A mixture model for random graphs

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ISRN INRIA/RR--5840--FR+ENG

a p p o r t

d e r e c h e r c h e

THÈME 4

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

A mixture model for random graphs

Jean-Jacques Daudin, Franck Picard, Stéphane Robin

N° 5840

22nd February 2006

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(18)

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(19)

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(20)

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(21)

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(22)

Unité de recherche INRIA Futurs

Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)

Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38330 Montbonnot-St-Martin (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)

A mixture model for random graphs

ISRN INRIA/RR--5840--FR+ENG

a p p o r t

d e r e c h e r c h e

THÈME 4

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

A mixture model for random graphs

Jean-Jacques Daudin, Franck Picard, Stéphane Robin

N° 5840

22nd February 2006

∗

∗

n

X ij

i

j

X ij = X ji =

{ i ↔ j } ,

{ A }

A

X ii = 0

K i

i

K i = X

j6=i

X ij .

p

{ X ij }

, X ij ∼ B (p).

n

p

λ = (n − 1)p

K i ∼ B (n − 1, p) ≈ P (λ).

Pr { K i = k } = c(ρ)k −(ρ+1) ,

k

ρ

c(ρ) = P

k≥1 k −(ρ+1) = 1/ζ(ρ + 1)

ζ (ρ + 1)

k = 0

Q

{ Z iq }

α q

i

q

α q = Pr { Z iq = 1 } = Pr { i ∈ q } ,

X

q

α q = 1.

{ Z iq = 1 }

{ i ∈ q }

i

q

K i |{ i ∈ q } ∼ P (λ q ).

Pr { K i = k } = X Q q=1

α q

e −λ q λ k q

k! .

n

λ q

n

Q

prior

{ α 1 , . . . α Q }

{ Z iq }

π q`

q

`

α q = Pr { Z iq = 1 } = Pr { i ∈ q } ,

X

q

α q = 1.

{ Z iq = 1 }

{ i ∈ q }

i

q

π q`

q

`

π q` = π `q .

Pr { K i = k } = c(ρ)k ^−(ρ+1) ,

k≥1 k ^−(ρ+1) = 1/ζ(ρ + 1)

e ^−λ ^q λ ^k _q

a ² ab ab b ²

(a ² + b ² ) ² (a + b) ²

1 + 3ε ²

(1 + ε) ²