So what, what the idea here is again we're
going to be looking at, at ways to easily incorporate
dependencies and networks and think about different kinds of
sub graphs that might be forming and allow those to
be generated and try and estimate how they're being generated.
And the idea of SUGMs the, the, the terminology is quite simple.
It's from subgraph generation model.
So we think of subgraphs as being generated.
Links, triangles small stars etcetera.
Those are the things that are going to be
generated, and then the network is a byproduct, right?
So what's happening is people are forming
different types of relationships, different kinds of
cliques, and the network bubbles up from that.
Okay?
So in particular, what we're going to think of is
think of whatever type of, of subnetwork you like.
So, maybe it's a link.
Maybe it's a triangle and what happens is these things will be independently
with some probability p, p sub j and then we end up with some number
of those. So, say s sub j, sub-networks.
So our 45 links or ten triangles, these things are
going to be generated and, the difficulty is that they may overlap.
Okay?
So, sometimes people form links. They also form a clique.
So, I end up partnering with somebody but I also form groups.
So if you wanted to look at
say a co-authorship network, I might have written
with one person.
And just one paper but then we also collaborated with somebody
else on a different paper, so we end up forming both the
link and the triangle and if we just look at the
co-authorship network then it's hard to see that directly in the data.
So we observed the resulting network.
And then what we're trying to do is infer these
different probabilities of different types of relationships they came out, okay?
So let's just go through a simple example.
We start with a bunch of nodes.
Let's think of just generating links and triangles.
So there's some probability that different links form.
So in this case, a bunch of
links form here, nine different sorry, triangles form.
So nine different triangles are formed at random.
So out of all the possible triangles, we form nine of them.
Then a bunch of links formed. So the links were dropped down afterwards.
And, what do we end up with?
We end up incidentally generating some triangles, right?
So when we look at this, there was these two edges had
already been there because of a triang, of triangles that were generated.
This one was generated as a link and it ends up generating a triangle and when
we actually see it in the end we
can't tell which ones were generated in which way.
So what we're left with is some links,
some triangles and we're trying to estimate then what is
the probability of these different types of sub-graphs being formed.
Okay.
So, that's the idea So these are sub-graph
generation models, so first of all let's try and
see is there a way to view these
sub-graph generation models as statistical exponential random graph models?
Now you remember when we went through exponential random graph models to begin
with, I showed you how you could take a a model of, of just an [INAUDIBLE]
type where you're generating links at random, and
you can represent that as an exponential function.
