Existence of a persistent hub in the convex preferential attachment - - PowerPoint PPT Presentation

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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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Barab´ asi-Albert random graph

V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12 V13

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Barab´ asi-Albert random graph

6/22 4/22 2/22 1/22 1/22 1/22 2/22 1/22 1/22 1/22 1/22 1/22 V13

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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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Two-dimensional random walk

A B (A, B)

A A+B B A+B Pavel Galashin (SPbSU) Persistent hub 03.07.2014 5 / 13

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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

A (m, m)

P[(A, 1) → diagonal ] ≤ P(A)/2A.

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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

∞

• n=1

P(C√n) 2C√n is convergent for any C.

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P´

• lya 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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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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Difficulties

Model Total weight of the vertices is random All pathes are equally likely Beta limiting distribution Basic No Yes Yes Linear No Yes Yes Convex Yes No No

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Comparison with the linear model

−1 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10

W(k)

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Thank you!

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