WEBVTT
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YUFEI ZHAO: So
today we are going
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to start a new chapter
on graph limits.
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So graph limits is a relatively
new subject in graph theory.
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So as the name
suggests, we're looking
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at some kind of an analytic
limit of graphs, which
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sounds kind of
like a strange idea
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because you think of graphs as
fundamentally discrete objects.
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But let me begin with
an example to motivate,
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at least pure mathematical
motivation for graph limits.
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There are several other ways
you can motivate graphs limits,
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especially coming from
more applied perspectives.
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But let me stick with
the following story.
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So suppose you lived
in ancient Greece
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and you only knew
rational numbers.
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You didn't know
about real numbers.
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But you understand
perfectly rational numbers.
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And we wish to maximize.
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So we wish to then minimize
the following polynomial,
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x cubed minus x, let's
say for x between 0 and 1.
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So you can do this.
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And suppose also the
Greeks knew calculus
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and take the derivative
and all of that.
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So you find that.
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You have a problem
because we know--
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so given our advanced
state of mathematics,
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we know that the maximum--
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so the minimizer is at x
equals to 1 over root 3.
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But that number doesn't
exist in the real numbers.
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So how might a civilization
that only knew rational numbers
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express this answer?
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They could say, the minimum
occurs not in Q. So there's
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not minimized in Q--
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but not minimized by a single
number, but by a sequence.
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And this is a sequence that
a more advanced civilization
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would know, a sequence that
converges to 1 over root of 3.
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But I can give you this sequence
through some other means.
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And this is one of the ways
of defining the complete set
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of real numbers, for instance.
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But you can define explicitly
a sequence of real numbers
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that converges.
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But of course, this is
all quite cumbersome
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if you have to
actually write down
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this sequence of real numbers
to express this answer.
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It will be much better if
we knew the real numbers.
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And we do.
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And the real numbers,
in some sense,
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in the very rigorous
sense, is a completion
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of the rational numbers.
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That's the story that
we're all familiar with.
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But now let's think
about graphs which
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are some kind of a
discrete set of objects,
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akin to the rational numbers.
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And the story now
is, among graphs,
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suppose I have a fixed
p between 0 and 1.
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And the problem
now is to minimize
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the 4-cycle density among graphs
with density, with edge density
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p.
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So this is some kind of
optimization problem.
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So I don't restrict
the number of vertices.
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You can use as many
vertices as you like.
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And I would like to minimize
the 4-cycle density.
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Now, we saw a few lectures
ago this inequality
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that tells us that--
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so we saw a few lectures ago
that this density is always
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at least p to the fourth.
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So in the lecture on
quasirandomness, so
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we saw this inequality.
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And we also saw
that this minimum
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is approached by a sequence
of quasirandom graphs.
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And in some sense, that-- so
the answer is p to the fourth.
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And there's not
a specific graph.
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There's no one graph
that minimizes.
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This 4-cycle density is
minimized by a sequence.
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And just like in the story
with the rational numbers
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and the real numbers,
it would be nice
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if we didn't have to
write out the answer
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in this cumbersome,
sequential way,
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but just have a single
graphical-like object that
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depicts what the
minimizer should be.
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And graph limits provides a
language for us to do this.
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So one of the goals
of the graph limits--
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this gives us a single
object for this minimizer
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instead of taking a sequence.
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So roughly that is the idea that
you have a sequence of graphs.
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And I would like
some analytic object
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to capture the behavior of
the sequence in the limit.
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And these graph limits
can be written actually
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in a fairly concrete form.
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And so now let me begin
with some definitions.
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The main object
that we'll look at
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is something called a graphon.
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So it merges the two
words graph, function.
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A graphon is by definition a
symmetric, measurable function,
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often denoted by the letter
W from the unit squared
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to the 0, 1 interval.
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And here being symmetric means
that if you exchange the two
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argument variables, this
function remains the same.
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So that's it.
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So that's the
definition of a graphon.
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And these are the objects that
will play the role of limits
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for sequences of graphs.
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And I will give you lots
of examples in a second.
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So that's the definition.
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This is the form of
the graphons that we'll
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be looking at mostly.
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But just to mention
a few remarks,
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that the domain can
be instead any product
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of any square of a
probability measure space--
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so instead of taking
the 0, 1 interval,
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I could also use any
probability measure space.
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So it's only slightly
more general.
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So there are some general
theorems in measure theory
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that tells us that most
probability measure
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spaces, if they're
nice enough, they
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are in some sense
equivalent or can
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be captured by this interval.
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So I don't want you to worry
too much about all the measure
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where there are technicalities.
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I think they are not so
important for the discussion
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of graph limits.
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But there are some subtle issues
like that just lurking behind.
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But I just don't want to
really talk about them.
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So for the most part,
we'll be looking
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at graphons of this one.
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And also the-- so instead of
the domain, so the values--
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so instead of 0, 1
interval, you could also
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take a more general space,
for example, the real numbers
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or even the complex numbers.
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I'm going to use
the word graphon
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to reserve this word for when
the values are between 0 and 1.
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And if it's in R, let me
call this just a kernel,
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although that will
not come up so much.
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So when I say
graphon, I just mean
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the values between 0 and 1.
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Although if you do look up
papers in the literature,
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sometimes they don't use
these words so consistently.
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So be careful what
they mean by a graphon.
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So that's the definition.
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But now let me give
you some examples
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on how do we think of
graphons and what do they
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have to do with graphs.
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So if we start with a
graph, I want to show you
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how to turn it into a graphon.
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So let's start with this graph,
which you've seen before.
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This is the half graph.
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So from this graph, I
can label the vertices
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and form an adjacency
matrix of this graph, where
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I label the rows and
columns by the vertices
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and put in zeros
and ones according
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to whether the
edges are adjacent.
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So that's the adjacency matrix.
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And now I want you
to view this matrix
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as a black and white picture.
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So think one of these
pixelated images, where I
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turn the ones into black boxes.
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Of course, on the blackboard,
black is white and white
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is black.
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So I turn the ones
into black boxes.
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And I leave the zeros
as empty white space.
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So I get this image.
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And I think of this
image as a function.
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And this is the function
going from 0, 1, squared to 0,
00:11:53.190 --> 00:11:55.920
1, interval, taking
only 0 and 1 values.
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So that's a function
on the square.
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But now, so this
is a single graph.
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So for any specific
graph, I can turn it
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into a graphon like this.
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But now imagine you have
a sequence of graphs.
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And in particular, consider
a sequence of half graphs.
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So here is H3.
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So Hn is the general half graph.
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And you can imagine
that, as n gets large,
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this picture looks like--
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instead of the
staircase you just
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have a straight line
connecting the two ends.
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And indeed, this function
here, this graphon,
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is the limit of the
sequence of half graphs
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as n goes to infinity.
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So one way you can
think about graphons
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is you have a
sequence of graphs.
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You look at their
adjacency matrix.
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You view it as a picture,
a pixelated image,
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black and white according
to the zeros and ones
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in its adjacency matrix.
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And as you take a sequence,
you make your eyes a little bit
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blurry.
00:13:35.750 --> 00:13:38.260
And then you think about
what the sequence of images
00:13:38.260 --> 00:13:40.120
converges to.
00:13:40.120 --> 00:13:44.110
So the resulting
limit is the limit
00:13:44.110 --> 00:13:47.120
of this sequence of graphs.
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So that's an
informal explanation.
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So I haven't done
anything precisely.
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And in fact, one
needs to be somewhat
00:13:53.950 --> 00:13:56.620
careful with this
depiction because let
00:13:56.620 --> 00:13:57.925
me give you another example.
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Suppose I have a sequence of
random or quasirandom graphs
00:14:16.080 --> 00:14:18.860
with edge density 1/2.
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So what does this look like?
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And I have this picture here.
00:14:29.860 --> 00:14:31.630
And I have a lot of--
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so I have a lot of--
00:14:37.990 --> 00:14:40.810
one-half of the
pixels are black.
00:14:40.810 --> 00:14:44.010
And the other half
pixels are white.
00:14:44.010 --> 00:14:45.930
And you can think,
from far away,
00:14:45.930 --> 00:14:48.570
I cannot distinguish necessarily
which ones are black and which
00:14:48.570 --> 00:14:49.610
ones are white.
00:14:49.610 --> 00:14:52.470
And in the limit, it looks
like a grayscale image,
00:14:52.470 --> 00:14:57.030
with a grayscale being
one-half density.
00:14:57.030 --> 00:15:01.330
And indeed, it converges to
the constant function, 1/2.
00:15:10.760 --> 00:15:15.190
So the limits represented
by this problem up here
00:15:15.190 --> 00:15:20.120
is the constant graphon
with the constant value p.
00:15:20.120 --> 00:15:22.370
But now let me give you
a different example.
00:15:25.350 --> 00:15:27.001
Consider a checkerboard.
00:15:36.340 --> 00:15:53.280
So here is a checkerboard, where
I color the squares according
00:15:53.280 --> 00:15:56.070
to, in this alternating
black and white manner,
00:15:56.070 --> 00:15:58.162
according to a
usual checkerboard.
00:16:02.890 --> 00:16:07.690
And as the number of
squares goes to infinity,
00:16:07.690 --> 00:16:10.330
what should this converge to?
00:16:13.210 --> 00:16:14.960
By the story I
just told you, you
00:16:14.960 --> 00:16:18.020
might think that
if you zoom out,
00:16:18.020 --> 00:16:21.030
everything looks density 1/2.
00:16:21.030 --> 00:16:24.210
And so you might guess
that the image, the limit,
00:16:24.210 --> 00:16:27.820
is the 1/2 constant.
00:16:27.820 --> 00:16:28.870
But what is this graph?
00:16:32.570 --> 00:16:34.090
It's a complete bipartite graph.
00:16:38.390 --> 00:16:44.053
It is a complete bipartite
graph between all the even rows.
00:16:44.053 --> 00:16:46.470
And there's a different way
to draw the complete bipartite
00:16:46.470 --> 00:16:48.590
graph--
00:16:48.590 --> 00:16:59.740
namely, that picture, just by
permuting the rows and columns.
00:16:59.740 --> 00:17:04.000
And it's much more
reasonable that this
00:17:04.000 --> 00:17:07.030
is the limit of the sequence
of complete bipartite
00:17:07.030 --> 00:17:08.470
graphs with equal parts.
00:17:11.010 --> 00:17:12.569
So one needs to be very careful.
00:17:12.569 --> 00:17:17.810
And so it's not necessarily
an intuitive definition.
00:17:17.810 --> 00:17:20.520
The idea that you
just squint your eyes
00:17:20.520 --> 00:17:22.770
and think about what
the image becomes,
00:17:22.770 --> 00:17:26.010
that works fine for
intuition for some examples,
00:17:26.010 --> 00:17:27.147
but not for others.
00:17:27.147 --> 00:17:28.980
So we do really need
to be careful in giving
00:17:28.980 --> 00:17:30.390
a precise definition.
00:17:30.390 --> 00:17:33.750
And here the rearrangement
of the rows and columns
00:17:33.750 --> 00:17:34.950
needs to be taken care of.
00:17:46.777 --> 00:17:47.860
So let me be more precise.
00:17:50.700 --> 00:17:57.680
Starting with a graph G, I can--
00:17:57.680 --> 00:18:03.280
so let me label the
vertices by 1 through n.
00:18:03.280 --> 00:18:15.490
I can denote by W sub G
this function, this graphon,
00:18:15.490 --> 00:18:20.292
obtained by the
following procedure.
00:18:24.630 --> 00:18:30.820
First, you partition
the interval
00:18:30.820 --> 00:18:39.090
into intervals of
length exactly 1 over n.
00:18:44.680 --> 00:18:54.330
And you set W of x
comma y to be basically
00:18:54.330 --> 00:18:57.780
what happened in
the procedure above.
00:18:57.780 --> 00:19:04.530
If x and y lie in the box
I sub I cross I sub J,
00:19:04.530 --> 00:19:11.300
then I put in 1 if I is
adjacent to J and 0 otherwise--
00:19:15.800 --> 00:19:18.390
so this picture,
where we obtained
00:19:18.390 --> 00:19:21.720
by taking the adjacency
matrix and transforming it
00:19:21.720 --> 00:19:22.980
into a pixelated image.
00:19:28.342 --> 00:19:29.800
What are some of
the things that we
00:19:29.800 --> 00:19:32.437
would like to do with graph
limits or graphs in general?
00:19:32.437 --> 00:19:32.937
Yeah?
00:19:32.937 --> 00:19:37.273
AUDIENCE: Is the range
also 0, 1, squared or 0, 1?
00:19:37.273 --> 00:19:38.190
YUFEI ZHAO: Thank you.
00:19:38.190 --> 00:19:39.060
The range is 0, 1.
00:19:42.730 --> 00:19:45.850
So here are some quantities
we are interested in when
00:19:45.850 --> 00:19:48.170
considering graph limits.
00:19:48.170 --> 00:19:58.954
So given two graphs, G and H, we
say that a graph homomorphism--
00:20:04.770 --> 00:20:13.390
so a graph homomorphism
between from G to H
00:20:13.390 --> 00:20:26.137
is a map of their vertexes such
that the edges are preserved.
00:20:30.910 --> 00:20:40.865
So you have-- and so
whenever uv is an edge of H,
00:20:40.865 --> 00:20:48.880
your image vertices get
mapped to an edge of G.
00:20:48.880 --> 00:20:52.810
And we are interested in the
number of graph homomorphisms.
00:20:52.810 --> 00:21:04.400
So often I use uppercase to
denote a set of homomorphisms G
00:21:04.400 --> 00:21:05.390
to H--
00:21:05.390 --> 00:21:07.580
and lowercase to
denote the number.
00:21:16.690 --> 00:21:21.480
So for example, the
number of homomorphisms
00:21:21.480 --> 00:21:24.570
from a single vertex--
00:21:24.570 --> 00:21:30.093
so a single vertex with no edge
to a graph G, that's just the--
00:21:30.093 --> 00:21:31.010
what is this quantity?
00:21:34.572 --> 00:21:36.530
So some number of
vertices of G--
00:21:40.730 --> 00:21:48.740
what about homomorphisms
from an edge to G?
00:21:48.740 --> 00:21:51.180
AUDIENCE: The number of edges?
00:21:51.180 --> 00:21:53.790
YUFEI ZHAO: Not quite the
number of edges, but twice
00:21:53.790 --> 00:21:54.690
the number of edges.
00:21:59.730 --> 00:22:02.810
What about the number
of homomorphisms
00:22:02.810 --> 00:22:04.310
from a triangle to G?
00:22:10.420 --> 00:22:12.170
AUDIENCE: 6 times the
number of triangles.
00:22:12.170 --> 00:22:14.570
YUFEI ZHAO: So yeah, you got
the idea-- so the 6 times
00:22:14.570 --> 00:22:16.088
the number of triangles.
00:22:22.172 --> 00:22:26.740
So now let me ask a slightly
more interesting question.
00:22:26.740 --> 00:22:28.890
What about the number
of homomorphisms from H
00:22:28.890 --> 00:22:29.665
to a triangle?
00:22:34.990 --> 00:22:37.010
What's a different name
for this quantity here?
00:22:45.630 --> 00:22:47.550
It's the number of
proper three colorings.
00:22:53.820 --> 00:22:57.380
So it's the number of
proper three colorings,
00:22:57.380 --> 00:23:05.160
the number of proper
colorings of H with three
00:23:05.160 --> 00:23:07.020
labeled colors, red,
green, and blue.
00:23:09.940 --> 00:23:11.640
So think about the
three vertices.
00:23:11.640 --> 00:23:13.510
That's red, green, and blue.
00:23:13.510 --> 00:23:16.790
And whichever vertex
of H can map to red,
00:23:16.790 --> 00:23:18.610
color that vertex red.
00:23:18.610 --> 00:23:21.150
So you see that there is a
one-to-one correspondence
00:23:21.150 --> 00:23:25.680
between such homomorphisms
and proper colorings.
00:23:25.680 --> 00:23:28.740
So many important graph
parameters, graph quantities,
00:23:28.740 --> 00:23:33.420
can be encoded in terms
of graph homomorphisms.
00:23:33.420 --> 00:23:35.100
And these are the
ones that we're
00:23:35.100 --> 00:23:36.920
going to be looking
at most of the time.
00:23:39.780 --> 00:23:42.490
When we're thinking
about very large graphs,
00:23:42.490 --> 00:23:45.490
often it's not the number of
homomorphisms that concern us,
00:23:45.490 --> 00:23:48.980
but the density
of homomorphisms.
00:23:48.980 --> 00:23:52.730
And the difference
between homomorphisms
00:23:52.730 --> 00:23:59.450
on one hand and subgraphs is
that the homomorphisms are not
00:23:59.450 --> 00:24:03.320
quite the same as
subgraphs, other
00:24:03.320 --> 00:24:06.620
than this multiplicity,
because you might have
00:24:06.620 --> 00:24:12.220
non-injective homomorphisms.
00:24:12.220 --> 00:24:14.770
But these non-injective
homomorphisms
00:24:14.770 --> 00:24:18.700
do not end up contributing
very much because they only
00:24:18.700 --> 00:24:25.540
have n to the number of
vertices of H minus 1
00:24:25.540 --> 00:24:30.070
on that border where I think
of n as the number of vertices
00:24:30.070 --> 00:24:33.690
of G. n is supposed to be large.
00:24:33.690 --> 00:24:37.170
So in terms of graph
limits when n gets large,
00:24:37.170 --> 00:24:40.350
I don't need to distinguish
so much between homomorphisms
00:24:40.350 --> 00:24:44.020
and subgraphs.
00:24:44.020 --> 00:24:57.380
We define the homomorphism
density, denoted by the letter
00:24:57.380 --> 00:25:04.060
t, from H to G, by--
00:25:04.060 --> 00:25:08.810
define it to be the
fraction of all vertex maps
00:25:08.810 --> 00:25:11.690
that are homomorphisms.
00:25:17.890 --> 00:25:22.720
So this is also equivalent to
be defined as the probability
00:25:22.720 --> 00:25:31.780
that a uniform random
map from the vertex set
00:25:31.780 --> 00:25:40.930
of H to the vertex set of G
is a homomorphism from H to G.
00:25:40.930 --> 00:25:43.460
So it's a graph homomorphism.
00:25:43.460 --> 00:25:45.680
And this quantity turns
out to be quite important.
00:25:45.680 --> 00:25:48.740
So we're going to be
seeing this a lot.
00:25:48.740 --> 00:25:55.170
And because of this remark
over here, in the limit,
00:25:55.170 --> 00:26:02.600
this quantity of graph
homomorphism densities
00:26:02.600 --> 00:26:06.960
in the limit as the number of
vertices G goes to infinity
00:26:06.960 --> 00:26:16.470
and H fixed, the
homomorphism densities
00:26:16.470 --> 00:26:20.467
approaches the same limit
as subgraph densities.
00:26:29.132 --> 00:26:30.840
So you should regard
these two quantities
00:26:30.840 --> 00:26:32.070
as basically the same thing.
00:26:35.575 --> 00:26:36.450
Any questions so far?
00:26:41.100 --> 00:26:45.480
So all of these quantities
so far defined are for--
00:26:45.480 --> 00:26:49.290
so everything is defined so
far for graphs, so what happens
00:26:49.290 --> 00:26:50.820
between graphs and graphs.
00:26:50.820 --> 00:26:53.590
So what about for graphons?
00:26:53.590 --> 00:26:56.730
I'll give you this limit
object, this analytic object.
00:26:56.730 --> 00:27:01.500
I can still define
densities by integrals now.
00:27:01.500 --> 00:27:06.560
So suppose I start with a
symmetric measurable function.
00:27:14.480 --> 00:27:19.250
So tell me, for
example, a graphon.
00:27:19.250 --> 00:27:23.780
But I can let my range
be even more generous.
00:27:23.780 --> 00:27:29.390
Starting with such a function,
I define the graph homomorphism
00:27:29.390 --> 00:27:35.030
density from a fixed graph
H to this graphon or kernel,
00:27:35.030 --> 00:27:45.223
more generally, to be the
following integral, where I'm--
00:27:45.223 --> 00:27:46.640
before writing
down the full form,
00:27:46.640 --> 00:27:48.015
let me first give
you an example.
00:27:48.015 --> 00:27:49.530
I think it will be more helpful.
00:27:49.530 --> 00:27:57.690
So if I'm looking at a triangle
going to W, what I would like
00:27:57.690 --> 00:28:01.725
is the integral that captures
the triangle density.
00:28:08.410 --> 00:28:19.120
So this quantity here, if I let
x, y, and z vary over 0 and 1,
00:28:19.120 --> 00:28:21.670
0 through 1, independently
and uniformly,
00:28:21.670 --> 00:28:27.540
then this quantity here captures
the triangle density in W.
00:28:27.540 --> 00:28:30.660
In fact, and I'll state this
more precisely in a second--
00:28:30.660 --> 00:28:33.180
if you look at the
translation from graph
00:28:33.180 --> 00:28:37.160
to graphon and combine
that translation
00:28:37.160 --> 00:28:41.240
with this definition here, you
recover the triangle density.
00:28:45.830 --> 00:28:51.500
More generally, for H
instead of a triangle,
00:28:51.500 --> 00:28:57.110
the H density in a graphon is
defined to be the integral of--
00:28:57.110 --> 00:29:01.010
instead of this
product here, I take
00:29:01.010 --> 00:29:05.880
a product corresponding to
the graph structure of H
00:29:05.880 --> 00:29:14.360
with one factor
for each edge of H.
00:29:14.360 --> 00:29:28.910
And the variables go
over the vertex set of H.
00:29:28.910 --> 00:29:33.000
So this is the definition
of homomorphism densities,
00:29:33.000 --> 00:29:37.250
not for graphs, but for
symmetric measurable functions,
00:29:37.250 --> 00:29:39.300
in particular, for graphons.
00:29:39.300 --> 00:29:40.970
And we define it
this way because--
00:29:40.970 --> 00:29:43.900
and we use the same symbols
because these two definitions
00:29:43.900 --> 00:29:44.400
agree.
00:29:48.970 --> 00:29:52.780
If you start with a graph and
look at the H density in G,
00:29:52.780 --> 00:29:58.780
then this quantity here
is equal to the H density
00:29:58.780 --> 00:30:02.590
in the graphon associated
to the graph G constructed
00:30:02.590 --> 00:30:03.520
as we did just now.
00:30:06.710 --> 00:30:09.110
So make sure you
understand why this is true
00:30:09.110 --> 00:30:11.030
and why we defined the
densities this way.
00:30:13.905 --> 00:30:14.780
Any questions so far?
00:30:24.990 --> 00:30:31.310
So we've given the definition
of graph homomorphism density.
00:30:31.310 --> 00:30:34.260
And we've defined these
objects, these graphons.
00:30:40.550 --> 00:30:43.800
And I mentioned even something
about the idea of a limit.
00:30:43.800 --> 00:30:46.980
But in what sense can we
have a limit of graphs?
00:31:04.430 --> 00:31:06.250
So here is an
important definition
00:31:06.250 --> 00:31:08.650
on the convergence of graphs.
00:31:08.650 --> 00:31:14.140
So in what sense can we say that
a sequence of graphs converge?
00:31:14.140 --> 00:31:28.680
So we say that a sequence
of graphs G sub n--
00:31:32.800 --> 00:31:36.310
graphs or graphons, so
these two definitions
00:31:36.310 --> 00:31:42.610
are interchangeable for what I'm
about to say regarding limits
00:31:42.610 --> 00:31:44.470
for graphons, in
which case I'm going
00:31:44.470 --> 00:31:46.450
to denote them by W sub n.
00:31:49.830 --> 00:31:57.370
So we say the
sequence is convergent
00:31:57.370 --> 00:32:06.590
if the sequence of
subgraph densities--
00:32:06.590 --> 00:32:08.390
of course, if you are
looking at graphons,
00:32:08.390 --> 00:32:14.600
then you should look at the
graphon, the subgraph density
00:32:14.600 --> 00:32:15.790
in--
00:32:15.790 --> 00:32:20.570
homomorphism density in
graphons if this sequence
00:32:20.570 --> 00:32:41.540
converges as n goes to
infinity for every graph H.
00:32:41.540 --> 00:32:43.460
So that's the
definition of what it
00:32:43.460 --> 00:32:46.220
means for a sequence
of graphs to converge,
00:32:46.220 --> 00:32:48.860
which so far looks actually
quite different from what
00:32:48.860 --> 00:32:50.810
we discussed intuitively.
00:32:50.810 --> 00:32:52.790
But I will state some
theorems towards the end
00:32:52.790 --> 00:32:56.450
of this lecture explaining
what the connections are.
00:32:56.450 --> 00:32:58.538
So intuitively
what I said earlier
00:32:58.538 --> 00:33:00.080
is that you have a
sequence of graphs
00:33:00.080 --> 00:33:05.600
that are convergent if you
have some vague notion of one
00:33:05.600 --> 00:33:09.260
image morphing into a
sequence of images morphing
00:33:09.260 --> 00:33:11.288
into this final image.
00:33:11.288 --> 00:33:12.830
Still hold that
thought in your mind.
00:33:12.830 --> 00:33:15.050
But that's not a
rigorous definition yet.
00:33:15.050 --> 00:33:18.470
The definition we will
use for convergence
00:33:18.470 --> 00:33:20.780
is if all the subgraph--
00:33:20.780 --> 00:33:23.480
all the homomorphism densities
were equivalently subgraph
00:33:23.480 --> 00:33:24.660
densities, they converge.
00:33:27.250 --> 00:33:28.960
So this is the definition.
00:33:28.960 --> 00:33:30.380
It's not required.
00:33:30.380 --> 00:33:33.910
So this is basically
rigorous as stated.
00:33:33.910 --> 00:33:38.890
Just as a remark,
it's not required
00:33:38.890 --> 00:33:46.350
that the number of
vertices goes to infinity,
00:33:46.350 --> 00:33:50.220
although you really should
think that that is the case.
00:33:57.140 --> 00:33:59.000
So just to put it
out there-- so I
00:33:59.000 --> 00:34:00.620
can have a sequence
of constant graphs
00:34:00.620 --> 00:34:02.037
and they will still
be convergent.
00:34:02.037 --> 00:34:03.283
And that's still OK.
00:34:03.283 --> 00:34:04.700
But you should
think of the number
00:34:04.700 --> 00:34:05.980
of vertices going to infinity.
00:34:05.980 --> 00:34:06.874
Yeah?
00:34:06.874 --> 00:34:09.560
AUDIENCE: What is F
in the definition?
00:34:09.560 --> 00:34:10.820
YUFEI ZHAO: F is H. Thank you.
00:34:14.670 --> 00:34:15.868
Any other questions?
00:34:32.380 --> 00:34:35.280
So there are some questions
that we'd like to discuss.
00:34:35.280 --> 00:34:37.980
And this will occupy
the next few lectures
00:34:37.980 --> 00:34:41.080
in terms of proving the
following statements.
00:34:41.080 --> 00:34:44.699
One is do you always
have graph limits?
00:34:44.699 --> 00:34:48.699
If you have a convergent
sequence of graphs,
00:34:48.699 --> 00:34:53.129
do they always approach a limit?
00:34:53.129 --> 00:34:55.440
And just because
something is convergent
00:34:55.440 --> 00:34:58.670
doesn't mean you can represent
the limit necessarily.
00:34:58.670 --> 00:35:01.080
So it turns out
the answer is yes.
00:35:01.080 --> 00:35:03.200
It turns out that--
and this makes
00:35:03.200 --> 00:35:05.400
it a good theory, a
good, useful theory,
00:35:05.400 --> 00:35:07.870
and an easy theory to use,
that there is always a limit
00:35:07.870 --> 00:35:11.530
object whenever you
have convergence.
00:35:11.530 --> 00:35:14.580
And the other
question is while we
00:35:14.580 --> 00:35:18.880
have described intuitively
one notion of convergence
00:35:18.880 --> 00:35:23.440
and also defined more
rigorously another definition
00:35:23.440 --> 00:35:27.370
of convergence, are these
two notions compatible?
00:35:27.370 --> 00:35:29.410
And what does this
even mean, this idea
00:35:29.410 --> 00:35:32.720
of image becoming closer
and closer to a final image?
00:35:32.720 --> 00:35:34.303
What does that even mean?
00:35:34.303 --> 00:35:35.720
So these are some
of the questions
00:35:35.720 --> 00:35:37.290
that I would like to address.
00:35:37.290 --> 00:35:42.440
So in the next few things
that I would like to discuss,
00:35:42.440 --> 00:35:44.870
first, I want to
give you a definition
00:35:44.870 --> 00:35:50.390
of a distance between two
graphons or two graphs.
00:35:50.390 --> 00:35:54.980
If I give you two graphs,
how similar or dissimilar
00:35:54.980 --> 00:35:56.860
are they--
00:35:56.860 --> 00:35:58.770
so that we have this metric.
00:35:58.770 --> 00:36:01.460
And then we can talk about
convergence in metric spaces.
00:36:04.040 --> 00:36:06.150
So let's take a quick break.
00:36:09.610 --> 00:36:11.700
So given this notion
of convergence,
00:36:11.700 --> 00:36:14.620
I would like to define
the notion of distance
00:36:14.620 --> 00:36:18.520
between graphs so that
convergence corresponds
00:36:18.520 --> 00:36:21.220
to convergence in
the metric space
00:36:21.220 --> 00:36:23.980
sense of distance going to 0.
00:36:23.980 --> 00:36:25.240
So how can we define distance?
00:36:36.742 --> 00:36:38.825
First, let me tell you
that there's a trivial way.
00:36:38.825 --> 00:36:41.200
And so there's a way in which
you look at that definition
00:36:41.200 --> 00:36:42.830
and produce a distance out.
00:36:42.830 --> 00:36:45.800
And here's what you can do.
00:36:45.800 --> 00:36:50.000
I can convert that
definition to a metric
00:36:50.000 --> 00:37:00.310
by setting the
distance between two
00:37:00.310 --> 00:37:10.150
graphs G and G prime to be the
following quantity, obtained
00:37:10.150 --> 00:37:11.920
by--
00:37:11.920 --> 00:37:12.920
what would I like to do?
00:37:12.920 --> 00:37:15.350
I would like to say the
distance goes to 0 if and only
00:37:15.350 --> 00:37:21.010
if the homomorphism
densities, they
00:37:21.010 --> 00:37:22.660
are all close to each other.
00:37:22.660 --> 00:37:30.290
And so I can sum up all
the homomorphism densities
00:37:30.290 --> 00:37:37.220
and look at their differences
between G and G prime.
00:37:37.220 --> 00:37:42.980
And I simply enumerate the
list of all possible graphs.
00:37:55.450 --> 00:37:57.950
I want to be just slightly more
careful with this definition
00:37:57.950 --> 00:38:00.390
here because I want
something which--
00:38:00.390 --> 00:38:03.380
so when I write
this, this number
00:38:03.380 --> 00:38:06.770
might be infinite for
all pairs G and G prime.
00:38:06.770 --> 00:38:12.400
So if I just add a scaling
factor here, then--
00:38:12.400 --> 00:38:15.140
and this is some distance.
00:38:15.140 --> 00:38:16.340
So this is some distance.
00:38:16.340 --> 00:38:18.890
And you see that it matches
the definition up there.
00:38:18.890 --> 00:38:21.050
But it's completely useless.
00:38:21.050 --> 00:38:22.040
It might as well--
00:38:22.040 --> 00:38:23.540
might as well not
have said anything
00:38:23.540 --> 00:38:29.080
because it's tautologically the
same as what happened up there.
00:38:29.080 --> 00:38:31.490
And if I give you two
graphs, it doesn't really
00:38:31.490 --> 00:38:32.930
tell you all that
much information
00:38:32.930 --> 00:38:35.360
except to encapsulate
that definition
00:38:35.360 --> 00:38:37.590
into a single number.
00:38:37.590 --> 00:38:38.090
Great.
00:38:38.090 --> 00:38:40.730
So I'm just-- the point of
this is just to tell you
00:38:40.730 --> 00:38:45.470
that there is always a trivial
way to define distance.
00:38:45.470 --> 00:38:49.160
But we want some more
interesting ways.
00:38:49.160 --> 00:38:51.920
So what can we do?
00:38:51.920 --> 00:38:55.310
So here is an attempt, which
is that of an edit distance.
00:38:58.690 --> 00:39:01.460
So we have seen this before when
we discussed removal lemmas.
00:39:01.460 --> 00:39:04.250
The edit distance is
the number of edges
00:39:04.250 --> 00:39:09.780
you need to change to go from
one graph to the other graph.
00:39:09.780 --> 00:39:12.060
And this seems like a pretty
reasonable thing to do.
00:39:12.060 --> 00:39:16.240
And it is an important
quantity for many applications,
00:39:16.240 --> 00:39:19.970
but turns out not the right
one for all application.
00:39:19.970 --> 00:39:21.840
And here is the reason.
00:39:21.840 --> 00:39:25.960
So this is why the
edit distance is--
00:39:25.960 --> 00:39:30.180
by edit distance, I
mean 1 over the number
00:39:30.180 --> 00:39:38.470
of vertex squared times the
number of edge changes needed.
00:39:43.148 --> 00:39:45.440
So there's normalization so
that the distance is always
00:39:45.440 --> 00:39:47.310
between 0 and 1.
00:39:47.310 --> 00:39:48.890
But this is not a
very good notion
00:39:48.890 --> 00:39:51.420
for the following reason.
00:39:51.420 --> 00:39:58.970
If I take two copies of the
Erdos-Reyni random graph G, n,
00:39:58.970 --> 00:40:04.180
1/2, what do you think is
the edit distance between two
00:40:04.180 --> 00:40:05.320
such random graphs?
00:40:10.440 --> 00:40:11.080
How many edges?
00:40:11.080 --> 00:40:11.580
Yeah?
00:40:11.580 --> 00:40:14.045
AUDIENCE: Isn't it roughly
one-half of the number of edges
00:40:14.045 --> 00:40:15.712
because there's like
a one-half probably
00:40:15.712 --> 00:40:21.440
that won't be there or
not be there [INAUDIBLE]??
00:40:21.440 --> 00:40:23.607
YUFEI ZHAO: So yeah, so
let me try to rephrase
00:40:23.607 --> 00:40:24.440
what you are saying.
00:40:24.440 --> 00:40:29.680
So suppose I have
this G and G prime
00:40:29.680 --> 00:40:37.850
both sitting on top
of the vertex set n.
00:40:37.850 --> 00:40:40.790
So if I'm not allowed to
rearrange the vertices,
00:40:40.790 --> 00:40:45.860
how many edge changes do I need
to go from one to the other?
00:40:45.860 --> 00:40:47.700
I need about 1/2.
00:41:01.890 --> 00:41:05.910
So one-half the time, I'm going
to have a wrong edge there.
00:41:05.910 --> 00:41:09.020
Now you can make this
number just slightly smaller
00:41:09.020 --> 00:41:11.330
by permuting the vertices.
00:41:11.330 --> 00:41:13.490
But actually you will
not improve that much.
00:41:13.490 --> 00:41:17.060
It is still going to be roughly
that edit distance, which
00:41:17.060 --> 00:41:18.560
is quite large.
00:41:18.560 --> 00:41:22.520
This is almost as large
as you can possibly get
00:41:22.520 --> 00:41:25.960
between two arbitrary graphs.
00:41:25.960 --> 00:41:30.510
So if we want to say
that random graphs,
00:41:30.510 --> 00:41:33.660
they approach a
limit, a single limit,
00:41:33.660 --> 00:41:36.630
then this is not a very
good notion because they are
00:41:36.630 --> 00:41:38.240
quite far apart for every n.
00:41:41.120 --> 00:41:45.110
So this is the reason why
the more obvious suggestion
00:41:45.110 --> 00:41:49.590
of an edit distance might
not be such a great idea.
00:41:49.590 --> 00:41:51.700
So what should we use instead?
00:41:51.700 --> 00:41:54.650
So we should take
inspiration from what we
00:41:54.650 --> 00:41:56.595
discussed in quasirandomness.
00:41:56.595 --> 00:41:58.416
You have a question.
00:41:58.416 --> 00:42:01.017
AUDIENCE: Is the edit
distance only for two graphs
00:42:01.017 --> 00:42:02.665
of the same vertex set?
00:42:02.665 --> 00:42:05.040
YUFEI ZHAO: So the question
is, is the edit distance only
00:42:05.040 --> 00:42:07.480
for two graphs with
the same vertex set?
00:42:07.480 --> 00:42:08.280
Let's say yes.
00:42:08.280 --> 00:42:10.500
So we'll see later
on, you can also
00:42:10.500 --> 00:42:15.580
compare graphs with
different number of vertices.
00:42:15.580 --> 00:42:18.540
So hold onto that thought.
00:42:18.540 --> 00:42:21.480
So I would like to come up
with a notion of distance
00:42:21.480 --> 00:42:24.660
between graphs that is
inspired by our discussion
00:42:24.660 --> 00:42:28.120
of quasirandomness earlier.
00:42:28.120 --> 00:42:34.940
So think about the discussion of
quasirandomness or quasirandom
00:42:34.940 --> 00:42:35.440
graphs.
00:42:42.920 --> 00:42:56.660
In what sense can G be close
to a constant, let's say p?
00:42:59.300 --> 00:43:03.050
And so this was the
Chung-Graham-Wilson theorem
00:43:03.050 --> 00:43:05.460
that we proved a
few lectures ago.
00:43:05.460 --> 00:43:07.790
So in what sense
can G be close to p?
00:43:07.790 --> 00:43:10.874
And one of those
definitions was discrepancy.
00:43:16.570 --> 00:43:25.840
And discrepancy says that
if the following quantity
00:43:25.840 --> 00:43:31.540
is small for all
subsets x and y, which
00:43:31.540 --> 00:43:34.370
are subsets of vertices of G--
00:43:34.370 --> 00:43:37.390
so you remember, all of
you remember, this part,
00:43:37.390 --> 00:43:41.550
the discrepancy hypothesis
for quasirandomness.
00:43:41.550 --> 00:43:43.240
And this is a kind
of definition that we
00:43:43.240 --> 00:43:45.590
would like to describe
when two graphs are
00:43:45.590 --> 00:43:50.750
similar to each other, when they
are close in this discrepancy
00:43:50.750 --> 00:43:52.270
sense.
00:43:52.270 --> 00:43:56.410
So now, instead of a
graph and a number,
00:43:56.410 --> 00:43:59.280
what if now I have two graphs?
00:43:59.280 --> 00:44:05.820
I'll give you two
graphs of G and G prime.
00:44:05.820 --> 00:44:11.750
And what I would like to
say is that, if for now,
00:44:11.750 --> 00:44:21.490
so if they have the
same vertex set,
00:44:21.490 --> 00:44:32.460
I want to say that there
are close if I have
00:44:32.460 --> 00:44:37.170
that the number of edges
between x and y in G
00:44:37.170 --> 00:44:42.390
is very close to
the number of edges
00:44:42.390 --> 00:44:46.860
between x and y in G prime.
00:44:46.860 --> 00:44:55.245
And I normalize by the
number of vertices squared,
00:44:55.245 --> 00:44:58.040
so n this number of vertices.
00:44:58.040 --> 00:45:01.800
And I would like to find out
the worst possible scenario, so
00:45:01.800 --> 00:45:07.130
overall, x and y subsets
of the vertex set.
00:45:07.130 --> 00:45:12.408
If this quantity
is small, then I
00:45:12.408 --> 00:45:14.950
would like to say that G and G
prime are close to each other.
00:45:18.240 --> 00:45:20.810
So this is inspired by
this discrepancy notion.
00:45:24.850 --> 00:45:28.142
Can you see anything wrong
with this definition here?
00:45:28.142 --> 00:45:28.642
Yeah?
00:45:28.642 --> 00:45:30.620
AUDIENCE: [INAUDIBLE]
00:45:30.620 --> 00:45:33.230
YUFEI ZHAO: So
permutations are vertices.
00:45:33.230 --> 00:45:37.140
So just like in the checkerboard
example we saw earlier,
00:45:37.140 --> 00:45:38.460
you have two graphs.
00:45:38.460 --> 00:45:40.860
And if they are
indeed labeled graphs
00:45:40.860 --> 00:45:43.920
in the same labeled
vertex set, then this
00:45:43.920 --> 00:45:46.470
is the definition more
or less what we used.
00:45:46.470 --> 00:45:48.600
I will define it more
precisely in a second.
00:45:48.600 --> 00:45:51.690
But if they are
unlabeled vertices,
00:45:51.690 --> 00:46:01.480
we need to possibly
optimize permutations
00:46:01.480 --> 00:46:12.470
over rearrangements of vertices,
which actually turns out
00:46:12.470 --> 00:46:13.630
to be quite subtle.
00:46:13.630 --> 00:46:16.160
So I'm going to give precise
definitions in a second.
00:46:16.160 --> 00:46:19.133
But this one here, so think
about permuting vertices.
00:46:19.133 --> 00:46:21.050
But it's actually a bit
more subtle than that.
00:46:26.680 --> 00:46:30.750
So here are some
actual definitions.
00:46:30.750 --> 00:46:34.630
I'm going to define this
quantity called a cut norm.
00:46:39.490 --> 00:46:43.360
So this chapter is all going to
be somewhat functional analytic
00:46:43.360 --> 00:46:44.570
in nature.
00:46:44.570 --> 00:46:47.680
So get used to the
analytic language.
00:46:47.680 --> 00:46:57.560
So the cut norm of W is defined
to be the following quantity
00:46:57.560 --> 00:47:02.510
denoted by this norm with a
box in the subscript, which
00:47:02.510 --> 00:47:04.860
is defined to be--
00:47:04.860 --> 00:47:11.910
if I look at this W, and
I integrate it over a box,
00:47:11.910 --> 00:47:15.420
and I would like to
maximize this quantity here
00:47:15.420 --> 00:47:19.590
over choices of
boxes S and T, they
00:47:19.590 --> 00:47:22.545
are subsets of the interval
measurable subsets.
00:47:27.390 --> 00:47:31.950
So choose your-- so over
all possible choices
00:47:31.950 --> 00:47:36.630
of measurable
subsets S and T, if I
00:47:36.630 --> 00:47:39.630
integrate W over
S cross T, what is
00:47:39.630 --> 00:47:41.760
the furthest I can get from 0?
00:47:47.117 --> 00:47:48.700
So this is the
definition of cut norm.
00:47:48.700 --> 00:47:51.550
And you can already see that
it has some relations to what
00:47:51.550 --> 00:47:53.100
were discussed up there.
00:47:53.100 --> 00:47:54.920
But while we're
talking about norms,
00:47:54.920 --> 00:47:56.690
let me just mention a
few other norms that
00:47:56.690 --> 00:48:00.950
might come up later on when
we discuss graph limits.
00:48:00.950 --> 00:48:03.920
So there will be a lot
of norms throughout.
00:48:03.920 --> 00:48:09.440
So in particular, the lp norm is
going to play a frequent role.
00:48:09.440 --> 00:48:15.170
So lp norm is defined by
looking at the peak norm
00:48:15.170 --> 00:48:17.690
of the absolute value,
integrated and then
00:48:17.690 --> 00:48:20.960
raised to 1 over p.
00:48:20.960 --> 00:48:26.500
And so the infinity norm--
00:48:26.500 --> 00:48:30.610
so this is almost, but not
quite the same as the sup--
00:48:34.940 --> 00:48:37.520
so almost the same
as the supremum,
00:48:37.520 --> 00:48:44.210
but not quite because I need
to ignore subsets of measure 0.
00:48:54.940 --> 00:48:57.760
So I can write down a formal
definition in a second.
00:48:57.760 --> 00:48:59.470
But I need to--
00:48:59.470 --> 00:49:01.447
if I change W on the
subset of the measure 0,
00:49:01.447 --> 00:49:03.030
I shouldn't change
any of these norms.
00:49:03.030 --> 00:49:07.090
And so the one way to define
this essential supremum--
00:49:07.090 --> 00:49:09.630
it's called an essential sup--
00:49:09.630 --> 00:49:14.360
is that it is the largest--
00:49:14.360 --> 00:49:20.920
so it is the smallest
lambda such that--
00:49:20.920 --> 00:49:25.210
so the smallest number
m such that the measure
00:49:25.210 --> 00:49:40.700
of the set taking value bigger
than m this set has measure 0.
00:49:40.700 --> 00:49:46.140
So it's the threshold
above which you--
00:49:46.140 --> 00:49:47.670
this, it has measure 0.
00:49:50.890 --> 00:49:53.983
And the l2 norm will play a
particularly special role.
00:49:53.983 --> 00:49:55.400
And for the l2
norm, you're really
00:49:55.400 --> 00:49:57.350
in the Hilbert
space, in which case
00:49:57.350 --> 00:50:00.770
we are going to
have inner products.
00:50:00.770 --> 00:50:04.380
And we denote inner
products using the square--
00:50:04.380 --> 00:50:06.430
using these brackets.
00:50:06.430 --> 00:50:07.590
So everything is real.
00:50:07.590 --> 00:50:10.231
I don't have to worry
about complex conjugates.
00:50:16.080 --> 00:50:17.910
So comparing with the
discussion up there,
00:50:17.910 --> 00:50:21.535
we see that a sequence of--
00:50:21.535 --> 00:50:32.328
so sequence Gn of
quasirandom graphs
00:50:32.328 --> 00:50:40.050
has a property that the
associated graphons converge
00:50:40.050 --> 00:50:42.470
to p in the cut norm.
00:50:55.610 --> 00:50:58.580
For quasirandom graphs,
there is no issue
00:50:58.580 --> 00:51:02.510
having to do with permutations
because the target is
00:51:02.510 --> 00:51:04.670
invariant upon permutations.
00:51:04.670 --> 00:51:06.410
But if I give you
two different graphs,
00:51:06.410 --> 00:51:09.110
then I need to think
about their permutations.
00:51:09.110 --> 00:51:12.530
And to study
permutations of vertices,
00:51:12.530 --> 00:51:15.290
the right way to do
this is to consider
00:51:15.290 --> 00:51:19.640
measure-preserving
transformations.
00:51:19.640 --> 00:51:27.870
So we say that phi from the
interval to the interval
00:51:27.870 --> 00:51:36.790
is measure-preserving
because first of all,
00:51:36.790 --> 00:51:38.040
it has to be a measurable map.
00:51:38.040 --> 00:51:40.500
And everything I'm going to
talk about are measurable.
00:51:40.500 --> 00:51:43.290
So sometimes I will
even omit mentioning it.
00:51:43.290 --> 00:51:53.480
So it is measure-preserving
if, for all measurable subsets
00:51:53.480 --> 00:52:07.160
A of this interval, one
has that the pullback of A
00:52:07.160 --> 00:52:09.200
has the same
measure as A itself.
00:52:13.720 --> 00:52:15.210
Let me give you an example.
00:52:15.210 --> 00:52:17.710
So you have to be also slightly
careful with this definition
00:52:17.710 --> 00:52:22.090
if you think about the
pushforward that's false.
00:52:22.090 --> 00:52:23.650
It has to be the pullback.
00:52:23.650 --> 00:52:30.930
So for example, the
map which sends--
00:52:30.930 --> 00:52:39.860
so an easy example, the map
which sends x to x plus 1/2--
00:52:39.860 --> 00:52:43.720
so think about a
circle as your space.
00:52:43.720 --> 00:52:47.900
And here I am just rotating the
circle by one-half rotation.
00:52:47.900 --> 00:52:49.625
So it's obviously
measure-preserving.
00:52:49.625 --> 00:52:53.350
I am not changing any measures.
00:52:53.350 --> 00:52:56.020
Slightly more
interesting example,
00:52:56.020 --> 00:53:00.340
quite a bit more interesting
example is setting x to 2x.
00:53:00.340 --> 00:53:02.350
This is also measure-preserving.
00:53:02.350 --> 00:53:04.940
And you might be
puzzled for a second why
00:53:04.940 --> 00:53:07.680
it's measure-preserving because
it sounds like it's dilating
00:53:07.680 --> 00:53:10.262
everything by a factor of 2.
00:53:10.262 --> 00:53:12.220
But if you look at the
definition-- and so here
00:53:12.220 --> 00:53:15.660
is again mod 1.
00:53:15.660 --> 00:53:25.790
If you look at the
definition, if you look at,
00:53:25.790 --> 00:53:32.300
let's say, a subset
A, which is--
00:53:35.020 --> 00:53:36.770
so what should I think?
00:53:40.018 --> 00:53:46.660
For example, so if that is my
A, so what's the inverse of A?
00:53:51.460 --> 00:53:54.250
So it's this set.
00:53:59.420 --> 00:54:03.105
So the measure is preserved
upon this pullback.
00:54:03.105 --> 00:54:05.360
And so if you
pushforward, then you
00:54:05.360 --> 00:54:06.680
might dilate by a factor of 2.
00:54:06.680 --> 00:54:08.857
But when you pullback, the
measure gets preserved.
00:54:15.400 --> 00:54:17.920
So these measure-preserving
transformations
00:54:17.920 --> 00:54:23.260
are going to play role of
permutations of vertices.
00:54:23.260 --> 00:54:25.900
So it turns out that these
things are actually-- they
00:54:25.900 --> 00:54:28.420
are quite subtle technically.
00:54:28.420 --> 00:54:31.510
And I am going to,
as much as I can,
00:54:31.510 --> 00:54:35.340
ignore some of the measure
theoretic technicalities.
00:54:35.340 --> 00:54:37.690
But they are quite subtle.
00:54:37.690 --> 00:54:40.180
So for example, so
now let me give you
00:54:40.180 --> 00:54:45.240
a definition for the distance
between two graphons.
00:54:45.240 --> 00:54:51.290
I write, starting with a
symmetric measurable function
00:54:51.290 --> 00:55:07.290
W, so I write W superscript phi
to denote the function obtained
00:55:07.290 --> 00:55:07.980
as follows.
00:55:07.980 --> 00:55:10.770
So I think of this as relabeling
the vertices of a graph.
00:55:14.480 --> 00:55:20.390
And now I define this distance.
00:55:20.390 --> 00:55:31.680
So this is going to be called
the cut distance between two
00:55:31.680 --> 00:55:34.860
symmetric measurable
functions, U and W,
00:55:34.860 --> 00:55:48.520
to be the infimum over all
measure-preserving bijections.
00:56:02.310 --> 00:56:04.340
So this is the definition
for the distance
00:56:04.340 --> 00:56:05.410
between two graphons.
00:56:09.648 --> 00:56:13.020
To take the optimal--
00:56:13.020 --> 00:56:15.440
and my question does it--
00:56:15.440 --> 00:56:16.930
I am looking at nth.
00:56:16.930 --> 00:56:18.700
So I haven't told
you yet whether you
00:56:18.700 --> 00:56:19.710
can take a single one.
00:56:19.710 --> 00:56:21.335
And it turns out
that's a subtle issue.
00:56:21.335 --> 00:56:23.050
And generally it doesn't exist.
00:56:23.050 --> 00:56:26.320
But I look over all
measure-preserving bijections
00:56:26.320 --> 00:56:27.010
phi.
00:56:27.010 --> 00:56:31.450
And I look at the
distance between W and Wv,
00:56:31.450 --> 00:56:35.061
optimized over the best possible
measure-preserving bijection.
00:56:39.680 --> 00:56:41.571
So this nth is really an nth.
00:56:45.128 --> 00:56:46.170
It's not always obtained.
00:56:49.270 --> 00:56:54.210
And actually, this example
here is a great example for--
00:56:54.210 --> 00:56:58.360
you can create an example for
why nth is not always obtained
00:56:58.360 --> 00:57:00.580
from the discussion over here.
00:57:00.580 --> 00:57:08.390
For example, if U is
the function x times y,
00:57:08.390 --> 00:57:18.020
this is a graphon
xy and W is Uv,
00:57:18.020 --> 00:57:28.793
where v is the map distance
x to 2x, then in your mind,
00:57:28.793 --> 00:57:31.210
you should think of these two
as really the same graphons.
00:57:31.210 --> 00:57:32.410
You are applying the
measure-preserving
00:57:32.410 --> 00:57:33.035
transformation.
00:57:33.035 --> 00:57:35.010
It's like doing a permutation.
00:57:35.010 --> 00:57:39.550
But because phi
is not bijective,
00:57:39.550 --> 00:57:43.980
you cannot putting phi here
to get these two things to be
00:57:43.980 --> 00:57:44.480
the same.
00:57:48.812 --> 00:57:50.020
So there are some subtleties.
00:57:50.020 --> 00:57:52.330
So this is really
an example just
00:57:52.330 --> 00:57:54.370
to highlight there's
some subtleties here,
00:57:54.370 --> 00:57:58.420
which I am going to try to
ignore as much as possible.
00:57:58.420 --> 00:58:02.067
But I will always give
you correct definitions.
00:58:02.067 --> 00:58:02.650
Any questions?
00:58:02.650 --> 00:58:03.150
Yeah?
00:58:06.533 --> 00:58:07.033
Yeah?
00:58:07.033 --> 00:58:09.575
AUDIENCE: So can we expect the
cut distance between these two
00:58:09.575 --> 00:58:10.733
sets to be 0 [INAUDIBLE]?
00:58:10.733 --> 00:58:12.150
YUFEI ZHAO: So the
question, do we
00:58:12.150 --> 00:58:14.640
expect the cut distance
between these two to be 0?
00:58:14.640 --> 00:58:16.280
And the answer is yes.
00:58:16.280 --> 00:58:19.510
So we do expect them to be 0.
00:58:19.510 --> 00:58:22.100
And they are 0.
00:58:22.100 --> 00:58:24.620
They are equal to 0.
00:58:24.620 --> 00:58:27.890
And let me just tell you
one something that is new.
00:58:27.890 --> 00:58:31.190
And this is one of
those statements
00:58:31.190 --> 00:58:34.250
that has a lot of measure
theoretic technicalities.
00:58:34.250 --> 00:58:42.650
For all graphons U and W, it
turns out that there exist
00:58:42.650 --> 00:58:43.940
measure-preserving maps--
00:58:52.020 --> 00:58:57.460
so not necessarily bijections,
but measure-preserving maps
00:58:57.460 --> 00:59:02.720
from 0, 1 interval to itself,
such that the distance between
00:59:02.720 --> 00:59:09.530
U and W, the cut distance,
is obtained by the cut norm
00:59:09.530 --> 00:59:10.730
difference between--
00:59:10.730 --> 00:59:14.600
the difference between
U phi and W psi.
00:59:27.314 --> 00:59:29.770
So don't worry about it.
00:59:40.840 --> 00:59:43.500
So far, we have defined
this notion of a cut
00:59:43.500 --> 00:59:47.170
distance between two graphons.
00:59:47.170 --> 00:59:49.750
But now I'll give
you two graphs.
00:59:49.750 --> 00:59:53.855
So what do you do
for two graphs?
00:59:53.855 --> 00:59:55.396
Or I can-- yeah?
00:59:55.396 --> 00:59:57.438
AUDIENCE: You can take
the graphon associated it.
00:59:57.438 --> 00:59:58.188
YUFEI ZHAO: Great.
00:59:58.188 --> 01:00:00.550
So take the graphon
associated with these graphs
01:00:00.550 --> 01:00:03.020
and consider their cut distance.
01:00:03.020 --> 01:00:12.310
So for graphs G and G
prime, and potentially even
01:00:12.310 --> 01:00:19.570
a different number
of vertices, I
01:00:19.570 --> 01:00:25.540
can define the distance, the
cut distance between these two
01:00:25.540 --> 01:00:34.650
graphs to be the distance
between the associated
01:00:34.650 --> 01:00:35.150
graphons.
01:00:40.100 --> 01:00:47.338
And similarly, if I have
a graph and a graphon,
01:00:47.338 --> 01:00:48.755
I can also compare
their distance.
01:01:01.660 --> 01:01:03.340
So what does this actually mean?
01:01:03.340 --> 01:01:05.190
So if I give you
two graphs, even
01:01:05.190 --> 01:01:08.240
with the same
number of vertices,
01:01:08.240 --> 01:01:12.620
it's not quite the same thing
as a permutation of vertices.
01:01:12.620 --> 01:01:14.930
It's a bit more subtle.
01:01:14.930 --> 01:01:17.520
Now why is it more subtle than
just permuting the vertices?
01:01:20.650 --> 01:01:23.180
So here we are using
measure-preserving
01:01:23.180 --> 01:01:27.020
transformations, which doesn't
see your atomic vertices.
01:01:27.020 --> 01:01:30.120
So we might split
up your vertices.
01:01:30.120 --> 01:01:33.510
So you might take a
vertex and chop it in half
01:01:33.510 --> 01:01:35.880
and send one half
somewhere and another half
01:01:35.880 --> 01:01:38.080
somewhere else
because these guys,
01:01:38.080 --> 01:01:41.060
they don't care about
your vertices anymore.
01:01:41.060 --> 01:01:47.130
So it's not quite the same
as permuting vertices.
01:01:52.450 --> 01:01:54.010
But it's some kind of--
01:01:54.010 --> 01:01:59.530
so you allow some kind of
splitting and rearrangement
01:01:59.530 --> 01:02:01.760
and overlays.
01:02:01.760 --> 01:02:04.510
So you can write
out this distance
01:02:04.510 --> 01:02:08.230
in this format, find out the
best way to split and overlay
01:02:08.230 --> 01:02:09.760
and to rearrange that way.
01:02:09.760 --> 01:02:13.250
But it's much cleaner to
define it in terms of graphons.
01:02:13.250 --> 01:02:13.750
Yes?
01:02:13.750 --> 01:02:15.792
AUDIENCE: Is this why we
take bijections up there
01:02:15.792 --> 01:02:16.972
[INAUDIBLE]?
01:02:16.972 --> 01:02:19.430
YUFEI ZHAO: The question is,
is that why we take bijections
01:02:19.430 --> 01:02:19.930
up there?
01:02:19.930 --> 01:02:22.370
And no, so up there,
if I wrote instead
01:02:22.370 --> 01:02:26.270
measure-preserving maps, it's
still a correct definition
01:02:26.270 --> 01:02:28.740
and it's the same definition.
01:02:28.740 --> 01:02:30.930
And the fact that these
two are equivalent
01:02:30.930 --> 01:02:35.660
goes to some measure
theory, which I will not--
01:02:35.660 --> 01:02:44.295
do not want to indulge yo Great.
01:02:44.295 --> 01:02:46.420
But the moral of the story
is you take two graphons
01:02:46.420 --> 01:02:50.620
and rearrange the vertices
in some way, in the best way,
01:02:50.620 --> 01:02:53.603
overlay them on top of each
other and take the difference
01:02:53.603 --> 01:02:54.645
and look at the cut norm.
01:02:54.645 --> 01:02:55.810
And so that's the distance.
01:02:59.180 --> 01:03:04.570
So I want to finish by
stating the main theorems
01:03:04.570 --> 01:03:07.890
that form graph limit theory.
01:03:07.890 --> 01:03:10.620
And these address the
questions I mentioned right
01:03:10.620 --> 01:03:11.940
before the break.
01:03:11.940 --> 01:03:14.960
So do there exist limits?
01:03:14.960 --> 01:03:18.620
And do these two
different notions of one
01:03:18.620 --> 01:03:20.900
having to do with
distance and another
01:03:20.900 --> 01:03:24.410
having to do with
homomorphism densities,
01:03:24.410 --> 01:03:26.400
how do they relate
to each other?
01:03:26.400 --> 01:03:27.590
Are they consistent?
01:03:50.070 --> 01:03:54.420
So the first theorem,
Theorem 1, has
01:03:54.420 --> 01:04:07.480
to do with the equivalence
of the convergence,
01:04:07.480 --> 01:04:22.800
namely, that if you have a
sequence of graphs or graphons,
01:04:22.800 --> 01:04:30.540
the sequence is convergent
in the sense, up there, if
01:04:30.540 --> 01:04:38.380
and only if they are
convergent in the sense
01:04:38.380 --> 01:04:40.440
of-- in this metric space.
01:04:40.440 --> 01:04:43.360
So remember what convergence
means in the metric space
01:04:43.360 --> 01:04:46.310
is that of a Cauchy sequence--
01:04:46.310 --> 01:04:55.600
so if and only if it is a
Cauchy sequence with respect
01:04:55.600 --> 01:05:02.470
to this cut distance.
01:05:02.470 --> 01:05:05.360
So it's just-- maybe
for many of you,
01:05:05.360 --> 01:05:07.140
it's been a while
since you took 18-100.
01:05:07.140 --> 01:05:10.800
So let remind you a Cauchy
sequence, in this case,
01:05:10.800 --> 01:05:20.310
it means that, if I look at the
distance between two graphs,
01:05:20.310 --> 01:05:22.920
if I look far enough
out, then I can
01:05:22.920 --> 01:05:28.500
contain the rest of the sequence
in an arbitrarily small ball.
01:05:28.500 --> 01:05:33.150
So the sup positive
m of this guy here,
01:05:33.150 --> 01:05:37.986
goes to 0 as n goes to infinity.
01:05:37.986 --> 01:05:42.173
But because we don't know
yet whether the limit exists,
01:05:42.173 --> 01:05:44.340
so I can't talk about them
getting closer and closer
01:05:44.340 --> 01:05:45.060
to a limit.
01:05:45.060 --> 01:05:47.110
But they mutually get
closer to each other.
01:05:50.320 --> 01:05:54.350
So Theorem 1 tells us that
these two notions, one having
01:05:54.350 --> 01:05:57.980
to do with homomorphism
densities, is consistent
01:05:57.980 --> 01:06:02.360
and in fact equivalent
to the appropriate notion
01:06:02.360 --> 01:06:03.450
in the metric space.
01:06:11.200 --> 01:06:16.380
So let's use a symbol.
01:06:16.380 --> 01:06:27.250
So we say that G sub n
converges to W, or in the case
01:06:27.250 --> 01:06:29.020
of a sequence of graphons.
01:06:29.020 --> 01:06:30.520
So we can do that as well.
01:06:34.630 --> 01:06:43.440
So here we say that G
sub n converges with W,
01:06:43.440 --> 01:06:48.960
if whenever you look at
the F density in G sub n,
01:06:48.960 --> 01:06:56.610
this sequence converges to the
corresponding f density in W
01:06:56.610 --> 01:07:01.360
for every f, and
similarly, if you have
01:07:01.360 --> 01:07:04.480
a graphon instead of a graph.
01:07:04.480 --> 01:07:08.160
So that definition was
just whether a sequence
01:07:08.160 --> 01:07:09.140
is convergent.
01:07:09.140 --> 01:07:16.190
Here it converges
to this graphon W.
01:07:16.190 --> 01:07:21.380
And the question is, if you
give me a convergent sequence,
01:07:21.380 --> 01:07:23.150
is there a limit?
01:07:23.150 --> 01:07:25.560
Does it converge to some limit?
01:07:25.560 --> 01:07:27.430
And the answer is yes.
01:07:27.430 --> 01:07:34.070
And that's the
second theorem, which
01:07:34.070 --> 01:07:37.840
tells us the existence of a
limit, of the limit object.
01:07:41.990 --> 01:07:49.830
So the statement is that
every convergent sequence
01:07:49.830 --> 01:08:04.650
of graph or graphons
has a limit graphon.
01:08:18.346 --> 01:08:22.930
So now I want you to imagine
this space of graphons.
01:08:22.930 --> 01:08:25.960
So we'll have this space
containing all the graphons.
01:08:25.960 --> 01:08:30.149
And let me denote this
space by this curly W0.
01:08:30.149 --> 01:08:32.279
So this, the 0 is--
don't worry about it.
01:08:32.279 --> 01:08:34.080
It's more just convention.
01:08:34.080 --> 01:08:37.600
But let me also put a tilde on
top for the following reason.
01:08:37.600 --> 01:08:51.630
Let this be the space of
graphons where we identify
01:08:51.630 --> 01:08:53.250
graphons with distance 0.
01:09:09.640 --> 01:09:17.290
So then the space combined with
this metric is a metric space.
01:09:22.149 --> 01:09:23.380
It is the space of graphons.
01:09:26.060 --> 01:09:32.859
And so the third
theorem is that it's
01:09:32.859 --> 01:09:35.529
the compactness of
the space of graphons,
01:09:35.529 --> 01:09:37.345
namely, that this
space is compact.
01:09:50.970 --> 01:09:53.819
Because we're in
the metric space,
01:09:53.819 --> 01:09:59.340
compactness in the usual
sense of every open cover has
01:09:59.340 --> 01:10:04.260
a finite subcover is equivalent
to the slightly more intuitive
01:10:04.260 --> 01:10:06.550
notion of sequential
compactness--
01:10:06.550 --> 01:10:10.170
every sequence has a
convergent subsequence.
01:10:10.170 --> 01:10:13.530
And then it's also,
if you have a limit,
01:10:13.530 --> 01:10:15.710
so it converges to some limit.
01:10:18.330 --> 01:10:20.670
So how should you
think of Theorem 3?
01:10:20.670 --> 01:10:24.720
So it's about compactness
and some tautological notion.
01:10:24.720 --> 01:10:27.720
But intuitively, you
should think of compactness
01:10:27.720 --> 01:10:29.170
as saying--
01:10:29.170 --> 01:10:31.050
and the English word,
the English meaning
01:10:31.050 --> 01:10:33.330
of the word compact is small.
01:10:33.330 --> 01:10:37.050
You should think of this
space as being quite small,
01:10:37.050 --> 01:10:40.140
which is rather
counterintuitive because we're
01:10:40.140 --> 01:10:43.970
looking at the space of
graphons, certainly at least as
01:10:43.970 --> 01:10:48.150
large as the space of graphs,
but really all functions
01:10:48.150 --> 01:10:49.875
from the square to the interval.
01:10:49.875 --> 01:10:53.010
This seems like a
pretty large space.
01:10:53.010 --> 01:10:55.290
But this theorem here says
that, in fact, that space
01:10:55.290 --> 01:10:57.950
is quite small.
01:10:57.950 --> 01:11:01.880
And where have we also seen
that philosophy before?
01:11:06.120 --> 01:11:09.920
So in Szemeredi's
Graph Regularity Lemma,
01:11:09.920 --> 01:11:12.350
the underlying
philosophy there is
01:11:12.350 --> 01:11:15.560
that, even though the
possibilities, the space
01:11:15.560 --> 01:11:19.400
of possibilities for
graph is quite large,
01:11:19.400 --> 01:11:22.360
once you apply Szemeredi's
Regularity Lemma,
01:11:22.360 --> 01:11:26.400
and once you are OK with
some epsilon approximations,
01:11:26.400 --> 01:11:29.510
there is only a
small description,
01:11:29.510 --> 01:11:32.280
this bounded
description, of a graph.
01:11:32.280 --> 01:11:35.060
And you can work with
that description.
01:11:35.060 --> 01:11:37.820
And these two philosophies,
it's no coincidence
01:11:37.820 --> 01:11:41.270
that they are consistent
with each other
01:11:41.270 --> 01:11:46.790
because we will use Szemeredi's
Regularity Lemma to prove
01:11:46.790 --> 01:11:49.710
this compactness.
01:11:49.710 --> 01:11:51.900
In fact, we will use a
slightly weaker version
01:11:51.900 --> 01:11:55.560
of Szemeredi's Regularity
Lemma to prove compactness.
01:11:55.560 --> 01:11:58.890
And then you will see
that, from the compactness,
01:11:58.890 --> 01:12:03.010
one can use properties
of the compactness
01:12:03.010 --> 01:12:07.750
to boost to a stronger
version of regularity.
01:12:07.750 --> 01:12:09.890
But the underlying
philosophy here
01:12:09.890 --> 01:12:19.570
is that this compactness is
in some sense a quantity.
01:12:19.570 --> 01:12:29.530
It's a qualitative
reformulation,
01:12:29.530 --> 01:12:38.770
analytic reformulation of
Szemeredi's Graph Regularity
01:12:38.770 --> 01:12:39.270
Lemma.
01:12:51.030 --> 01:12:53.780
OK, so--
01:12:53.780 --> 01:12:55.560
So this topic,
this graph limits,
01:12:55.560 --> 01:12:57.840
which we'll explore for
the next few lecturers,
01:12:57.840 --> 01:13:01.620
including giving a proof of all
three of these main theorems,
01:13:01.620 --> 01:13:05.250
nicely encapsulates the past
couple of topics we have done.
01:13:05.250 --> 01:13:08.040
So on one hand, Szemeredi's
Regularity Lemma,
01:13:08.040 --> 01:13:09.930
or some version of
that, will be used
01:13:09.930 --> 01:13:14.520
in proving the existence of the
limit and also the compactness.
01:13:14.520 --> 01:13:17.370
And also it's philosophically
and in some sense
01:13:17.370 --> 01:13:19.830
related and very much
equivalent in some sense
01:13:19.830 --> 01:13:22.830
and related to these notions.
01:13:22.830 --> 01:13:25.710
It is also related
to quasirandomness--
01:13:25.710 --> 01:13:28.320
in particular,
quasirandom graphs
01:13:28.320 --> 01:13:31.680
that we did a few lectures ago,
where in quasirandom graphs,
01:13:31.680 --> 01:13:35.030
we are really looking
at the constant graphon
01:13:35.030 --> 01:13:36.890
in this language.
01:13:36.890 --> 01:13:39.340
And now we expand our horizons.
01:13:39.340 --> 01:13:43.570
And instead of just looking
at the constant graphon,
01:13:43.570 --> 01:13:47.840
we can now consider
arbitrary graphons.
01:13:47.840 --> 01:13:51.970
They are also this model
for a very large graph.
01:13:54.530 --> 01:13:56.110
Any questions?
01:13:56.110 --> 01:13:56.610
Yeah?
01:13:56.610 --> 01:14:00.530
AUDIENCE: Can we prove the
theorem analytically and then
01:14:00.530 --> 01:14:02.490
deduce the Regularity
Lemma with it?
01:14:02.490 --> 01:14:03.950
YUFEI ZHAO: The
question is, can we
01:14:03.950 --> 01:14:07.520
prove Theorem 3 analytically
and deduce the Regularity Lemma?
01:14:07.520 --> 01:14:09.990
So you will see once
you see the proof.
01:14:09.990 --> 01:14:11.600
It depends on what you mean.
01:14:11.600 --> 01:14:12.990
But roughly, the answer is yes.
01:14:12.990 --> 01:14:14.840
But there's a very
important caveat.
01:14:14.840 --> 01:14:17.930
It's that, because we
are using compactness,
01:14:17.930 --> 01:14:20.390
any argument
involving compactness
01:14:20.390 --> 01:14:23.750
gives no quantitative bounds.
01:14:23.750 --> 01:14:28.940
So you will have a proof of
the Szemeredi Regularity Lemma
01:14:28.940 --> 01:14:33.020
that tells you there is
a bound for each epsilon.
01:14:33.020 --> 01:14:37.305
But it doesn't tell
you what the bound is.
01:14:37.305 --> 01:14:38.778
Yeah?
01:14:38.778 --> 01:14:40.742
AUDIENCE: Doesn't
Theorem 3 imply Theorem 1
01:14:40.742 --> 01:14:43.197
because of the [INAUDIBLE]?
01:14:46.150 --> 01:14:48.280
YUFEI ZHAO: Does Code
Theorem 3 imply Theorem 1?
01:14:48.280 --> 01:14:50.800
And the answer is no
because in Theorem 1,
01:14:50.800 --> 01:14:55.800
the notion of convergence is
about homomorphism densities.
01:14:55.800 --> 01:14:59.000
So Theorem 1 is about
these two different notions
01:14:59.000 --> 01:15:02.730
of convergence and that they
are equivalent to each other.
01:15:02.730 --> 01:15:05.420
Theorem 3 is just
about the metric.
01:15:05.420 --> 01:15:07.506
It's about the cut metric.
01:15:07.506 --> 01:15:10.340
And so Theorem 1 is-- the point
of Theorem 1 is that you have
01:15:10.340 --> 01:15:11.470
these two--
01:15:11.470 --> 01:15:13.670
you have these two
notions of convergence,
01:15:13.670 --> 01:15:16.340
one having to do with subgraph
densities and the other
01:15:16.340 --> 01:15:18.052
having to do with
a cut distance.
01:15:18.052 --> 01:15:19.760
And in fact, they are
equivalent notions.
01:15:23.140 --> 01:15:24.870
So all great questions--
01:15:24.870 --> 01:15:25.959
any others?
01:15:25.959 --> 01:15:27.875
AUDIENCE: And for
that F, is F a graphon
01:15:27.875 --> 01:15:30.270
because the [INAUDIBLE]?
01:15:30.270 --> 01:15:32.838
Is F a graphon or a graph?
01:15:32.838 --> 01:15:35.130
YUFEI ZHAO: The question is,
is F a graph or a graphon?
01:15:35.130 --> 01:15:37.300
F is always a graph.
01:15:37.300 --> 01:15:45.880
So in t F, W, I do not define
this quantity for graphon F.
01:15:45.880 --> 01:15:48.000
So this quantity
here, I have only
01:15:48.000 --> 01:15:50.580
allowed the second
argument to be a graphon.
01:15:50.580 --> 01:15:52.740
The first argument is not
allowed to be a graphon.
01:15:52.740 --> 01:15:53.657
It doesn't make sense.
01:15:56.277 --> 01:15:56.777
Yeah?
01:15:56.777 --> 01:16:00.810
AUDIENCE: Doesn't Theorem 1
and 2 together imply Theorem 3?
01:16:00.810 --> 01:16:03.310
YUFEI ZHAO: The question is,
doesn't Theorem 1 and Theorem 2
01:16:03.310 --> 01:16:05.240
together imply Theorem 3?
01:16:05.240 --> 01:16:07.780
So first of all,
Theorem 1 is really--
01:16:07.780 --> 01:16:10.090
it's not about compactness.
01:16:10.090 --> 01:16:11.590
So it's really about
the equivalence
01:16:11.590 --> 01:16:13.452
of two different
notions of convergence.
01:16:13.452 --> 01:16:15.160
It's like you have
two different metrics.
01:16:15.160 --> 01:16:16.618
I am showing that
these two metrics
01:16:16.618 --> 01:16:18.300
are equivalent to each other.
01:16:18.300 --> 01:16:21.430
Theorem 2 and Theorem 3 are
quite intimately related.
01:16:21.430 --> 01:16:22.530
So Theorem 2 is about--
01:16:25.590 --> 01:16:27.788
Theorem 2, so they
are quite related.
01:16:27.788 --> 01:16:29.080
But they're not quite the same.
01:16:29.080 --> 01:16:31.290
So let me just give you
the real line analogy,
01:16:31.290 --> 01:16:33.140
going back to what we
said in the beginning.
01:16:33.140 --> 01:16:37.410
So Theorem 2 is kind of like
saying that the real numbers is
01:16:37.410 --> 01:16:38.490
complete.
01:16:38.490 --> 01:16:40.870
Every convergent
sequence has a limit,
01:16:40.870 --> 01:16:43.060
whereas Theorem 3
is more than that.
01:16:43.060 --> 01:16:45.295
It's also bounded in some sense.
01:16:45.295 --> 01:16:48.118
But here, there is
no notion of bounded.
01:16:48.118 --> 01:16:48.660
It's compact.
01:16:51.570 --> 01:16:54.390
But the main-- you
should think of these two
01:16:54.390 --> 01:16:55.890
are very much related
to each other.
01:16:55.890 --> 01:16:59.347
But here it's-- but
they are not equivalent.
01:17:02.647 --> 01:17:03.230
Anything else?
01:17:05.910 --> 01:17:06.410
Great.
01:17:06.410 --> 01:17:08.410
So that's all for today.