The long-tailed degree distribution
structure of real world complex networks
has implications for their robustness.
For example, these networks are largely
structured around hubs. They are
vulnerable to targeted hub failure.
An example of this is on the web, when
large highly connected web sites are
targeted by malicious people via attacks
called distributed denial of service
attacks. If my web page goes down, nobody
is going to get really upset because it
doesn't have a lot of links coming into it
Whereas, if a big web site like Yahoo goes
down, a very highly connected web site,
it is going to create a lot more havoc on
the web. That's true in all networks
that have this kind of hub structure.
We talked about this for instance with
respect to airline networks. Where if a
hub city is having bad weather, causing
flights to be delayed, that's going to
have effects that percolate throughout the
system. Similarly, in biological networks
the extinction of a hub species, whether
it be a predator or prey species, the
food web is going to have implications
for the entire food web. The same thing
goes for other kinds of biological
networks. And, it may be that a network
approach to thinking about biology is
going to be more important for
understanding health and well-being than
the genetic sequencing of the human genome
project. Although, these networks are
vulnerable to targeted hub failure, they
are robust to random node failure.
If a single node becomes unusable for
some reason, since most nodes have very
low degree, it's not going to have a big
effect on the network. In the internet
for example, internet servers are going
down all the time, being taken out of
commission for temporary periods, but
because of the long-tail structure of
these networks such random node failure
often doesn't have much effect, unless
nodes can cause other nodes to fail, then
we can get something called cascading
failure. An example, the power grid.
Here's a satellite picture of a region of
the northeast US and Canada before and
after a huge electrical black-out that
occurred August 2003. Evidently, a power
station in Ohio got overloaded because of
a downed power line. It transferred its
load to another power station, that itself
got overloaded and shut-down in a domino
effect, a cascade of failures, bringing
down a lot of the electrical grid of the
northeastern US. You can see before the
black-out lots of lights, here much
diminished, here on the day of the
black-out. That's a common example of a
cascading failure. Another example, is
economic systems such as banks. Here in
years from 2007 to 2010, a sort of domino
effect of banks failing. Once one bank
fails, it causes an effect throughout the
bank network if the bank network is
closely inter-connected and has this
cascading failure possibility. We can
get this cascade of bank failures. We see
similar patterns of cascading failures in
biological systems, ecological systems,
computer and communication networks, wars,
and so on. Now we want to talk about the
implications of a long-tailed distribution
for thinking about risk. It's been
traditional to model risk using normal
distributions or bell curve distributions.
These are different shapes of these normal
distributions. You can see some of them
have high peaks, some have lower peaks,
wider, and so on, but all of them have
the feature that events in the tail of the
distribution are highly unlikely; however
if you look at a long-tailed distribution
like this one, it turns out that events in
the tail are more likely than in a normal
distribution, sometimes much more likely.
So, if you are thinking about risk, for
instance, in a financial market, a
housing market, in any kind of economic
system, or even earthquake risk,
or something like that, if your underlying
model is a normal distribution, a bell
curve, you are going to assume that risks
out in the tail have very low probability,
but if your model was one of these
long-tail distributions, you would be much
more concerned about possible risks
because they would have a higher
probability. Some people have called this
situation, the long-tailed distribution,
more normal than normal because they are
so common in complex networks that are
ubiquitous in all kinds of different
domains. This sort of thinking was behind
Paul Krugman's quote in the New York Times
in 2009, Paul Krugman being a Nobel prize
winning economist, and he said, "Few
economists saw our current crisis coming,
but this predictive failure was the least
of the field's problems. More important
was the professions blindness to the very
possibility of catastrophic failures in a
market economy. So, he's talking right
there about what people think about the
tails of these distributions. These
risk distributions. How much probability
is down here. Well, there is a lot more
if you assume that you have a power law
or other log-tail distribution than if you
have a normal distribution. And, what
Krugman is saying is that the profession
of economics was probably more fixated on
modeling risk and other factors using
normal or Gaussian distributions whereas
they should have been using these more
long-tailed distributions.
You have all seen the phrase,
Too Big To Fail, with respect to large
banks or other financial institutions.
But, an alternative idea, put forth in an
editorial by Duncan Watts is that instead
of worrying about things that are too big
to fail, we should worry about
institutions that are too complex to exist
because complexity seems to produce such
power law or long-tailed distributions
rather than normal distributions and in
this article Watts talks about how the
power grid, economic systems, bank
networks and so on, can become so complex
that catastrophic failure is sometimes
inevitable because of the long-tail of
risk. Let's end with another quote from
Duncan Watts, so far in this lecture we
have focused on the structure of networks
We've looked at small world structure,
scale-free, and long-tail degree
distributions and so on. We haven't
really talked much about dynamics on
networks. So, Duncan says, "Next to the
mysteries of dynamics on a network --
whether it be epidemics of disease,
cascading failures in power systems, or
the outbreak of revolutions - the problems
of networks that we have encountered up
to now are just pebbles on the seashore."
Duncan said this 10 years ago, and within
the last 10 years a huge amount of work
has been done on studying how dynamics
occur on networks - how information
spreads, how epidemics occur, how
cascading failures happen. We're not
going to cover those topics in this
course, but hopefully in a future course.
I would also like to mention that on the
course materials page, there are many
references that talk about what's been
happening in the last 10 years of network
science, in terms of dynamics particularly
that you might find interesting.