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arXiv:0902

.0878v1

[q-fin.GN

]5Feb2009

http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://arxiv.org/abs/0902.0878v1http://sbattiston@ethz.ch/http://jglattfelder@ethz.ch/http://arxiv.org/abs/0902.0878v1

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x y

s

h

x

y

y

8/14/2019 The Backbone of Complex Networks of Corporations: Who is Controlling Whom?

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i1 i2 i3

jsj

Wi1j

Wi2j

Wi3j

sj

j

sj = kinj k

inj j

j

i

ki

kin kout

kwi :=j

Wij.

kinw koutw

kin/outw

j

sj :=

kinj

i=1 Wij

2kin

j

i=1 W2ij

.

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i1

j1

j2

j3

i2

i3

Hi1j1

Hi1j2

Hi1j3

Hi2j3

Hi3j3

hi1

hi

i

Hij

sj j

i j

i j

Hij :=W2ij

kinjl=1 W2lj

.

(0, 1] Hij 1 i

j

Hij

i j Hij

hi :=

koutij=1

Hij.

i hi

i

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s h

s h

sj

kouti

hi

hi

hi

hi = 1

1 5 10 20

103

102

101

100

JP

1 5 10 20

102

101

100

DE

1 5 10 20

102

101

100

FR

1 5 10 20 40

103

102

101

100

GB

1 5 10 20

102

101

100

SG

1 5 10 20 40

102

101

100

IT

1 5 10 20

103

102

101

100

TW

1 5 10 20 40 70

103

102

101

100

US

1 5 10 2010

3

102

101

100

IN

ln sln sln s

sj

8/14/2019 The Backbone of Complex Networks of Corporations: Who is Controlling Whom?

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100

101

102

103

103

10

2

101

100

JP

100

101

102

103

10

2

101

100

DE

100

101

102

103

104

103

102

101

100

US

101

100

101

102

103

102

101

100

101

100

103

102

101

100

101

100

101

102

104

103

102

101

100

101

100

101

102

103

102

101

100

101

100

103

102

101

100

101

100

101

102

104

103

102

101

100

ln hln hln h

ln hln hln h

ln koutln koutln kout

kouti

hi hi

vj vj

Wij

i j j

i

pi :=

kouti

j=1

Wijvj.

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Wij

Hij

ci :=

koutij=1

Hijvj.

ci

Hij

Hij

i j

n

Aij i, j = 1, 2,...,n, Wij Hij i

j A = [Aij ] n

ni=1

Aij 1; j = 1,...,n.

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i n

Aij := Aij +n

AinAnj,

A = A + AA,

A = (I A)1A.

(I A)

(A) < 1

S

j

iS Aij < 1

k (k = 1, . . . , n)

k

Hij Hij

vj

ci :=

koutij=1

Hijvj.

i Wij j i

Wij

8/14/2019 The Backbone of Complex Networks of Corporations: Who is Controlling Whom?

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Wij

0.5

F : (0, 1]N (0, 1]N N {Wij }

F1({Wij}), . . . , F N({Wij})

j Wij

Hij Hij

j Hij = 1 j

Hij

Hij Hij Hij

sj hi

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ci

x x%

y

x

y

ci

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Hij

Wij > 0.5

ci

ci

Uin(n)

n Ucu(n)

Uin(n)Ucu(n) =

n Uin(n) Ucu(n)

vcu(n) := jUin(n)

vj + jUcu(n)

vj .

j

Wij

ntot vtot

(n) :=n

ntot, (n) :=

vcu(n)

vtot,

, (0, 1]

x

(103, 0.2) 0.1%

20%

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ci

104

103

102

101

100

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

US

GB

JP

DE

IT

CN

KR

IN

ci

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BB c1, . . . , cn

c sort

descending(c1, . . . , cn)

c get largest(c)

I I index(c)

P F stocks controlled by(I)

P F V value of portfolio(P F)

c c \ {c}

P F V total market value

prune network(I , P F )

n

= 0.8

ci

0.5

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j

sj

j

sj

vj 0 vj > 0

i j Wij j i

vj

vj vj

t = 1 i i j t Wij

j

j

i(t + 1) =j

Wijvj +j

Wiji(t).

iWij = 1 i

Wij = 0

= W(v + ).

= (1 W)1W v,

(W) < 1

S

j

iS Wij < 1 S

S

t = 1

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c

c = Hv = (1 H)1Hv,

i

ci =

j Hijvj i

U0 E0

U U0

U = U0 E E0

U

Hij Wij

Hij Wij

Hij

(, )

x

i

U y

k

(k) = k/|U| (k) = (k

i=1 i)/tot

ci

B = (UB , EB ) UB U EB E

(U, E)

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= 60%

nst nsh

sj

s =

nstj=1 sj

nst,

s

sj s

h =

nshi=1 hinsh

=nstnsh

,

h

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nst nsh

= 0.8

n100

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s h

vtot

:=

n100.

s

h

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10% 20%

76.6% p

3.2 106 s 10%

s

60.0%

9.3 104

20%

s

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0 0.5 1 1.5 2 2.52

1

0

1

2

3

4

AE

AR

AT

AU

BE BM

CA

CH

CLCN

DE

DK

ESFI

FR

GB

GR

HK

ID

IE

IL

IN

IS

IT

JO

JP

KR

KW

KYLU

MX

MY

NLNO

NZ

OM

PH

PT

SA

SE

SG

TH

TN

TR

TW

US

VG

ZA

ln(s)

ln(h)

s

h

h

80%

s

h

s

0 0.5 1 1.5 2 2.51

0

1

2

3

4

5

AE

AR

AT

AU

BE BM

CA

CH

CL

CN

DE

DK

ESFI

FR

GB

GRHKID

IE

ILIN

IS

IT

JO

JPKR

KW

KYLU

MX

MY

NL

NO

NZ

OM

PH

PTSA

SE

SGTH

TNTR

TW

US

VG

ZA

ln(s)

ln(

)

s

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s

s

h

stot htot

s < stot.

sj

s

stot s 0.5

h > htot.

htot

[0.06, 1.09] h [0.3, 34.26]

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s h

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i j

j

Wij vj

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