午夜福利1000集合

VILLAGERS in a remote part of central Africa start to fall victim to a
mysterious fever. Within hours they are on the brink of death, bleeding from
every pore, their bodies under attack from a lethal virus. A few weeks later,
the virus strikes again, claiming another victim鈥攖his time in rural
Wiltshire. Suddenly a nightmare from far away has come to life in southern
England.

Fiction? Not at all. This is precisely what happened in the autumn of 1976,
when the deadly Ebola virus found its way from outbreaks in Zaire and Sudan to
England, via a scientist at the Centre for Applied Microbiology and Research at
Porton Down. Fortunately, the scientist survived, but the incident remains one
of the most dramatic examples of a phenomenon attracting increasing interest
among researchers in a host of different fields: the small world effect.

Until recently, it seemed like little more than folklore: the notion that
everyone on the planet is just six handshakes away from everyone else. But now a
small army of scientists is finding the small world effect in situations as
diverse as the spread of disease, corporate globalisation and even the workings
of our brains.

This ubiquity stems from a notion at the heart of the small world: the
mathematical concept of a network. From flow charts to family trees, networks
pop up in many guises, but they all have two basic components: points
representing the basic units鈥攏eurons, people, species or
whatever鈥攁nd lines between them showing which points are related.

The folklore of the six handshakes highlights the astonishing ability of some
huge networks to be spanned in just a few steps. A bit of simple maths suggests
that the six handshakes story actually underplays the true situation. Each of us
has on average several hundred people that we know. Say you are one handshake
from a pool of around 300 people, then you鈥檙e two from about 90 000 and so on.
With a world population of six billion, we鈥檙e typically just four handshakes
away from Bill Clinton.

But there鈥檚 a catch in this back-of-the-envelope calculation. It assumes
that the connections between our friends are utterly random, so that we鈥檙e as
likely to know Congressmen as postmen, Aborigines as eskimos. It鈥檚 a nice,
Utopian assumption, but it鈥檚 false. We all tend to inhabit our own little
cliques鈥攃lusters of people that aren鈥檛 obviously linked to everyone
else.

So how can the folklore be even remotely true? The puzzle deepens when you
redo the back-of-the-envelope calculation, this time assuming that networks of
friends are utterly regular, so everyone knows only those 300 people who live
nearest to them. In that case, we鈥檇 typically be 10 million handshakes from the
President of the United States.

But we鈥檝e all met strangers who turn out to be friends-of-a-friend, and often
enough to show that we鈥檙e more connected to each other than this regular network
calculation suggests.

This observation put Duncan Watts, a former PhD student at Cornell
University, New York, and his supervisor Steve Strogatz at the centre of an
explosion of interest in small world theory. For it made them realise that in
the real world, networks are neither entirely random nor entirely regular, but
somewhere in between.

In 1996 they set about exploring this grey area using a computer model of
networks. What they found was stunning. The computer was programmed to simulate
a network of 1000 people, each with 10 friends. Wired up as a perfectly regular
network, the simulations confirmed that it would typically take several dozen
steps鈥攔oughly 500 divided by 10鈥攖o plod from one community to
another. With a totally random network, it took an average of just three steps
to jump from any one community to another, again just as one would expect.

It was when Watts and Strogatz began to add random links to an otherwise
regular network that they got their big surprise. The average path length didn鈥檛
just fall鈥攊t plummeted. They had discovered a point where the plodding
from cluster to cluster of regular networks turns into the vast leaps that
appear in random networks鈥攁 kind of phase transition, like the switch from
ice to water as you raise the temperature.

And you don鈥檛 need many random links to reach this point. According to the
computer, if just 1 in 100 of all the connections is allowed to connect to some
random point in the network, the average number of steps needed to get around
the network falls by a factor of 10. The tiny number of random links turns the
vast network into one in which whole communities can be reached via very few
steps. In short, it becomes a small world.

But do any small worlds exist in real life, outside the realm of computer
simulations? The big challenge for Watts and Strogatz was finding a network
whose structure was documented well enough to be compared with theory. 鈥淎
typical social network would certainly be large enough,鈥� says Watts. 鈥淏ut the
connections and extent of social networks are notoriously difficult to pin down.
Imagine, for instance, trying to map out the friendship network of New York
颁颈迟测.鈥�

The two physicists found that movie buffs had provided them with the perfect
test bed for their research: the so-called Kevin Bacon game. The aim of the game
is to link Bacon to any other actor via the smallest number of films. For
example, he can be linked to Charlie Chaplin in just three steps: Bacon played
in a film with Laurence Fishburne, who in turn was in a film with Marlon Brando,
who once appeared with Chaplin.

There are more than 220 000 actors who can be connected to Bacon in this way,
about 90 per cent of all the actors who have ever appeared in feature films.
Within this network, most actors tend to stick to certain genres of films, and
yet the Kevin Bacon game can almost always be won in just a few moves. This led
Watts and Strogatz to suspect that the vast film industry is actually a small
world.

They confirmed their suspicions by analysing the links and clusters within
the industry. On average, one actor can be reached from another in 3.65
steps鈥攃lose to the 3 expected if the network were utterly randomly
connected (see 鈥淎 small introduction to small world theory鈥�). Yet the
industry is still highly clustered. Because most actors work within genres or
eras, cliques form in which almost everybody has worked with almost everybody.
So in short, lots of clusters are joined by surprisingly direct links.

What turns the global film industry into a small world, explains Watts, are
the 鈥渓inchpins鈥�: prolific actors who transcend genres and eras, and thus
short-circuit the network. 鈥淎n example is Eddie Albert, who has appeared in over
eighty films spanning a sixty-year career,鈥� says Watts. 鈥淗e links together such
greats as Bogart, Brando, Richard Burton, John Travolta鈥攁nd, of course,
Kevin Bacon.鈥�

Watts and Strogatz went on to identify two other real small worlds: the
electricity grid of the western US, and the nervous system of the microscopic
worm Caenorhabditis elegans. Their paper in Nature last year
(vol 393, p 440) put Watts and his supervisor in the media spotlight. The
headline-catching clarity of their discovery has captured the imagination of
many other researchers. 鈥淲e were contacted by dozens of researchers from
virtually every discipline鈥攅ven English literature鈥攚ho have all been
struck by the relevance of small world networks to something in their field,鈥�
says Watts.

Watts and Strogatz were not the first scientists to approach the small world.
Since the 1950s, researchers in fields from biophysics to sociology had been
looking at networks which often seemed to be mysteriously well connected.

Since the early 1990s, Olaf Sporns and his colleagues at the Neurosciences
Institute in San Diego have been using network theory to probe the secrets of
that most imposing of complex networks, the brain. Their aim has been to
understand how the brain performs so many different functions in different
parts, while integrating them all so efficiently.

Sporns and his colleagues have found hints that the brain may exploit small
world effects to pull off this trick. In mammal brains, there is a very low
average path length from one neuron to another鈥攖hat is, all areas can be
reached from all other areas in only a few steps, even though areas with related
functions are still clustered together.

鈥淪mall world networks lack an organisational centre, yet one gets global
interactions,鈥� says Sporns. 鈥淭his is what we see in the organisation of the
肠辞谤迟别虫.鈥�

Nature may take advantage of another benefit of small world networks: their
amazing compactness: 鈥淲iring is `expensive鈥� because it takes up precious
cortical volume, and it turns out that cortical networks use very little
wiring,鈥� says Sporns. To make the brain a small world network, most connections
can remain short, local ones; only a few need be long. Sporns says he鈥檚 in no
doubt that small world theory will cast light on the working of the brain.

Big business is another small world. Bruce Kogut of the Wharton School of the
University of Pennsylvania, and Gordon Walker at the Edwin L. Cox School of
Business at the Southern Methodist University, Texas, have been studying who
owns what among Germany鈥檚 biggest corporations. Here the firms are the network
points, and people who own a piece of two firms form the links. Despite the vast
range of businesses covered, some of which only have a few owners, any firm can
typically be linked to any other via roughly four intermediaries. Kogut and
Walker have thus shown that the corporate world is another small world鈥攁nd
one of the handful, so far, to have its properties fully quantified.

That may help to explain why firms with apparently tenuous links to one
another still show similar corporate behaviour, say Kogut and Walker. And it may
also dispel fears about 鈥済lobalisation鈥濃�攖he spread of worldwide corporate
links, and the spectre of crises in faraway economies wreaking havoc back home.
There may already be enough such links to have turned the whole corporate world
into a small world. And if it has already happened, without disastrous
consequences, there doesn鈥檛 seem to be much to worry about.

While this might make some chief executives sleep more easily at night, the
small world has a darker side. At Cambridge University鈥檚 department of zoology,
Matt Keeling is investigating its role in the spread of disease. Just a few
infectious people travelling from place to place, or from social group to social
group, can be enough to trigger a full-blown epidemic. It鈥檚 the classic 鈥渟hort
circuiting鈥� behaviour seen in small worlds. And with the rise of air travel, the
number of links is getting greater and greater, increasing fears about global
epidemics. In the film Outbreak, an imaginary Ebola-like virus spread
across the US in a matter of days. To invade the rest of the world would,
presumably, have taken little longer.

Keeling is using small world theory to investigate how to fight such
epidemics. Already the theory has pointed to a possible reason why the measles
vaccination programme in England and Wales 30 years ago was less successful than
expected in preventing outbreaks. Around 60 per cent of vulnerable people were
covered鈥攁 sizeable proportion, but according to small world theory, not
enough to stop the 鈥渟hort circuiting鈥� effect. 鈥淚t鈥檚 not until the level of
vaccination is much higher鈥攁round 90 per cent鈥攖hat the measles
epidemic rapidly declines,鈥� says Keeling, who is now working on tailoring
vaccination programmes to the small world features of various diseases.

Small world networks can also wreak havoc on everyday tasks such as arranging
school timetables and scheduling air flights. Computer scientist Toby Walsh at
the University of York has discovered that such problems can be thought of as
networks. In a school, the lessons are the points, and lines between a pair of
points mark clashes, such as students having to be in two places at the same
time, or teachers having to teach two classes simultaneously.

The standard way of finding clash-free timetables is to use computer programs
designed to sniff out the best possible timings as fast as possible. The trouble
is, says Walsh, timetables often prove to be small worlds, with lots of
clustering but also nasty long-range effects. For example, a seemingly harmless
choice for introductory chemistry has knock-on effects for advanced biology.
鈥淭he small world effect means there are no local decisions,鈥� says Walsh. 鈥淵ou
either solve the whole problem all at once, or forget it.鈥�

Small world theory should help computer scientists design better programs,
says Walsh. 鈥淭he best already try to take advantage of the small world topology
by looking for clusters鈥攚hich is intriguing, as they were developed before
small world topologies were recognised.鈥�

Even the sprawling Internet turns out to be a small world. In a
Nature paper in September (vol 401, p 130), a group from the University of
Notre Dame, Indiana, worked out that you can get from one document to another on
the Web in only 19 clicks, on average. That is, of course, if you know where to
click.

Despite seeing his Nature paper cited in so many different fields,
Watts himself remains modest about his own contribution, which he insists
is just a small part of something much bigger. 鈥淪mall world networks are neat,
but there are still some key issues to sort out,鈥� he says. 鈥淔or example, what
sorts of assumptions can we make in our network models and still get the right
answers? Theoretically, we鈥檙e in the awkward position of not really even knowing
what it is that we don鈥檛 know.鈥�

His new book Small Worlds sets out the basics, and will be seized on
by those seeking a first rough map of this fascinating new mathematical land.
Those entering can expect to find some amazing connections between areas of
research with apparently nothing in common, such as neurology to business
studies.

But then, it鈥檚 a small world.

AT THE heart of small world theory is the idea of a huge network of
things鈥攁ctors, nerve cells, whatever鈥攖hat are somehow linked to each
other; actors are linked, for example, by working in a film together. Suppose
that each member is linked, on average, to k others; then, for a
network with a total of N members, you can calculate two numbers of
crucial importance in small world theory: L, the average number of
steps needed to get from one member of the network to another, and C,
which reflects the level of clustering within the network鈥攈ow likely two
friends of yours are to be friends of each other.

These two numbers capture the differences between regular and random
networks. For regular networks, like the atoms in a crystal, every point is
connected to every other by precisely the same number of neighbours, and small
world theory shows that L is N/2(k+1) while C
is around 1. At the other extreme, a random network鈥攚hose members are
connected by random links, some short, some very long鈥攈as L= log
N/log k and C = k/N.

A small world has aspects of both extremes: any member can reach any other
almost as easily as in a random network, but there is almost as much clustering
as in a regular network.

For the film industry, the total number of actors, N, is 225 000
while k, the typical number of actors any one actor has worked with, is
61. Computer analysis of all the links gives L= 3.65 and C =
0.79. In comparison, theory gives L = 3.00 and C = 3 脳 10^-4
for a random network, and L = 1800 and C is around 1 for a
regular network. So the film industry truly is a small world.