Okay, so just to sort of remind you, this is a picture from Bear,

Moody, and Stovel's 2004 data as high school romances.

This would be something, if we were thinking about transmission of

mononucleosis or something, we could think about a network like this.

And what we end up with is the component structure will actually tell us a lot.

So if an initially infected individual ends up being in one of these nodes,

then if they end up being in the large component,

then things can spread quite extensively.

If they end up being in the small component,

then things can be quite limited.

And so, looking at the component structure will help us answer two questions.

First of all, what's the probability that we start a contagion?

And that's going to be the probability that we end up sort of hitting one of

these large components, the large component,

in this case the giant component, and then how extensive should it be.

And in this case, if we did hit somebody in the giant component,

then the reach of it could potentially be the size of the giant component.

So understanding what the component structure is will help us understand

both the probability of starting and the eventual reach conditional on that.

So, we'll think about getting nontrivial diffusion if somebody in

the giant component is infected adopts, so I'll use the word infected but

it could be adopting a new technology, and so forth.

And the size of the giant components can determine both likelihood and its extent.

And random network models are going to allow for giant component calculations.

And now, in terms of what we want to be thinking about in links now.

We could can say, okay, well, a lot of networks we actually look at in the real

world might be very well-connected and have links so that everyone can reach

everyone else in the world, and so the world is one giant component.

But the component structure we actually want to be thinking about are going to

have link probabilities that are associated with the likelihood that

one individual actually infects another.

So it might be that somebody just doesn't catch the flu because they

don't have interactions with people at the right times, and so forth.

So, the network again we're going to looking at is, our people going to

interact within a given time period when they're infected enough to transmit.

And that can actually have a much more fragmented network structure

than an overall long-term network that we would look at normally.