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Existence of a persistent hub in the convex preferential
attachment model
Pavel Galashin
St Petersburg State University
pgalashin@gmail.com
03.07.2014
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page.2
Barabási-Albert random graph
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Barabási-Albert random graph
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Persistent hub
Question
Is there always a vertex which has the maximal degree for all but finitely
many steps?
Definition
Such vertex is called a persistent hub.
Theorem
A persistent hub appears with probability 1.
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page.5
Two-dimensional random walk
B
A+B
A
A+B
B
(A, B)
A
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page.6
Probabilities of paths
Lemma
Let (A1 , B1 ) and (A2 , B2 ) be two points. Then for any path S connecting
these points its probability equals
P(S) =
A1 · (A1 + 1) · ... · (A2 − 1) · B1 · (B1 + 1) · ... · (B2 − 1)
.
(A1 + B1 ) · (A1 + B1 + 1) · ... · (A2 + B2 − 1)
Remark
This probability does not depend on S.
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The probability of crossing the diagonal
(m, m)
A
P[(A, 1) → diagonal ] ≤ P(A)/2A .
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page.8
Finite number of leaders
Lemma
P[(A, 1) → diagonal ] ≤ P(A)/2A .
Lemma
√
The degree of the first vertex after n-th step grows like C n for some
random non-zero constant C .
Corollary
For almost all vertices their degree will always be lower than the degree of
the first vertex, because the series
√
∞
X
P(C n)
√
2C n
n=1
is convergent for any C .
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page.9
Pólya urn
Proposition
If our random walk starts at the point (A, 1) then the quantity
Ak /(Ak + Bk ) tends to some random variable H(A) as k tends to infinity.
Moreover, H(A) has a beta probability distribution:
H(A) ∼ Beta(1, A) .
Corollary
The random walk crosses the diagonal only finitely many times.
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page.10
Weighted preferential attachment
Generalized model
Probability of assigning new edge to a vertex of degree k is proportional to
W(k) for some weight function W : N → R>0 .
Linear model
W(k) = k + β, β > −1.
Convex model
W(k) is convex and unbounded.
Theorem
The persistent hub appears with probability 1 also in the linear and convex
models.
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page.11
Difficulties
Model
Total weight of the
vertices is random
All pathes
equally likely
Basic
No
Yes
Yes
Linear
No
Yes
Yes
Convex Yes
No
No
Pavel Galashin (SPbSU)
Persistent hub
are
Beta
limiting
distribution
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page.12
Comparison with the linear model
10
9
8
7
W(k)
6
5
4
3
2
1
−1
0 1
2
3
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8
9
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Thank you!
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