Here's a cycle of length four. All the cycles are even.

All cycles are 2, 4, 6, 8, etc. Greatest common divisor, 2.

So, this one is periodic. It's not aperiodic, not aperiodic.

Okay? So when we, when we're looking at, at the

structure of these matrices, what turns out to be true is that this aperiodicity

is going to be what defines whether or not you get convergence of this process.

So in particular, there's a theorem, which you can get out of a variety of

sources. But basically can be derived from work in

Markov chain theory and, and more generally in linear algebra.

Suppose that we have Tb strongly connected.

And let me say a little about what strongly connected means.

Strongly connected is going to say that from every given individual, there exists

a path, a, a directed path from i to j. So there exists a directed path from i to

j for all i, j. Okay.

So basically, we're not in a situation where some people could never end up

getting information from other individuals, everybody could eventually

hear from everybody else. Okay?

If you, if you have non-strong connection, the con, the characterization

here gets a little more complicated. if you're interested in that, I have a

paper with Ben Golub in 2010 which gives the complete characterization of

convergence for non-strongly connected networks.

We'll just deal with the strongly-connected ones, and that's most

of the intuition. Okay.

So what's the theorem say? The theorem says that you get convergence

if and only if you get aperiodicity. So that aperiodic aperiodicity is

necessary and sufficient for convergence. and separate parts, so first part is

aperiodicity is going to give you convergence.

And in the second part of, that we learn is if things are convergent, then we're

also going to have that the limit of this matrix actually looks like every row is

going to converge to having, so if we look at, at what T raised and, and a

limit raised to the infinite power looks like.

What it looks like is a vector. A set of each row has entries S1 to Sn,

the same S1 to Sn. So, everybody ends up putting this in the

limit the same weights on other people's initial beliefs indirectly.

Where s is the unique left hand side eigenvector that has eigen value 1.

Okay, so that's a mouthful, but it says that in fact, what this is, is the same

eigenvector that we were talking about when we talked about eigenvectors

centrality. So this an eigenvector that has

eigenvalue 1. in this case it's going to be the unique

one if we've got strongly connected and aperiodicity.

this thing's going to be convergent and it's going to have nicely-defined weights

here S1 through Sn, and in fact, they will all be non-negative, they'll all be

positive, in fact. So, so these is a very powerful set of

results here. It tells us, aperiodicity, gives us

convergence, moreover, convergence occurs to a very specific limit and that limit's

given by the left hand side unit eigenvector of the matrix.

And that's where those 2 sevenths and 2 sevenths, 2 sevenths came from and the

other numbers that we found in those examples before.

Okay. So, what we're going to look at next is

I'll go through how you prove this. it involves a bit linear algebra and some

theorems from that, so if you're you can skip that if you like.

If you want to see the details I'll explain how you can derive that kind of

theorem. And then afterwards we'll begin to put

this in process. So what we've got is convergence from

aperiodicity plus we know what the limit looks like so we'll then talk a little

more in detail about what this S vector actually means; what it comes from.

And look at some examples and, and try and understand how we might use this,

this model a bit.