So, how can we solve this dangling node problem?

We refer to these nodes with no out degrees, or out degree being zero, as

dangling nodes. One possible solution is to say, well,

since you don't point to any other nodes, I will assume that you point to all nodes.

I will force you through a mandatory score spreading.

You have to spread your importance score and since you don't tell me which ones you

point to, I'm going to say, you point to all nodes, okay?

Either all the N minus one nodes, or for simplicity let's say, all the N nodes

including yourself, the self loop. For example, in the last graph here,

instead of this zero, zero, zero, zero, we'll say, it must be one fourth, one

fourth, one fourth, one fourth. Now, here's a shorthand notation to denote

this action of forced importance score spreading to all the nodes.

What we saw is that, we basically say that there is one over four times one, one,

one, one, okay? This is a vector of ones so we write it as

a bold face one, a vector one. Since, by convention of this

research field, I have to flip it over, so this is a row vector, okay?

So, it's one over N, this one-fourth here, times row vector of all ones, okay?

This is what we want to use but only for the dangling nodes.

So, I'll have to identify the dangling nodes.

In this case, the dangling node with a binary indicator is the last ones, the

fourth node. So, we write down one here in a four by

one vector. It's the indicator of all the dangling

nodes. If nodes one, two and three are not

dangling, dangling nodes, we'll just write down zero.

If it is a dangling node, we write down one.

And now, we can look at an outer product between this column vector and this row

vector, okay? Usually, when we write down two vectors

multiplying, a transpose b, we mean inner product, okay?

So, this column vector flipped, therefore it's one by N, and b is N by one column

vector, so the inner product is a scalar, one by one.

But in this case, we're talking about a column vector producting with a row

vector, not the other way around. So, we actually end up with a four by four

matrix. It's a little funny operation, but soon

you'll get used to this shorthand notation. In this four by four matrix, the first

entry is simply this times this. So, that's zero times one was still zero.

And then, you can see that this whole thing would be zero.

Similarly this second row is all zero, third row, all zero.

That's because, in this outer product, the indicator vector's first three entries all zeros,

so it doesn't matter what you have over there.

The last one is one times one fourth, one times one fourth, and so on.

So, you got, exactly what you want. Again, this is just the short hand matrix

notation. And we call this indicator function,

vector w, this vector is just one. So, we have w multiply one transpose, and

don't forget to normalize by the number of webpages one over N.