And then we'll actually show that if you're going to arrange people,

you're best off doing it in a star.

And then you don't want to have multiple stars,

you'd be better off having one star.

And then we can just compare whether it's better to have

a big star with everybody in it, or no star at all, okay?

And that'll be the difference between the medium cost and the really high cost, so

when costs get high enough, a star is not even valuable.

So basically what we're going to show, is that if you're going to arrange people

when c is bigger than delta- delta squared, you better do it in a star.

And then is a question of whether a star is valuable, okay?

So value of a star with k players, what is it?

Well, with k different players, then we've got 1 person in the middle,

k- 1 other individuals out here, so we've got k- 1 links in total.

And so each one of those links gives a value of delta- c

to each of its participants.

There's two people in it, k- 1 links.

So the direct value of connections is 2 (k- 1) [delta- c].

And then the indirect values that we're getting wre coming from the fact that each

one of these indirect people, there's k- 1 of them, right?

They have k- 2 other neighbors, each at a distance of 2,

so each of these k- 1 people have k- 2 neighbors,

each one of those gives a benefit of delta squared.

So the overall value of a star Is 2(k -1)[delta-

c] + (k -1)(k- 2) delta squared.

Okay, so that's the value of a star.

Now let's look at the value of some other configuration,

that involves k players and m links,

where m has to be at least k- 1 in order to connect these k players together.

Okay, so if you've got m links, first of all,

what's the value you can get out of the links directly?

Well again, same kind of calculation, you've got m links, 2 people in each link,

delta- c, so that's going to be the value there, of the direct value.

And the most you could be getting indirectly is,

you've got k players,

k- 1 other people that they could be connected to,

2m of those connections are direct connections.

So the remaining connections, this is how many remaining indirect connections there

can be, and at most they could be with delta squared, okay?

So this is the maximum possible value we can imagine for

some other component with m links, okay?

So let's take the difference between these two, so let's take this,

we'll take the difference between these two different, right?

So take this expression, subtract off this expression and what do we get?

If you subtract this from that, well, you can do the arithmetic.

The difference is going to turn out to be 2(m-(k-1)[delta

squared- (delta- c)], okay?