
everyone in the world.

Now try the circle on the right, the 鈥渙pen world鈥�, where the connections are
random and can leap across and around the circle without any limitation. This is
a totally open society where friends can appear at random. Even if you still
keep to the rule that on average everyone knows four people, you can travel all
over the circle with only a very few links.

Most interesting is the middle situation, the 鈥渟mall world鈥�, where people are
still clustered mostly in cliques but a few people have connections to a distant
place (an analogy might be that if you are living in Britain, you have one
friend who has another friend who lives in Australia). Surprisingly, Watts and
Strogatz show that just a few of these long-distance connections drastically
reduces the number of steps that is needed to travel around the ring.

That鈥檚 an important general mathematical conclusion that could help to create
better designs for cellular phone networks, improve our understanding of the
spread of infections and even explain why the brain is wired the way it is.

It also helps to answer the original question asked above鈥� the real
world does seems rather like the 鈥渟mall world鈥� and this suggests that there is a
good chance that seven links will connect you to almost everyone else in the
world.

An experimental test is needed. You can see how this has been partially
achieved鈥攗sing a data base of movie stars and the films they have appeared
in by playing the 鈥淜evin Bacon鈥� game at http://www.cs.virginia.edu/oracle/,
and you can join in tracing your own connections at http://www.sixdegrees.com/.
For further information on this subject see
New 杏吧原创 (This Week, 6 June 1998, p 7)
andDiscover Magazine(December 1998)鈥擡d

This week鈥檚 question

Sundial shenanigans: I left my watch exposed to the sunlight for a long time
and afterwards its LCD (liquid-crystal display) was completely black.

However, when I removed it from the sunlight and it cooled down, after a few
minutes it was back to normal again.

Could anyone tell me what happened to the display and how it recovered?

Guillem de Vera Almenar

Aig眉es, Spain