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Models based on preferential attachment
Analysis of PA-models
Problems and conclusion

Generalized preferential attachment
Liudmila Ostroumova
Yandex
Lomonosov Moscow State University
Joint work with A. Ryabchenko and E. Samosvat
October, 2013

Liudmila Ostroumova



Plan

1

2

3

4

Models based on the preferential attachment
Experimental illustrations
Theoretical analysis of the general case

Liudmila Ostroumova



Properties of interest
Examples

Degree distribution

Real-world networks often have the power law degree
distribution:
#{v : deg(v) = d}
c
≈ γ,
n
d
where 2 < γ < 3.

Liudmila Ostroumova



Examples

Clustering coeﬃcient
Global clustering coeﬃcient of a graph G:
C1 (n) =

3#(triangles in G)
.
#(pairs of adjacent edges in G)

Liudmila Ostroumova



Examples

Clustering coefficient
Global clustering coefficient of a graph G:
C1 (n) =

3#(triangles in G)
.
#(pairs of adjacent edges in G)

Average local clustering coefficient
T i is the number of edges between the neighbors of a vertex i
i
P2 is the number of pairs of neighbors
Ti
i
P2
1
=n

C(i) =
C2 (n)

is the local clustering coefficient for a vertex i
n
i=1 C(i)

– average local clustering coefficient

Liudmila Ostroumova



Examples

Preferential attachment
Idea of preferential attachment [Barab´si, Albert]:
a
Start with a small graph
At every step we add new vertex with m edges
The probability that a new vertex will be connected to a vertex
i is proportional to the degree of i

Liudmila Ostroumova



Examples

Preferential attachment
Idea of preferential attachment [Barab´si, Albert]:
a
Start with a small graph
At every step we add new vertex with m edges
The probability that a new vertex will be connected to a vertex
i is proportional to the degree of i
Theorem[Bollob´s, Riordan]
a
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 (m) → ∞
as n → ∞. Then there is a random graph process T (n) satisfying
the conditions of Barab´si and Albert such that, with probability 1,
a
T (n) has exactly f (n) triangles for all suﬃciently large n.

Liudmila Ostroumova



Examples

P A-class of models
Start from an arbitrary graph Gn0 with n0 vertices and mn0
m
edges

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Examples

P A-class of models
m
edges
We make Gn+1 from Gn by adding a new vertex n + 1 with
m
m
m edges

Liudmila Ostroumova



Examples

P A-class of models
m
edges
m
m
m edges
PA-condition: the probability that the degree of a vertex i
increases by one equals
A

1
deg(i)
+B +O
n
n

Liudmila Ostroumova

(deg(i))2
n2



Examples

P A-class of models
m
edges
m
m
m edges
A

1
deg(i)
+B +O
n
n

(deg(i))2
n2

The probability of adding a multiple edge is O

Liudmila Ostroumova

(deg(i))2
n2



Examples

P A-class of models
m
edges
m
m
m edges
A

1
deg(i)
+B +O
n
n

(deg(i))2
n2

The probability of adding a multiple edge is O

(deg(i))2
n2

2mA + B = m, 0 ≤ A ≤ 1
Liudmila Ostroumova



Examples

T -subclass

Triangles property:
The probability that the degree of two vertices i and j
eij

D
+O
mn

dn dn
i j
n2

Here eij is the number of edges between vertices i and j in
Gn and D is a positive constant.
m

Liudmila Ostroumova



Examples

Bollob´s–Riordan, Buckley–Osthus, M´ri, etc.
a
o
Fix some positive number a – "initial attractiveness".
(Bollob´s–Riordan model: a = 1).
a
Start with a graph with one vertex and m loops.
At n-th step add one vertex with m edges.
We add m edges one by one. The probability to add an edge
n → i at each step is proportional to deg(i) + a.

Liudmila Ostroumova



Examples

Bollob´s–Riordan, Buckley–Osthus, M´ri, etc.
a
o
Fix some positive number a – "initial attractiveness".
(Bollob´s–Riordan model: a = 1).
a
Start with a graph with one vertex and m loops.
At n-th step add one vertex with m edges.
We add m edges one by one. The probability to add an edge
n → i at each step is proportional to deg(i) + a.
Outdegree: m
Triangles property: D = 0
PA-condition: A =

1
1+a

Degree distribution: Power law with γ = 2 + a
Global clustering:

(log n)2
n

Liudmila Ostroumova

(a = 1),

log n
n

(a > 1)



Examples

Holme–Kim model
Idea: to mix PA steps with the steps of triangle formation.

Liudmila Ostroumova



Examples

Holme–Kim model
Add a new vertex v with m edges
Perform one PA step
Then perform a triangle formation step with the probability Pt
or a PA step with the probability 1 − Pt
Triangle formation: if an edge between v and u was added in the
previous PA step, then add one more edge from v to a randomly
chosen neighbor of u.

Liudmila Ostroumova



Examples

Holme–Kim model
Add a new vertex v with m edges
Perform one PA step
Then perform a triangle formation step with the probability Pt
or a PA step with the probability 1 − Pt
Triangle formation: if an edge between v and u was added in the
previous PA step, then add one more edge from v to a randomly
chosen neighbor of u.
Outdegree: m
Triangles property: D = (m − 1)Pt
1
PA-condition: A = 2
Degree distribution: Power law with γ = 3
Average local clustering: constant
Global clustering: tends to zero
Liudmila Ostroumova



Examples

Random Apollonian networks

Liudmila Ostroumova



Examples

Random Apollonian networks

Outdegree: m = 3
Triangles property: D = 3
1
PA-condition: A = 2
Degree distribution: Power law with γ = 3
Global clustering: tends to zero
Liudmila Ostroumova



Examples

Polynomial model
Put m = 2p

Liudmila Ostroumova



Examples

Polynomial model
Put m = 2p
Fix α, β, δ ≥ 0 and α + β + δ = 1

Liudmila Ostroumova



Examples

Polynomial model
Put m = 2p
Fix α, β, δ ≥ 0 and α + β + δ = 1
Add a new vertex i with m edges. We add m edges in p steps

Liudmila Ostroumova



Examples

Polynomial model
Put m = 2p
Fix α, β, δ ≥ 0 and α + β + δ = 1
α – probability of an indegree preferential step
β – probability of an edge preferential step
δ – probability of a random step

Liudmila Ostroumova



Examples

Polynomial model
Put m = 2p
Fix α, β, δ ≥ 0 and α + β + δ = 1
Edge preferential: cite two endpoints of a random edge

Liudmila Ostroumova



Examples

Polynomial model
Put m = 2p
Fix α, β, δ ≥ 0 and α + β + δ = 1
Edge preferential: cite two endpoints of a random edge
Outdegree: 2p
Triangles property: D = βp
PA-condition: A = α + β .
2
2
Degree distribution: Power law with γ = 1 + 2α+β
Global clustering: constant for A > 1/2 (γ > 3), tends to
zero for A ≤ 1/2 (2 < γ ≤ 3)
Liudmila Ostroumova



Theoretical analysis: degree distribution
Theoretical analysis: clustering coeﬃcient

Global clustering
α: indegree preferential step
β: edge preferential step
γ = 3.5
b)
α = 0.4, β = 0
α = 0, β = 0.8
0,4

0,2

0

101

102

103

Liudmila Ostroumova

104

105

106

107




Average local clustering
α: indegree preferential step
β: edge preferential step
γ = 3.5
1

c)
α = 0.4, β = 0
α = 0, β = 0.8

0,8

0,6

0,4

0,2

0

101

102

103

Liudmila Ostroumova

104

105

106

107




Global and average local clustering depending on n
α = 0.5, β = 0.2 ⇒ γ = 8/3
1 b)
Global clustering

0,8

0,6

0,4

0,2

0
101

102

103

Liudmila Ostroumova

104
n

105

106

107




Global and average local clustering depending on A
β = 0.5 – probability of edge preferential step
γ = 1 + 1/A
a)

0,4 a)
Global clustering

0,35
0,3
0,25
0,2
0,15
0,1
0,05
0
0,3

0,4

0,5

0,6

A
Liudmila Ostroumova


0,7



Degree distribution
Let Nn (d) be the number of vertices with degree d in Gn .
m

Liudmila Ostroumova




Degree distribution
Let Nn (d) be the number of vertices with degree d in Gn .
m
Expectation
For every d ≥ m we have
1

ENn (d) = c(m, d) n + O d2+ A

,

where
c(m, d) =

Γ d+
AΓ d +

B
B+1
A Γ m+ A
B+A+1
Γ m+ B
A
A

∼

Γ m+

B+1
A

AΓ m +

and Γ(x) is the gamma function.

Liudmila Ostroumova


1

d−1− A
B
A



Idea of the proof
p1 (d) := P dn+1 = d + 1 | dn = d = A
n
v
v
pj (d) := P dn+1 = d + j | dn = d = O
n
v
v

d
1
+B +O
n
n
d2
n2

m

P(dn+1 = m + k) = O
n+1

pn :=
k=1

Liudmila Ostroumova

d2
n2

, 2≤j≤m
1
n




Idea of the proof
p1 (d) := P dn+1 = d + 1 | dn = d = A
n
v
v
pj (d) := P dn+1 = d + j | dn = d = O
n
v
v

d
1
+B +O
n
n
d2
n2

m

P(dn+1 = m + k) = O
n+1

pn :=
k=1

d2
n2

, 2≤j≤m
1
n

m

pj (d)
n

pn (d) :=
j=1

Liudmila Ostroumova




Idea of the proof
p1 (d) := P dn+1 = d + 1 | dn = d = A
n
v
v
pj (d) := P dn+1 = d + j | dn = d = O
n
v
v

d
1
+B +O
n
n
d2
n2

m

P(dn+1 = m + k) = O
n+1

pn :=
k=1

d2
n2

, 2≤j≤m
1
n

m

pj (d)
n

pn (d) :=
j=1

E(Ni+1 (d) | Ni (d), Ni (d − 1), . . . , Ni (d − m)) = Ni (d) (1 − pi (d)) +
m

Ni (d − j)pj (d − j) + O(pi ) .
i

+ Ni (d − 1)p1 (d − 1) +
i
j=2
Liudmila Ostroumova




Degree distribution

Concentration
For every d = d(n) we have
√
P |Nn (d) − ENn (d)| ≥ d n log n = O n− log n .
Therefore, for any δ > 0 there exists a function ϕ(n) = o(1) such
that
A−δ

lim P ∃ d ≤ n 4A+2 : |Nn (d) − ENn (d)| ≥ ϕ(n) ENn (d) = 0 .

n→∞

Liudmila Ostroumova




Idea of the proof
Azuma, Hoeﬀding
Let (Xi )n be a martingale such that |Xi − Xi−1 | ≤ ci for any
i=0
1 ≤ i ≤ n. Then
P (|Xn − X0 | ≥ x) ≤ 2e

−

2

x2
n
c2
i=1 i

for any x > 0.

Liudmila Ostroumova




Idea of the proof
Azuma, Hoeﬀding
Let (Xi )n be a martingale such that |Xi − Xi−1 | ≤ ci for any
i=0
1 ≤ i ≤ n. Then
P (|Xn − X0 | ≥ x) ≤ 2e

−

2

x2
n
c2
i=1 i

for any x > 0.
Xi (d) = E(Nn (d) | Gi ), i = 0, . . . , n.
m
Note that X0 (d) = ENn (d) and Xn (d) = Nn (d).
Xn (d) is a martingale.
For any i = 0, . . . , n − 1: |Xi+1 (d) − Xi (d)| ≤ M d, where
M > 0 is some constant.
Liudmila Ostroumova




Idea of the proof
Fix 0 ≤ i ≤ n − 1 and some graph Gi .
m
E Nn (d) | Gi+1 − E Nn (d) | Gi
m
m
≤

max

˜m
Gi+1 ⊃Gi
m

˜m
E Nn (d) | Gi+1

Liudmila Ostroumova

≤

− min

˜m
Gi+1 ⊃Gi
m

˜m
E Nn (d) | Gi+1


.



Idea of the proof
m
E Nn (d) | Gi+1 − E Nn (d) | Gi
m
m
≤

max

˜m
Gi+1 ⊃Gi
m

˜m
E Nn (d) | Gi+1

≤

− min

˜m
Gi+1 ⊃Gi
m

˜m
E Nn (d) | Gi+1

.

ˆ
˜
m
m
¯ i+1 = arg min E(Nn (d) | Gi+1 ).
˜
Gm
m
For i + 1 ≤ t ≤ n put
i
ˆ
¯
δt (d) = E(Nt (d) | Gi+1 ) − E(Nt (d) | Gi+1 ).
m
m
i
i
δt+1 (d) = δt (d) (1 − pt (d)) +
i
+ δt (d − 1)p1 (d − 1) + O
t
Liudmila Ostroumova

ENt (d)d2
t2

+O


1
t

.



Local clustering

Whp
1
C2 (n) ≥
n

i:deg(i)=m

Liudmila Ostroumova

C(i) ≥

2cD
.
m(m + 1)




Global clustering

Let P2 (n) be the number of all path of length 2 in Gn .
m
P2 (n)
(1) If 2A < 1, then whp P2 (n) ∼ 2m(A + B) +

m(m−1)
2

(2) If 2A = 1, then whp P2 (n) ∝ n log(n) .
(3) If 2A > 1, then whp P2 (n) ∝ n2A .

Liudmila Ostroumova


n
1−2A

.



Global clustering

Let P2 (n) be the number of all path of length 2 in Gn .
m
P2 (n)
(1) If 2A < 1, then whp P2 (n) ∼ 2m(A + B) +

m(m−1)
2

(2) If 2A = 1, then whp P2 (n) ∝ n log(n) .
(3) If 2A > 1, then whp P2 (n) ∝ n2A .
Triangles
Whp the number of triangles T (n) ∼ D n .

Liudmila Ostroumova


n
1−2A

.



Global clustering

Global clustering

(1) If 2A < 1 then whp C1 (n) ∼

3(1−2A)D

(2m(A+B)+ m(m−1) )
2
(2) If 2A = 1 then whp C1 (n) ∝ (log n)−1 .
(2) If 2A > 1 then whp C1 (n) ∝ n1−2A .

Liudmila Ostroumova

.



P2 (n) and T (n) in real networks
Retweet graph
108

Number of P2
200·(number of triangles)
8x107

6x107

4x107

2x107

0
0

105

2x105

3x105

4x105

5x105

Number of vertices

Liudmila Ostroumova



P2 (n) and T (n) in real networks
Retweet graph
1010

Number of P2
200·(number of triangles)
108

106

104

102
103

104

105

106

Number of vertices

Slope: 2.3
Liudmila Ostroumova



Conclusion
Generalized preferential attachment:
Power law degree distribution with any exponent γ > 2
Constant average local clustering coeﬃcient
Constant global clustering coeﬃcient only for γ > 3

Liudmila Ostroumova



Conclusion
Generalized preferential attachment:
Power law degree distribution with any exponent γ > 2
Constant average local clustering coeﬃcient
Constant global clustering coeﬃcient only for γ > 3
Ways to overcome this obstacle:
The number of added edges is a random variable (C. Cooper,
2006)
A new vertex added at time t generates tc edges (C. Cooper,
P. Pralat, 2011)
Adding edges between already existing nodes (e.g., the
Cooper–Frieze model)

Liudmila Ostroumova



Thank You!
Questions?

Liudmila Ostroumova


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