1. Web graph models and applications
Andrei Raigorodskii
CSEDays 2012
Andrei Raigorodskii Web graph models and applications
2. Barab´si, Albert
a
World-wide web graph
Andrei Raigorodskii Web graph models and applications
3. Barab´si, Albert
a
World-wide web graph
Experimental observations [Barab´si, Albert (1999)]:
a
Andrei Raigorodskii Web graph models and applications
4. Barab´si, Albert
a
World-wide web graph
Experimental observations [Barab´si, Albert (1999)]:
a
Sparse graphs (n vertices, mn edges)
Andrei Raigorodskii Web graph models and applications
5. Barab´si, Albert
a
World-wide web graph
Experimental observations [Barab´si, Albert (1999)]:
a
Sparse graphs (n vertices, mn edges)
Small world (diameter ≈ 5-7)
Andrei Raigorodskii Web graph models and applications
6. Barab´si, Albert
a
World-wide web graph
Experimental observations [Barab´si, Albert (1999)]:
a
Sparse graphs (n vertices, mn edges)
Small world (diameter ≈ 5-7)
Power law
|{v : deg(v) = d}| c
≈ λ, λ ∼ 2.4
n d
Andrei Raigorodskii Web graph models and applications
7. 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)
Andrei Raigorodskii Web graph models and applications
8. 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.
Andrei Raigorodskii Web graph models and applications
9. 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.
Andrei Raigorodskii Web graph models and applications
10. 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.
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
Andrei Raigorodskii Web graph models and applications
11. 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.
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.
Andrei Raigorodskii Web graph models and applications
12. 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 .
Andrei Raigorodskii Web graph models and applications
13. Bollob´s–Riordan model
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
Andrei Raigorodskii Web graph models and applications
14. Bollob´s–Riordan model
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
#n (d)
In particular, whp for all d in this range we have m
n = Θ d−3
Andrei Raigorodskii Web graph models and applications
15. Bollob´s–Riordan model
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
#n (d)
In particular, whp for all d in this range we have m
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
Andrei Raigorodskii Web graph models and applications
16. Bollob´s–Riordan model
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
#n (d)
In particular, whp for all d in this range we have m
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
In particular if n = 20 · 106 we have log n/ log log n ≈ 5.96.
Andrei Raigorodskii Web graph models and applications
17. Bollob´s–Riordan model
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
#n (d)
In particular, whp for all d in this range we have m
n = Θ d−3 vs c · d−2.4 in
World-wide web.
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
In particular if n = 20 · 106 we have log n/ log log n ≈ 5.96.
Andrei Raigorodskii Web graph models and applications
18. Bollob´s–Riordan model
a
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
Andrei Raigorodskii Web graph models and applications
19. Bollob´s–Riordan model
a
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
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.
Theorem (Grechnikov)
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)
Andrei Raigorodskii Web graph models and applications
20. 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"
Andrei Raigorodskii Web graph models and applications
21. 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)
As for Gm , a process Ha,m is defined by identifying vertices in groups of m.
Andrei Raigorodskii Web graph models and applications
22. Buckley–Osthus model
#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
Andrei Raigorodskii Web graph models and applications
23. Buckley–Osthus model
#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
Andrei Raigorodskii Web graph models and applications
24. Buckley–Osthus model
#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
Andrei Raigorodskii Web graph models and applications
25. Buckley–Osthus model
Theorem (Grechnikov)
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)
Andrei Raigorodskii Web graph models and applications
26. Buckley–Osthus model
Theorem (Grechnikov)
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)
Andrei Raigorodskii Web graph models and applications
27. Buckley–Osthus model
Theorem (Grechnikov)
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)
Theorem (Grechnikov)
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 )
Andrei Raigorodskii Web graph models and applications
28. Buckley–Osthus model
Theorem (Grechnikov)
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)
Theorem (Grechnikov)
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 )
Theorem (Grechnikov)
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)
Andrei Raigorodskii Web graph models and applications
29. Buckley–Osthus model
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
Andrei Raigorodskii Web graph models and applications
30. 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.
Andrei Raigorodskii Web graph models and applications
31. 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 )
Andrei Raigorodskii Web graph models and applications
32. 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
Andrei Raigorodskii Web graph models and applications
33. 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.
Andrei Raigorodskii Web graph models and applications
34. Link rings
Andrei Raigorodskii Web graph models and applications
35. Link rings
Andrei Raigorodskii Web graph models and applications
36. 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
Andrei Raigorodskii Web graph models and applications
37. 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
If N ≤ N0 , this structure can be a natural formation.
Andrei Raigorodskii Web graph models and applications
38. 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
If N ≤ N0 , this structure can be a natural formation.
If N > N0 , this structure is probably a real link farm with some buyers.
Andrei Raigorodskii Web graph models and applications
39. Total number of edges between vertices with fixed degree in Buckley-Osthus
model
Theorem (Grechnikov)
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
Andrei Raigorodskii Web graph models and applications
40. Total number of edges between vertices with fixed degree in Buckley-Osthus
model
Theorem (Grechnikov)
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
Theorem (Grechnikov)
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
Andrei Raigorodskii Web graph models and applications
41. Total number of edges between vertices with fixed degree in Buckley-Osthus
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.
Andrei Raigorodskii Web graph models and applications
42. Total number of edges between vertices with fixed degree in Buckley-Osthus
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.
Theorem (Grechnikov)
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)
Andrei Raigorodskii Web graph models and applications
43. Total number of edges between vertices with fixed degree in
Bollob’as-Riordan model
Theorem (Grechnikov)
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
Andrei Raigorodskii Web graph models and applications
44. 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
Andrei Raigorodskii Web graph models and applications
45. 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
Da (x) - normalized sum of squared deviations of the observed values of x from the
values predicted.
min Da (x) ⇒ a = 0.4357
a
Andrei Raigorodskii Web graph models and applications
46. 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
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(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
Andrei Raigorodskii Web graph models and applications
47. 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
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(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
Andrei Raigorodskii Web graph models and applications
48. 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
Andrei Raigorodskii Web graph models and applications
49. 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
Theorem (Grechnikov–Ostroumova)
For any k ≥ 1
4n k2 log2 k
E (Xn (k)) = 1+O 1+O .
k2 n k
Andrei Raigorodskii Web graph models and applications
50. 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
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
Andrei Raigorodskii Web graph models and applications
51. 2nd degrees
Let Yn (k) denote the number of vertices with second degree ≥ k in Gn .
a,1
Andrei Raigorodskii Web graph models and applications
52. 2nd degrees
Let Yn (k) denote the number of vertices with second degree ≥ k in Gn .
a,1
Theorem (Ostroumova)
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
Andrei Raigorodskii Web graph models and applications
53. 2nd degrees
Let Yn (k) denote the number of vertices with second degree ≥ k in Gn .
a,1
Theorem (Ostroumova)
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
Corollary
1
n
EYn (k) = Θ ka for k = O n 1+a .
Andrei Raigorodskii Web graph models and applications
54. 2nd degrees
Let Yn (k) denote the number of vertices with second degree ≥ k in Gn .
a,1
Theorem (Ostroumova)
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
Corollary
1
n
EYn (k) = Θ ka for k = O n 1+a .
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).
¯
Andrei Raigorodskii Web graph models and applications
55. 2nd degrees
Let Yn (k) denote the number of vertices with second degree ≥ k in Gn .
a,1
Theorem (Ostroumova)
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
Corollary
1
n
EYn (k) = Θ ka for k = O n 1+a .
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).
¯
Corollary
1
Let δ > 0 and k = O n 4+a+δ . Then for some > 0 we have
1−
P |Xn (k) − E(Xn (k))| > E(Xn (k)) = o(1).
¯
Andrei Raigorodskii Web graph models and applications
56. (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
Andrei Raigorodskii Web graph models and applications
57. (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
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
Andrei Raigorodskii Web graph models and applications
58. (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
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
Andrei Raigorodskii Web graph models and applications
59. (n)
Number of copies of H in Gm
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
Andrei Raigorodskii Web graph models and applications