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>> Kamal Jain: Our speaker today is Milena Mihail. And let me tell one short story she might remember
about. So one thing common between me and my advisor was our wives' birthdates. So there was this
thing -- I won't tell about it -- it was my wife's birthday. I was newly married. So I was taking off. And so
supposedly we [indiscernible] at home. Milena didn't come to office today because it was his wife's
birthday. Then Milena told me the next day he didn't remember it was my birthday, too, [laughter].
Anyway, here is Milena.
>> Milena Mihail: Thank you. So I'll be talking about -- so the title is conductance of random inner product
graphs. This is joint work with my former student Steven young who is a post-doc at UCSD.
So this is a really random graph model. It's strict for generalizations, and the more general context in
exploring their conductance is that we are obtaining these generalizations trying to get to mathematical
models. But whose properties are closer to the properties of real graphs as we observe them in complex
networks.
And so now something fun is going to happen. I don't know why. There's some animation. I didn't put it
there. Anyway, I hope it's the only place that it happens. I don't know how it happened.
So this is the outline of the talk. Well, usually one starts with context. But I don't want to start by giving
pictures. So I will just start with two theorems and I will go through the conductance primitives very, very
fast because I think everybody in this room knows them.
And then I will say in which context sort of we're trying to get this more general theorems. And it's the
context of what kind of random graphs are modeling real complex networks. And in particular, then I will go
into say how existing random graphs have discrepancies from real networks.
And then I will introduce the random inner product graph model as one candidate to extend the parameters
of random graphs and allow them to come closer to real networks, and then I will talk what tools technical
tools do we use to analyze their conductance and give a flavor of the theorems further directions.
So all right. So the conductance is of a set is how many edges cross from the set to its complement,
divided by the sum of the degrees inside the set. And it's a very well known quantity, and in different
contexts you want to show different things about it. Many times we want to show that the conductance is
very large so that algorithms work very fast.
In social networks, you're actually interested in particular sets that have small conductance, because
they're indicative of communities. And they exist. And perhaps you want to model them.
So just one thing to remember is that just what is the conductance of a set. In the context of real networks,
we need to see, you know, what is the size of the sets that exceed worse conductance, how many they are.
These are questions that sort of some of them are folklore and some of them we've touched, we sort of
know anecdotally. But as we're getting more into the detail of the networks, there will be -- these will be
questions that will be coming up. And of course the conductance of the entire graph is the conductance of
the worst, is the conductance of the sort of worst set.
And there's a classic theorem that I mean of course this is a hard quantity to compute. But it can be
bounded in a pretty tight way from above and below by the second eigenvalue of a certain matrix. I will
come to this, to the matrix later.
But the point is that there is -- there's a second eigenvalue of a certain matrix that bounds the conductance
of a graph from above and below. And so here is one theorem. So now here I come to my two theorems.
Here's one theorem. It is due to Chung Lu Vu and it's one generalization of [indiscernible] graphs.
So they call this the expected degree sequence model. And it goes like this. So you have targeted
degrees W1 up to WN. So the smallest degree has to be large enough. I mean, one would have hoped for
C log N. But, okay, it's C -- it's some constant log squared N. So you want the smallest degree to be large
enough at least to get connectivity.
And the largest degree is, has to be less than the square root of the sum of the degrees. The volume of
the graph. So if, for example, if you had N log squared N volume, the largest degree would have been
bounded by some square root of N times log N.
It's a strong condition. It is not necessarily the case that -- I mean why wouldn't you allow, why wouldn't
you have the model where you would want -- you would allow the largest degree to be larger.
So anyway, so we're given -- a sequence which is a target of degrees. And this is how we form a random
graph that the probability of an edge between I and J is proportional to the expected degree of I, expected
degree of J, divided by the sum of the degrees.
So the justification of this constraint was that so that this is a probability. But it's really a little bit more
serious. It comes in a crucial way in the proof.
And this is a little bit troublesome. So, okay, but in this model, which is the generalization of the
Dershraney [phonetic], you get lambda too small and you get constant conductance with high probability,
all the statements are with high probability.
And here's another way to think about this theorem. So why should I have target degrees? Right? So
instead I can just say that there are points, you know, N points, and they have weights. Right? So the
lowest point has very small significance and the highest point -- okay. So this should be less than or equal
to 1, could have significance as much as one, right? And then you connect to points with probabilities
proportional to their significance.
It's equivalent to the previous model. I'm just stating it in terms of real numbers. And then, again, you can
get a bound that the second eigenvalue is bounded away from one, the conductance is large.
Okay. Now what is missing from this picture, if we think of them as weights? Well, why should the highest
weight be one? Well, perhaps the highest weight should be you can go all the way to, it could be N or
something. And then here you could take the minimum of one and the product or you could just bound it by
something that makes sense. Sort of maybe N over log squared N. And allow for the whole range of
graphs that you can get.
And, I mean, intuitively, you know, allowing more connectivity on the higher degree nodes should not harm
your conductance -- I mean, should only improve the situation. But -- no. So the known techniques have
this constraint. So it's another way of thinking of the Chung Lu Vu theorem. And now if we start thinking
about these real numbers as weights, well, then maybe they're not necessarily on a line. So let's think of
them as lying, maybe, in two dimensions. In a two -- in two dimensional space. And each point has a
weight, but the probability that they're connected is proportional to the product of the products of their
weights and the cosign of the angle that the corresponding vectors form. Right? So that says that if we are
in the same direction, then, yes, it is the product of our relative weights. But maybe we have something -we have some differences also.
So we will connect with lower probability. And one would expect that if the angle of this cone is small
enough so that -- so that the cosign is large enough, one would get a similar theorem. And the answer is
yes, of course.
So for an angle that's small enough so that the cosign is at least, I don't know, a little bit more than
one-half. So if the angle is small enough, then, again, you get constant conductance. And of course you
pay a little bit of penalty because you had these -- yeah, because the probability -- the connectivity
probability is where there was a cosign of angles of vectors involved.
Okay. And now I want to say a little bit about the proof techniques of how one gets ->>: About the signature, do you expect this inverse code and ratio to be sharp, this here?
>>: Do I expect it to be sharp? No. No. No.
>>: So anything positive should be okay?
>>: Any probability lower bounds?
>>: Yes. So whenever the angle, and this is essentially the final theorem that I will show. So as long as
we do not have complete orthogonality. Right? So as long as you're within a cone bounded away, of
course you'll get conductance. But the frustrating thing, and this is what I'm trying to project, I'm not sure
I'm doing a good job, is that here is -- you know, an elementary case, an elementary sort of extension of
Dershraney. And we have a lower -- okay. So let me put it here.
So, first of all, look at the lower bound. Why should it be log squared N. Log N is good enough. I mean,
why should it be that there is no reason that having larger and larger degrees should -- you should pay a
penalty on the conductance or the second eigenvalue, because the denser parts of the graph, the larger
clicks are in the denser parts of the graph. And it's just indicative of perhaps the limitations of our
techniques rather than anything else. And, plus, there's a very frustrating upper bound. Why does the
maximum degree have to be square root of the volume.
>>: You need the upper bound for your result.
>> Milena Mihail: It is implicit, yes. Yeah, yeah, yeah, sure. It's at most one, I'm saying. Right? And I
think that the results should hold for anything and just take the minimum of one and XI times XJ. And this
limitation is not only sort of the expected degree sequence or inner product model or whatever, it's also in
the configurational models.
We have -- we seem to be bounded by the largest degree being the square root of the volume of the graph.
And I think it's an indication of the limitations of the techniques.
>>: But you do some truncation threw away the large amount, reduced number -- the conductance is
monotone.
>> Milena Mihail: It's not monotone if you increase the number of edges.
>>: Because the denominator has ->> Milena Mihail: Yes. But I would be very interested to talk about it, in particular, with you. I think these
are just -- they speak about the limitations of our techniques rather than the graphs themselves. And, okay.
So let's keep going. Okay. So let me just say what proof techniques are not for this.
So the quantity that is relevant to conductance is the second eigenvalue of the adjacency matrix of the
generated random graph. You know, D to the minus 1. D is the diagonal of the degrees. This is pretty
standard.
And then in order to get a bound on this quantity, it's the same as, because we know what is -- well, first of
all, we ortho normalize, so the relative quantity now becomes whatever this expression is.
But it has to do with the diagonal of the degrees and the adjacency matrix, and then we know what the first
eigenvector is. So we project to a space vertical to the space of the first eigenvector. So we just remove it.
Now they're looking at the first eigenvalue in the projected space, which is the second eigenvalue in the
original space.
All right. So the difficulty with dealing with these qualities is that there are random variables everywhere.
So the degrees are random variables, but the adjacency metrics, the entries are random variables.
So -- okay. So everybody -- I mean, when we're dealing with these things, the first thing is we try to control
variability in different ways. So let me tell you sort of the ultimate control of variability. Namely to deal with
the schtick matrix instead of the adjacency matrix that was generated from the PIJs, let me take the matrix
of dealing with probabilities peak. And instead of the degrees, let me take the expected degrees. So all
deterministic. And let me remove the corresponding spaces, would have been a wonderful quantity. It's all
deterministic. So the question is how closely does it -- how does it relate with the second -- with the first
eigenvalue of this quantity, which is what we really want to analyze? And so we have one way of rewriting
the matrix that involves the random variables in a way that the variability is smaller. So we can write it as
the sum of two matrices. One is this completely deterministic matrix. And another matrix, which is
completely stochastic matrix. And it is whether the edge appeared or not. So AIJ of the adjacency matrix,
minus PIJ, the generating probability. And we normalize by the square root of the expected degrees. And
then we multiply left and right with whatever we have to multiply to make the whole thing work. So the
point is that this is a new as far as I know way of expressing the matrix, and it is kind of crucial. So this
normalization is kind of crucial, and this is the bottleneck in making the results go through for general, for
large weights.
And okay so this is step two. And once we have this, well, step three is -- so it's a very nice bound by Vu.
And it says if we have a matrix where the expectation of each entry is zero, which is like the C, the
variability matrix that we have. And if we can get an upper bound on the absolute value of each entry, and
if the variance is large enough compared to how the entries vary, then you can get a bound on the first
eigenvalue of this matrix.
So technically the limitations of the methods come, I think, from this way of bounding the second
eigenvalue or the first eigenvalue of the stochastic matrix. And it is because -- so Vu's theorem is really
tailored for regular graphs. So I mean, by the way, it can be extended when you don't have regularity. But
you know it assumes a uniform upper bound on all the entries of the matrix and a sigma that is large
enough and the point is that you could have edges in -- with probability one in the very dense part of the
graph, which would make this sigma zero. So it's a limitation of the bounding method. And this is a very
sort of detailed and technical theorem. It improves on the Freudian cumlash theorem [phonetic] of '83
or '84, and its proof is based on very detailed enumerative combinatorics computing the trace of certain
matrices and it's not at all clear how these combinatorics would go if the viabilities were larger.
So I'm just pointing to where technically the limitation is coming from. Okay. And so these were the two
theorems. Now why are we looking for these theorems? This is the context. So let me say a little bit about
the context of complex networks and what discrepancies are observed. Okay. So context of complex
networks in one picture that we've been seeing for the last 10 years. Vastly homogenous degrees and
vastly literature so far.
And within a few months, after people started modeling random graphs within homogeneous degrees, you
know, there were objections. So people from networking came and they said, oh, no, in real networks -- so
if you see the Internet of Microsoft, they say, no, the large degrees are not in the center of the graph
connected to each other. The large degrees are the ones that take the signal and split it in the end many,
many different customers, clients, and the small degrees are the high banded nodes that are in the center
of the network. Okay. So and then there's a whole bunch of mathematicians trying to mathmetize this
notion as [indiscernible] something was missing there was semantics. So in this graph there's clear
semantics. High banded node. Mid bandwidth, low bandwidth. Right? So networks do not form to satisfy
positive or negative assertivity they form for other reasons. And how to capture the semantics.
So I'm just giving a reference. Actually, you know with today's methodology we can capture it very nice
mathematically, but just to point out that immediately after that degrees can vary vastly came the first
objections that: But, look, please don't keep giving me random graphs. Here is discrepancy with respect
to -- okay. So the first discrepancy was with respect to hierarchy. So you have hierarchy in the bandwidth
of the nodes. So the second discrepancy is with respect to you can call it -- so with respect to existence of
sparse cuts. So you can also talk about sort of clastoefficient triangles. I think conductance is much more
interesting, because it's a global property and so this is again pictures. This is a picture for Flickr, what do
you get if you input Flickr the word graph. So some people mean something as a graph. Some people
mean something else. Some people -- very strong clustering. And it can -- very sparse cuts and it can
become arbitrarily sparse. This is some kind of social network of collaborations and patterns in the Boston
area.
So the point is that there is semantics. There's underlying communities. And this will not be captured by a
random graph, if you only model the degrees. And finally -- so in 2008, actually this was a -- it has become
a very well-cited paper. At Yahoo! Leskovec, Mahoney, Faloutsos measured a whole bunch of real
networks. And this is what they did. I mean, they were trying to see what was the sparsity of the cut. So
conductance is on one axis, right? And they were trying to see how many nodes the corresponding set
involved. And there was major effort in developing algorithms, how do you find the worst cut for this many
nodes. But in a nutshell, a random graph that has a certain degree sequence, I mean we all know it. It
looks like this. So you have, you know, as the sets become large enough, you get your constant
conductance and pretty much know what that constant is, three quarter, whatever lambda two, and you get
your sparsest cut around log N.
So it is a small set that present the difficulty. What they measure is not only deeper cuts, sort of sparser
cuts, but also repeated sparse cuts, larger and larger, involving larger and larger number of vertices of the
graph. And, again, these are unquantified. I don't know if this is log squared N or N to the 1 over 100.
Who knows, so it's mindboggling if you know all random graph models behave like this. In practice,
repeated for different methods and for different networks, you see something so vastly different, it's
something worth exploring.
All right. So, again, within the context of complex networks and the inhomogenous degrees for which
there's vast literature, here are two properties, heirarchy and clustering. Communities that have very clear
semantics. And ordinary, standard random graph models do not capture them. And so this is speaking
towards expanding the parameters of our random graph models to start capturing in a mathematical way
such properties.
And I put here the work of Kleinberg. Seminal work of Kleinberg on the small world phenomenon. So what
was Kleinberg really saying?
So yes you can navigate a graph, and that you can do it in an algorithmically efficient way. But then if you
look at his mathematics, what was his mathematics really saying. His mathematics was saying look there's
further parameterization to be done and please play very close attention to the parameters because it
makes a big difference, in retrospect.
So we want to do something like this as people are addressing discrepancies from networks of random
graphs from real networks.
So here is what I think are, this is the random graph model. Of course it should be predictive sort of as
much as possible. Sort of it should match the structure, if you're talking about it should match the function.
It should be explanatory.
It should perhaps fit the semantics. Okay. Predictive. It's obvious. It should also be mathematical.
Enable to analysis. So I think that by now we've seen way too many anecdotal models.
So you know the real challenge is it has to be mathmetized because only then can you be really predictive
and only then can you point to what are the parameters and how if you trick them you're going to get
different function and -- and of course they should be efficiently implemented.
If you're going to generate graph on nodes it shouldn't take N squared time if we're talking about millions.
And the graphs that I'm going to talk about are efficient to implement. So here are three random graph
models that generalize G and P. The inner product, random inner product graphs, which was all
[indiscernible] in 2008 and his publications before. And this is what I will focus.
There a special case of generally inhomogenous random graphs which were introduced by Béla Bollobás
and Riordan in 2007. This was sort of independent work. Essentially the Béla Bollobás and Riordan model
is whatever I'm going to talk to you about except you can replace inner product with a much more general
kernel function.
So it's a special case. And then there's another -- I mean, all generalizations of [indiscernible] and there's
another class that I'm not going to talk about stochastic chronicle graphs, which are also generalization of
[indiscernible]. So we're exploring all options to see what one can get.
So this is the model. All right. So the vertices are in D dimensional space, and D is fixed, independent of
N, right? So it means that no matter how large the graph grows, every node can remember D things. I can
remember 100 things. It doesn't matter how big the population becomes.
So there's like D relevant parameters, and every vector is a point in D dimensional space. So one
dimension for each relevant parameter.
So the vertices are going to be vectors in D dimensional space, and how are they going to be generated?
Well, there's a probability distribution mu in D dimensional space. And mu is also fixed independent of N.
All right.
And this is essentially the only new parameter that -- only one, only one parameter that we're introducing
that points are generated according to distribution in D dimensional space.
And how do vertices connect? There's a probability between two, there's an edge between two points,
probability proportional to the inner product.
Okay. So I'm making some assumptions here that the vectors have positive entries and okay so it didn't
have to -- I could say the minimum of one and the inner product of two vectors.
So this is the nature of the model. Okay? And, again, the more general model would be, if instead of the
inner product, you could have some other kernel function.
All right. And we have to introduce a couple more things. So one is that you realize that because mu is
fixed in D dimensional space, independent of N, this could generate very dense graphs which are not
particularly interesting.
So, for example, if mu was focusing on one point, one-half, all of mu was on one-half, you would be
generating [indiscernible] graphs with degree, average degree. That's not interesting.
So we are going to divide by all right so then we have a sparsification function. That's the other parameter
of the model. So we are going to divide by a sparsification function, we can choose whatever we want,
whatever we think it's appropriate. But something to make the graphs interesting, to make them sparse
and the ones that I'm using is at least log squared N over N.
So it is so that the line theorem and the thin cone theorem I told you before pass, and it's also, I need log N
over N to get connectivity in the first place.
All right. So this is the model. Mu fixed distribution in D dimensional space. Sparsification function. G of
N, which is not too small. And N points in D dimensional space generated according to mu and connected
according to the inner product, and output one more condition.
And this is that -- so these points have some weight. So that I don't get disconnected for trivial reasons. If
the distribution goes to, if mu generating distribution goes to zero too fast, you won't get connectivity,
because points are just not interested in the network that they're participating.
So it's a technical -- it's just to ensure connectivity. And that's it. That's the model. And what would be the
strategy outline to get conductance? Well, I can show conductance for cones, right, for thin cones as I said
before. So I want to I'm going to decompose the graph into D cones. I'll say how this is possible and why it
is possible.
And in fact the thin cones that I'm going to decompose will depend on mu. What is the generating
distribution. And I will -- I already know what is the conductance inside a cone, and then I need to argue
about connection across the cones and then apply sort of relatively standard Markov chain decomposition
or graph decomposition theorems.
All right. So the point is that because -- so the point is that the decomposition can be done so that we have
a fixed number of cones. So for the random graph model, there's a representative graph R, which is the
red graph that you see here which has a fixed number of vertices. So there's a very small description of
how the entire model behaves, and it is a weighted graph.
I will say what the weights are. And it is the conductance of this very small graph that characterizes the
conductance of the entire, of the random graph that is generated.
And now you can see why I say that it's a little bit closer to real networks. Because you can imagine that,
you know, we could just simply have two points very far away. And then already you would be getting your
sparsest cuts at very large values and they could have -- and they could involve cut sparsity much larger,
so much deeper, worse cuts than you would get in [indiscernible] graphs. Still, I'm working within
constants. But they're different constants. That's the point.
Now, how do we do the decomposition? So mu is the fixed distribution in D dimensional space. And so
what I'm going to do is that I'm going to get regents, cones, actually, in D dimensional space, such that the
measure of the distribution on each one of these cones is a constant. And there's a fixed number of them.
Fixed independent of -- there could be many. But they're going to be fixed.
And, okay, so here's a fact that unless mu is trivial, meaning a point or a line, in which case it's a refrenny
[phonetic] or a thin cone, you can do arbitrarily fine but fixed decompositions of mu. Here's the reason why.
Well, suppose that mu is bounded. So suppose it's on a sphere. Everybody knows we can cover a sphere
with smaller spheres of small radius, more of a fixed number of them. And we can do the same with cones
in D dimensional space.
And then we can use this covering to get the decomposition that we want, right? So we start with the first
relevant sphere or cone and then -- and that has constant -- that attracts a constant fraction of mu, and
then we can get to the next one, just subtract what you've done, what you've already counted. And then
we get to the next one and subtract what you've already counted. It's sort of straightforward.
Okay.
>>: Where is mu I?
>> Milena Mihail: Where is mu I?
>>: Right there. Mu I.
>> Milena Mihail: That must be a typo. I don't know why I would have written mu I. It should have been
mu or omega I. Is it a typo? No it's mu and omega. Mu I is a typo. Mu. Now I'm going up. Oh, no.
Okay. Now I messed up. Okay. There you go. How does this decomposition of mu. Now, I talked about
decomposition of the generating distribution. How does this suggest decomposition of the graph? Well, in
expectation, each class has a constant number of vertices.
And so it is constant N. So I get the conductance that I need or the bounded lambda two that I need for
each one of the classes inside the class. And then, okay, so these are sort of the restriction graphs to
follow Markov chain Monte Carlo terminology. So these are the restriction graphs inside each cone, and
then I get the projection graph, which is what happens across the cones.
And the representative graph that I was talking about is going to have one vertex for each partition class
which was dictated by mu. It's going to have a self loop which is going to be the volume of the number of
vertices of this cone within itself and it's going to have weights on the edges going across partition classes,
which is the expected number of edges that go across partition classes. And here's a theorem. If mu is
bounded, meaning inside some large sphere, or mu does not go to infinity too fast, the reason I need this is
so that I can apply the cone theorem and the cone theorem was saying that the largest XI has to be one.
This is artificial. This shouldn't be there.
Okay. So this condition shouldn't be here. Mu is pairs of points that have inner product close to zero is
zero. So you are away from the axis, right? Then the representative graph, which we already said is a
graph on a fixed number of vertices connected, and the weights on its edges is independent of N.
And the conductance of this graph, which is independent of N, but of course could be very different than
the conductance of random graphs, lower bounds, conductance of the random graph in the random inner
product graph model.
And as I said the specific techniques is just to decompose into thin cones and consider the restriction
graphs and the thin cones and then Markov chain decomposition theorems to argue across the cones.
All right. So I think it's important to characterize more general mu. The more general mu being the one on
the line. Forget D dimensions where you get points that have higher weight. We need finer distinction of
behavior of conductance. If you remember in one of the pictures, the cuts were not just sparse.
They were very sparse. So it seems to be that the general status of affairs in this sort of area of
characterizing conductance is that we have sharp -- so either the conductance is zero or the conductance
is a constant. I mean, in real networks there seems to be a much wider range of behaviors. And, by the
way, this is also the case in stochastic chronicle graphs in Leskovec Kleinberg. And I believe that if we
start including orthogonality, sort of the notion of how mu approaches the axis or touches the axis, then that
might be a starting point of getting much finer behavior on conductance.
Perhaps mu could be a variable, could be a function of N. Though that would challenge very much the
decomposition methods. Okay. So I think this is an important direction to follow, to characterize mu that
includes almost full orthogonality and let mu be unbounded.
And then in a different direction, okay, so as our random graph models become richer and richer, we see
graphs generated from matrices of probability, P. So how can we relate the properties of P with the
properties of the generated graph? And I'm raising this because, okay. So I mean in this case it so
happened that the relative, that the generating matrix, you know, was some P, and then you know there
was a quantity involving so many random variables, and then we were able to control this variance
essentially saying that it is really P that determines in this case, sort of the behavior and can this be
generalized. But just an indication towards looking for stronger methods.
I guess sparse graphs -- well, yeah? I mean so these graphs have average degree log N. So we want to
go to sparser graphs, in which case we either have to look at the giant components, the behavior of the
giant components or we have to go to different models the configuration model for which we get
connectivity at much lower degrees. I think the reason I really put this last bullet on is because it is really
folklore, sort of even in sparse graphs, even in the range at sort of constant degree [indiscernible].
So look at the giant component. I mean, once it has formed. And I mean, okay, so there are sparse cuts
but what is the sparsity, how many, of what kind. How many nodes do they include? These were things
that are not important when we were only looking at showing that graphs have good conductance and all
eigen values are bounded away from one. But if you start looking at existence of sparse cuts, there has to
be a systematic characterization of how many cuts, what sparsity for random graph so that we start
comparing them with models.
And I guess that's all I had to say.
[applause]
>> Kamal Jain: Any questions?
>>: For the last little -- so asking about just the question about G and P, MP as a constant.
>> Milena Mihail: Okay. Okay. Okay. So let me tell you now the full story. So this is a nice picture to use
in talks for explanatory reasons, but when I asked these people -- actually John Kleinberg and I
interrogated them to exactly see what graphs were going on here. These were sparse graphs. So they
had average degree. They told us sometimes it was 5, ten, these are very, very sparse graphs.
So they were measuring and they were measuring large cuts of deep conductance. But these were sparse
graphs. So what can we say? Perhaps what they measured was consistent with what exists at very -- but
the point is that -- I mean, we know that even when we have constant expansion, conductance in any
random graph model, I mean we know it. We see it in the calculations. We can prove it, that with high
probability there will be a set that has pretty -- and it's going to be a small one. This is where the variances
are large around size log N. Of course, if you go to large sets you don't have a problem.
So I'm asking how many of them are there? What size? So now these people were working in the sparse
range. But if people start drawing pictures like this saying there's discrepancy we might as well understand
in full detail where are the sparse cuts, what kind of sparse cuts we have, and that's a project in its own
right.
>>: So your methods are based on looking at the second item and your matrix. Has anyone tried to study
conductance looking directly at the old subsets and the estimating the number of edges they use?
>> Milena Mihail: Yeah, so this is sort of always the first approach, probabilistic counting union bound. It
does not work in this case. I mean, it works very well if you're dealing with a configurational model. It does
not work well in this case. And in fact you need very detailed trace methods.
Where you need to count the probability that you return -- these are the only methods that we know for ->>: Configuration model.
>> Milena Mihail: Yes.
>>: More steps.
>> Milena Mihail: The configurational model has much less variance than this model. That's the problem.
Yeah? So I mean, you have lots of dependencies and the dependency for the configurational model help
you. And here you have independent which you can use. But in this case it does not help.
No, I do not think that as you generalize the model so much, I mean, I think the trace method has
limitations. Either that or we have to get inside its enumerative combinatorics, which are nontrivial and fix
them. So I don't know. It's something frustrating, maybe conductance is not the correct measure. But it's
very frustrating that we clearly see better behavior in graphs, but just this inhomogeneity creates this
problem. What you suggested formal [indiscernible] it doesn't go per se conductance. But some
monotonicity. Not any monotonicity because you can start plugging, clicks in wrong places. So there's
minimization in technique.
I think the inner product model is good, because it sort of has the right amount of difficulty. Like not too
much. But enough that you can start working and say something -- I mean, it's a good test point.
>>: One option to emphasize besides conductance to look at what the spectrum [indiscernible] profile for
sets of different sizes, what does the less cuts ->> Milena Mihail: This is another thing. And this is a discussion. It comes up actually -- yes, actually when
you're talking to sort of social networking. People that work on social networks. So they think we have a
fixation with the second eigen value. And we've tried to measure it. So they keep coming back and so
what did we learn?
Okay. So per se we don't learn very much, right? But the point is what you're really interested in is the
eigenvectors and how they behave. You're interested in the entire spectrum and how does it behave.
These are much harder problems. The second eigen value becomes an indication that you can put ->>: Conductance size, maybe you have small conductance because there's small sets that have even
smaller boundary but you could say I'm only worried about sets from a certain size on.
>> Milena Mihail: Oh, then it's wonderful. I mean, yeah, if you're interested about large sets only, there's
no problem. But, no, the ->>: What threshold you start being interested in.
>> Milena Mihail: Okay. Okay. So, yeah, sure. [indiscernible] mixing type estimates.
>> Milena Mihail: Yeah. So I mean I would not exclude the log N area from an interesting area, let me put
it that way. Or poly log N. I wouldn't say some constant and above. I mean, or some N to the 1 over 100
or N to the epsilon and above.
I think the poly log N area is important. And that is precisely where the methods break down, because of
the variance.
>> Kamal Jain: Any other questions? Then let's thank the speaker.
[applause]