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>> Yuval Peres: Hello, everyone. We're very happy to have Andrei from Yandex in
Moscow here to tell us about Web graph models, properties and applications.
>> Andrei Raigorodskii: Okay. Thank you very much for this presentation and for having
invited me here. It's a great pleasure for me to speak on this subject. I'm sorry I have
some problems with my throat, probably I'll try to be as loud as possible, but I'm not sure.
>>: You have a microphone.
>> Andrei Raigorodskii: I don't hear it. So, okay. Yes, you hear me.
Okay. So I'd like to speak about Web graph models, properties and applications. I'd like
to present here our recent results concerning classical Web graph models and maybe
some new conjectures and possible applications of these results.
First of all, I'd like to present here some history of the problems. Of course, all of you, I
think, know the history, but I'd like to present it anyways, since probably it's necessary to
remind you some.
Some steps, all this history. So by World Wide Web graph, I mean, you know, the
network which consists of sites, of hosts and hyperlinks between them. So these are
sites and hyperlinks between them.
And, of course, it's interesting to consider some, to invent, to discover, to consider some
models which are very close to the experimental observations.
So, first of all, I'd like to present some experimental observations which were done in the
end of '90s by Barabasi and Albert. I think you know, of course, that experimental
observations by Barabasi and Albert in 1999 were, of course, as follows.
First of all, the Web graph is a sparse graph who has N vertices and MN edges. So M is
a constant and the number of vertices is multiplied by M. So this is the number of edges.
The graph is of course sparse in this sense. Also we have the well-known effect of the
small world. The domain of this graph is not precisely but approximately equal to five to
seven.
And, finally, the well-known property of almost any network of a Web graph, of course, as
well, is the probability distribution of its degrees.
So it means that the number of vertices who have degree D where D is a constant or a
function on N over the number of all the vertices of this graph is something like some
constant over D to the power lambda, where lambda is by experimental observations
equal to something like 2.1. I have to look at the audience. You should know that, of
course.
>>: Actually, people work on many diverse things. So everyone works on this.
>> Andrei Raigorodskii: Yes, that's why I remind you this history, of course. This is also
useful. So these are observations, experimental observations due to Barabasi-Albert in
1999. The graph is part of the small world. Diameter was something like that, the power
law of degree distribution.
And now it's interesting to construct some Web graph model which has almost the same
properties. So this was suggested by Barabasi-Albert and they also suggested some
way to do that. So they proposed a very natural idea of preferential, the so-called
preferential attachment. We consider at the nth step of the procedure we add a new
vertex N with M edges from E to a probability of edge to vertex I proportional to the
degree of I. So if we have some graph, which is already constructed, which is already,
which has already appeared during the history, the story of its development, when we
add any U vertex, which is called here N, we send M edges to some points to some
vertices from the previous graph. According to their degrees, proportional, with
probability proportional to their degrees. So the probability of an edge from N to I is,
should be equal to the degree of I in the previous graph over the sum of all degrees of G
over all G in these graphs.
This is the idea. But unfortunately Barabasi-Albert, who proposed to consider such a
preferential attachment, they did not say any words about the distribution of these
targets. So it's almost -- it's also necessary to say what is the distribution of these M
targets here.
Since it's possible to distribute them uniformly or to consider some dependencies
between them, so there is a problem with the dependencies between the targets, the
targets, the M targets of these new vertex.
And it's very funny that these problems, it leads to the following theorem which is due to
Bollobas. So the problem with formalization when M is greater to 1 is described here in
this theorem by Bollobas, which says that given an arbitrary function F of N for N greater
than or equal to 2, an arbitrary integer value function with F of 2equal to 0 and F of N less
than or equal to F of N plus 1, less than or equal to F of N plus 1, here you see such that,
of course, F of N goes to infinity as N tends to infinity. Then there's a random graph
process of this kind which is depicted here with these probabilities, such that with
probability tending to 1, T of N has exactly F of N triangles for all sufficiently large N. So
it's possible to consider such dependencies here that here we have an arbitrary given
number of triangles in this graph process.
So it's absolutely necessary to say what are the dependencies between edges here, such
targets. And the problem of doing that was solved by Bollobás and Riordan who
proposed the following well-known Web graph model.
Let us consider GMN graph with N vertices and M and N edges with M which is a natural
number. So we are just trying to realize the idea that the number of vertices should be M
times smaller than the number of edges of our Web graph.
Let us denote by DG of degree of vertex V in the graph G. Now we describe the graph
process for M equal to 1. So let G 1 1 be a graph with one vertex V sub 1 in one loop,
just one vertex and one loop in it. This is the first step of our graph process. Given a
graph G sub 1 N minus 1 we can make G sub 1 M and U vertex VM and an edge from it
to vertex VI, picked from V sub 1 V sub N. You see here I can be equal to N. So we
admit loops as well here, and the probablity that I equals to S is given here due to
preferential attachment just. So this is this ratio whenever S lies between 1 and 10 minus
1, and this probability equals 1 over 2 N minus 1 for S equal to N.
So there is a non-0 probability of a loop here. It's possible to obtain a loop.
But it's more natural to obtain some link to a previous vertex of this process.
In the case of M, which is greater, streak less than 1, we take G1MN which is described
here, which is found by this probabilistic process here, and we can make from GMN by
gluing the first M vertices of this graph into one big vertex into, say, V sub 1 prime, then
we glue the next M vertices of this graph into the second vertex V sub 2 prime and so on.
So we obtain the graph whose number of vertices is exactly equal to M and whose
number of edges is already M multiplied by N, so we realize all -- everything we wanted
to realize.
So this is the Bollobas and Riordan model well known, which has already a ten-year-old
history, which is well-studied, of course. It's possible to reformulate the same model, this
model, in the following static way. It's possible to consider the so-called linearized core
diagrams. Linearized core diagram with two M vertices and MN edges is described here.
We take two MN vertices, one, two, three, and so on two MN, we consider pairings of
those vertices, and then we obtain a graph from such a construction in the following way:
We go from the left to the right. We find the first right-hand endpoint of an edge here, of
a cord here.
And these four vertices on this picture, they are glued into one vertex in our graph. And
so on. This is the second vertex. This is the third one. This is the fourth one, the fifth
one and so on.
So it's possible to consider this static model. We just take a uniformly linearized core
diagram. We consider a graph which corresponds to this random linearized core
diagram. And this graph has the same distribution as the graph on the previous slide.
This one, I mean. It's possible to understand what I say? Right now if you have some
questions, please.
So it's well known, of course. I'm just reminding you some previous results in some
previous models. Now I'd like to proceed to some results which were obtained by
different authors.
So the first theorem, which is due to Bollobas Riordan Spencer and Tusnady, it says the
following: If we consider, if we take an arbitrary natural number M greater than or equal
to 1, and if we fix an epsilon which is also greater than or equal to 0, then we can set
alpha MD equal to this value to M multiplied by N plus 1 over this product.
And then with high probability, I mean with the probability which is tending to 1, when N
tends to infinity, the number of vertices of our graph, then the ratio of the number of
vertices of degree D, this is the number of vertices of degree D in GMN graph over the
whole number of vertices in this graph, is tightly concentrated around this value.
So with high probability, this value lies between 1 minus epsilon multiplied by alpha MD
and 1 plus epsilon multiplied by the same value.
But for every D only in the range which is depicted here, D is greater than or equal to 0
and less than or equal to N to the power 115th. I think you understand that this is not
very good since for the real networks, for the real Web, the number of vertices is
something like ten billions or something like that.
But if you take this root of this value, this is smaller than 10. So it's not interesting to
consider this. Okay. But in particular with high probability for all D in this range, we have
this value equal to some constant multiplied by D to the power minus 3.
Here D multiplied by D multiplied by D gives D to the power minus 3. And another
theorem by Bollobas and Riordan says that for a fixed integer M greater than or equal to
2 and the positive real number epsilon, with high probability, this graph GMN is
connected and has diameter which satisfies this relation with high probability the
diameter of our random graph lies between log N over log log N multiplied by 1 minus
epsilon and the same value multiplied by 1 plus epsilon.
This is also very good, since, in particular, if we take N equal to 20 multiplied by 20
millions, just, we have log N over log log N equal to something very close to 6. And if you
remember here, it was 5 to 7. So this is very close to the expected value, which is just
obtained by experimental observations by Barabasi-Albert. This is very good.
However, as I've already said to you, there are some problems. The first problem is that
we can take only this range, which is very small, from practical purposes, practical point
of view, and the second problem is that here we have minus 3 versus C multiplied D
minus 2.1 in the World Wide Web as it was considered by Barabasi and Albert, and I've
already talked about it.
So we have some problems in this theorem, and we have almost no problems in this one.
>>: But you're saying the problem is it's inconsistent with the ->> Andrei Raigorodskii: Yes, it's a little bit inconsistent here. Yes. And also we have
this technical -- technical bounds which are not very good, of course, from practical point
of view, from mathematical point of view as well.
Okay. First of all, I'd like to speak about this problem. It's possible to remove it, to
remove this gap. It's possible to remove this gap completely. And one of our
researchers, Grechnikov, succeeded in doing that. Once again, the theorem by Bollobas
Riordan Spencer and Tusnady, and here I put new theorems by Grechnikov.
You see the first theorem says that the expectation of the number of vertices having
degree D equals just the same value as it's written here, minus something small. And
plus big OM of D over N.
So if you look at this theorem, you understand that, first of all, there are no limitations like
here in it. So it's not necessary to consider any limitations on D, if D is small O of N. If D
is asymptotically small in comparison with N, then, of course, we have here an
asymptotic formula up to the third theorem. Three theorem asymptotic formula without
any limitations on D.
This is a very, very exact, very precise formulation of the result concerning the
expectation of the number of vertices hidden degree D in the Bollobas and Riordan
model.
Also, it's possible to prove here a concentration result of this kind. So if D depends on N
and C of N goes to infinity when N tends to infinity, then with high probability we have a
difference between the number of vertices of degree D and it's expectation. The absolute
value, of course, of this difference is less than or equal to this value which is small, of
course.
So for every D, not for D less than or equal to N to the power 1 over 15, we can prove a
three term asymptotic formula and we can prove a concentration result around this mean
value.
I think in this case we have removed all the gaps which could be removed here. So now
it's interesting to speak about the second problem. What shall we do concerning D to the
power minus 3 or D to the power 2.1? How can we ->>: How can we try to reduce ->> Andrei Raigorodskii: How can we try to produce such a model which is more closed,
which is closer to the reality? So for this purpose many different models have been
proposed. For example, there was a model by Buckley and Osthus for the case M equal
to 1. M, I remind you, is the ratio between the number of edges and the number of
vertices.
So for the case M equal to 1, and for a fixed positive integer A, we define a process
exactly as G1N is defined above. But we replace the probability of an edge going to
some previous vertex, say V sub S by this value. Here we plus this number A, which is
called initial attractiveness.
So we consider some kind of initial attractiveness of any vertex. It's constant. It's the
same for all the vertices. And it's denoted by A which is greater than or equal to 0 and
this probability is defined, here it's defined correctly, of course, since the sum of all such
probabilities equals 1, certainly. And what can be said about this model?
Of course, in the case of M greater than 1, we just consider the same process HAMN.
We identify the vertices in groups of N, groups of M.
And Buckley and Osthus, they succeeded in proving the following theorem. I don't know.
Maybe it's rather difficult to read it since it's very big. But be advised a little bit later I will
show you much more difficult theorems. Much more. So it's not a problem to consider
this. It should be a problem to consider further progress here.
So let M be a natural number which is greater than or equal to 0 and they be greater than
or equal to 1. So both numbers are fixed integers. And let's consider this value.
This value is rather cumbersome, but it's possible to consider it. Let epsilon greater than
0 be fixed. Then with high probability the number of vertices of degree D in our new Web
graph model lies between 1 minus epsilon multiplied by this cumbersome value and 1
plus epsilon multiplied by the same value.
So we have here a concentration result. But for all D in the range, which is depicted
here. So N is not to the power 1 over 15. But now N to the power 1 over 100 multiplied
by A plus 1. 100. Not 15.
In principle, in particular with high probability for all D in this range we have this ratio
equal to some constants multiplied by D to the power minus 2 minus 3. It would be great,
of course, if we could put here A equal to 0.1, since if we take here A equal to 0.1 minus
2, minus 0.1 is exactly minus 2.1 and this is very close to what we have on the Internet, in
the real network.
>>: Isn't it possible that by 99 this is not the right constant?
>> Andrei Raigorodskii: Huh?
>>: 2.1 could it be true 10 years ago but now ->> Andrei Raigorodskii: No, it's possible to check it right now, and the same constant is
true. So we checked it. And the constant is still 2.1. Huh?
>>: How did you perform the experimental ->> Andrei Raigorodskii: It's some statistics over the Web. We have ->>: If you want to start changing a type of graph and ->> Andrei Raigorodskii: In the index there is a lot of information concerning real Web
graphs. So it's possible to make some statistics, some confidential intervals and so on.
>>: Of course, it's very tricky to get an unbiased sample because ->> Andrei Raigorodskii: Yes, but it's possible to do so. So the constant is almost the
same. But the problem is that, just a second, the problem is that E here is a fixed integer.
So it's impossible to take E equal to 0.1. There's a problem here.
So it's necessary to remove this problem concerning the integers, and the second
problem concerning this very small range. And it's possible to do so. Grechnikov, the
same Grechnikov as before, he succeeded in removing all the problems here. Here is
the first theorem that is due to him. A lot AB greater than 0, a fixed real. It's a real
number. Then the expectation of the number of vertices in this model which have degree
D looks like that. This is an asymptotic formula, which is not so good as the previous
one, since here we have not three term asymptotic formula, this is only two term
asymptotic.
But here we have the asymptotic behavior of this coefficient, which is exactly -- which is
asymptotically something like that, which is equal to this.
And now we can take any real number A, for example, 1.0, we can remove this problem
concerning the integers here.
Also it's possible to prove a concentration result, and in order to prove here a
concentration result, we don't use [inaudible] in equality. We don't use Martingale
technique. We just use the second moment method, and we calculate covariances here.
It's possible to calculate just covariances between the number of vertices of degree D1
and number of vertices of degree D2. It's given here asymptotically these covariance.
And by using this result it's possible to prove this concentration formula so the number of
vertices of degree D in our model minus this first term of our asymptotic formula is less
than or equal to this value.
It's possible to describe some consequences of this asymptotic result, this concentration
result. So here I am trying to make it more clear.
It's rather difficult to understand what happens in this right-hand side. But it is possible to
look at some consequences when D is asymptotically equal to some constant C
multiplied by N to the power 1 over A plus 2, compares this to the N to the power A plus 1
over 100, 100. Only N to the power of 1 over A plus 2, 100, with some constant C.
The expectation of the number of vertices of degree D is a constant and this value is also
a constant. But if D is already small O of N to the power 1 over A plus 2, also known
100s, then with high probability this value is asymptotically equal to the first term. Not
expectation, but the same value, with high probability.
If D is integer of N to the power of 1 over A plus 2, then with high probability this number
is equal to small O of 1.
Since this number is an integer by definition, then, of course, with high probability we
have here 0.
So everything is depicted here. No problem with anything like that. Only here we have
not three term asymptotic formula, only two term asymptotic formula.
I don't know whether it's so necessary to obtain so strict formulas. Maybe it's not
necessary to try to take here and to get here a formula with three terms. Two terms is
already enough.
Okay. Now I'd like to proceed to another characteristic of the Web, which is also very
important. It's not only interesting for mathematical, from a mathematical point of view,
but it's also interesting from some practical points of view. So I'd like to present before
some motivations for considering it and then I will define it exactly in mathematical terms.
So let's consider the so-called link rings, which appear in the Internet which are formed
artificially by some people. Wait. Sorry. So they look like bipartite graphs which are
close to complete bipartite graphs.
Here we have a link farm, and here we have some link buyers who want to obtain many
links from somewhere from a link farm or from anywhere. We denote the sites which are
on a link farm as a sub one A sub two and A sub N, and let's denote link buyers by B sub
one B sub two and so on and B sub M. So N sellers and N buyers. Suppose we have
found such a thing. Suppose we have found such a construction in the Internet. How to
understand whether this is just a link ring or probably this is some natural construction,
which is not artificially done. Maybe it's possible to understand somehow statistically.
So let us consider the number of edges between farm and buyers, let us denote by N the
complete number of those edges. The first one, the second one, the third one, the fourth
one and so on.
This is the number of edges between the left side, the left part of this bipartite graph, and
N and the right part of this bipartite graph. Let us denote by D of A2, for example, the
number of edges coming to this vertex A sub 2, D of B2 is the same, the number of
edges which are coming to the vertex denoted by V sub 2. And let's consider the value
which is denoted by X of D1, D2N, the total number of edges, link and node with degree
D1 and the node with degree D2. When 1 is equal to D2, we count every edge twice but
do not count loops which can appear in our model.
Okay. So the expected value of O over N, if we consider some model of a graph, say
Bollobas and Riordan model or Buckley and Osthus model we can try to calculate the
expected number of edges between this part and this part.
And it's given by the following formula. N0 is equal to the summation over all possible
vertices from the left-hand side, and over all possible vertices from the right-hand side.
Of this value, X of D of AID of BJN denotes the expected value of none of edges in our
model between A sub I and B sub J. This is the expected number of edges between
those vertices.
Is it possible to understand it? Yes. So N0 is the expected value for N, which is written
here. So if N is substantially smaller in some sense than N0, this structure can be natural
formation.
Wait. Sorry. And if it's greater than N0, this structure is probably a real link farm with
some buyers. So one should only determine what should be a difference between N and
0, in order to say with high probabilities that the structure can be a natural formation, so
on.
>>: [inaudible].
>> Andrei Raigorodskii: Uh-huh.
>>: So you have -- you're looking over the Web, there are many, many possible
formations. So --
>> Andrei Raigorodskii: Yes, I use, for example, I use some algorithm in order to find
some structures which are suspected to be link rings.
>>: But any ->> Andrei Raigorodskii: For example, I found 100, 1,000 of such structures, yes.
>>: What I don't understand N0 is the expectation once you fix the candidate nodes.
>> Andrei Raigorodskii: Yes, yes, I found a lot of formations. I take one of them, for
example, and I'm trying to understand it automatically whether this structure is really link
ring, not a natural formation.
>>: Right. But N0 is not an exception of value, just the expectation.
>> Andrei Raigorodskii: Yes. It's some expectation, which is given by our graph model.
>>: But let's suppose the graph model is completely accurate.
>> Andrei Raigorodskii: Yes.
>>: Then, first of all, random -- I mean, why can't the random variable exceed the
expectation? And, second, it could be the sample -- there are many, many possible
formations. So some of them surely the number would exceed the target expectations.
I'm missing something here.
>>: The algorithm you are using is picking the places where this number is exceptionally
high, significantly.
>> Andrei Raigorodskii: If we use an algorithm who tries to obtain all possible bipartite
graphs, almost complete bipartite graphs, this algorithm would not work, not only for
linear time, but also for -- it will take exponential time.
>>: We're not looking at heuristic problem. Fist suppose you have infinite resources,
don't you have to understand why, I mean, why you think of anything that exceeds N0 as
something unnatural, since the world is so big, you have many examples naturally where
in your model that exceed the expectation.
>> Andrei Raigorodskii: But we have that concentration result. So if something exceeds
expectation substantially, not like here, but substantially, not just N greater than N0 but
many times greater than N0 of course we can suspect that ->>: So we didn't see them the many times there is invisible print.
>> Andrei Raigorodskii: Yes, yes, I tried to -- yes, I'm sorry, this is not so good reading
here. Yes.
>>: Should depend on the size of the network.
>> Andrei Raigorodskii: Yes, yes.
>>: The number of the whole ->> Andrei Raigorodskii: Of course it's some statistics. It's necessary to put here some
constant C and here some constant C, which also depends on not even constant. Yes,
yes, not even a constant. But some degrees depends on the number of these vertices, of
course.
>>: That seems to be the whole point, I mean, how to determine the value.
>>: It's not just the number of vertices it's the number of possible choices of a bipartite
graph.
>> Andrei Raigorodskii: You are absolutely right. It's a big statistical problem and it's
possible to solve it. But now I don't want to speak about this statistical problem.
I'm just giving some motivation for the consideration of this value X of D1, D2 and N. So
it's interesting to consider the distribution of the number of edges between links with
given -- with given degrees.
>>: Certainly [inaudible].
>> Andrei Raigorodskii: It's interesting, since it's possible to consider this value and
another question, of course, is how to put here what to put here, in order to statistically
decide between two hypothesis, the first one is that this is a natural structure, and the
second one is that the structure is probably a real link farm.
So I have 15 minutes or -- okay. So I'd like to speak about this value. In this value it's
much more complicated than the number of vertices of a given degree. It's much more
difficult to study it from a mathematical point of view. However, it's also possible to do it.
And, in fact, the results concerning the degree distributions are just some simple
corollaries from the consideration, from the theorems which are due to the consideration
of such values.
So the first theorem here is given in a rough form, not in a very precise form. But it's
written down here there exists a function of CX of D1 D2 such that the expectation of this
value equals CX of D1 D2 multiplied by N plus something small. So this is also an
asymptotic formula, and CX is given here by this rather cumbersome formula. This is not
absolutely the most cumbersome formula from this talk.
I am sorry. This is a concentration result.
>>: We are in the competition.
>> Andrei Raigorodskii: Yes, it is possible to compete here. We are very close to that
record, yes. So here the formula which is depicted here doesn't give an asymptotic
behavior only if D2 over D1 tends to C, a constant C which is not equal to 0. The precise
bounds, however, show that the term, this one, this is the main term here still gives the
correct order of growth for CX, but the coefficient can be different and in fact the
coefficient differs. This is almost the record. This is the exact formulation of this
theorem. This is some kind of the power and the glory. I don't know how to -- what to
say about it. It's just -- here it's understandable but here already probably not. But this is
the correct formula here. The expectation of this value is given by this formula, where
such precise bounds.
And another record concerns Bollobas and Riordan model, since here we considered
Buckley and Osthus, but here we consider Bollobas and Riordan model and in this case
we can prove even more precise asymptotic formula which is still three term asymptotic
formula. I don't know whether it's possible to understand something on this slide, but this
is once again the power and the glory of this method.
Now I'd like to proceed to some other definitions and to some other characteristics of the
Web graph. Here's the so-called second degrees. We define the second degree of a
vertex T as the number of edges which go to the vertices which are adjacent to the vertex
T. So this is the number of I and J such that I is not equal to T and J is not equal to T.
The pair IT belongs to the set of vertices, of edges of G1N. IJ lies in the same set of
edges.
We defined by XN of K number of vertices with second degree equal to K in this graph.
You should -- I should mention here that we consider now only the case of M equal to 1.
So here we have index 1, not an arbitrary M. But just the graph whose set of vertices
equal the number of vertices equals to the number of edges.
We succeeded in calculating the exact expectation, the asymptotic formula for the
expectation of this value, of the number of vertices of the second degree K in the graph G
sub 1N. It also obeys the kind of power law degree distribution, power law distribution.
This sum remainders and this is the first term of this asymptotic.
And the theorem by Ostroumova says that XM of K is tightly concentrated around this
mean value which is asymptotically found.
So it's possible also to use such results in Ryabchenko since it's expected to know the
third and fourth degree of a vertex so on in our Web graph.
And some other calculations here, this is probably the most interesting things concerning
the Internet concerning the Web graph models, considered in our work.
And here it's also possible to consider the number of concrete graphs, of concrete graphs
in a random Web graph. For example, in Bollobas and Riordan model, here we can see
number of copies of H in G and M in the Bollobas and Riordan model, and theorem, even
three theorems which are proposed by Bollobas are written down here, expected number
of K3 of triangles in GMN is asymptotically equal to this value. So this is some constant
depending on M and this is the logarithm to the power 3 of N.
The expected number of triangles in the Bollobas and Riordan model equals something
multiplied by logarithm of N cubed. As for cycles, expected number of cycles,
asymptotically equals to this value.
And for the number of pairs of adjacent edges, it's also possible to find the asymptotic
formula which is depicted here. It's, of course, interesting to understand how to unify all
these results. In principle, in Bollobas's work, it's possible to find some unification of this
kind, but a much more readable and a much more understandable formula is given on
the left slide, a theorem about arbitrary subgraph, which is due to Ryabchenko and
Samosvat. For arbitrary graph H, fixed graph H, we have the following formula for the
expectation of the number of copies of H in the Bollobas and Riordan graph GMN. This
is some constant multiplied by N to the power.
Here was the number of vertices of this graph H whose degree is equal to 0. Here is the
number of vertices of this graph whose degree is equal to 1, and here is the number of
vertices of this graph whose degree is equal to T. I'm sorry, to 2. So, for example, if we
consider K3 a triangle instead of H, then, of course, it has three vertices of degree 2.
Three vertices of degree 2 means logarithm to the power of 3, 0 vertices of degree 1 and
0 vertices of degree 0.
So here we have the same formula as on the previous slide. And it's very easy to
calculate any asymptotic here for an arbitrary graph H which is fixed. Probably I'd like to
finish by that. Thank you very much for your attention.
[applause]
>>: So high degrees don't appear here?
>> Andrei Raigorodskii: Don't appear here, yes. Only those degrees, yes.
>>: Is it possible to sort of see intuitively why that?
>> Andrei Raigorodskii: This is not very good in fact since in the real Internet this is not
the case.
>>: Can you see intuitively why it's the case for this model, why degree three?
>> Andrei Raigorodskii: Hmm, maybe it's rather difficult to say it in a few words, I don't
know how to explain it. In this model, yes, yes. Degree three doesn't work, but I'm sorry
I don't know how to say it in a few words.
>> Yuval Peres: No more questions? Let's thank Andrei.
[applause]