[00:00:05.14]
Ok so well comes.
[00:00:07.14]

[00:00:08.15]
First still of all that
thank you Richard for
[00:00:13.12]

[00:00:13.12]
introduction and so we'll get started.
[00:00:17.02]

[00:00:19.01]
So the title is faster approximation all
boredom sand complex here that this is
[00:00:23.11]

[00:00:23.11]
for the.
[00:00:23.23]

[00:00:25.07]
Design of network systems.
[00:00:26.18]

[00:00:29.11]
First of all introduce what is the control
of network control systems and
[00:00:34.04]

[00:00:34.04]
what are the design problems so we always.
[00:00:37.14]

[00:00:39.17]
So these.
[00:00:41.19]

[00:00:43.04]
These this litle off works was
dungarees my former colleagues in for
[00:00:49.03]

[00:00:49.03]
the university.
[00:00:50.02]

[00:00:51.04]
And the chat and my Ph d. advisor and.
[00:00:55.01]

[00:00:56.11]
These works are also short for
Greece Stacey part of my.
[00:01:00.17]

[00:01:03.12]
Postdoc advisor and also Richard Ok so
[00:01:09.10]

[00:01:09.10]
did the talk consists of 3 parts 1st.
[00:01:12.17]

[00:01:15.09]
Introduced for
what his network control system and
[00:01:19.05]

[00:01:19.05]
what are the statistics we
care about such systems and
[00:01:23.13]

[00:01:23.13]
how the problems are formulated and
then I will talk about 2 of our works
[00:01:29.12]

[00:01:29.12]
which hard leaders which deals
with which deal with these 2
[00:01:34.14]

[00:01:34.14]
set of different problems one is leader
selection and our why is education.
[00:01:39.13]

[00:01:40.23]
So I would talk about 2 works one
is to minimize coherence which is
[00:01:46.10]

[00:01:46.10]
kind of the sum of variances awful
nodes Sue Niederer selections other
[00:01:51.10]

[00:01:51.10]
wind is revising entropy
through education and
[00:01:57.14]

[00:01:57.14]
finally I will conclude by talking
about sound which are directions.
[00:02:02.01]

[00:02:04.04]
Of oak.
[00:02:06.14]

[00:02:06.14]
So 1st of all network systems
are system composed of dynamic units
[00:02:11.17]

[00:02:11.17]
that interact with over a network so by
that I mean I mean they there are states.
[00:02:17.12]

[00:02:18.12]
Change their state change by time so for
[00:02:22.15]

[00:02:22.15]
example here's a sensor network here's
some formation it work here some.
[00:02:27.05]

[00:02:29.04]
Are composed of different components
seen a robot so in these networks.
[00:02:35.06]

[00:02:36.15]
These networks can be large scale and
specially distribute it so
[00:02:40.11]

[00:02:40.11]
we usually we need some kind of.
[00:02:42.06]

[00:02:43.12]
Decentralized McCann ism to
spread out the information in
[00:02:48.09]

[00:02:48.09]
off each node to everybody
know what you're doing.
[00:02:52.11]

[00:02:53.16]
So so average consensus algorithms
are actually the most commonly used.
[00:02:59.19]

[00:03:01.00]
Protocol to spread out
once information for
[00:03:04.17]

[00:03:04.17]
example here we have
a simplest form of it so for.
[00:03:08.08]

[00:03:09.23]
Vertex Exidy it has a state and
[00:03:14.12]

[00:03:14.12]
it opt out operates his state according
to the state of its neighbors and
[00:03:20.01]

[00:03:20.01]
itself so if so yes this is easy
to understand if the state is.
[00:03:25.12]

[00:03:26.21]
A value is lower than its neighbors
to increase these states.
[00:03:31.12]

[00:03:32.22]
And here w i j is just edge
way toward cutting streets and
[00:03:38.21]

[00:03:38.21]
this absolute here's a state step size.
[00:03:41.02]

[00:03:43.05]
So Judy we also studied these continuing.
[00:03:47.02]

[00:03:48.21]
Off.
[00:03:49.10]

[00:03:50.17]
Often question so.
[00:03:52.15]

[00:03:54.08]
So why do we care about these
continuity version because they are.
[00:03:58.11]

[00:03:59.13]
Simpler and.
[00:04:00.14]

[00:04:01.15]
There are some minor differences between
them but it looks like most of the time is
[00:04:06.21]

[00:04:06.21]
enough to study this so from now on I will
talk about these companies for action.
[00:04:11.13]

[00:04:13.23]
But in reality.
[00:04:15.09]

[00:04:16.22]
Always are subject to noises or
[00:04:21.10]

[00:04:21.10]
disturbances or some Amado dynamic.
[00:04:24.17]

[00:04:26.19]
In the system so
we just see some we some noise
[00:04:32.05]

[00:04:32.05]
with some noise process here so
usually this is modeled by a white
[00:04:36.23]

[00:04:36.23]
voice which is a mathematical
abstraction in nearing.
[00:04:41.08]

[00:04:44.11]
So in this way we can write
these equation as these compact
[00:04:49.09]

[00:04:49.09]
will this is time derivative of
the vector x. is equal to the minus.
[00:04:54.03]

[00:04:55.08]
Matrix times hex process noise vector so
the law question is just defined as the.
[00:05:01.23]

[00:05:03.01]
Degree matrix minus.
[00:05:04.11]

[00:05:05.16]
The Matrix so on the bag no its degree.
[00:05:08.19]

[00:05:10.23]
Matrix and off.
[00:05:12.11]

[00:05:13.20]
Entries are just equal to
the negative adjacency matrix and
[00:05:18.14]

[00:05:18.14]
this matrix can be written as the sum
of the passions of every edges
[00:05:23.14]

[00:05:23.14]
of every edge So for example here it can
be written as these the sum of these 3.
[00:05:28.16]

[00:05:30.05]
Beatrice's and the speedy is just defined
vector with only 2 non-zero entries
[00:05:38.10]

[00:05:38.10]
being corresponding to 2 and
vertices off the edge and
[00:05:43.23]

[00:05:43.23]
we assign wind and
negative one to them opportunity and.
[00:05:49.00]

[00:05:50.20]
From these 2 we see that it's p s d so so
[00:05:55.04]

[00:05:55.04]
we can defined by just inverting
all non-zero matters so
[00:06:00.19]

[00:06:00.19]
we've been talking about these
equations so how does this.
[00:06:04.13]

[00:06:05.20]
System evolve so on so
[00:06:09.07]

[00:06:09.07]
this is a simulation we can see that's
an average of the whole system treats by
[00:06:13.14]

[00:06:13.14]
the time it's like a random
walking one dimensional space.
[00:06:16.04]

[00:06:17.14]
Wow the different colors
indicates different nodes so
[00:06:22.01]

[00:06:22.01]
they're all the nodes are banking
near is its current Herridge right it
[00:06:26.14]

[00:06:26.14]
behaves like this so
we want to study these formally So
[00:06:31.16]

[00:06:31.16]
this is again the equation so it has
been shown that under some conditions.
[00:06:36.07]

[00:06:38.01]
X. t. is a Gaussian process so
the condition is like.
[00:06:42.08]

[00:06:43.14]
X. the initial state is a constant or.
[00:06:46.17]

[00:06:48.19]
Are.
[00:06:49.07]

[00:06:51.08]
Sample from a.
[00:06:52.06]

[00:06:54.11]
Distribution.
[00:06:55.03]

[00:06:58.12]
So so expectation of this state vector and
[00:07:03.16]

[00:07:03.16]
the covariance matrix are defined just
as true to us like this and it was fine.
[00:07:09.21]

[00:07:11.01]
It is so
it has been shown that these these
[00:07:14.23]

[00:07:14.23]
statistics they follow these
deterministic different equations so
[00:07:20.00]

[00:07:20.00]
it's enough to study these equations
to characterize their behavior.
[00:07:23.22]

[00:07:27.17]
So for this system we can find that
expectation has a steady state
[00:07:32.21]

[00:07:32.21]
it comes to the average
value of the initial state.
[00:07:36.19]

[00:07:36.19]
But from this equation we can see
that the covariance matrix is
[00:07:43.13]

[00:07:43.13]
bounded by time bonded I
mean value of it to its
[00:07:49.04]

[00:07:49.04]
own body so we can consider
some quadratic form of the use
[00:07:54.11]

[00:07:54.11]
of corresponds to all of it or
we can find this is always positive.
[00:07:59.00]

[00:08:02.03]
So but we've seen that
the behavior of the system so
[00:08:05.13]

[00:08:05.13]
we can defines another like this why t.
[00:08:10.20]

[00:08:10.20]
as projection to like
the deviation from off the east
[00:08:16.05]

[00:08:16.05]
every denotes States to the average of the
system so again this is a Gaussian process
[00:08:23.08]

[00:08:23.08]
we define its expectation covariance
matrix that followed these equations.
[00:08:28.06]

[00:08:30.08]
And this time we get a steady state for
the expectation and the and
[00:08:35.18]

[00:08:35.18]
covariance matrix here so
the covariance matrix should set his
[00:08:40.22]

[00:08:40.22]
steady state of the covariance
matrix at his face these equation
[00:08:46.05]

[00:08:46.05]
this is also called North equation in
control and in your control theory.
[00:08:50.22]

[00:08:55.10]
Are So given these equations so
[00:08:58.20]

[00:08:58.20]
we want to optimize the system
what do we care about what So
[00:09:02.15]

[00:09:02.15]
1st of all we care about convergence
rate which is given by the.
[00:09:07.01]

[00:09:10.15]
Connectivity of the graph and
from this equation we can see.
[00:09:14.21]

[00:09:15.23]
And we we care about steady state
variance which is defined as the sum of
[00:09:21.02]

[00:09:21.02]
variances off each node so
the this matrix is just solution.
[00:09:26.18]

[00:09:28.13]
For the east for this equation so
it's the covariance.
[00:09:31.21]

[00:09:33.18]
Or.
[00:09:34.06]

[00:09:35.09]
Not.
[00:09:35.21]

[00:09:37.05]
I will get there we can design Yeah
we can design the nose to control.
[00:09:43.06]

[00:09:45.10]
Our.
[00:09:45.22]

[00:09:48.00]
Yeah yeah.
[00:09:48.12]

[00:10:00.09]
So also from this covariance
matrix you can define
[00:10:04.20]

[00:10:04.20]
We can then we know we
we know we know it's.
[00:10:09.08]

[00:10:10.13]
Got in process so
given time it's a distribution so
[00:10:14.10]

[00:10:14.10]
this determinant gives a volume.
[00:10:17.06]

[00:10:19.08]
Of.
[00:10:19.20]

[00:10:21.04]
Uncertainty volume so
it's given by determining the days and.
[00:10:28.01]

[00:10:29.21]
Sousa coffee maker street
serum we know that it's
[00:10:34.18]

[00:10:34.18]
correlated to the number of
spending trees but question and
[00:10:40.07]

[00:10:40.07]
since given the graphs given a number
of nodes these are decided so
[00:10:46.12]

[00:10:46.12]
the network structure
affects only these part so.
[00:10:50.12]

[00:10:52.12]
This is searching we want to optimize so
sometimes they define
[00:10:57.16]

[00:10:57.16]
these has network and p. as a negative
log of that number of spending trees
[00:11:02.23]

[00:11:04.11]
so here's another kind of systems
code leader follower systems in
[00:11:09.09]

[00:11:09.09]
such a system some nodes are controlled so
they have some assigned.
[00:11:13.22]

[00:11:15.03]
Value So in these or pick a chanst
it has different meanings for
[00:11:21.04]

[00:11:21.04]
example here these nodes can be operated
you know opinion it works these
[00:11:26.09]

[00:11:26.09]
those can have some fixed value 0 or
one meaning
[00:11:32.05]

[00:11:32.05]
they have some polarizing opinions
without opinion leaders and.
[00:11:36.11]

[00:11:38.11]
So far worse just updates
it states according to
[00:11:43.01]

[00:11:43.01]
its neighbors whether it's the euro or.
[00:11:45.10]

[00:11:46.21]
Follower.
[00:11:47.15]

[00:11:50.17]
So this is a question for
the for the for the system.
[00:11:55.12]

[00:11:56.16]
So.
[00:11:57.04]

[00:11:58.06]
This is the simplest case that
this leaders have all the leaders
[00:12:03.08]

[00:12:03.08]
have the same state.
[00:12:04.15]

[00:12:05.21]
We can discuss about they have different
states but it starts from here so.
[00:12:13.11]

[00:12:14.19]
The followers just operates
it states according to
[00:12:18.10]

[00:12:18.10]
the State of the followers and
leaders since here are neater so
[00:12:23.12]

[00:12:23.12]
I have a value of 0 so
it says this equation so I hope.
[00:12:27.22]

[00:12:28.23]
The whole question is given here
again we have this noise factor here.
[00:12:33.15]

[00:12:36.09]
So on so we get an Amex for
the follower or the foreign news it's
[00:12:43.07]

[00:12:43.07]
given by this question again we
study the code congregants rate and
[00:12:49.14]

[00:12:49.14]
internal energy which is given by
the variance some of variance is and
[00:12:54.06]

[00:12:54.06]
the uncertainty value
off of the covariance.
[00:12:58.06]

[00:13:01.09]
Ok I think this is a background so
I will soon get your
[00:13:06.19]

[00:13:06.19]
definition of the problems and
our approach use so
[00:13:12.10]

[00:13:12.10]
the 1st set of problems
are co-leaders action so
[00:13:17.11]

[00:13:17.11]
given the graph g.
we want to choose the subset of at most k.
[00:13:21.17]

[00:13:21.17]
vertices assets leaders such that for
the rest of the nodes.
[00:13:26.02]

[00:13:27.04]
We have these.
[00:13:28.05]

[00:13:29.12]
Matrix l. with the rows and
[00:13:32.18]

[00:13:32.18]
columns corresponds to these
the rest of the nodes.
[00:13:35.20]

[00:13:36.22]
And we want to optimize
these quantities for
[00:13:41.02]

[00:13:41.02]
is a matrix so
they are then that you offer.
[00:13:44.13]

[00:13:47.23]
These subtly tricks and trace of
the inverse of this operation truth and
[00:13:53.03]

[00:13:53.03]
and determine the toffs inverse
of the separate tricks just.
[00:13:57.22]

[00:13:59.04]
Want to do.
[00:13:59.16]

[00:14:01.04]
The math.
[00:14:01.16]

[00:14:03.22]
On the.
[00:14:05.17]

[00:14:07.18]
Yeah yeah yeah.
[00:14:09.15]

[00:14:11.20]
It's the graph that you
combine all the beaters.
[00:14:15.00]

[00:14:19.12]
And.
[00:14:20.00]

[00:14:23.07]
It's it's the sub matrix
off all columns and
[00:14:28.00]

[00:14:28.00]
rows corresponds to the rest of the note
so you did it always a needle to use.
[00:14:33.17]

[00:14:35.12]
Yeah so basically.
[00:14:37.04]

[00:14:41.01]
You.
[00:14:41.13]

[00:14:43.15]
Are Not one grid duce this case
it's it's it's you could use
[00:14:48.19]

[00:14:48.19]
a number of spending trees in the graph
that you combine always admired you.
[00:14:54.12]

[00:15:10.22]
Yes this is related to some column in a
row section problems I will discuss these
[00:15:16.20]

[00:15:16.20]
near the end of the top are so
I will talk about why and.
[00:15:22.13]

[00:15:23.14]
How to minimize these
trace of the inverse here.
[00:15:26.23]

[00:15:29.05]
So the problem is defined as given g.
we want to choose.
[00:15:32.15]

[00:15:33.16]
K. nodes as leaders as such that for
[00:15:39.07]

[00:15:39.07]
the rest of the nodes that
these 7 matrix inverse of sup
[00:15:44.21]

[00:15:44.21]
trace of the inverse of the subway
creaks is minimized so when as is.
[00:15:52.21]

[00:15:54.13]
The has only one node here one vertex
here this is actually the Sam off
[00:16:00.04]

[00:16:00.04]
effective resistance is from all our other
nodes to use a bigger vertex at the and
[00:16:08.01]

[00:16:08.01]
one which in your ass is
asking to have to have a has.
[00:16:12.22]

[00:16:15.02]
Many nodes so in this case it's the sum of
effect a reason to the combined leader.
[00:16:19.16]

[00:16:22.10]
So but we don't really use the seen
that this worked but that's a.
[00:16:25.14]

[00:16:26.23]
Good way to think about this problem so
[00:16:30.22]

[00:16:30.22]
effective resist in the cities defined
as voted difference between s. and t.
[00:16:36.03]

[00:16:36.03]
when you need current is
applied from AC to t..
[00:16:40.23]

[00:16:42.02]
And we have these 6 brush in.
[00:16:44.20]

[00:16:47.12]
This is known for a long time and on and
[00:16:50.16]

[00:16:50.16]
this is also related to commute
time by a factor of 2 m..
[00:16:55.01]

[00:16:58.01]
So the algorithms that I talk
about today all rely on these.
[00:17:04.13]

[00:17:06.04]
These frame or at least monitor the city
and super model are the properties skips
[00:17:11.02]

[00:17:11.02]
us the algorithm of we
swam this one over a year
[00:17:15.23]

[00:17:15.23]
proximate ratio so
this is a classic result.
[00:17:20.13]

[00:17:23.21]
Ok.
[00:17:24.09]

[00:17:27.20]
So.
[00:17:28.08]

[00:17:29.08]
We 1st show that the this trace
of the universe of the sub matrix
[00:17:34.11]

[00:17:34.11]
is monotone as you promote it or.
[00:17:38.04]

[00:17:39.16]
Yeah yeah we prove that.
[00:17:41.13]

[00:17:42.19]
Yes.
[00:17:43.07]

[00:17:46.08]
Yes yes yes.
[00:17:49.15]

[00:17:58.12]
So this multi-city is.
[00:18:01.15]

[00:18:01.15]
It is kind of straightforward in
treaty but we were asking for a.
[00:18:06.13]

[00:18:07.22]
Different approach here so.
[00:18:11.04]

[00:18:12.21]
The supermodel Artie's also
know from these commute time
[00:18:18.02]

[00:18:18.02]
perspective but
here we give our back approach which.
[00:18:23.01]

[00:18:24.05]
Has a stronger result so
we actually show that this matrix
[00:18:29.07]

[00:18:29.07]
in this universe of the sub matrix
is Enco I supermodel her and
[00:18:34.11]

[00:18:34.11]
why is this useful because this implies
that these is also untrue I super
[00:18:39.17]

[00:18:39.17]
model her and this this can be used for
side orders or.
[00:18:45.09]

[00:18:46.17]
As a functional or
as a function of ast note ass.
[00:18:50.22]

[00:18:52.08]
But with.
[00:18:52.20]

[00:18:54.16]
Every entry of it is often either
a trio of them we're talking.
[00:19:00.10]

[00:19:01.21]
With Yeah yeah.
[00:19:03.10]

[00:19:09.08]
We're different.
[00:19:09.23]

[00:19:12.17]
Than.
[00:19:13.05]

[00:19:21.15]
What.
[00:19:22.05]

[00:19:27.04]
You owe.
[00:19:28.21]

[00:19:30.02]
It it's defined for
the next in the set for
[00:19:34.21]

[00:19:34.21]
one yeah yeah that's the big one but.
[00:19:38.16]

[00:19:47.14]
Yeah yeah f. f. has different
different sizes we've got a lot.
[00:19:51.18]

[00:19:54.07]
Of that you the.
[00:19:54.22]

[00:19:56.10]
Verse 4 and.
[00:19:58.00]

[00:20:00.11]
4 So given the ass.
[00:20:04.01]

[00:20:04.01]
So we have a fixed hour fast.
[00:20:07.02]

[00:20:08.07]
And.
[00:20:09.18]

[00:20:09.18]
Then take the inverse and
every entry of it it's a function of s.
[00:20:13.11]

[00:20:20.05]
1.
[00:20:20.17]

[00:20:22.17]
1111 and a yes.
[00:20:25.03]

[00:20:26.08]
Or yes yes but it was.
[00:20:29.20]

[00:20:31.21]
As I'm alone we so
[00:20:36.18]

[00:20:36.18]
we need we studied that after
the entry that stealing af.
[00:20:40.19]

[00:20:53.03]
Yes yes.
[00:20:53.22]

[00:20:59.02]
Yes yes.
[00:20:59.19]

[00:21:01.07]
One.
[00:21:01.19]

[00:21:02.23]
With the 110000.
[00:21:05.01]

[00:21:06.17]
0000 l.
[00:21:12.10]

[00:21:13.21]
entry $11.00 of this hour of his.
[00:21:16.09]

[00:21:17.14]
It's a $11.00 entry of the us.
[00:21:20.14]

[00:21:25.17]
I mean I mean the for the rest of the note
[00:21:28.16]

[00:21:28.16]
it's the now that one is not
correspond to a leader it's for the.
[00:21:33.07]

[00:21:42.04]
Maybe yeah.
[00:21:42.20]

[00:21:44.10]
Yeah Ok so how to how do we do in these
we 1st define this matrix that you
[00:21:51.19]

[00:21:51.19]
Punk this matrix valued
function of these vectors e so.
[00:21:57.21]

[00:22:00.19]
So if.
[00:22:01.14]

[00:22:03.08]
On the Bagnall It's the oppression
Matrix it's it's it's the same
[00:22:08.10]

[00:22:08.10]
as the tricks and the often Bagnall
entries are defined as these.
[00:22:13.19]

[00:22:15.02]
One menace these are the it's the is
a vector with every entry corresponds to.
[00:22:19.11]

[00:22:20.12]
2 to one who'd So it's defined as these so
[00:22:25.02]

[00:22:25.02]
this matrix has these form if a z.
is equal to one.
[00:22:29.01]

[00:22:30.13]
We have a back know here which
is given by these l. and
[00:22:35.02]

[00:22:35.02]
if it's not and we have 0 here
[00:22:39.13]

[00:22:39.13]
because these entries are time
times with the arrow and
[00:22:45.03]

[00:22:45.03]
this this part if if we
desist see is the indicator
[00:22:50.01]

[00:22:50.01]
vector of who are the leaders and
who are the followers then
[00:22:55.04]

[00:22:55.04]
this is this part is is
the same as the sub matrix.
[00:23:00.15]

[00:23:04.10]
We keep so we consider so
so when we find like.
[00:23:10.16]

[00:23:12.07]
The change of them
inverse inversed matrix.
[00:23:15.14]

[00:23:16.17]
So we care about this part so.
[00:23:20.06]

[00:23:30.14]
Yeah yeah I just defined for
the paragraph and the is n.
[00:23:37.03]

[00:23:37.03]
and vector and then.
[00:23:42.15]

[00:23:44.09]
This is the change of
the inverse matrix and
[00:23:47.23]

[00:23:47.23]
we can write it has
the integral of the tif and
[00:23:51.19]

[00:23:51.19]
after expanding these we can find that
each of these matrices are untrue eyes and
[00:23:57.09]

[00:23:57.09]
on their t.v.
So this gives one of them the city.
[00:24:01.05]

[00:24:04.07]
And twice my holy city of
these are matrix and then 4.
[00:24:09.18]

[00:24:12.09]
Similarly we can find out that for
each of these part these are the same.
[00:24:18.11]

[00:24:19.14]
Lists for.
[00:24:20.13]

[00:24:22.09]
Sat that includes as t. a t.
[00:24:25.13]

[00:24:25.13]
that includes as this is also smaller so
and
[00:24:30.05]

[00:24:30.05]
this is not related to ass it's just for
always avert old or
[00:24:36.05]

[00:24:36.05]
diverted his so this gives a darn
supermodel arity that we need more.
[00:24:41.23]

[00:24:46.03]
Yeah.
[00:24:46.15]

[00:24:51.09]
So then this part is l o f f.
[00:24:53.18]

[00:25:13.01]
E.
[00:25:13.13]

[00:25:16.04]
What is it and it here.
[00:25:17.18]

[00:25:21.23]
Yet it's a indexing problem you mean.
[00:25:24.02]

[00:25:31.13]
Like.
[00:25:32.01]

[00:25:44.21]
I mean every row and
columns are assigned a node right
[00:25:49.13]

[00:25:52.14]
before for the last question but.
[00:25:54.20]

[00:26:02.08]
Yet you hear this part.
[00:26:03.23]

[00:26:19.11]
But I guess we use the sale.
[00:26:21.19]

[00:26:23.16]
Yet.
[00:26:24.04]

[00:26:34.09]
Not.
[00:26:34.21]

[00:26:39.13]
So much.
[00:26:40.01]

[00:27:03.15]
Yes.
[00:27:04.03]

[00:27:23.16]
But but these are just that that the this
is just a Dyna Bagnall with the degrees.
[00:27:29.21]

[00:27:36.22]
Ok this problem is and
you hardly proof did in the prior paper it
[00:27:42.09]

[00:27:42.09]
follows from these particular on through
very little graph so I won't explain.
[00:27:47.01]

[00:27:48.03]
It here.
[00:27:48.20]

[00:27:50.11]
So once we get these.
[00:27:51.13]

[00:27:53.12]
Montana a city and
supermarket I repeat we.
[00:27:56.03]

[00:27:57.08]
We have these deterministically
de Our than for
[00:28:00.02]

[00:28:00.02]
the 1st step we don't have that property
but when and how we steal choose.
[00:28:05.17]

[00:28:06.23]
The best choice that minimizes
the sum of effective resistance
[00:28:12.06]

[00:28:12.06]
to the node so remember I said
these quantities are equal.
[00:28:17.11]

[00:28:20.08]
And in every following steps we.
[00:28:24.05]

[00:28:25.10]
We choose know that gives us the.
[00:28:27.12]

[00:28:28.18]
Biggest marginal gain and
include that there would here and.
[00:28:34.11]

[00:28:35.23]
Once we know the best choice we
update the inverse of the law to me
[00:28:40.02]

[00:28:40.02]
off the subway tricks by taking
this step matrix and this is.
[00:28:44.21]

[00:28:46.05]
This is given by the rock wide blocking
verse and this marginal gain is directly
[00:28:53.09]

[00:28:53.09]
derived from this equation and
this hour the wrong thing
[00:29:00.05]

[00:29:00.05]
order of time by Missy and
super ordinary 2 we know that.
[00:29:06.07]

[00:29:06.07]
It has the separate summation ratio which
is essentially why minus one will vary
[00:29:10.07]

[00:29:10.07]
with these factors and.
[00:29:13.11]

[00:29:14.21]
So you order to accept this we want
to compute these marginal gang and
[00:29:20.03]

[00:29:20.03]
1st step which is some of a very
good wrist resistance it's fast.
[00:29:25.13]

[00:29:27.12]
So the algorithm ingredients we use
are fasts s.t.d.m. Matrix over and.
[00:29:35.05]

[00:29:37.01]
And.
[00:29:37.14]

[00:29:38.18]
Johnson into straws to map it
to a lower direction of space so
[00:29:42.20]

[00:29:42.20]
we can solve fewer equations so
[00:29:47.12]

[00:29:47.12]
again this is a deterministic
gritty with them and
[00:29:52.04]

[00:29:52.04]
this is the solver we use so
it takes a matrix.
[00:29:56.08]

[00:29:57.10]
As just as d.d.
a matrix as and a vector b.
[00:30:01.09]

[00:30:01.09]
that solves it solves it which
gives the x. that is close to the.
[00:30:06.02]

[00:30:07.03]
Assay inverse p..
[00:30:08.06]

[00:30:10.05]
And this is error parameter and in terms
of complexity it's in the law terms here.
[00:30:16.20]

[00:30:19.01]
And this is.
[00:30:19.19]

[00:30:21.05]
Like random projection we use
instead of so instead of solving.
[00:30:27.01]

[00:30:28.03]
And more and.
[00:30:29.02]

[00:30:30.10]
Equations we only need to solve
these number of equations
[00:30:34.23]

[00:30:34.23]
to approximate speeds are the result.
[00:30:37.09]

[00:30:39.03]
So from the to this is just a win
that we used to estimate so
[00:30:44.09]

[00:30:44.09]
much no gain so origin of you we
want to compute these and these for
[00:30:50.02]

[00:30:50.02]
any any you right now we approximate by.
[00:30:55.03]

[00:30:56.05]
By solving every road of these.
[00:30:58.13]

[00:31:00.02]
By Proxy making at the rows of these
the these matrices so we solve these.
[00:31:07.02]

[00:31:08.17]
3 times these number of equations
with these every parameter
[00:31:12.20]

[00:31:16.11]
and similarly we can compute to
some off effectively distance.
[00:31:20.03]

[00:31:21.23]
In similar ways and
this gives us and approximation.
[00:31:28.11]

[00:31:29.21]
The approximate greedy algorithm so
[00:31:33.09]

[00:31:33.09]
even here in the 1st line we give out each
the sum off the effective resistance to.
[00:31:38.13]

[00:31:41.05]
Choose.
[00:31:41.17]

[00:31:44.07]
To true the need or
note for it every node and
[00:31:48.01]

[00:31:48.01]
choose the best then we repeat
the following steps to choose.
[00:31:53.02]

[00:31:54.12]
Every time we should
note that gives us the.
[00:31:57.12]

[00:31:59.00]
Biggest margin again and
then we get up Ok time so to die in our.
[00:32:04.14]

[00:32:08.23]
Running time algorithm.
[00:32:10.02]

[00:32:11.11]
Which has these approximation retreats.
[00:32:13.13]

[00:32:14.15]
Essentially why Minnes is k.
these factor times while over a year.
[00:32:20.22]

[00:32:22.16]
Or so you start as a best choice for
[00:32:27.10]

[00:32:27.10]
your 1st No because the 1st step
you don't have one of the city and
[00:32:32.08]

[00:32:32.08]
super what effect it has actually
increases the value of the trace from.
[00:32:38.02]

[00:32:39.06]
Interfacing.
[00:32:46.08]

[00:32:47.14]
So we studied this in term of these
statistics in the control systems but
[00:32:53.21]

[00:32:53.21]
this is also the same quantity
also arse studied in.
[00:32:58.23]

[00:33:00.14]
Social networks it's called current flow
close they subtract he or informations and
[00:33:04.16]

[00:33:04.16]
ready so
I won't go through the definition so
[00:33:09.18]

[00:33:09.18]
it's essentially defined as
an over the quantity we study so
[00:33:14.11]

[00:33:14.11]
minimizing these kind of equivalent
to 1000000 to maximizing
[00:33:20.01]

[00:33:20.01]
their that quantity so
we do some we did some experiments.
[00:33:24.12]

[00:33:26.03]
On maximizing the same gravity so
we see that
[00:33:30.02]

[00:33:30.02]
these greedy algorithms either it's
exactly a random the problem perform well.
[00:33:35.00]

[00:33:36.02]
And this is optimal
sometimes they overlap.
[00:33:39.02]

[00:33:40.20]
And these obviously are to perform so
random strategy and
[00:33:44.22]

[00:33:44.22]
this is some experiments for
real world networks.
[00:33:47.23]

[00:33:49.20]
These are on experiments for
larger case that we know now we don't
[00:33:54.18]

[00:33:54.18]
have all people on but obviously
8 of performs other curious ticks
[00:33:59.18]

[00:34:01.14]
and this is a running time
of the hour with so for
[00:34:05.07]

[00:34:05.07]
these graphs these graphs have.
[00:34:08.17]

[00:34:09.18]
Several 1000 of nodes and vertices so
[00:34:14.13]

[00:34:14.13]
we see that the exact Grady County
deal with these graphs and
[00:34:19.02]

[00:34:19.02]
these graphs have millions of nodes and
edges so
[00:34:24.01]

[00:34:24.01]
approximation the approximate Grady can.
[00:34:28.04]

[00:34:30.01]
Can solve these problems efficiently so
we did all these experiments
[00:34:35.10]

[00:34:35.10]
on a single 3 year compare the speed so
you can see that it actually runs
[00:34:40.09]

[00:34:40.09]
faster and the results are close for
these small graphs.
[00:34:45.06]

[00:34:49.10]
It's not exact it's exact greedy it's
not it's not unique that I'll get there.
[00:34:56.20]

[00:34:58.19]
That's not proximity ratio
that's just their ratio Ok.
[00:35:03.12]

[00:35:07.13]
Ok so here so now.
[00:35:09.22]

[00:35:13.04]
So this is the 2nd set
of problems we study.
[00:35:15.18]

[00:35:17.00]
It's called education so
given the connected graph g.
[00:35:20.14]

[00:35:20.14]
we want to add a set of
edges to the graph g.
[00:35:24.11]

[00:35:24.11]
So we have additional set of edges p.
[00:35:29.04]

[00:35:29.04]
that we add to the graph g. and.
[00:35:32.22]

[00:35:34.00]
Such that the performance measures of the
system now it's not just up Matrix it's
[00:35:38.19]

[00:35:38.19]
the whole matrix It's
the system without the beaters.
[00:35:42.18]

[00:35:43.21]
Are optimized So 1st of all we study
the congregants rate which is given
[00:35:49.04]

[00:35:49.04]
by the number 2 off base matrix and
internal energy
[00:35:54.17]

[00:35:54.17]
this is the sum of variances
of all of our own nodes is and
[00:35:59.23]

[00:35:59.23]
always given by the suing Merce
trace of the su doing worse and so
[00:36:03.20]

[00:36:03.20]
on a certain volume is given
by these determine if of this
[00:36:07.20]

[00:36:07.20]
matrix so.
[00:36:13.10]

[00:36:14.12]
So this matrix we need to recode that it's
related to the number of spending increase
[00:36:19.17]

[00:36:19.17]
we had a.
[00:36:20.06]

[00:36:30.02]
Yes yes P's Bunny by Ok so
we are at that most k.
[00:36:35.13]

[00:36:35.13]
edges so
this is this this is one of the problem.
[00:36:41.23]

[00:36:43.15]
So for a graph g. we add.
[00:36:45.16]

[00:36:46.23]
A set of edges q. So this cute.
[00:36:50.05]

[00:36:52.05]
Has some one to constrain.
[00:36:54.10]

[00:36:55.19]
So we want to.
[00:36:56.13]

[00:36:57.18]
Maximize the number of
spending trees of these g.p.s.
[00:37:03.13]

[00:37:03.13]
this is defined as adding
the edges to that graph.
[00:37:06.07]

[00:37:11.11]
So we also have a harness result for
[00:37:13.22]

[00:37:13.22]
these that shows there is no
this this is this they are so
[00:37:18.09]

[00:37:18.09]
this is the I'm sorry so are we then
that gives then to die plus and
[00:37:24.07]

[00:37:24.07]
prosecutors q.
is a can of a sample pages over absent.
[00:37:28.23]

[00:37:32.12]
Negative wind and
the need gives approximation ratio of y.
[00:37:37.22]

[00:37:37.22]
minutes while remind us.
[00:37:39.13]

[00:37:40.23]
To close to one nice one very Ok so
we also give a harness result for
[00:37:49.09]

[00:37:49.09]
the use that say's there exists a c.
that you cannot approximate within this.
[00:37:54.20]

[00:37:56.01]
Absolute constant.
[00:37:58.06]

[00:38:02.07]
It's given.
[00:38:02.19]

[00:38:03.23]
It is a paper it.
[00:38:05.05]

[00:38:07.09]
All Yeah yeah it's kind
of we did did not give it
[00:38:12.01]

[00:38:12.01]
the explicitly we give it by
defining some meaning or some
[00:38:18.01]

[00:38:18.01]
of a few different
parameters we are all on.
[00:38:22.23]

[00:38:24.00]
Constant.
[00:38:24.12]

[00:38:30.23]
Us So then what is weighted number of
spending freeze is defined as a sum
[00:38:37.03]

[00:38:37.03]
of the weights of all spending trees all
possible spanning trees in the graph and
[00:38:41.17]

[00:38:41.17]
at the weight of every spending tree
is defined as a product of all the edge
[00:38:46.13]

[00:38:46.13]
weights so
from kickoffs matrix trees theorem we.
[00:38:50.19]

[00:38:51.22]
We know that it is a determinant
of any and minus one times and
[00:38:56.16]

[00:38:56.16]
minus one principle may treat
a sub matrix of a whole so
[00:39:02.01]

[00:39:02.01]
we just arbitrary fixie to be in.
[00:39:05.06]

[00:39:08.00]
So from the kickoffs.
[00:39:09.18]

[00:39:10.23]
And a matrix determine the lemma.
[00:39:12.19]

[00:39:15.12]
Which is a 101 update for
the determinant would know that
[00:39:20.03]

[00:39:20.03]
the number spent in trees in the new
graph after adding this edge
[00:39:25.02]

[00:39:25.02]
is the number of the number of spending
through seen the or original graph time
[00:39:29.19]

[00:39:29.19]
swine plus the weight off the edge
times the effective or distance.
[00:39:35.07]

[00:39:36.09]
Between these un v e
before you at that age so
[00:39:41.13]

[00:39:41.13]
by picking log this this is
this this is their relation so.
[00:39:49.08]

[00:39:51.03]
So it's obvious that from this equation we
can all observe it's some order and model.
[00:39:56.05]

[00:39:57.10]
One at home because these
termes always non-negative and
[00:40:03.10]

[00:40:03.10]
and if we add more edges to the graph
this term can on a decrease so.
[00:40:11.11]

[00:40:12.15]
In order to get a fast how were we
want to approximate these quantity so
[00:40:18.10]

[00:40:18.10]
we would 1st show that it's enough to
approximate the c factor of resistance
[00:40:23.22]

[00:40:23.22]
to approximates the whole thing so
we need these.
[00:40:27.02]

[00:40:28.20]
These 2 true.
[00:40:29.16]

[00:40:33.06]
So this is.
[00:40:34.06]

[00:40:35.15]
The fast some order.
[00:40:38.19]

[00:40:39.21]
Greedy routine that we use it's
given in the speaker and for so
[00:40:48.04]

[00:40:48.04]
1st of all we need to find entry that
has the largest margin 0 gain and
[00:40:54.16]

[00:40:54.16]
define the marginal gain
as as a stray showed and
[00:40:59.02]

[00:40:59.02]
then every in every terrorist we
actually decrease this ritual and
[00:41:04.03]

[00:41:04.03]
add always the entries that gives.
[00:41:06.18]

[00:41:08.01]
That gives a larger
marginal gain than these.
[00:41:12.01]

[00:41:13.16]
And this survey showed So
this is algorithms
[00:41:19.05]

[00:41:19.05]
uses this number of queries so
Curie is the number of
[00:41:24.14]

[00:41:24.14]
candidate candidate edges so
this number of queries for
[00:41:29.12]

[00:41:29.12]
effective reason is the distance to
get this approximation ratio ratio
[00:41:35.16]

[00:41:37.09]
so if we can do this
these queries fast enough
[00:41:42.15]

[00:41:42.15]
then we get a faster algorithm then k.
times.
[00:41:47.08]

[00:41:49.12]
Yeah then there Kate and then k.
the k. k. round k. wrong.
[00:41:54.05]

[00:41:59.07]
Agreed so.
[00:42:00.22]

[00:42:02.08]
In order to did do this we use these we
need these notion of sure a complement So
[00:42:09.07]

[00:42:09.07]
what we really need here is that
sure company give it meant.
[00:42:12.23]

[00:42:14.04]
We're off that supported on the subset.
[00:42:16.15]

[00:42:18.11]
Of the graph the subset
the vertex at his c. and the.
[00:42:23.13]

[00:42:25.10]
It's a like a graph supported on
these subset offered his is that that
[00:42:31.23]

[00:42:31.23]
meant hands always the effective
resistance between all pairs in the or
[00:42:37.04]

[00:42:37.04]
take set and.
[00:42:39.20]

[00:42:41.01]
Just But a should come into tends to be.
[00:42:44.01]

[00:42:45.18]
Dense graph so we need these approximate
to a complement to make it sparse and
[00:42:51.09]

[00:42:51.09]
the Tony has these number of
add non-zero entries which are.
[00:42:56.07]

[00:42:58.07]
Edges see the graph and it gives us these.
[00:43:02.13]

[00:43:04.10]
The supports mission for say this s.c.
[00:43:09.17]

[00:43:09.17]
is a sure complement of
a 0 on the vertex at a c..
[00:43:14.16]

[00:43:16.01]
And it can be computed in
the near the near time.
[00:43:20.03]

[00:43:22.23]
So this is the algorithm
that in every wronged
[00:43:28.13]

[00:43:28.13]
$48.00 every I mean for
every search showed we check at all q.
[00:43:34.06]

[00:43:34.06]
nodes to decide whether
to absent the No Q Q To
[00:43:40.05]

[00:43:40.05]
decide whether to at the edges to the to
the networking this wrong so we divide it.
[00:43:46.02]

[00:43:47.03]
Always carry the edge evenly.
[00:43:49.15]

[00:43:51.07]
We size sizes and
these that we have to interview.
[00:43:57.02]

[00:43:58.14]
Each one comes.
[00:43:59.13]

[00:44:01.00]
To over 2 edges so.
[00:44:03.20]

[00:44:04.23]
So we find the vertices
vertex said you want and
[00:44:10.17]

[00:44:10.17]
beat you as and vertices of
edges seemed f. one and f. 2 so.
[00:44:15.18]

[00:44:17.11]
If one of these has at most q.
but has this
[00:44:22.07]

[00:44:23.21]
then we do sure a complement
of hung these vertex set.
[00:44:29.14]

[00:44:31.23]
To get to get these a proper
proximate true complement.
[00:44:36.04]

[00:44:38.06]
So we recursively do this.
[00:44:40.04]

[00:44:42.00]
On these.
[00:44:42.16]

[00:44:44.05]
On these.
[00:44:44.19]

[00:44:46.04]
Yes such as this if one and.
[00:44:49.11]

[00:44:50.17]
And it returns the edges
that we need to add so
[00:44:54.23]

[00:44:54.23]
we add these edges to the graph and
then do the same thing for the 2nd.
[00:45:00.17]

[00:45:01.18]
For the rest of the edges and then.
[00:45:05.06]

[00:45:06.10]
Finally we return like the old the edges
we need to need to add to that
[00:45:12.12]

[00:45:12.12]
graph and we keep recursing on to q.
equals one then we get this.
[00:45:17.08]

[00:45:18.14]
We can again compute approximate
true accompaniment and
[00:45:23.04]

[00:45:23.04]
to get the 2 by 2 matrix essentially and
which gives us.
[00:45:26.22]

[00:45:28.05]
The effective resistance
between the end and
[00:45:31.22]

[00:45:31.22]
points of these edge so
in this way we get.
[00:45:36.23]

[00:45:39.23]
In this way we get these odds
effective resistance computed.
[00:45:43.23]

[00:45:46.21]
So this is a recurring tree so
here we have q.
[00:45:50.14]

[00:45:50.14]
edges and every time we divide by hov and.
[00:45:55.00]

[00:45:56.07]
We set set error ever permit or
[00:46:00.16]

[00:46:00.16]
on every of these snowed in the recursion
tree to be absent over log q.
[00:46:05.19]

[00:46:05.19]
because that there is that sits q. And so
[00:46:10.04]

[00:46:10.04]
that on every leaf vertex we
have we have to guarantee that.
[00:46:13.23]

[00:46:15.08]
Error is at most Absalom and the running
time meets these for every wrong
[00:46:20.12]

[00:46:21.23]
so this algorithm success
was hyperbole by by pound
[00:46:27.09]

[00:46:27.09]
them by unity among So
here's the algorithm So
[00:46:32.15]

[00:46:32.15]
again this is greedy that we
that we use a threshold to add.
[00:46:37.12]

[00:46:39.10]
So the edges in.
[00:46:41.23]

[00:46:44.18]
Uniloc lot Absalom like negative log
Absalom over absolutely one wrong
[00:46:51.05]

[00:46:51.05]
and here in every wrong we
use these to add up all of
[00:46:56.10]

[00:46:57.10]
to decide whether to add the edges
to the graphene is wrong.
[00:47:01.21]

[00:47:04.18]
And this gives us an algorithm that wrong
seeing nearly time of the number of
[00:47:10.01]

[00:47:10.01]
edges in here is the number
of ages in the origin of
[00:47:13.13]

[00:47:13.13]
curious candidate number of a just
see in the county they set.
[00:47:17.03]

[00:47:18.19]
And the test they saw proximately racial
So we also have a harness proof so do.
[00:47:24.23]

[00:47:26.05]
This is the instance that we use to
reduce it so for for a graph that
[00:47:31.19]

[00:47:31.19]
this off a star graph and the obvious
require off supported on its Neves.
[00:47:36.23]

[00:47:38.08]
We know that number of spending
freeze is maximized if we
[00:47:43.03]

[00:47:43.03]
have a Hamilton house here but
obvious that it's that's hard.
[00:47:48.01]

[00:47:51.10]
So we reduce it we can reduce.
[00:47:54.19]

[00:47:57.11]
This the and
the hard problem to our problem so.
[00:48:01.08]

[00:48:02.14]
So actually we can get own will we we was
able to get some stronger result that.
[00:48:10.11]

[00:48:11.19]
We get these 3 duction from one to t.s.p.
to minimum spending meet minimum
[00:48:16.13]

[00:48:16.13]
pass past cover problem and
2 to the number of
[00:48:21.03]

[00:48:21.03]
smacks in my the number of spending freeze
so do you want me to expand this a bit or
[00:48:25.23]

[00:48:27.06]
can I say I can't skip this
part time I think so this
[00:48:32.17]

[00:48:32.17]
is a result that we have a constant that
we cannot approximate with the least cost.
[00:48:36.13]

[00:48:46.01]
That.
[00:48:46.13]

[00:48:55.16]
Is presented.
[00:48:56.10]

[00:48:57.22]
To them it's one of my most why worry.
[00:49:00.11]

[00:49:05.15]
You.
[00:49:06.03]

[00:49:07.15]
Know none of the this see
is some small number.
[00:49:10.06]

[00:49:12.22]
Ok.
[00:49:13.10]

[00:49:20.13]
Yeah yeah I think that's
interesting that's.
[00:49:22.12]

[00:49:26.17]
Ok so I will conclude by comparing
it with some related works.
[00:49:31.01]

[00:49:32.05]
So this is the setting for
your 2nd we want given a graph g.
[00:49:36.16]

[00:49:36.16]
we've choose a subset of at most
cave vertices as such that for
[00:49:41.02]

[00:49:41.02]
the rest of the nodes I guessed
something is missing here so
[00:49:46.08]

[00:49:46.08]
for the sub matrix software
passion this matrix we want to
[00:49:52.06]

[00:49:52.06]
maximize the minimize the sand minimize
base so I don't I'm not aware of any.
[00:49:58.03]

[00:49:59.22]
Day algorithm was guaranteed for
the system that you problem now.
[00:50:04.13]

[00:50:05.17]
And for these.
[00:50:07.04]

[00:50:09.02]
Yes this is our the op in
the that we improve upon So
[00:50:13.11]

[00:50:13.11]
this is our result and
for this is actually the.
[00:50:18.15]

[00:50:20.18]
Largest simplex problem in some sense but.
[00:50:23.17]

[00:50:25.07]
But in the sense that you
are choosing the followers so so
[00:50:31.06]

[00:50:32.20]
I have a slightly I could sure it made
her so so did gives this strange.
[00:50:38.07]

[00:50:39.13]
Bond but I think to someone that
assists we can get better better than
[00:50:44.05]

[00:50:44.05]
maybe that's possible but I don't.
[00:50:47.01]

[00:50:48.19]
Want.
[00:50:49.07]

[00:50:51.15]
To post I think.
[00:50:52.12]

[00:50:54.22]
And this is at the edge problem is
obviously has more related work
[00:51:00.12]

[00:51:00.12]
from experimental design so
on so far I'm not you
[00:51:05.19]

[00:51:06.19]
this is our only are within that
we know for an arbitrary k..
[00:51:12.06]

[00:51:12.06]
It gives these spawn and this problem
has been show shown to be n.p. hard
[00:51:17.22]

[00:51:19.23]
and this is our all or our result and
[00:51:24.21]

[00:51:24.21]
these are just experiments you know for
general experimental design result.
[00:51:29.01]

[00:51:30.14]
I'm sure that are there are more
recent result that I did
[00:51:34.06]

[00:51:34.06]
did knowledge include here.
[00:51:35.18]

[00:51:38.03]
But that's that kind of like the context.
[00:51:40.18]

[00:51:44.06]
So conclusion so we give and
to Doc am all for them for.
[00:51:48.19]

[00:51:49.23]
And hard and harness for
minimizing days and.
[00:51:53.18]

[00:51:55.12]
And the new need a time out for
them for and harness result for
[00:52:00.23]

[00:52:00.23]
minimizing the determinant and
future directions
[00:52:05.05]

[00:52:05.05]
here are some of the some some some of
the things that we can think about.
[00:52:09.21]

[00:52:11.02]
First of all is to improve d.c. if we can
maintain these fighting for every few.
[00:52:16.02]

[00:52:17.23]
Efficiently and we also want to
improve their proximity to reshow.
[00:52:22.11]

[00:52:29.11]
And also as a Parness result would
mean I mean these are not tight for
[00:52:34.19]

[00:52:34.19]
now and these genes can fall and
can find a pic you
[00:52:40.07]

[00:52:40.07]
can see in social networks optimization
and sparse country designs with some some
[00:52:45.22]

[00:52:46.23]
like every problem has some
different things to consider Ok.
[00:52:52.03]

[00:52:54.20]
Thank you.
[00:52:55.08]

[00:52:56.19]
Thank.
[00:52:58.06]

[00:53:06.03]
You.
[00:53:07.21]

[00:53:12.17]
That's that's it if you can change
it I'll be true to its comics.
[00:53:19.08]

[00:53:25.04]
Yeah yeah.
[00:53:25.20]

[00:53:29.16]
Well let me just follow.
[00:53:31.05]

[00:53:39.01]
Up on.
[00:53:39.15]

[00:53:48.05]
That.
[00:53:48.17]

[00:53:50.20]
Yes yes I think.
[00:53:51.15]

[00:53:52.17]
All of these works are based on the comics
optimize the shing and then rounding by
[00:53:58.06]

[00:53:58.06]
some different because for
the most part of getting these y.
[00:54:02.13]

[00:54:02.13]
and then the these results are all
based on the comics up in mileage.
[00:54:07.06]

[00:54:17.01]
So Ok so
this result says that you can get.
[00:54:19.18]

[00:54:20.19]
One process approximation
ratio if the number of edges
[00:54:25.22]

[00:54:25.22]
that you add is bigger than the number so
it's not for I have a trick a.
[00:54:30.22]

[00:54:35.16]
Yeah.
[00:54:37.03]

[00:54:37.03]
And this is this is for any k. but you.
[00:54:40.23]

[00:54:42.08]
Hear.
[00:54:42.20]

[00:54:46.18]
What.
[00:54:47.06]

[00:54:49.06]
You thought.
[00:54:50.08]

[00:54:53.15]
Yeah.
[00:54:54.04]

[00:55:03.14]
Yeah yeah yeah yeah you can
you can you can figure out.
[00:55:06.10]

[00:55:07.14]
So with noise you can't
really say a whole.
[00:55:10.06]

[00:55:12.02]
Life you know if we don't consider
noise that we have a steady state for
[00:55:17.08]

[00:55:17.08]
for the whole.
[00:55:17.22]

[00:55:26.04]
Life.
[00:55:26.16]

[00:55:31.21]
What.
[00:55:32.09]

[00:55:35.04]
Is it possible to define something I mean.
[00:55:37.15]

[00:55:39.07]
So so for so I have father thought that.
[00:55:42.20]

[00:55:44.22]
Yeah.
[00:55:45.10]

[00:55:48.17]
Yes.
[00:55:50.04]

[00:55:50.04]
Yes yes for the leader for
systems that has.
[00:55:53.00]

[00:55:54.21]
So this say this access is
arbitrary you assign for
[00:56:00.09]

[00:56:00.09]
all these years so
we have these equations for 4 dart
[00:56:05.13]

[00:56:05.13]
the expectation and covariance matrix so
different for these we can see that
[00:56:10.13]

[00:56:10.13]
it's it's not related to this for
these we have these equation and
[00:56:15.07]

[00:56:15.07]
what's what is convergence or
it is it's given by these so.
[00:56:20.05]

[00:56:22.02]
Yeah yeah yeah it's it's different on.
[00:56:24.00]

[00:56:27.19]
Worst of.
[00:56:28.09]

[00:56:32.21]
That dice you need.
[00:56:34.14]

[00:56:36.04]
That that are Ok with.
[00:56:38.21]

[00:56:39.22]
Or.
[00:56:41.09]

[00:56:41.09]
Without.
[00:56:41.21]

[00:56:46.03]
It I haven't thought about.
[00:56:47.18]