WEBVTT
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YUFEI ZHAO: So we've been
discussing graph limits
00:00:20.240 --> 00:00:21.790
for a couple of lecturers now.
00:00:21.790 --> 00:00:23.970
In the first lecture
on graph limits,
00:00:23.970 --> 00:00:28.340
so two lectures ago, I stated
a number of main theorems.
00:00:28.340 --> 00:00:30.830
And today, we will
prove these theorems
00:00:30.830 --> 00:00:34.310
using some of the tools that
we developed last time, namely
00:00:34.310 --> 00:00:35.720
the regularity lemma.
00:00:35.720 --> 00:00:40.130
And also we proved this
Martingale convergence theorem,
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which will also come in.
00:00:42.800 --> 00:00:45.920
So let me recall what were the
three main theorems that we
00:00:45.920 --> 00:00:49.160
stated at the end
of two lectures ago.
00:00:49.160 --> 00:00:53.150
So one of them was the
equivalence of convergence.
00:00:53.150 --> 00:00:56.570
On one hand, we defined
a notion of convergence
00:00:56.570 --> 00:01:04.190
where we say that Wn
approaches W, by definition,
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if the F densities converge.
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We can say convergence even
without a limit in mind,
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where we see a sequence
converges if all of these F
00:01:22.640 --> 00:01:25.400
densities converge.
00:01:25.400 --> 00:01:28.040
So the first main theorem
was that the two notions
00:01:28.040 --> 00:01:34.280
of convergence are equivalent,
so one notion being convergence
00:01:34.280 --> 00:01:38.120
in terms of F densities,
and the second notion
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being convergence in the
sense of the cut norm, the cut
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distance.
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There was a second
term that tells us
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that limits always exist.
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If you have a
convergent sequence,
00:01:53.650 --> 00:01:58.730
then you can represent
a limit by a graphon.
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And the third statement
was about compactness
00:02:01.300 --> 00:02:03.452
of the space of graphons.
00:02:03.452 --> 00:02:05.410
So we're actually going
to prove these theorems
00:02:05.410 --> 00:02:06.380
in reverse order.
00:02:06.380 --> 00:02:09.590
We're going to start with the
compactness and work backwards.
00:02:09.590 --> 00:02:12.100
So this is not how these
theorems were originally
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proved, or not in that
order, but it will be helpful
00:02:15.610 --> 00:02:18.640
for us to do this
by first considering
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the compactness statement.
00:02:20.950 --> 00:02:23.480
So remember we're
compactness says.
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I start with this
space W tilda, which
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is the space of graphons,
where I identify
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graphons that have distance 0.
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So if they have cut distance
0, then I refer to them
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as the same point.
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So this is now a metric space.
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And the theorem is that
this space is compact.
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So I think this, it's
a really nice theorem.
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It's a beautiful theorem
that encapsulates
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a lot of what we've been talking
about so far with similarities,
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regularity, and what not, in
a qualitatively succinct way,
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just that this space
of graphons is compact.
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You may not have some intuition
about what the space looks
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like at the moment,
but we'll see the proof
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and hopefully that will
give you some more intuition
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I first learned
about this theorem
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when Laszlo Lovasz, who was one
of the pioneers in the subject,
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when he came to
MIT to give a talk
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when I was a graduate student.
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And he said that
analysts thought
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that they pretty much knew
all the naturally occurring
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compact spaces out there.
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So there are lots of spaces
that occur in analysis
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and topology that are compact.
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I mean, the first one you
learn in analysis undergraduate
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is probably that an
interval is compact.
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But there are also
many other spaces.
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But this one here
doesn't seem to be
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any of these classical
notions of compactness.
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So it's, in some sense,
a new compact space.
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So let's see how the proof goes.
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Now, because we are
working in a metric space,
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it suffices to show, due to the
equivalence between compactness
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in the sense of
finite open covers
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and sequential compactness
in the metric space,
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so it suffices to show
sequential compactness,
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that every sequence of graphons
has a convergent subsequence
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with respect to this
cut metric and also will
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produce a limit as a convergence
subsequence with a limit point.
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So that's what we'll do.
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I give you an arbitrary
sequence of graphons.
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I want to construct by taking
subsequences a convergent
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sequence.
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And I will tell you
what that limit is.
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So here is what we're going
to do, given this sequence.
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As I hinted before, it has to
do with the regularity lemma.
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So we're going to
apply the regularity
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lemma in the above form,
which we did last time.
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So apply the weak
regularity lemma,
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which will tell us
that for each Wn,
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there exists a partition, in
fact, a sequence of partitions,
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each one refining the next.
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So what's going to happen is,
I'm going to start with Wn
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and starting with a trivial
partition, apply that lemma,
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and obtain a
partition P sub n, 1.
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And then starting
with that as my P0,
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I'm going to apply regularity
lemma again and obtain
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a refinement.
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I will have this
sequence of partitions,
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each one refining the next.
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So all of these are
going to be partitions
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of the 0, 1 interval.
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And as I mentioned
last time, everything's
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going to be measurable.
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I'm not going to even
mention measurability.
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Everything will
be measurable such
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that they satisfy the
following conditions.
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So the first one
is what I mentioned
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earlier, is that you have
a sequence of refinements.
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So each P sub n k plus 1 refines
the previous one for all n
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and k.
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And the second condition, as
given by the regularity lemma,
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you get to control
the number of parts.
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So I will say in the third part
what the error of approximation
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is.
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But you get to control
the number of parts.
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So in particular, I can make
sure that this number here,
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the number of parts in the k'th
partition depends only on k.
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Now, you might
complain somewhat,
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because the regularity lemma
only tells you an upper bound
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on the number of parts.
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But that's OK.
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I can allow empty parts.
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So now I make sure that the k'th
partition has exactly nk parts.
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And the third one has to do
with the area of approximation.
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OK, so suppose we write W sub
nk as the graphon obtained
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by applying the
stepping operator.
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So this is the averaging
operator on this partition,
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corresponding to
the k'th partition.
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I apply that partition, do a
stepping averaging operator
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on the n'th graphon.
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I get W sub nk.
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The third condition is that the
k'th partition approximates--
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it's a good approximation in the
cut norm up to error 1 over k.
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So 1 over k is some
arbitrary sequence going to 0
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as k goes to infinity.
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So I obtained a
sequence of partitions,
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so by applying the
regularity lemma
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to each to each graphon
in the sequence.
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Now, these graphons,
I mean, they
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each have their own vertex set.
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And so far, they're not
related to each other.
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But to make the visualization
easier and also in order
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to do the next
step in the proof,
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I am going to do some
measure-preserving bisection So
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think of this as permuting
the vertex labels.
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So by replacing each
Wn by some W sub
00:11:04.760 --> 00:11:15.350
n of phi, where phi is a
measure-preserving bisection,
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we can assume that
all these partitions
00:11:24.080 --> 00:11:27.940
are partitions into intervals.
00:11:36.660 --> 00:11:38.670
So initially, you
might have a partition
00:11:38.670 --> 00:11:41.640
into arbitrary measurable sets.
00:11:41.640 --> 00:11:47.120
Well, what I can do is to push
over the first set to the left,
00:11:47.120 --> 00:11:50.400
and so on, so do a
measure-preserving bisection
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in a way so that I can maintain
that all the partitions are
00:11:55.740 --> 00:11:58.189
visually chopping
up into intervals.
00:11:58.189 --> 00:11:58.689
Yeah?
00:11:58.689 --> 00:12:01.179
AUDIENCE: So at some point,
we need just one measure
00:12:01.179 --> 00:12:04.920
for the projection, like
all of them be in k?
00:12:04.920 --> 00:12:07.110
YUFEI ZHAO: OK, so
the question is,
00:12:07.110 --> 00:12:11.510
it may be the case
that, for a given k,
00:12:11.510 --> 00:12:15.640
I can do this
arrangement, but it's not
00:12:15.640 --> 00:12:18.040
clear to you at the
moment why you can
00:12:18.040 --> 00:12:21.790
do this uniformly for all k.
00:12:21.790 --> 00:12:24.610
So one way to get
around this is, for now,
00:12:24.610 --> 00:12:26.500
just think of for each given k.
00:12:26.500 --> 00:12:29.290
And then you'll see at the end
that that's already enough.
00:12:32.898 --> 00:12:34.512
OK, any more questions?
00:12:38.170 --> 00:12:41.940
So now assume all of these
P sub nk's are intervals.
00:12:41.940 --> 00:12:45.210
So in fact, what you said
may be a better way to go.
00:12:45.210 --> 00:12:48.430
But to make our life
a little bit easier,
00:12:48.430 --> 00:12:50.430
let's just assume for
now that you can do this.
00:12:53.670 --> 00:12:56.190
OK, and what's
going to happen next
00:12:56.190 --> 00:12:59.490
is some kind of a
diagonalization argument.
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We're going to be
picking subsequences.
00:13:05.760 --> 00:13:10.740
So I'm going to be
picking subsequences
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so that they are
going to have very
00:13:12.810 --> 00:13:14.850
nice convergence properties.
00:13:14.850 --> 00:13:18.150
And so I'm going to
repeatedly throw out
00:13:18.150 --> 00:13:19.930
a lot of the sequence.
00:13:19.930 --> 00:13:21.930
So this is a
diagonalization argument.
00:13:21.930 --> 00:13:25.390
And basically what
happens is that,
00:13:25.390 --> 00:13:27.370
by passing two subsequences--
00:13:31.564 --> 00:13:35.450
and we're going to do this
repeatedly, many times--
00:13:35.450 --> 00:13:51.420
we can assume, first, that
the end points of P sub n1,
00:13:51.420 --> 00:13:59.070
they converge as n
goes to infinity.
00:13:59.070 --> 00:14:14.250
So each P sub n1 is some
partition of interval
00:14:14.250 --> 00:14:17.370
into some fixed number of parts.
00:14:17.370 --> 00:14:19.320
So by passing to
a subsequence, I
00:14:19.320 --> 00:14:22.005
make sure that the division
points all converge.
00:14:28.850 --> 00:14:34.420
And now, by passing
one more time,
00:14:34.420 --> 00:14:36.860
so by passing to
subsequence one more time,
00:14:36.860 --> 00:14:46.020
let's assume that also, W sub
n1 converges to some function,
00:14:46.020 --> 00:14:49.080
some graphon u1, point-wise.
00:15:00.580 --> 00:15:12.100
So initially, I
have these graphons.
00:15:12.100 --> 00:15:15.710
Each one of them
is an m by n block.
00:15:15.710 --> 00:15:19.060
They have various
division points.
00:15:19.060 --> 00:15:21.590
By passing to a
subsequence, I assume
00:15:21.590 --> 00:15:24.560
that the points of
division, they converge.
00:15:24.560 --> 00:15:27.130
And now by passing to an
additional subsequence,
00:15:27.130 --> 00:15:32.600
I can make sure the individual
values, they converge.
00:15:32.600 --> 00:15:37.700
So as a result, W sub
n, 1 converges to W1--
00:15:37.700 --> 00:15:42.382
converges to some graphon, u1,
point-wise, almost everywhere.
00:15:46.030 --> 00:15:54.990
And we repeat for W
sub nk for each k.
00:15:59.970 --> 00:16:01.520
So do this sequentially.
00:16:01.520 --> 00:16:03.490
So we just did it
for k equals to 1.
00:16:03.490 --> 00:16:05.890
Now do it for 2,
3, 4, and so on.
00:16:08.537 --> 00:16:10.120
So this is a
diagonalization argument.
00:16:10.120 --> 00:16:12.670
We do this countably many times.
00:16:12.670 --> 00:16:15.310
At the end, what do we get?
00:16:15.310 --> 00:16:20.970
We pass down to the
following subsequence.
00:16:26.850 --> 00:16:29.820
And just to make my life
a bit more convenient,
00:16:29.820 --> 00:16:34.900
instead of labeling the
indices of the subsequence,
00:16:34.900 --> 00:16:37.380
I'm going to
relabel the sequence
00:16:37.380 --> 00:16:40.560
so that it's still labeled
by 1, 2, 3, 4, and so on.
00:16:40.560 --> 00:16:47.320
So we now pass to a
sequence W1, W2, W3,
00:16:47.320 --> 00:16:54.630
and so on, such that if you
look at the first partition,
00:16:54.630 --> 00:16:59.620
the first weak regularity
partition, they produce W1,1,
00:16:59.620 --> 00:17:05.079
W2,1, W3,1, and so on.
00:17:05.079 --> 00:17:09.981
And these guys, they
converge to u1, point-wise.
00:17:15.849 --> 00:17:19.695
The second level, W2,1--
00:17:19.695 --> 00:17:32.310
sorry, W1,2, W2,2, W3,2, they
converge to u2, point-wise,
00:17:32.310 --> 00:17:32.810
and so on.
00:17:46.820 --> 00:17:48.893
OK, so far so good?
00:17:48.893 --> 00:17:49.800
Question?
00:17:49.800 --> 00:17:52.310
AUDIENCE: Sorry,
earlier, why did that
00:17:52.310 --> 00:17:54.110
converge to u1 point-wise?
00:17:54.110 --> 00:17:56.620
YUFEI ZHAO: OK, so the
question is, why is this true?
00:17:56.620 --> 00:18:03.490
Why is Wn,1 converge
to u1 point-wise?
00:18:03.490 --> 00:18:05.200
Initially, it might not.
00:18:05.200 --> 00:18:07.900
But what I'm saying is, you
can pass to a subsequence.
00:18:07.900 --> 00:18:08.663
AUDIENCE: Yes.
00:18:08.663 --> 00:18:10.330
YUFEI ZHAO: You can
pass to subsequence,
00:18:10.330 --> 00:18:15.210
because there are only n1 parts.
00:18:17.800 --> 00:18:24.500
So it's an n1 by m1
matrix of real numbers.
00:18:24.500 --> 00:18:27.720
And so you only have finite
bounded many of them.
00:18:27.720 --> 00:18:30.080
So you can pick a subsequence
so that they converge.
00:18:30.080 --> 00:18:30.580
Yeah?
00:18:30.580 --> 00:18:31.955
AUDIENCE: So how
do you make sure
00:18:31.955 --> 00:18:35.420
that your subsequence is
not empty at the end-- like,
00:18:35.420 --> 00:18:37.800
could you fix the first k?
00:18:37.800 --> 00:18:39.760
YUFEI ZHAO: OK,
so you're asking,
00:18:39.760 --> 00:18:44.080
if we do this slightly
not so carefully,
00:18:44.080 --> 00:18:46.180
we might end up with
an empty sequence.
00:18:46.180 --> 00:18:48.430
So this is why I say, you
have to do a diagonalization
00:18:48.430 --> 00:18:49.240
argument.
00:18:49.240 --> 00:18:53.170
Each step, you keep the
first term, the sequence,
00:18:53.170 --> 00:18:56.440
so that you always
maintain some sequence.
00:18:56.440 --> 00:19:00.870
You have to be slightly
careful with diagonalization.
00:19:00.870 --> 00:19:01.976
Any more questions?
00:19:06.080 --> 00:19:07.900
So by passing to
a subsequence, we
00:19:07.900 --> 00:19:11.680
obtain this very nice sequence,
this nice subsequence,
00:19:11.680 --> 00:19:16.960
such that each row corresponding
to each level of regularization
00:19:16.960 --> 00:19:20.740
converges point-wise to some u.
00:19:20.740 --> 00:19:23.370
So what do this u's look like?
00:19:23.370 --> 00:19:24.800
So they are step graphons.
00:19:27.460 --> 00:19:29.740
So let's explore the
structure of u a bit more.
00:19:36.050 --> 00:19:43.880
OK, so since we have that each--
00:19:43.880 --> 00:19:47.070
OK, so we have the property
that each partition
00:19:47.070 --> 00:19:49.190
refines the previous partition.
00:19:53.190 --> 00:20:00.680
And as a result, if you look
at the k plus 1'th stepping,
00:20:00.680 --> 00:20:07.490
and I step it by the previous
partition in the sequence,
00:20:07.490 --> 00:20:14.460
I should get back, I should
go back one in the sequence.
00:20:14.460 --> 00:20:19.490
So this was this graphon
obtained by averaging over
00:20:19.490 --> 00:20:20.870
the k'th partition.
00:20:20.870 --> 00:20:26.000
And this is the graphon obtained
by averaging over the k'th plus
00:20:26.000 --> 00:20:28.070
1st partition.
00:20:28.070 --> 00:20:33.000
So if I go back one more, I
should go back in the sequence.
00:20:35.640 --> 00:20:41.530
And since the u's are the
point-wise limit of these
00:20:41.530 --> 00:20:50.530
W's, the same relationships
should also hold
00:20:50.530 --> 00:21:06.820
for the u's, namely that u sub
k should equal to u sub k plus 1
00:21:06.820 --> 00:21:17.210
if I step it with
Pk, where Pk is the--
00:21:20.020 --> 00:21:22.870
so if you look at, all
these endpoints converge.
00:21:22.870 --> 00:21:28.550
And these partitions,
they converge to P1.
00:21:28.550 --> 00:21:36.760
So if you look at the partitions
that correspond to P1,1, P2,1,
00:21:36.760 --> 00:21:41.480
and so on, I want these
partitions to converge to P1.
00:21:41.480 --> 00:21:51.460
I want these partitions
to converge to P2.
00:21:51.460 --> 00:21:55.313
So all these partitions, they
are partitions into intervals.
00:21:55.313 --> 00:21:56.980
So I'm just saying,
if you look at where
00:21:56.980 --> 00:21:59.350
the intervals, where the
divisions of intervals go,
00:21:59.350 --> 00:22:00.250
they converge.
00:22:00.250 --> 00:22:02.830
And then I'm calling the
limit partition P sub k.
00:22:07.810 --> 00:22:16.240
And here we're using that P
sub k plus 1 refines P sub k,
00:22:16.240 --> 00:22:18.712
because the same is
true for each end.
00:22:18.712 --> 00:22:20.670
So in the limit, the same
must be true as well.
00:22:25.080 --> 00:22:27.620
So you have this column of u's.
00:22:27.620 --> 00:22:32.852
So let me draw you a picture of
what these u's could look like.
00:22:32.852 --> 00:22:34.810
So here is an illustration
that may be helpful.
00:22:40.980 --> 00:22:43.830
So what could these
u's look like?
00:22:43.830 --> 00:22:51.390
Each one of them is represented
by values on the unit square.
00:22:51.390 --> 00:22:53.520
And I write this
in matrix notation
00:22:53.520 --> 00:22:55.480
so that inversion is
in the top left corner.
00:23:00.180 --> 00:23:05.400
Well, maybe P1 is just
the trivial partition,
00:23:05.400 --> 00:23:08.910
in which case u1 is going
to be a constant graphon.
00:23:08.910 --> 00:23:14.630
Let's say it has value 0.5.
00:23:14.630 --> 00:23:19.300
u2 came from u1 by
some partitioning.
00:23:19.300 --> 00:23:21.290
And suppose just for the
sake of illustration,
00:23:21.290 --> 00:23:23.210
there was only a
partitioning into two parts.
00:23:26.950 --> 00:23:30.123
And OK, so it doesn't
have to be at the origin.
00:23:30.123 --> 00:23:31.540
It doesn't have
to be at midpoint.
00:23:31.540 --> 00:23:33.550
But just for illustration,
suppose the division
00:23:33.550 --> 00:23:36.800
were at the midpoint.
00:23:36.800 --> 00:23:40.520
Because u1 needs to have--
00:23:40.520 --> 00:23:44.220
so this 0.5 value should
be the average value
00:23:44.220 --> 00:23:46.530
in all of these four squares.
00:23:46.530 --> 00:23:59.020
So for instance, the points
may be like 0.6, 0.6, 0.4, 0.4,
00:23:59.020 --> 00:24:01.900
so for example.
00:24:01.900 --> 00:24:08.820
And in u3, the partition,
the P3 partition-- so here
00:24:08.820 --> 00:24:10.120
are the partition is P1.
00:24:10.120 --> 00:24:11.740
The partition is P2.
00:24:11.740 --> 00:24:13.070
There's two parts.
00:24:13.070 --> 00:24:18.142
And suppose P3 three
now has four parts.
00:24:18.142 --> 00:24:19.600
And again, for
illustration's sake,
00:24:19.600 --> 00:24:22.120
suppose it is equally
dividing the interval
00:24:22.120 --> 00:24:25.000
into four intervals.
00:24:25.000 --> 00:24:28.960
It could be that now
each of these parts
00:24:28.960 --> 00:24:32.950
is split up into four
different values in a way
00:24:32.950 --> 00:24:37.471
that so you can obtain the
original numbers by averaging.
00:24:40.560 --> 00:24:44.010
So that's one possible example.
00:24:44.010 --> 00:24:50.514
Likewise, you can have
something like that.
00:24:54.888 --> 00:25:07.220
Sorry, 4, 7-- so and I
should maintain symmetry
00:25:07.220 --> 00:25:08.990
in this matrix.
00:25:08.990 --> 00:25:11.520
And the last one, I'm just
going to be lazy and say
00:25:11.520 --> 00:25:15.960
that it's still 0.4 throughout.
00:25:15.960 --> 00:25:20.080
OK, so this is what the sequence
of u's are going to look like.
00:25:20.080 --> 00:25:25.660
Each one of them splits up a box
in the previous u in such way
00:25:25.660 --> 00:25:31.700
that the local averages, the
step averages are preserved.
00:25:31.700 --> 00:25:32.590
Any questions so far?
00:25:35.410 --> 00:25:41.270
All right, so now we
get to Martingales.
00:25:41.270 --> 00:25:45.870
So I claim that this is
basically a Martingale.
00:25:45.870 --> 00:25:56.550
So and suppose you let x, y
be a uniform point in the unit
00:25:56.550 --> 00:25:57.050
square.
00:25:59.930 --> 00:26:05.220
And consider this sequence.
00:26:05.220 --> 00:26:07.190
So this is now a
random sequence,
00:26:07.190 --> 00:26:09.290
because x, y are random.
00:26:12.030 --> 00:26:17.752
I evaluate these u's on this
uniform random point, x, y.
00:26:17.752 --> 00:26:18.960
So this is a random sequence.
00:26:23.800 --> 00:26:27.279
And the main observation is
that this is a Martingale.
00:26:34.320 --> 00:26:38.450
So remember the definition of
a Martingale from last time.
00:26:38.450 --> 00:26:42.990
Martingale is one
where, if you look
00:26:42.990 --> 00:26:47.430
at the value of u
sub k conditioned
00:26:47.430 --> 00:26:53.730
on the previous
values, the expectation
00:26:53.730 --> 00:26:56.190
is just the previous term.
00:26:56.190 --> 00:26:58.493
And I claim this is
true for the sequence,
00:26:58.493 --> 00:27:00.035
because of the way
we constructed it,
00:27:00.035 --> 00:27:03.750
it's splitting up each box in
an averaging-preserving way.
00:27:07.380 --> 00:27:09.177
A different way to see
this, and for those
00:27:09.177 --> 00:27:11.010
of you who actually
know what the definition
00:27:11.010 --> 00:27:13.920
of a random variable is in the
sense of probability theory,
00:27:13.920 --> 00:27:21.160
is that you should view this
0, 1 squared as the probability
00:27:21.160 --> 00:27:27.115
space, in which case u itself
is the random variable.
00:27:30.050 --> 00:27:34.480
And this partitioning gives
you a filtration of the space.
00:27:34.480 --> 00:27:37.450
It's a sequence of sigma
algebras dividing up the space
00:27:37.450 --> 00:27:39.590
into finer and finer pieces.
00:27:39.590 --> 00:27:43.280
So this is really
what a martingale is.
00:27:43.280 --> 00:27:45.710
So we have a Martingale.
00:27:45.710 --> 00:27:56.770
It's bounded because the
values take place in 0, 1.
00:27:56.770 --> 00:28:00.340
So by the Martingale
convergence theorem,
00:28:00.340 --> 00:28:15.370
which we proved last time,
we find that this sequence
00:28:15.370 --> 00:28:18.760
must converge to some limit.
00:28:21.550 --> 00:28:24.970
So this sequence of
Martingale converges,
00:28:24.970 --> 00:28:29.490
which means, so if you think
about the interpretation
00:28:29.490 --> 00:28:33.480
up there, so there
exists a u which
00:28:33.480 --> 00:28:45.210
is a graphon such that uk
converges to u point-wise
00:28:45.210 --> 00:28:51.086
almost everywhere as
k goes to infinity.
00:28:57.410 --> 00:28:59.000
That's the limit.
00:28:59.000 --> 00:29:00.897
So this is the limit.
00:29:00.897 --> 00:29:02.980
And we're going to show
that it is indeed a limit.
00:29:02.980 --> 00:29:05.110
But you see, this is a
construction of the limit,
00:29:05.110 --> 00:29:08.950
where we took regularity,
got all these nice pieces,
00:29:08.950 --> 00:29:11.140
found convergent
subsequences, and then
00:29:11.140 --> 00:29:14.140
applied the
martingale convergence
00:29:14.140 --> 00:29:17.290
theorem to produce
for us this candidate
00:29:17.290 --> 00:29:19.170
for the limit, this u.
00:29:19.170 --> 00:29:28.520
So now let us show that it is
indeed the limit that we're
00:29:28.520 --> 00:29:33.650
looking for in the subsequence.
00:29:33.650 --> 00:29:37.500
So again, I've tossed
out all the terms
00:29:37.500 --> 00:29:41.510
which we removed in
passing to subsequences.
00:29:41.510 --> 00:29:43.290
So in the remaining
subsequence, I
00:29:43.290 --> 00:29:48.920
want to show that the
Wn's indeed converge to u.
00:29:48.920 --> 00:29:52.130
And this is now a fairly
straightforward three epsilons
00:29:52.130 --> 00:29:55.910
argument, the standard
analysis type argument.
00:29:55.910 --> 00:29:57.630
But OK, so let's
carry it through.
00:29:57.630 --> 00:30:01.370
So for every epsilon
bigger than 0,
00:30:01.370 --> 00:30:08.360
suppose you pick a
sufficiently large k.
00:30:08.360 --> 00:30:11.170
There exists a
sufficiently large k.
00:30:11.170 --> 00:30:17.010
And we make sure
k is large enough
00:30:17.010 --> 00:30:22.880
such that u differs
from u sub k in l1 norm
00:30:22.880 --> 00:30:28.400
by, at most, epsilon
over 3, because the uk's,
00:30:28.400 --> 00:30:32.090
they converge to u
point-wise almost everywhere.
00:30:32.090 --> 00:30:33.150
So we find this k.
00:30:33.150 --> 00:30:34.060
So let's fix this k.
00:30:39.220 --> 00:30:47.080
Then there exists an n0 such
that if you look at this u sub
00:30:47.080 --> 00:30:51.930
k, it does not--
00:30:51.930 --> 00:31:03.540
it is very close to
W sub nk for all n
00:31:03.540 --> 00:31:13.150
large enough because of
what happened up there.
00:31:18.300 --> 00:31:25.435
So we can now compute
the difference between--
00:31:27.950 --> 00:31:30.770
in fact, let's do it this way.
00:31:35.400 --> 00:31:42.585
So now let's compute
the difference,
00:31:42.585 --> 00:31:45.810
the cut norm of the
difference between the term
00:31:45.810 --> 00:31:48.060
in the sequence W sub n and u.
00:31:48.060 --> 00:31:53.510
So by triangle
inequality, we have
00:31:53.510 --> 00:31:54.900
that the following is true.
00:32:11.060 --> 00:32:15.940
The cut norm is
upperbounded by the l1 norm.
00:32:15.940 --> 00:32:17.332
Look at the definitions.
00:32:22.420 --> 00:32:25.710
So I'm going to replace the
first couple of these cut norms
00:32:25.710 --> 00:32:28.290
by l1 norms and leave
the last one in tact.
00:32:35.500 --> 00:32:41.000
The first term, I claim is,
at most, epsilon over 3,
00:32:41.000 --> 00:32:44.810
because up there.
00:32:44.810 --> 00:32:47.690
The second term is going
to be at least epsilon
00:32:47.690 --> 00:32:52.040
over 3, because over here.
00:32:52.040 --> 00:32:56.510
And the third term is going
to be also, at most, epsilon
00:32:56.510 --> 00:33:02.660
over 3, because well, from
the regularity approximation,
00:33:02.660 --> 00:33:07.010
I know that it is,
at most, 1 over k.
00:33:07.010 --> 00:33:09.970
And I chose k large enough
so that there is also,
00:33:09.970 --> 00:33:12.410
at most, epsilon over 3.
00:33:12.410 --> 00:33:17.285
Put everything together, we
find that these two are--
00:33:17.285 --> 00:33:21.770
they different by, at most,
epsilon if n is large enough.
00:33:21.770 --> 00:33:30.380
But now, since epsilon
can be arbitrarily small,
00:33:30.380 --> 00:33:38.460
we find that you indeed have
convergence, as claimed.
00:33:41.750 --> 00:33:46.160
And this finishes the
proof of compactness.
00:33:46.160 --> 00:33:47.700
So there are a few components.
00:33:47.700 --> 00:33:48.910
One is passing to--
00:33:48.910 --> 00:33:53.970
so applying regularity,
passing to subsequences,
00:33:53.970 --> 00:33:58.630
and obtaining this limit from
the regularity approximations,
00:33:58.630 --> 00:34:00.140
these u's.
00:34:00.140 --> 00:34:03.840
And then we observe that these
u's, they form a Martingale.
00:34:03.840 --> 00:34:08.340
So we can apply the Martingale
convergence theorem to get
00:34:08.340 --> 00:34:12.280
us a candidate for the limit.
00:34:12.280 --> 00:34:14.449
And then the rest is
fairly straightforward,
00:34:14.449 --> 00:34:17.074
because all the steps
are good approximations.
00:34:17.074 --> 00:34:19.960
You put them together,
you prove the limit.
00:34:23.090 --> 00:34:24.163
Any questions?
00:34:27.790 --> 00:34:30.980
All right, so you may ask,
well, now we have compactness.
00:34:30.980 --> 00:34:33.179
What is compactness good for?
00:34:33.179 --> 00:34:36.022
So it may seem like a
somewhat abstract concept.
00:34:36.022 --> 00:34:37.730
So in the second half
of today's lecture,
00:34:37.730 --> 00:34:41.870
I want to show you how to use
this compactness claim combined
00:34:41.870 --> 00:34:46.790
with the first definition of
compactness that you've seen,
00:34:46.790 --> 00:34:51.110
namely every open cover
contains a finite sub cover,
00:34:51.110 --> 00:34:56.179
and to use that to
prove many consequences
00:34:56.179 --> 00:34:58.460
about the space of graphons.
00:34:58.460 --> 00:35:02.030
And some things that we
had to work a bit hard at,
00:35:02.030 --> 00:35:06.790
but they turn out to fall from
the compactness statement.
00:35:06.790 --> 00:35:09.910
So let's take a quick break.
00:35:09.910 --> 00:35:11.420
In the first part
of this lecture,
00:35:11.420 --> 00:35:14.270
we proved that a space
of graphons is compact.
00:35:14.270 --> 00:35:17.840
So now let me show you what
we can reap as consequences
00:35:17.840 --> 00:35:20.450
from the compactness
result. So I
00:35:20.450 --> 00:35:24.140
want to show you how
to apply compactness
00:35:24.140 --> 00:35:25.820
and prove some consequences.
00:35:35.540 --> 00:35:38.450
As I mentioned earlier,
the compactness result
00:35:38.450 --> 00:35:40.550
is related to regularity.
00:35:40.550 --> 00:35:43.040
And in fact, many of the
results I'm going to state,
00:35:43.040 --> 00:35:47.870
you can prove maybe with some
more work using the regularity
00:35:47.870 --> 00:35:49.010
lemma.
00:35:49.010 --> 00:35:51.320
But I also want to show you
how to deduce them directly
00:35:51.320 --> 00:35:52.190
from compactness.
00:35:52.190 --> 00:35:54.650
In fact, we'll deduce
the regularity lemma
00:35:54.650 --> 00:35:55.850
from compactness.
00:35:55.850 --> 00:35:58.550
So these two ideas,
compactness and regularity,
00:35:58.550 --> 00:36:01.830
they go hand-in-hand.
00:36:01.830 --> 00:36:05.390
So first, more so as a warm
up, but as also an interesting
00:36:05.390 --> 00:36:07.880
result, statement, an
interesting statement
00:36:07.880 --> 00:36:11.548
on its own, so let me
prove the following.
00:36:14.970 --> 00:36:21.580
So here is a statement that we
can deduce from compactness.
00:36:21.580 --> 00:36:26.720
So for every epsilon,
there exists some number N
00:36:26.720 --> 00:36:36.530
which depend only on epsilon,
such that for every W graphon,
00:36:36.530 --> 00:36:49.440
there exists a graph
G with N vertices,
00:36:49.440 --> 00:37:02.770
such that the cut
distance between G and W
00:37:02.770 --> 00:37:05.100
is, at most, epsilon.
00:37:08.440 --> 00:37:10.540
So think about what this says.
00:37:10.540 --> 00:37:12.930
So for every epsilon,
there is some
00:37:12.930 --> 00:37:17.100
bound N such that every
graphon-- so a graphon is
00:37:17.100 --> 00:37:22.900
some real-value function, so
taking values between 0 and 1.
00:37:22.900 --> 00:37:24.990
You can approximate
it in the distance
00:37:24.990 --> 00:37:28.560
that we care about by a
graph with a bounded number
00:37:28.560 --> 00:37:31.180
of vertices.
00:37:31.180 --> 00:37:33.700
This is kind of like
regularity lemma.
00:37:33.700 --> 00:37:38.210
If you are allowed edge
weights on this graph G,
00:37:38.210 --> 00:37:43.220
then it immediately follows
from the weak regularity lemma
00:37:43.220 --> 00:37:44.570
that we already proved.
00:37:48.500 --> 00:37:50.930
And from that weak
regularity lemma
00:37:50.930 --> 00:37:55.370
which allows you to get
some G with edge weights,
00:37:55.370 --> 00:37:57.440
you can think
about how you might
00:37:57.440 --> 00:38:03.100
turn an edge-weighted graph
into an unweighted graph.
00:38:03.100 --> 00:38:04.968
So that can also be done.
00:38:04.968 --> 00:38:07.010
But I want to show you a
completely different way
00:38:07.010 --> 00:38:10.700
of proving this result that
follows from compactness.
00:38:13.570 --> 00:38:16.930
And so I say it's a warm
up, because it's really
00:38:16.930 --> 00:38:19.918
a warm up for the next
thing we're going to do.
00:38:19.918 --> 00:38:21.460
This is an easier
example showing you
00:38:21.460 --> 00:38:24.690
how to use compactness.
00:38:24.690 --> 00:38:26.440
So the idea is, I have
this compact space.
00:38:26.440 --> 00:38:32.380
I'm going to cover this space
by open sets, by open balls.
00:38:32.380 --> 00:38:38.230
So the open balls are going
to be this B sub epsilon
00:38:38.230 --> 00:38:41.200
G. So for each
graph, G, I'm going
00:38:41.200 --> 00:38:53.005
to consider the set of graphons
that are within epsilon of G.
00:38:53.005 --> 00:38:54.755
So this is you have
some topological space
00:38:54.755 --> 00:38:55.810
or some metric space.
00:38:55.810 --> 00:38:58.260
I have a point G. And I
look at its open ball.
00:39:07.930 --> 00:39:09.580
This is the ball.
00:39:09.580 --> 00:39:22.753
So I claim that these open
balls, they form an open cover,
00:39:22.753 --> 00:39:25.620
of the space.
00:39:30.300 --> 00:39:31.410
Where is that?
00:39:31.410 --> 00:39:35.010
So I want to show every
point W is covered.
00:39:39.220 --> 00:39:55.930
So this follows from the claim
that every W is the limit
00:39:55.930 --> 00:39:57.340
of some sequence of graphs.
00:40:04.940 --> 00:40:08.600
So we didn't technically
actually prove this claim.
00:40:08.600 --> 00:40:11.895
I said that if you take W
random graphs, you get this.
00:40:11.895 --> 00:40:15.320
So we didn't
technically prove that.
00:40:15.320 --> 00:40:16.870
But OK, so it turns
out to be true.
00:40:16.870 --> 00:40:18.780
There are easier ways
to establish it as well
00:40:18.780 --> 00:40:20.790
by taking l1 approximations.
00:40:20.790 --> 00:40:24.780
But the point is that, if
you use this claim here,
00:40:24.780 --> 00:40:28.460
you do not get a bound on
the number of vertices.
00:40:28.460 --> 00:40:32.660
It could be that for very
bizarre-looking W's, you
00:40:32.660 --> 00:40:37.690
might require much more
number of vertices.
00:40:37.690 --> 00:40:40.320
And a priori, you do not
know that it is bounded
00:40:40.320 --> 00:40:43.510
as a function of epsilon.
00:40:43.510 --> 00:40:47.110
But now we have this open cover.
00:40:47.110 --> 00:40:55.870
So by compactness of
this space of graphons,
00:40:55.870 --> 00:41:03.340
we can find an open cover
using a finite subset
00:41:03.340 --> 00:41:10.510
of these graphs, so G1
to Gk, so a finite subset
00:41:10.510 --> 00:41:13.740
to do an open cover.
00:41:13.740 --> 00:41:22.330
And now we let N to be the
least-common multiple of all
00:41:22.330 --> 00:41:25.105
of these vertex set sizes.
00:41:49.130 --> 00:41:59.140
So all of these graphs,
they are within--
00:42:03.010 --> 00:42:05.270
so for each of these
graphs, I can replace it
00:42:05.270 --> 00:42:07.550
by a graph on
exactly N vertices.
00:42:16.300 --> 00:42:28.440
There exists a graph Gi
prime of exactly N vertices,
00:42:28.440 --> 00:42:34.208
such that they
represent the same point
00:42:34.208 --> 00:42:35.250
in the space of graphons.
00:42:38.436 --> 00:42:40.740
So why is this?
00:42:40.740 --> 00:42:43.950
Think about the representation
of a graphon using
00:42:43.950 --> 00:42:45.630
from a graph.
00:42:45.630 --> 00:43:00.650
If I start with G and I blow
up each vertex into some k
00:43:00.650 --> 00:43:04.530
vertices, then it turns out--
00:43:04.530 --> 00:43:06.530
I mean, you should think
about why this is true.
00:43:06.530 --> 00:43:09.580
But it's really not hard to
see if you draw the picture.
00:43:09.580 --> 00:43:12.662
So remember, this black and
white picture, that actually,
00:43:12.662 --> 00:43:13.620
they're the same point.
00:43:13.620 --> 00:43:17.360
They are represented
by the same graphon.
00:43:17.360 --> 00:43:18.890
OK, and that's it.
00:43:18.890 --> 00:43:23.790
So we found these G's.
00:43:23.790 --> 00:43:27.540
All of them have
exactly N vertices such
00:43:27.540 --> 00:43:32.700
that their epsilon open
balls form an open cover
00:43:32.700 --> 00:43:33.870
of the space of graphons.
00:43:33.870 --> 00:43:36.000
So every graphon
can be approximated
00:43:36.000 --> 00:43:37.172
by one of these graphs.
00:43:40.070 --> 00:43:44.030
So you get that
from compactness.
00:43:44.030 --> 00:43:46.760
The statement says, for every
epsilon, their exists an N.
00:43:46.760 --> 00:43:50.650
So N is a function of
epsilon.l What's the function?
00:43:54.590 --> 00:43:57.090
This proof doesn't tell
you anything about that.
00:43:57.090 --> 00:44:06.860
So this proof gives
no information
00:44:06.860 --> 00:44:18.750
about the dependence
of N on epsilon.
00:44:18.750 --> 00:44:21.420
So in some sense, it's even
worse than some of the things
00:44:21.420 --> 00:44:23.700
we've seen in the
earlier discussion
00:44:23.700 --> 00:44:25.950
on Szemerédi's regularity
lemma where there were tower
00:44:25.950 --> 00:44:27.030
or Wowzer-types.
00:44:27.030 --> 00:44:29.340
Here there is no
information, because it comes
00:44:29.340 --> 00:44:32.850
from a compactness statement.
00:44:32.850 --> 00:44:39.316
So you just know there exists
a finite open cover, no bounds.
00:44:39.316 --> 00:44:43.970
OK, any questions about
this warm-up application?
00:44:43.970 --> 00:44:45.160
So it feels a bit magical.
00:44:45.160 --> 00:44:46.160
So you have compactness.
00:44:46.160 --> 00:44:51.216
And then you have all
of these consequences.
00:44:51.216 --> 00:44:54.080
So now let me show you how
you can deduce the regularity
00:44:54.080 --> 00:44:56.900
lemma itself from compactness.
00:44:56.900 --> 00:45:03.010
In fact, in the proof
of the existence,
00:45:03.010 --> 00:45:06.840
in the proof of compactness,
we only used weak regularity.
00:45:06.840 --> 00:45:09.590
And now let me show you how
you can use the weak regularity
00:45:09.590 --> 00:45:11.780
consequence of
namely compactness
00:45:11.780 --> 00:45:16.650
to bootstrap itself
to strong regularity.
00:45:16.650 --> 00:45:20.150
So we saw a version of strong
regularity in the earlier
00:45:20.150 --> 00:45:23.510
chapter when we discussed
Szemerédi's regularity lemma.
00:45:23.510 --> 00:45:27.770
So let me state it in a
somewhat different-looking form,
00:45:27.770 --> 00:45:32.640
but that turns out to
be morally equivalent.
00:45:32.640 --> 00:45:35.590
Suppose I have a
vector of epsilons.
00:45:41.740 --> 00:45:44.090
So all of these are
positive real numbers.
00:45:49.680 --> 00:45:55.470
The claim is that
there exists an M which
00:45:55.470 --> 00:46:10.310
only depends on this vector
such that for every graphon W,
00:46:10.310 --> 00:46:10.950
one can--
00:46:10.950 --> 00:46:25.010
so every graphon W can be
written as the following,
00:46:25.010 --> 00:46:26.260
decomposing the following way.
00:46:26.260 --> 00:46:31.970
We write W as a sum
of a structured part,
00:46:31.970 --> 00:46:38.540
a pseudo-random part,
and a small part,
00:46:38.540 --> 00:46:51.060
where the structured part is
a step function with k parts,
00:46:51.060 --> 00:46:57.450
but k is, at most, M,
this claimed bound M.
00:46:57.450 --> 00:47:03.640
The pseudo-random part
has a very small cut norm,
00:47:03.640 --> 00:47:08.520
so its cut norm, very
small, even compared
00:47:08.520 --> 00:47:10.870
to the number of parts.
00:47:10.870 --> 00:47:22.117
And finally, the small part has
l1 norm bounded by epsilon 1.
00:47:22.117 --> 00:47:22.950
So that's the claim.
00:47:22.950 --> 00:47:25.530
You can always--
there exists some
00:47:25.530 --> 00:47:28.200
bound M in terms of
these error parameters
00:47:28.200 --> 00:47:30.667
so that you have
this decomposition.
00:47:30.667 --> 00:47:32.250
So we saw some version
of this earlier
00:47:32.250 --> 00:47:35.280
when we discussed the spectral
proof of regularity lemma.
00:47:35.280 --> 00:47:38.520
And I don't want to go into
details of how these two
00:47:38.520 --> 00:47:41.880
things are related, but just
to comment that depending
00:47:41.880 --> 00:47:44.010
on your choice of the
epsilon parameters,
00:47:44.010 --> 00:47:46.110
it relates to some of
the different versions
00:47:46.110 --> 00:47:48.820
of regularity lemma
that we've seen before.
00:47:48.820 --> 00:47:54.600
So for example, if epsilon
k is roughly epsilon,
00:47:54.600 --> 00:47:57.240
some fixed epsilon
over k squared,
00:47:57.240 --> 00:48:08.400
then this is basically the
same as Szemerédi's regularity
00:48:08.400 --> 00:48:16.920
lemma, whereas if all the
k's are the same epsilon,
00:48:16.920 --> 00:48:20.236
then this is roughly the same
as the weak regularity lemma.
00:48:27.540 --> 00:48:30.820
All right, so how
to prove this claim?
00:48:30.820 --> 00:48:32.430
We're going to use
compactness again.
00:48:38.070 --> 00:48:43.670
So first, there always
exists an l1 approximation
00:48:43.670 --> 00:48:55.870
so that every W has some
step function u associated
00:48:55.870 --> 00:49:02.010
to it such that the l1
distance between W and u
00:49:02.010 --> 00:49:06.060
is, at most, epsilon 1.
00:49:06.060 --> 00:49:08.730
So again, this is one of
these more measured theoretic
00:49:08.730 --> 00:49:10.610
technicalities I don't
want to get into,
00:49:10.610 --> 00:49:12.567
but so it's not hard to prove.
00:49:12.567 --> 00:49:14.400
So roughly speaking,
you have some function.
00:49:14.400 --> 00:49:16.820
You can approximate
it using steps.
00:49:26.860 --> 00:49:29.950
So similar to what
we did just now,
00:49:29.950 --> 00:49:33.220
if you just do that,
the number of steps
00:49:33.220 --> 00:49:36.840
might not be a
function of epsilon,
00:49:36.840 --> 00:49:40.480
so you might need much
more steps just doing
00:49:40.480 --> 00:49:45.570
that if your W looks
more pathological.
00:49:45.570 --> 00:49:48.120
So now what we're
going to do is consider
00:49:48.120 --> 00:49:52.050
the following
function, k of W. And I
00:49:52.050 --> 00:49:57.390
define it to be
the minimum k such
00:49:57.390 --> 00:50:10.660
that there exists a k step
graphon u such that u you
00:50:10.660 --> 00:50:20.290
minus W is, at most, epsilon 1.
00:50:20.290 --> 00:50:26.030
So among all the step
function approximations,
00:50:26.030 --> 00:50:29.330
pick one that has the
minimum number of steps
00:50:29.330 --> 00:50:34.755
and call the number of
steps k of W. So now,
00:50:34.755 --> 00:50:36.130
as before, we're
going to come up
00:50:36.130 --> 00:50:38.780
with an open cover of
the space of graphons.
00:50:41.380 --> 00:50:44.770
So the open cover is
going to be consisting
00:50:44.770 --> 00:50:50.722
of the cut norm balls of--
00:50:50.722 --> 00:50:52.750
actually, what notation
did I use over there?
00:51:01.200 --> 00:51:03.990
So this is a ball
centered around W
00:51:03.990 --> 00:51:08.660
with radius epsilon sub kW.
00:51:08.660 --> 00:51:15.972
This is an open cover
of the space of graphons
00:51:15.972 --> 00:51:19.490
as W ranges over all graphons.
00:51:23.980 --> 00:51:26.490
So I'm literally looking
at every point in the space
00:51:26.490 --> 00:51:28.020
and putting an open
ball around it.
00:51:28.020 --> 00:51:31.750
So obviously, this
is an open cover.
00:51:31.750 --> 00:51:33.850
And because of
compactness, there
00:51:33.850 --> 00:51:35.340
exists a finite sub cover.
00:51:41.080 --> 00:51:46.910
So there exists a
finite set, we write
00:51:46.910 --> 00:52:06.940
curly s, of graphons such that
these balls, as I range over W
00:52:06.940 --> 00:52:13.010
and curly s, they cover
the space of graphons.
00:52:22.550 --> 00:52:25.430
Now the goal is,
given the W, I want
00:52:25.430 --> 00:52:26.870
to approximate it in some way.
00:52:26.870 --> 00:52:31.580
So having a finite set
of things to work with
00:52:31.580 --> 00:52:36.010
allows us to do some
kind of approximations.
00:52:36.010 --> 00:52:45.950
So thus, for every W graphon,
there exists a W prime in s
00:52:45.950 --> 00:52:53.420
whose ball in that collection
covers the point W, such that W
00:52:53.420 --> 00:53:06.400
is contained in this ball.
00:53:06.400 --> 00:53:16.670
And OK, so given this W prime,
because of this definition
00:53:16.670 --> 00:53:22.460
over here, so there
exists a u which
00:53:22.460 --> 00:53:35.570
is a k step graphon
with k, at most,
00:53:35.570 --> 00:53:48.450
the maximum over all such
possible number of steps,
00:53:48.450 --> 00:53:59.830
such that W and W prime,
they are close in cut norm
00:53:59.830 --> 00:54:03.250
because you have
this open cover.
00:54:03.250 --> 00:54:11.380
And furthermore, W prime
is close to a graphon
00:54:11.380 --> 00:54:13.180
with a small number of steps.
00:54:21.970 --> 00:54:32.860
So suppose we now write W as u
plus W minus W prime and then
00:54:32.860 --> 00:54:37.870
plus W prime minus u.
00:54:37.870 --> 00:54:39.850
We find that this is the
decomposition that we
00:54:39.850 --> 00:54:44.290
are looking for because the u--
00:54:44.290 --> 00:54:46.380
so this is the
structural component--
00:54:46.380 --> 00:54:58.635
has k steps, where k is less
than this quantity here.
00:54:58.635 --> 00:55:03.890
And that quantity there is
just some function of epsilons.
00:55:03.890 --> 00:55:05.900
So it's, at most, some
function of the epsilons.
00:55:12.030 --> 00:55:18.700
It doesn't depend on the
specific choice of W.
00:55:18.700 --> 00:55:22.570
The second term, this is
this pseudo-random piece,
00:55:22.570 --> 00:55:35.586
because it's cut norm is
small, so what we have here.
00:55:35.586 --> 00:55:38.508
Yeah, so this entire
thing should be subscript.
00:55:45.690 --> 00:55:52.620
And finally, the third term
here is the small term,
00:55:52.620 --> 00:55:54.810
because it's l1 norm is small.
00:56:01.307 --> 00:56:03.557
So putting them together,
we get the regularity lemma.
00:56:06.480 --> 00:56:10.140
So again, the proof gives
you no information whatsoever
00:56:10.140 --> 00:56:13.230
about the bound M as a
function of the input
00:56:13.230 --> 00:56:16.180
parameters, the epsilons.
00:56:16.180 --> 00:56:18.890
So it turns out you can
use a different method
00:56:18.890 --> 00:56:20.030
to get the bounds.
00:56:20.030 --> 00:56:22.700
Namely, we actually more
or less did this proof
00:56:22.700 --> 00:56:25.830
when we discussed regularity
lemma, the strong regularity
00:56:25.830 --> 00:56:26.330
lemma.
00:56:26.330 --> 00:56:29.120
So we did a different proof
where we iterated an energy
00:56:29.120 --> 00:56:30.780
increment argument.
00:56:30.780 --> 00:56:33.740
And that gave you some concrete
bounds, some bounds which
00:56:33.740 --> 00:56:36.020
iterates on these epsilons.
00:56:36.020 --> 00:56:37.565
But here is a different proof.
00:56:37.565 --> 00:56:41.150
It gives you less
information, but it elegantly
00:56:41.150 --> 00:56:43.690
uses this compactness feature
of the space of graphons.
00:56:47.440 --> 00:56:48.660
Any questions?
00:56:54.310 --> 00:56:56.040
OK, so we proved compactness.
00:56:56.040 --> 00:56:59.130
So now let's go on to
the other two claims,
00:56:59.130 --> 00:57:01.210
namely the existence
of the limit
00:57:01.210 --> 00:57:04.080
and that equivalences
of convergence.
00:57:04.080 --> 00:57:05.800
The existence of the
limit more or less
00:57:05.800 --> 00:57:07.410
is a consequence of compactness.
00:57:15.380 --> 00:57:23.690
So you have this sequence of
graphons, W1, W2, and so on.
00:57:23.690 --> 00:57:34.260
And the claim is that, if
this sequence of F densities
00:57:34.260 --> 00:57:47.750
converges for each
F, then there exists
00:57:47.750 --> 00:57:57.680
some limit W such that all of
these sequences of F densities
00:57:57.680 --> 00:58:01.140
converge to the limit density.
00:58:01.140 --> 00:58:05.810
So that was the claim, so
nothing about cut norms
00:58:05.810 --> 00:58:10.040
in at least as far as
the statement goes.
00:58:10.040 --> 00:58:15.540
Well OK, from
compactness, we know
00:58:15.540 --> 00:58:19.680
that you can produce always
a subsequential limit.
00:58:19.680 --> 00:58:28.030
So by compactness or
sequential compactness,
00:58:28.030 --> 00:58:36.810
there exists some limit
point which we call W.
00:58:36.810 --> 00:58:40.950
And this W has
the property that,
00:58:40.950 --> 00:58:49.230
for some subsequence,
the cut distance
00:58:49.230 --> 00:58:54.120
from the subsequence
converges to W.
00:58:54.120 --> 00:59:07.660
So for some subsequence n0
as ni going to infinity.
00:59:07.660 --> 00:59:20.785
But now, by the counting lemma,
the sequence of F densities--
00:59:20.785 --> 00:59:22.410
so the counting Lemma
tells you, if you
00:59:22.410 --> 00:59:26.040
have cut distance going
to 0, then the F density
00:59:26.040 --> 00:59:28.120
should also go to 0.
00:59:28.120 --> 00:59:29.620
So indeed, that's
what we have here.
00:59:38.450 --> 00:59:40.840
So this is so far just
for the subsequence.
00:59:40.840 --> 00:59:45.340
But we assumed already
that the entire sequence
00:59:45.340 --> 00:59:49.550
converges in respect
to every F densities.
00:59:56.940 --> 00:59:59.310
So it must be the same limit.
01:00:04.130 --> 01:00:08.660
And that finishes the
proof of convergence,
01:00:08.660 --> 01:00:11.830
so proof of the
existence of the limit.
01:00:11.830 --> 01:00:16.260
So we obtain this
limit from compactness.
01:00:16.260 --> 01:00:19.590
Next, let's prove the
equivalence of convergence.
01:00:19.590 --> 01:00:21.310
And this one is
somewhat trickier.
01:00:21.310 --> 01:00:24.150
So what happens here
is that we would
01:00:24.150 --> 01:00:27.840
like to show that these
two notions of convergence,
01:00:27.840 --> 01:00:30.750
one having to do
with F densities
01:00:30.750 --> 01:00:33.320
and another having to
do with cut distance,
01:00:33.320 --> 01:00:36.100
that these two notions are
equivalent to each other.
01:00:41.250 --> 01:00:51.700
So the goal here is to show that
this F density convergence is
01:00:51.700 --> 01:01:01.390
equivalent to the statement that
W sub n is Cauchy with respect
01:01:01.390 --> 01:01:04.630
to the cut distance.
01:01:07.410 --> 01:01:11.360
All right, claim one of
the directions is easy.
01:01:11.360 --> 01:01:12.360
Which direction is that?
01:01:22.070 --> 01:01:24.298
So which direction is
the easy direction?
01:01:28.590 --> 01:01:31.230
So which way, left,
going left, going right?
01:01:34.520 --> 01:01:36.240
OK, so going left?
01:01:36.240 --> 01:01:40.620
So I claim that this
is easy, because it
01:01:40.620 --> 01:01:41.830
follows from counting lemma.
01:01:50.633 --> 01:01:53.050
Counting lemma, remember the
spirit of the counting lemma,
01:01:53.050 --> 01:01:56.130
at least qualitatively, is that
if you have two graphons that
01:01:56.130 --> 01:01:59.760
are close in cut distance, then
they are close in F densities.
01:01:59.760 --> 01:02:03.030
So if you have Cauchy with
respect to cut distance,
01:02:03.030 --> 01:02:06.764
then they are Cauchy, and hence,
convergent in F densities.
01:02:09.370 --> 01:02:12.760
And it's the other direction
that will require some work.
01:02:12.760 --> 01:02:16.130
And this one is actually
genuinely tricky.
01:02:16.130 --> 01:02:21.160
So and it's almost kind
of a miraculous statement,
01:02:21.160 --> 01:02:25.630
that somehow if you only
knew the F densities--
01:02:25.630 --> 01:02:28.480
so somebody gives you this
very large sequence of graphs
01:02:28.480 --> 01:02:30.760
and only tells you that
the triangle densities,
01:02:30.760 --> 01:02:34.090
the C4 densities, all of these
graph densities, they converge.
01:02:34.090 --> 01:02:37.600
Somehow from these
small statistics,
01:02:37.600 --> 01:02:41.140
you conclude that the
graphs globally look
01:02:41.140 --> 01:02:43.370
very similar to each other.
01:02:43.370 --> 01:02:44.870
That's actually,
if you think about,
01:02:44.870 --> 01:02:49.035
this is an amazing statement.
01:02:49.035 --> 01:02:50.160
OK, so let's see the proof.
01:02:54.030 --> 01:02:56.100
The proof method
here is somewhat
01:02:56.100 --> 01:03:00.450
representative of these
graph-limit-type arguments.
01:03:00.450 --> 01:03:03.870
So it's worth paying attention
to see how this one goes.
01:03:03.870 --> 01:03:07.380
So by compactness, if--
01:03:07.380 --> 01:03:11.550
OK, we're going to set
up by contradiction.
01:03:11.550 --> 01:03:24.290
If the sequence is
not Cauchy, then there
01:03:24.290 --> 01:03:26.380
exists two limit points.
01:03:31.930 --> 01:03:34.830
So there exists at least
two distinct limit points.
01:03:34.830 --> 01:03:40.200
And call them u and
W, and such that--
01:03:46.370 --> 01:03:50.110
so because you have two
separate limit points,
01:03:50.110 --> 01:03:54.590
you must have that
this sequence,
01:03:54.590 --> 01:03:57.990
at least along a subsequence
that converges to W,
01:03:57.990 --> 01:04:05.970
converges in F densities
to W. So initially, this
01:04:05.970 --> 01:04:07.260
is true along subsequence.
01:04:07.260 --> 01:04:09.160
But the left-hand
side is convergent,
01:04:09.160 --> 01:04:12.790
so this is true
along the sequence.
01:04:12.790 --> 01:04:15.370
But u is also a limit point.
01:04:15.370 --> 01:04:21.990
So the same is true for u.
01:04:26.337 --> 01:04:34.670
And therefore, the F density in
W must equal to the F density
01:04:34.670 --> 01:04:45.670
in u for all F.
01:04:45.670 --> 01:04:49.420
So we would be done if we can
prove that the F densities,
01:04:49.420 --> 01:04:51.670
the collection of all
these F densities,
01:04:51.670 --> 01:04:53.705
they determine the graphon.
01:04:53.705 --> 01:04:54.830
And that's indeed the case.
01:04:54.830 --> 01:04:56.540
And so this is the next claim.
01:04:56.540 --> 01:05:00.140
So it's what I
will call a moment
01:05:00.140 --> 01:05:17.450
lemma, is that if u and W
are graphons such that the F
01:05:17.450 --> 01:05:27.380
densities agree for all F,
then the cut distance between u
01:05:27.380 --> 01:05:31.910
and W is equal to 0.
01:05:31.910 --> 01:05:34.880
Somehow the local statistics
tells you globally
01:05:34.880 --> 01:05:39.420
that these two graphons
must agree with each other.
01:05:39.420 --> 01:05:42.360
Does anyone know why I
call it a moment lemma?
01:05:42.360 --> 01:05:46.730
There is something else which
this should remind you of.
01:05:46.730 --> 01:05:50.040
So there are some classical
results in probability that
01:05:50.040 --> 01:05:53.240
tells you, if you have two
probability distributions,
01:05:53.240 --> 01:05:57.830
both, assume are nice enough,
then if they have the same
01:05:57.830 --> 01:06:01.490
k'th moment for every k, so
first moment, second moment,
01:06:01.490 --> 01:06:03.770
third moment, if all
the moments agree,
01:06:03.770 --> 01:06:07.260
that these two probability
distributions should agree.
01:06:07.260 --> 01:06:10.130
And this is some
graphical version of that.
01:06:10.130 --> 01:06:12.600
So instead of looking at the
probability distribution,
01:06:12.600 --> 01:06:15.590
we're looking at graphons,
which are two-dimensional.
01:06:15.590 --> 01:06:17.135
These are
two-dimensional objects.
01:06:17.135 --> 01:06:19.760
And this moments lemma tells you
that in these two-dimensional,
01:06:19.760 --> 01:06:21.927
in the corresponding
two-dimensional moments, namely
01:06:21.927 --> 01:06:25.040
these F moments,
if they agree, then
01:06:25.040 --> 01:06:27.380
the two graphons must agree.
01:06:30.240 --> 01:06:33.810
So it's the analog of
the probability theory
01:06:33.810 --> 01:06:37.860
statement about moments.
01:06:37.860 --> 01:06:39.540
The proof is actually
somewhat tricky.
01:06:39.540 --> 01:06:41.190
So I'm only going to
give you a sketch.
01:06:44.880 --> 01:06:49.590
And the key here is to consider
the W random graph, which
01:06:49.590 --> 01:06:51.090
we saw last lecture.
01:06:57.326 --> 01:07:00.320
So this is W random
graph with k vertices
01:07:00.320 --> 01:07:06.070
sampled using the graphon W.
01:07:06.070 --> 01:07:09.010
So a key observation
here is that,
01:07:09.010 --> 01:07:19.140
for every F, the probability
that the sampled W
01:07:19.140 --> 01:07:26.560
random graph agrees with F--
01:07:26.560 --> 01:07:29.960
and here, there is a
bit of a technicality.
01:07:29.960 --> 01:07:31.910
I want them to agree
as labeled graphs.
01:07:35.880 --> 01:07:39.340
So the vertices of W are a
priori labeled 1 through k.
01:07:39.340 --> 01:07:43.330
And this kW random graph
is generated with vertices
01:07:43.330 --> 01:07:44.890
labeled 1 through k.
01:07:44.890 --> 01:07:48.640
They agree with some
probability that is completely
01:07:48.640 --> 01:07:54.170
determined by the F densities.
01:07:54.170 --> 01:07:54.670
Yes?
01:07:54.670 --> 01:07:56.740
AUDIENCE: Is k the
number of vertices of F?
01:07:56.740 --> 01:08:02.390
PROFESSOR: Yeah, so k is
the number of vertices of F.
01:08:02.390 --> 01:08:04.867
And the specific formula
is not so important.
01:08:04.867 --> 01:08:05.950
Let me just write it down.
01:08:05.950 --> 01:08:09.470
But the point is that, if you
know all the F densities, then
01:08:09.470 --> 01:08:12.500
you have all the information
about the distribution
01:08:12.500 --> 01:08:15.110
of this W random graph.
01:08:15.110 --> 01:08:18.810
And the way you can calculate
the actual probability
01:08:18.810 --> 01:08:21.173
is via an inclusion exclusion.
01:08:23.840 --> 01:08:26.090
And the reason we have to
do this inclusion exclusion
01:08:26.090 --> 01:08:28.340
is just because this
is more like counting
01:08:28.340 --> 01:08:29.760
induced subgraphs.
01:08:29.760 --> 01:08:31.939
And this is counting
actual subgraphs.
01:08:31.939 --> 01:08:33.770
So there is an extra step.
01:08:33.770 --> 01:08:36.880
But the point is that, if you
knew this data, the moment's
01:08:36.880 --> 01:08:40.026
data then you immediately
know the distribution of the W
01:08:40.026 --> 01:08:40.609
random graphs.
01:08:51.800 --> 01:08:57.520
OK, so if I have two graphons
for which I know that their F
01:08:57.520 --> 01:09:00.350
densities agree,
then I should be
01:09:00.350 --> 01:09:05.660
able to conclude that
the corresponding W
01:09:05.660 --> 01:09:19.580
random graphs also have the same
distribution, in particular,
01:09:19.580 --> 01:09:25.340
this, the W random graph
and the u random graph
01:09:25.340 --> 01:09:27.324
have the same distribution.
01:09:34.010 --> 01:09:37.939
I am going to create a
variant of the W random graph
01:09:37.939 --> 01:09:42.979
which is something
called an H random graph.
01:09:42.979 --> 01:09:44.720
It's kind of like
the W random graph
01:09:44.720 --> 01:09:46.800
except I forget
the very last step.
01:09:46.800 --> 01:09:51.319
So I only keep a weighted--
01:09:51.319 --> 01:10:06.310
think of it as, so think this
is an edge-weighted graph where
01:10:06.310 --> 01:10:16.860
you sample x1 through xk
uniformly between 0 and 1.
01:10:16.860 --> 01:10:26.680
And I put edge weight between
i and j to be W of xi xj.
01:10:26.680 --> 01:10:29.580
So the difference between this
H version and the G version
01:10:29.580 --> 01:10:31.110
is that the G
version is obtained
01:10:31.110 --> 01:10:34.120
by turning this weight
into an actual edge
01:10:34.120 --> 01:10:37.320
with that probability, but
if I don't do the last step,
01:10:37.320 --> 01:10:41.150
I obtain this
intermediate object.
01:10:41.150 --> 01:10:46.920
So the following are true.
01:10:46.920 --> 01:10:49.430
And this is where I'm
going to skip the proofs.
01:10:52.700 --> 01:11:02.430
If I look at this H random
graph and the G random graph,
01:11:02.430 --> 01:11:05.663
they are very close
in cut distance.
01:11:05.663 --> 01:11:07.080
You can think of
this as the claim
01:11:07.080 --> 01:11:11.250
that G and P is very close
to G in cut distance.
01:11:11.250 --> 01:11:12.930
So they are very
close in cut distance.
01:11:18.160 --> 01:11:23.550
As k going to infinity
with probability 1--
01:11:23.550 --> 01:11:25.050
so now I'm going
to do the proof.
01:11:25.050 --> 01:11:27.720
But it's some kind of a
concentration argument.
01:11:33.990 --> 01:11:41.930
And the second claim is that
the H random graph is actually
01:11:41.930 --> 01:11:46.660
very close to the original
graphon W, as well.
01:11:51.260 --> 01:11:57.770
This is also little l1 in
distance as k goes to infinity.
01:11:57.770 --> 01:12:01.180
So this one is,
again, not so obvious.
01:12:01.180 --> 01:12:08.870
But it's easier
in the case when W
01:12:08.870 --> 01:12:16.520
itself is a step function,
in which case, the produced
01:12:16.520 --> 01:12:20.180
H is almost the same as W,
except the boundaries are
01:12:20.180 --> 01:12:24.150
slightly shifted, perhaps.
01:12:24.150 --> 01:12:30.890
And so you first approximate
W by a step function,
01:12:30.890 --> 01:12:33.530
and prove this up to an
epsilon approximation,
01:12:33.530 --> 01:12:37.470
and then let the
steps go to infinity.
01:12:37.470 --> 01:12:41.060
So if you have these
two claims, so then
01:12:41.060 --> 01:12:58.640
we see that this one here is
identically distributed as ku.
01:12:58.640 --> 01:13:06.200
So it should follow that the
corresponding H random graph
01:13:06.200 --> 01:13:14.765
for u, if you place the same
inequalities by the u versions,
01:13:14.765 --> 01:13:16.550
it should also be true.
01:13:16.550 --> 01:13:21.730
So because these two are
the same distribution,
01:13:21.730 --> 01:13:24.220
if you follow this
chain, you obtain
01:13:24.220 --> 01:13:30.590
that the cut distance between
u and w is equal to 0.
01:13:38.410 --> 01:13:45.790
I want to close by
mentioning, in some sense--
01:13:45.790 --> 01:13:50.580
so here, you have two graphons
that have exactly the same F
01:13:50.580 --> 01:13:51.770
moments.
01:13:51.770 --> 01:13:54.790
But what if I give you two
graphons which have very
01:13:54.790 --> 01:13:56.500
similar moments to each other?
01:13:56.500 --> 01:13:59.140
Can you conclude
that the two graphons
01:13:59.140 --> 01:14:01.370
are close to each other?
01:14:01.370 --> 01:14:04.510
And that will be some kind
of an inverse counting lemma.
01:14:04.510 --> 01:14:06.670
And in fact, it does
follow as a corollary.
01:14:17.070 --> 01:14:20.340
And the statement is
that, for every epsilon,
01:14:20.340 --> 01:14:32.330
there exists k and eta such that
if the two graphons u and W are
01:14:32.330 --> 01:14:42.080
such that the F densities do
not differ by more than eta
01:14:42.080 --> 01:14:51.460
for every F on, at
most, k vertices,
01:14:51.460 --> 01:15:01.120
then the cut distance between
u and W is, at most, epsilon.
01:15:01.120 --> 01:15:03.910
So the counting lemma tells you,
if the cut distance is small,
01:15:03.910 --> 01:15:07.120
then all the F moments
are close to each other.
01:15:07.120 --> 01:15:09.430
And the inverse tells
you this converse.
01:15:09.430 --> 01:15:13.090
So it tells you this, if
you have similar F moments,
01:15:13.090 --> 01:15:18.690
up to a certain point,
then this is small.
01:15:18.690 --> 01:15:20.720
You can deduce the
inverse counting lemma
01:15:20.720 --> 01:15:23.730
from the moments lemma
via a compactness argument
01:15:23.730 --> 01:15:27.420
similar to the one that
we did in class today.
01:15:27.420 --> 01:15:30.240
And I want to give you a chance
to practice with that argument.
01:15:30.240 --> 01:15:33.120
So this will be on the
homework, for the next homework.
01:15:33.120 --> 01:15:34.860
I'll give you some
practice with using
01:15:34.860 --> 01:15:37.780
these compactness arguments.
01:15:37.780 --> 01:15:42.670
But you see, just with the
other compactness statements,
01:15:42.670 --> 01:15:46.860
it doesn't tell you anything
about the k and the epsilon
01:15:46.860 --> 01:15:48.210
as a function of--
01:15:48.210 --> 01:15:51.600
the k and eta as a
function of epsilon.
01:15:51.600 --> 01:15:54.400
So there are other proofs that
gives you concrete bounds,
01:15:54.400 --> 01:15:58.270
but this proof here
is much simpler
01:15:58.270 --> 01:16:04.370
if you assume the corresponding
results about compactness.
01:16:04.370 --> 01:16:11.780
And finally, I want to mention
that in the moments lemma,
01:16:11.780 --> 01:16:15.980
in order to deduce that
u and w have the same--
01:16:15.980 --> 01:16:18.320
that they are basically
the same graphon,
01:16:18.320 --> 01:16:23.060
we need to consider F
moments for all F's.
01:16:23.060 --> 01:16:25.640
So you might ask,
could it be the case
01:16:25.640 --> 01:16:31.800
that we only need some
finite set of F's to deduce--
01:16:31.800 --> 01:16:34.220
to recover the graphon?
01:16:34.220 --> 01:16:40.400
Is it the case that
you can recover W
01:16:40.400 --> 01:16:44.410
from only a finite
number of F moments?
01:16:44.410 --> 01:16:47.000
And this is, it's actually
a very interesting problem
01:16:47.000 --> 01:16:50.660
for which we already
saw one instance.
01:16:50.660 --> 01:16:54.710
Namely, when we discussed
quasi-random graphs,
01:16:54.710 --> 01:17:01.910
we saw that if you know
that the k2 moment is p
01:17:01.910 --> 01:17:12.860
and also the C4 moment
is p to the 4, then
01:17:12.860 --> 01:17:16.880
we can deduce that
the graphon must
01:17:16.880 --> 01:17:18.942
be the constant graphon, p.
01:17:18.942 --> 01:17:20.930
OK, so we didn't do
it in this language,
01:17:20.930 --> 01:17:23.060
but that's what the proof does.
01:17:23.060 --> 01:17:27.050
And likewise, you can use this
to deduce a qualitative version
01:17:27.050 --> 01:17:40.670
where you have an extra slack
and an extra slack over here.
01:17:40.670 --> 01:17:45.410
So you might ask, except
for the constant graphons,
01:17:45.410 --> 01:17:48.050
are there other graphons
for which you can similarly
01:17:48.050 --> 01:17:49.880
deduce--
01:17:49.880 --> 01:17:53.840
recover this graphon from just
a finite amount of moments data?
01:17:53.840 --> 01:17:56.510
And such graphons are
known as finitely forcible.
01:18:03.350 --> 01:18:08.000
So finitely forcible
graphons W such
01:18:08.000 --> 01:18:20.690
that a finite number of
moments can uniquely recover--
01:18:25.350 --> 01:18:28.620
can uniquely identify
this graphon, W.
01:18:28.620 --> 01:18:30.510
And a very interesting
question is,
01:18:30.510 --> 01:18:34.130
what is the set of all
finitely forcible graphons?
01:18:34.130 --> 01:18:36.290
And it turns out, this
is not at all obvious.
01:18:36.290 --> 01:18:38.790
And let me just give you some
examples, highly non-trivial,
01:18:38.790 --> 01:18:42.240
that turned out to
be finitely forcible.
01:18:42.240 --> 01:18:47.040
For example, anything
which is a step graphon
01:18:47.040 --> 01:18:47.915
is finitely forcible.
01:18:52.120 --> 01:18:55.350
The half graphon
which corresponds
01:18:55.350 --> 01:18:58.870
to the limit of a
sequence of half graphs
01:18:58.870 --> 01:19:00.450
is finitely forcible.
01:19:00.450 --> 01:19:03.630
I mean, already, I think
neither of these two examples
01:19:03.630 --> 01:19:06.570
are easy at all.
01:19:06.570 --> 01:19:10.140
And this example here
can be generalized where
01:19:10.140 --> 01:19:12.000
you have any polynomial curve.
01:19:20.820 --> 01:19:22.905
I think this has to be--
01:19:22.905 --> 01:19:28.660
so if it's a polynomial curve,
it's also finitely forcible.
01:19:28.660 --> 01:19:30.370
But turns out finitely
forcible graphons
01:19:30.370 --> 01:19:32.920
can get quite complicated.
01:19:32.920 --> 01:19:38.150
And there is still rather quite
a bit of mystery around them.
01:19:38.150 --> 01:19:43.810
OK, so next time, I want to
discuss some inequalities that
01:19:43.810 --> 01:19:45.640
come out of--
01:19:45.640 --> 01:19:49.034
you can state between
different F densities.
01:19:49.034 --> 01:19:49.940
OK, great.
01:19:49.940 --> 01:19:51.820
That's all for today.