So we're talking about exponential random graph models

and some of the issues in estimating those.

And so now what I want to do is, is talk

about another class of models

called Statistical Exponential Random Graph Models.

So SERGMs for short.

And so what we've been through is we had

sets of models that are, are linked based, other models

which are still link by link based which can

begin to capture different things like clustering and so forth.

We brought in attributes

in the Stochastic block models.

And then we said okay, now there's the class of models which allow us

to capture richer things where there's two

dependencies and trying to estimate those things statistically.

Exponential Random Graph Models, but as we

saw there is difficulties in estimating those.

So what I want to talk about now is a

class of new ones called Statistical Exponential Random Graph Models.

That'll allow us to keep the track of these local features

and dependencies and actually do some accurate and fast and easy estimation.

'Kay.

So, the Exponential Random Graph Models are

not accurately estimable in, in many cases.

There's basically just too many alternative networks to consider.

So what's the idea here?

What's the way out?

The way out is going to be that many networks lead to the same statistics.

So for instance in the, in the simulations we did in

the last video what we had was a series of you know,

45 links, 10 triangles and 20 isolated nodes.

And in fact, under the statistical, under an Exponential Random Graph

formulation, any network that had exactly

those characteristics would be equally likely.

They'd all lead to the same function, they'd lead to the same probability.

So what we can do is we can just say,

okay look, even though there's many different networks to search over,

In fact, what we really need to keep track of is just the number of different

types of statistics, because all the networks that

have the same statistic have the same probability.

And so we can look at those as equivalent ones.

Collapse all those equivalent networks.

And that makes the summation and the area that we have

to be searching over much smaller than what it was before.

Okay.

And, and part of this came out, you know, so, so why am I

going to talk about this, this is coming out of a paper with Arun Chandrasekhar.

And effectively, the way I got interested in this

was in projects where I had to be using Exponential

Random Graph Models, or, or something that really accounted

for dependencies, and finding that the software just didn't work.

And, and I couldn't find any models that were actually working.

And so, the idea was, okay, we need some models, so let's go develop them.

So, so here I am going to tell you about a class of

models which I have been working with, which I think solve a lot

of the problems that we have there in the literature.

When you collapse these equilvalent networks and so

lets just go through the, the idea here.

So we start with our exponential random graph model, right?

So we have got some vector of statistics.