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.