So this log node plot is the best way for you to distinguish when a particular

graph, or when a particular network is either power law, or heavy tailed, or

exponential, or something else.

You draw the log node plot.

If we'd just trade down your sloping line, it's the power law graph.

If it has a heavy tail, if it has straight tail, that's heavy tailed.

And otherwise, it is some other distribution.

So a lot of small world networks are also power law graphs.

For instance, the internet backbone graph, the telephone call graph,

protein networks, the worldwide world graph is, in fact, a small-world graph.

And also, power law graph has a value of alpha equals somewhere between 2.1 and

2.4, remember that.

You have in our power law graph a degree of

with probable deeper version of they key minus alpha.

The Gnutella p2p system has a heavy-tailed degree distribution.

It's not necessarily power law but it is heavy tail, so

it has a straight downward sloping tail in the log log point.

Power law networks are often called a scale-free networks, for instance,

in Gnutella, you have about 3.4 edges per vertex independent of

the number of vertices that are present in your network or in your graph.

But then, not all small world networks are power-law graphs and vice versa.

For instance, co-authorship networks are small world but

they're not necessarily power-law.

And also, not all power-law graphs are small-world.

You can generate power-law networks using the data distribution,

where you have disconnected components of the graph, where they're not necessarily

reachable from each other and so the graph is not a small-world network.

But many small world networks are in fact power-law graphs as far as you find them

in reality, and because a lot of these graphs,

a small world and power-law, their resilience comes into question.

Most nodes have a small degree in these graphs, but

a few nodes have a very high degree in these graphs.

So if you launch an attack on one of these graphs, if the attack is random, for

instance, you kill a very large number of randomly chosen nodes,

this will not disconnect the graph, okay?

This is why attacking a power-law small world graph randomly is not really

effective at all.

But if you target some of these very high degree nodes in the power-law

small world graph, this has a much higher chance of disconnecting the graph, okay?

And this is essentially why, in the human body for

instance, because the graph of proteins in the human body or chemicals in

the human body is a small world network and also a power-law network,

a few of the nutrients are very high degree nodes in the graph.

So, essentially edges in this protein graph,

edges are joined to chemicals that react with each other.

And so some nutrients happen to have a much higher degree than

other nutrients because they react with a lot of other chemicals in the body.

And these nutrients are essentially, the vitamins in the human body,

the minerals that the human body acquires like calcium and potassium.

So, if you have a shortage of any of these vitamins or nutrients in your body,

or minerals, your likely to be in trouble.

But if you have a shortage of some of these other random chemicals that

are there in your human body,

then it's not likely that you're going to be affected all that much.

Similarly, networks like the internet as well as the electric power grid,

if you target a few high degree vertices, that's much more likely to disconnect

the graph and cause a partition, and cause outages in the network.

But if you target randomly chosen vertices,

that's not likely to have any effect on the network itself.