лекция райгородский слайды версия 1.1

Web graph models and applications

Andrei Raigorodskii

CSEDays 2012

Andrei Raigorodskii Web graph models and applications

Barab´si, Albert
a

World-wide web graph


Barab´si, Albert
a


Experimental observations [Barab´si, Albert (1999)]:
a


Barab´si, Albert
a


a
Sparse graphs (n vertices, mn edges)


Barab´si, Albert
a


a
Small world (diameter ≈ 5-7)


Barab´si, Albert
a


a
Small world (diameter ≈ 5-7)
Power law
|{v : deg(v) = d}| c
≈ λ, λ ∼ 2.4
n d


Preferential attachment

At the n-th step we add a new vertex n with m edges from it, with probability of edge
to a vertex i proportional to deg(i)

deg(i)
P(edge from n to i) =
j deg(j)


Problems with formalization when m > 1

Theorem (Bollob’as)
Let f (n), n ≤ 2, be any integer valued function with f (2) = 0 and
f (n) ≤ f (n + 1) ≤ f (n) + 1 for every n ≥ 2, such that f (n) → ∞ as n → ∞. Than
there ia s random graph process of Barab’asi and Albert T n such that, with
probability 1, T n has exactly f (n) triangles for all suﬃciently large n.


Bollob´s–Riordan model
a

(n)
Gm – graph with n vertices and mn edges, m ∈ N.
dG (v) – degree of vertex v in graph G.


a

(n)

Case m =1
(1)
G1 – graph with one vertex v1 and one loop.
(n−1) (n)
Given G1 we can make G1 by adding vertex vn and edge from it to vertex vi ,
picked from {v1 , . . . , vn } with probability

d (n−1) (vs )/(2n − 1) 1≤s≤n−1
P(i = s) = G1
1/(2n − 1) s=n


a

(n)

Case m =1
(1)
G1 – graph with one vertex v1 and one loop.
(n−1) (n)
Given G1 we can make G1 by adding vertex vn and edge from it to vertex vi ,
picked from {v1 , . . . , vn } with probability

d (n−1) (vs )/(2n − 1) 1≤s≤n−1
P(i = s) = G1
1/(2n − 1) s=n

Case m >1
(mn) (n)
Given G1 we can make Gm by gluing {v1 , . . . , vm } into v1 , {vm+1 , . . . , v2m }
into v2 , and so on.


Bollob´s–Riordan model, Static description
a

Linearized chord diagrams (LCD) model
LCD with 2mn vertices and mn edges.

Denote by φ(L) graph obtained after gluing.
(2mn)!
If L is chosen uniformly from all (mn)!2mn LCDs with mn edges, then φ(L) has the
(n)
same distribution as Gm .


a

#n (d) - number of vertexes with degree d in G(n)
m m

Theorem (Bollob´s–Riordan–Spencer–Tusn´dy)
a a
Let m ≥ 1 and ≥ 0 be ﬁxed, and set
2m(m + 1)
αm,d = .
(d + m)(d + m + 1)(d + m + 2)
Then whp we have
#n (d)
m
(1 − )αm,d ≤ ≤ (1 + )αm,d
n
1
for every d in the range 0 ≤ d ≤ n 15


a

m m

a a
2m(m + 1)
αm,d = .
(d + m)(d + m + 1)(d + m + 2)
Then whp we have
#n (d)
m
(1 − )αm,d ≤ ≤ (1 + )αm,d
n
1

#n (d)
In particular, whp for all d in this range we have m
n = Θ d−3


a

m m

a a
2m(m + 1)
αm,d = .
(d + m)(d + m + 1)(d + m + 2)
Then whp we have
#n (d)
m
(1 − )αm,d ≤ ≤ (1 + )αm,d
n
1

#n (d)
n = Θ d−3

Theorem (Bollob´s–Riordan)
a

Fix an integer m ≥ 2 and a positive real number . Then whp G(n) is connected and has
m
diameter diam G(n)
m satisfying

(n)
(1 − ) log n/ log log n ≤ diam Gm ≤ (1 + ) log n/ log log n


a

m m

a a
2m(m + 1)
αm,d = .
(d + m)(d + m + 1)(d + m + 2)
Then whp we have
#n (d)
m
(1 − )αm,d ≤ ≤ (1 + )αm,d
n
1

#n (d)
n = Θ d−3

a

m
diameter diam G(n)
m satisfying

(n)

In particular if n = 20 · 106 we have log n/ log log n ≈ 5.96.


a

m m

a a
2m(m + 1)
αm,d = .
(d + m)(d + m + 1)(d + m + 2)
Then whp we have
#n (d)
m
(1 − )αm,d ≤ ≤ (1 + )αm,d
n
1

#n (d)
n = Θ d−3 vs c · d−2.4 in
World-wide web.

a

m
diameter diam G(n)
m satisfying

(n)

In particular if n = 20 · 106 we have log n/ log log n ≈ 5.96.


a

a a
2m(m + 1)
αm,d = .
(d + m)(d + m + 1)(d + m + 2)
Then whp we have
#n (d)
m
(1 − )αm,d ≤ ≤ (1 + )αm,d
n
1


a

a a
2m(m + 1)
αm,d = .
(d + m)(d + m + 1)(d + m + 2)
Then whp we have
#n (d)
m
(1 − )αm,d ≤ ≤ (1 + )αm,d
n
1

Theorem (Grechnikov)

n (2mn + 1)(m + 1) I{d = 0} d
E(#m (d)) = I{d ≥ 0} − + Om
(d + m)(d + m + 1)(d + m + 2) m n

I{X} — indicator of event X.

If d = d(n) and ψ(n) → ∞ when n → ∞, then whp we have
n n
√ −1
E(#m (d)) − #m (d) ≤ d−2 n + d ψ(n)


Buckley–Osthus model

Case m = 1
(n) (n)
For a fixed positive integer a define a process Ha,1 exactly as G1 is defined above,
but replacing probability of edge with
 d
 Ha,1 (vs )+a−1
(n−1)

P(i = s) = (a+1)n−1
1≤s≤n−1
a

(a+1)n−1
s=n

where a is called "initial attractiveness"



Case m = 1
(n) (n)
For a fixed positive integer a define a process Ha,1 exactly as G1 is defined above,
but replacing probability of edge with
 d
 Ha,1 (vs )+a−1
(n−1)

P(i = s) = (a+1)n−1
1≤s≤n−1
a

(a+1)n−1
s=n

where a is called "initial attractiveness"

Case m > 1
(n) (n)
As for Gm , a process Ha,m is defined by identifying vertices in groups of m.



#n (d) - number of vertexes with indegree d.
a,m

Theorem (Buckley–Osthus)
Let m ≥ 1 and a ≥ 1 be ﬁxed integers, and set
d + am − 1 d!
αa,m,d = (a + 1)(am + a)! .
am − 1 (d + am + a + 1)!
Let > 0 be ﬁxed. Then whp we have
#n (d)
a,m
(1 − )αa,m,d ≤ ≤ (1 + )αa,m,d
n
for all d in the range 0 ≤ d ≤ n1/100(a+1) . In particular, whp for all d in this range we have
#n (d)
a,m −2−a
=Θ d
n



Let a > 0 be ﬁxed real, then
n B(d + ma, a + 2) 1
E #a,m (d) = n + Oa,m
B(ma, a + 1) d
The asymptotic behavior of the coeﬃcient when d grows is
B(d + ma, a + 2) Γ(a + 2) −2−a Γ(ma + a + 1) −2−a
∼ d = (a + 1) d
B(ma, a + 1) B(ma, a + 1) Γ(ma)



n B(d + ma, a + 2) 1
E #a,m (d) = n + Oa,m
B(ma, a + 1) d
B(d + ma, a + 2) Γ(a + 2) −2−a Γ(ma + a + 1) −2−a
∼ d = (a + 1) d
B(ma, a + 1) B(ma, a + 1) Γ(ma)

Let d1 > 0 and d2 > 0. Than
n n −2−a −2−a −1 −1
cov(#a,m (d1 ), #a,m (d2 )) = Oa,m ((d1 + d2 )n + d1 d2 )



n B(d + ma, a + 2) 1
E #a,m (d) = n + Oa,m
B(ma, a + 1) d
B(d + ma, a + 2) Γ(a + 2) −2−a Γ(ma + a + 1) −2−a
∼ d = (a + 1) d
B(ma, a + 1) B(ma, a + 1) Γ(ma)

Let d1 > 0 and d2 > 0. Than
n n −2−a −2−a −1 −1
cov(#a,m (d1 ), #a,m (d2 )) = Oa,m ((d1 + d2 )n + d1 d2 )

If d = d(n) and ψ(n) → ∞ when n → ∞, then whp we have

n B(d + ma, a + 2) √ −1
#a,m (d) − n ≤ d−a−2 n + d ψ(n)
B(ma, a + 1)



Consequences
1
When d ∼ Cn a+2 with some constant C,
n
√ −1
E(#a,m (d)) = O(1), d−a−2 n + d = O(1).
If
1
d = o(n a+2 ),
then whp
n (a + 1)Γ(ma + a + 1) −2−a
#a,m (d) ∼ d n.
Γ(ma)
If
1
d = ω(n a+2 ),
than whp #n (d) = o(1); since #n (d) is an integer number by deﬁnition, in this case whp
a,m a,m
n
#a,m (d) = 0


Copying model

The linear growth copying model is parametrized by a copy factor α ∈ (0, 1) and a
constant out-degree m ≥ 1.

We start with one vertex and m loops. At the n-th step we add a new vertex n with
m edges from it.

To generate the out-links, we begin by choosing a ‘prototype’ vertex p uniformly at
random from the old vertices. The i-th out-link of n is then chosen as follows. With
probability α, the destination is chosen uniformly at random from {1, 2, . . . , n}, and
with the remaining probability the out-link is taken to be the i-th out-link of p.


Copying model

Theorem
For d > 0, the limit
#n (d)
α,m
Pd = lim
n→∞ n
exists, and satisﬁes
1 + α/(i(1 − α))
Pd = P0 d
i=0
1 + 2/(i(1 − α))

and
2−α
− 1−α
Pd = θ(d )


Directed scale-free graphs

We consider a graph which grows by adding single edges at discrete time steps.

Let α, β, γ, δin and δout be non-negative real numbers, with α + β + γ = 1. Let G0
be any ﬁxed initial graph, for example a single vertex without edges, and let t0 be the
number of edges of G0 .

We set G(t0 ) = G0 , so at time t the graph G(t) has exactly t edges, and a random
number n(t) of vertices.

For t ≥ t0 we form G(t + 1) from G(t) according the the following rules:
With probability α, add a new vertex v together with an edge from v to an
din (v )+δin
existing vertex w, where w is chosen with probability t+δ i ·n(t) .
in
With probability β, add an edge from an existing vertex v to an existing vertex w,
d (v )+δout
where v and w are chosen independently, v with probability out i ·n(t) , and w
t+δ out
din (wi )+δin
with probability t+δin ·n(t)
.
With probability γ, add a new vertex w and an edge from an existing vertex v to
d (v )+δout
w, where v is chosen with probability out i ·n(t) .
t+δ out


Directed scale-free graphs

#in (d) - a number of vertices with indegree d.
#out (d) - a number of vertices with outdegree d. Let

α+β β+γ
c1 = , c2 =
1 + δin (α + γ) 1 + δout (α + γ)

Theorem
Let i > 0, There exists pi , qi such that whp
#in (i) = pi n + o(n), #out (i) = qi n + o(n). If αδin + γ > 0, γ < 1 then

−1− c1
pi ∼ Cin i 1

as i → ∞, Cin - some positive constant.
If γδout + γ > 0, α < 1 then
−1− c1
qi ∼ Cout i 2

as i → ∞, Cin - some positive constant.


Link rings


Total number of edges between vertices with ﬁxed degree

X(d1 , d2 , n) – total number of edges linking a node with degree d1 and a node with
degree d2 . When d1 = d2 , we count every edge twice, but do not count loops.

The expected value for N given d(Ai ) and d(Bj ) is

I J
X(d(Ai ), d(Bj ), n)
N0 =
i=1 j=1
#n (d(Ai ))#n (d(Bj ))
m m





I J
N0 =
i=1 j=1
#n (d(Ai ))#n (d(Bj ))
m m

If N ≤ N0 , this structure can be a natural formation.





I J
N0 =
i=1 j=1
#n (d(Ai ))#n (d(Bj ))
m m

If N ≤ N0 , this structure can be a natural formation.
If N > N0 , this structure is probably a real link farm with some buyers.


Total number of edges between vertices with ﬁxed degree in Buckley-Osthus
model

There exists a function cX (m + d1 , m + d2 ) such that
EX(m + d1 , m + d2 , n) = cX (d1 , d2 )n + Oa,m (1).
When both d1 and d2 grow, the asymptotic behaviour of cX is

Γ(ma + a + 1) (d1 + d2 + 2m)1−a
cX (d1 , d2 ) = ma(a + 1) ·
Γ(ma) (d1 + m)2 (d2 + m)2
1 1 d1 d2
· 1 + Oa,m + + .
d1 d2 (d1 + d2 )2


model

There exists a function cX (m + d1 , m + d2 ) such that
EX(m + d1 , m + d2 , n) = cX (d1 , d2 )n + Oa,m (1).
When both d1 and d2 grow, the asymptotic behaviour of cX is

Γ(ma + a + 1) (d1 + d2 + 2m)1−a
cX (d1 , d2 ) = ma(a + 1) ·
Γ(ma) (d1 + m)2 (d2 + m)2
1 1 d1 d2
· 1 + Oa,m + + .
d1 d2 (d1 + d2 )2

Let c > 0. Then
√ c2
P |X(m + d1 , m + d2 , n) − EX(m + d1 , m + d2 , n)| ≥ c(d1 + d2 ) mn ≤ 2 exp −
8
√
In particular, if c(n) → ∞ when n → ∞, then whp |X − EX| < c(n)(d1 + d2 ) mn


model
The formula for cX (d1 , d2 ) does not give an asymptotic behaviour if
d2
→ c = 0.
d1
The precise bounds show that term
(d1 + d2 )1−a
d2 d2
1 2

still gives the correct order of growth for cX , but the coefficient can be different. And in fact
the coefficient differs.


model
The formula for cX (d1 , d2 ) does not give an asymptotic behaviour if
d2
→ c = 0.
d1
The precise bounds show that term
(d1 + d2 )1−a
d2 d2
1 2

still gives the correct order of growth for cX , but the coefficient can be different. And in fact
the coefficient differs.

There exists a function cX (d1 , d2 ) such that
EX(m + d1 , m + d2 , n) = cX (d1 , d2 )n + Oa,m (1)
and

Γ(d1 + ma)Γ(d2 + ma)Γ(d1 + d2 + 2ma + 3)
cX (d1 , d2 ) = ·
Γ(d1 + ma + 2)Γ(d2 + ma + 2)Γ(d1 + d2 + 2ma + a + 2)
Γ(ma + a + 1) (d1 + ma + 1)(d2 + ma + 1)
· ma(a + 1) 1 + θ(d1 , d2 )
Γ(ma) (d1 + d2 + 2ma + 1)(d1 + d2 + 2ma + 2)
where
2 Γ(ma + 1)Γ(2ma + a + 3)
−4 + ≤ θ(d1 , d2 ) ≤ a
1 + ma Γ(2ma + 2)Γ(ma + a + 2)


Total number of edges between vertices with ﬁxed degree in
Bollob’as-Riordan model

If d1 + d2 = 0, then X(m + d1, m + d2, n) = 0. if d1 + d2 ≥ 1, then

m(m + 1)
EX(m + d1 , m + d2 , n) = ·
(d1 + m)(d1 + m + 1)(d2 + m)(d2 + m + 1)
 
m+1 d
C2m+2 Cd 1+d
· 1 − d +m+11 2  (2mn + 1)−
Cd 1+d +2m+2
1 2

m d +m−i
Cd 1+d +2m−2i (2i)! m + 1 (2m)!
1 2
− d +m
+ [i = m] −
i=1 (d1 + m)(d2 + m)Cd 1+d +2m i!(i + 1)! 2m 2(m − 1)!2
1 2
(m − 1)(m + 1) (m − 1)(m + 1)
− [d1 = 0] − [d2 = 0] +
2m(d2 + m)(d2 + m + 1) 2m(d1 + m)(d1 + m + 1)
1
+ Om,d1 ,d2
n


Experimental validation

From theory for x(d) - number of vertexes with degree d we have,

x(d) = #n (d − m) ∼ Cd−2−a n
a,m




x(d) = #n (d − m) ∼ Cd−2−a n
a,m

Da (x) - normalized sum of squared deviations of the observed values of x from the
values predicted.
min Da (x) ⇒ a = 0.4357
a




x(d) = #n (d − m) ∼ Cd−2−a n
a,m

values predicted.
min Da (x) ⇒ a = 0.4357
a

X(d1 −m,d2 −m)
Let ρ(d1 , d2 ) = x(d1 )x(d2 )
. From theory we have,

ρ(d1 , d2 ) = c1 (d1 + d2 )1−a (d1 d2 )a n−1




x(d) = #n (d − m) ∼ Cd−2−a n
a,m

values predicted.
min Da (x) ⇒ a = 0.4357
a

X(d1 −m,d2 −m)
Let ρ(d1 , d2 ) = x(d1 )x(d2 )
. From theory we have,

ρ(d1 , d2 ) = c1 (d1 + d2 )1−a (d1 d2 )a n−1

Dρ (x) - normalized sum of squared deviations of the observed values of ρ from the
values predicted.
min Da (ρ) ⇒ a = 0.44079
a


2-nd degrees

Deﬁne 2-nd degree of vertex t as
(n) (n)
d2 (t) = #{i, j : i = t, j = t, it ∈ E(G1 ), ij ∈ E(G1 )}

(n)
Deﬁne by Xn (k) number of vertices with 2-nd degree equal to k in G1


2-nd degrees

(n) (n)

(n)

Theorem (Grechnikov–Ostroumova)
For any k ≥ 1

4n k2 log2 k
E (Xn (k)) = 1+O 1+O .
k2 n k


2-nd degrees

(n) (n)

(n)

Theorem (Grechnikov–Ostroumova)
For any k ≥ 1

4n k2 log2 k
E (Xn (k)) = 1+O 1+O .
k2 n k

Theorem (Ostroumova)
For any ε > 0 there is such a function ϕ(n) = o(n), that for any 1 ≤ k ≤ n1/6−ε ,
whp we have
ϕ(n)
|Xn (k) − E (Xn (k)) | ≤
k2


2nd degrees
Let Yn (k) denote the number of vertices with second degree ≥ k in Gn .
a,1


2nd degrees
a,1

For any k > 1 we have

(a + 1)Γ(2a + 1) (ln k) a+1 k1+a
EYn (k) = n 1+O +O .
Γ(a + 1)ka k n


2nd degrees
a,1


(a + 1)Γ(2a + 1) (ln k) a+1 k1+a
EYn (k) = n 1+O +O .
Γ(a + 1)ka k n

Corollary
1
n
EYn (k) = Θ ka for k = O n 1+a .


2nd degrees
a,1


(a + 1)Γ(2a + 1) (ln k) a+1 k1+a
EYn (k) = n 1+O +O .
Γ(a + 1)ka k n

Corollary
1
n

Theorem (Kupavskii, Ostroumova, Tetali)
1
Let δ > 0 and k = O n 2+a+δ . Then for some > 0 we have

1−
P |Yn (k) − E(Yn (k))| > E(Yn (k)) = o(1).
¯


2nd degrees
a,1


(a + 1)Γ(2a + 1) (ln k) a+1 k1+a
EYn (k) = n 1+O +O .
Γ(a + 1)ka k n

Corollary
1
n

Theorem (Kupavskii, Ostroumova, Tetali)
1

1−
P |Yn (k) − E(Yn (k))| > E(Yn (k)) = o(1).
¯

Corollary
1

1−
P |Xn (k) − E(Xn (k))| > E(Xn (k)) = o(1).
¯


(n)
Number of copies of H in Gm

Theorem about triangles (Bollob´s)
a
(m − 1)m(m + 1) 3
E(#(K3 , G(n) )) = (1 + o (1))
m ln (n)
48


(n)

a
(m − 1)m(m + 1) 3
E(#(K3 , G(n) )) = (1 + o (1))
m ln (n)
48

Theorem about cycles (Bollob´s)
a
l
E(#(l-cycles, G(n) )) = (1 + o(1))Cm,l (ln n)
m

where Cm,l is a positive constant, Cm,l = Θ ml


(n)

a
(m − 1)m(m + 1) 3
E(#(K3 , G(n) )) = (1 + o (1))
m ln (n)
48

Theorem about cycles (Bollob´s)
a
l
E(#(l-cycles, G(n) )) = (1 + o(1))Cm,l (ln n)
m

where Cm,l is a positive constant, Cm,l = Θ ml

Theorem about pairs of adjacent edges (P2 ) (Bollob´s)
a

m(m + 1) m(m + 1)
(1 − ) n ln n ≤ #(P2 , G(n) ) ≤ (1 + )
m n ln n
2 2
holds whp as n → ∞ where > 0 be ﬁxed


(n)

Theorem about arbitrary subgraph (Ryabchenko – Samosvat)
for arbitrary graph H

(n) √ di
E # H, Gm = Θ(1) n#(di =0) ( n)#(di =1) (ln n)#(di =2) m 2

where di — is degree of node i in H
or
(n)
E # H, Gm =0


лекция райгородский слайды версия 1.1

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Viewers also liked

Viewers also liked (8)

Similar to лекция райгородский слайды версия 1.1

Similar to лекция райгородский слайды версия 1.1 (20)

More from csedays

More from csedays (8)

лекция райгородский слайды версия 1.1