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As we part, I'd like to congratulate you
for sticking with this course throughout

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its eight weeks.
I hope that you have a good sense about

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networks.
What I, what I hope you got out of it is

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understanding what are the key properties
of empirically observed networks, what are

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the processes that shape them that help us
to understand what those empirical numbers

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mean, and what implications this has for
various functions that the network is

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supposed to be carrying out.
For example, whether it's enabling people

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to communicate or to search and, and find
one another,

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Or, you know, some of the cool and unusual
applications where the nodes might be

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different products in a product space that
different countries participate and

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compete in.
At the same time, we've in a sense just

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scratched the surface of all of the
wonderful research and learning that you

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can do in this area.
One thing that I haven't covered at all

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are biological networks and this is one of
the largest application areas with many

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fascinating findings, ranging from
understanding how different functions have

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evolved to how cells work to I mean, it
just goes on and on.

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It's, it's really,
I mean, things like the relationship

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between genetics and disease, all of this
now is heavily influenced by our

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understanding of how things are networked
together.

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Similarly, there's so many other
application areas that we haven't, that

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you wouldn't even imagine.
For example, network analysis of space

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that architects use to understand which
spaces are going to be conducive to

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individuals flowing through, which spaces
are going to allow them to communicate.

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Just things of that sort.
We also have not used some of the tools

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that are heavily used in some domains.
So, for example, there is a most excellent

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book by Mark Newman, who's also faculty at
the University of Michigan, called

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Introduction to Networks and, it's really
a tome, it's a, it's a very thick book,

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where he shows you how using, you know,
some heavy-duty math and analytics you can

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derive many network properties from
knowledge of simple things such as the

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degree distribution.
So you can actually derive what the size

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of the giant component will be given the
degree distribution of the network, for

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example.
It also looks a lot at different

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algorithms for detecting community
structure and really grounds such

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approaches on, you know, kind of, in a, in
a statistical way where you're looking at

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ensembles of many different network
instantiations and looking at whether what

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you observe is actually significantly
different from what you would see at

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random.
On the economic side of things, Matthew

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Jackson, who's faculty at Stanford, has
written an excellent book Social and

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Economic Networks,
And things that we really didn't get into

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that much are, but that the book covers,
are things like, what strategically, if

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you have, if the nodes are self-interested
and maximizing the utility that they get

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from the network,
How will the network be wired?

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Or if you're playing certain games,
I mean we did this simple cascade model

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but what if you were playing prisoner's
dilemma on a network.

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.
How does the network topology affect the

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outcomes?
.

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Then Laszlo Barabasi and his collaborators
are writing more chapters of the wonderful

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Network Science Book,
That which is aimed more at, at a beginner

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audience. So I think you would you could
probably understand, no matter your, your

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background, a great portion of that book.
So other topics that we didn't get to this

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time around, include how networks evolve
over time.

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So, Gephi actually has some functionality
for visualizing how networks evolve.

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But questions that you might be interested
in is,

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Does the community structure teach?
Does the network densify that is this will

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become denser over time?
You can track centralities,

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So are there different nodes that emerge
as being important during different time

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periods.
So that's a whole area, which, it's sort

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of easy to extrapolate from where we are
now to doing it over time.

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But, I'm just sorry, I didn't get a chance
to do that with you this time around.

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So,
No, another very fruitful part are these

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p-star models where you statistically
determine factors that are affecting

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network formations, so rather than saying
oh, okay it's going, it'll, it's going to

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have this probability of preferential
attachment,

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You're going to actually run an analysis
that's going to extract that from the

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observed data.
You know, is the network going

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preferentially or not?
Is triadic closure a large factor in how

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this network has formed?
And finally, you know, cool and unusual

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applications, they're happening every day
all the time,

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Whether you want to understand what's
happening in politics,

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In, education.
[laugh].

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Really, there, there's no limit to where
you can apply, network analysis.

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So I do hope you, have found this class
useful and that now you will go forth and

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apply social network analysis to whatever
might interest you.

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So, thanks for sticking with me through
these weeks.