Emergence

of

Scaling

in

Random

Networks

Albert-L

szl

Barab

si

et

al.

Science

-LRB-

-RRB-

DOI

science

The

following

resources

related

to

this

article

are

available

online

at

-LRB-

this

information

is

current

as

of

May

-RRB-

Updated

information

and

services

including

high-resolution

figures

can

be

found

in

the

online

version

of

this

article

at

A

list

of

selected

additional

articles

on

the

Science

Web

sites

related

to

this

article

can

be

found

at

This

article

cites

articles

of

which

can

be

accessed

for

free

This

article

has

been

cited

by

article

-LRB-

s

-RRB-

on

the

ISI

Web

of

Science

This

article

has

been

cited

by

articles

hosted

by

HighWire

Press

see

This

article

appears

in

the

following

subject

collections

Physics

Information

about

obtaining

reprints

of

this

article

or

about

obtaining

permission

to

reproduce

this

article

in

whole

or

in

part

can

be

found

at

Science

-LRB-

print

ISSN

online

ISSN

-RRB-

is

published

weekly

except

the

last

week

in

December

by

the

American

Association

for

the

Advancement

of

Science

New

York

Avenue

NW

Washington

DC

Copyright

by

the

American

Association

for

the

Advancement

of

Science

all

rights

reserved

The

title

Science

is

a

registered

trademark

of

AAAS

Downloaded

from

on

May

REPORTS

that

proposed

to

explain

in

terms

of

sp-d

bonding

the

positive

polarization

at

the

Co-ALO

interface

-LRB-

-RRB-

However

there

is

no

general

theory

predicting

the

trend

of

the

experimental

results

for

Co

that

is

a

negative

polarization

with

oxides

of

d

elements

-LRB-

STO

CLO

-RRB-

and

a

positive

one

when

there

are

only

s

and

p

states

-LRB-

ALO

-RRB-

It

is

likely

that

the

spin

polarization

should

also

depend

on

the

position

of

the

Fermi

level

with

respect

to

the

electronic

levels

of

each

character

above

and

below

the

gap

of

the

insulator

In

addition

as

an

evanescent

wave

in

an

insulator

is

a

Bloch

wave

with

an

imaginary

wave

vector

one

can

expect

different

decay

lengths

for

Bloch

waves

of

different

character

This

means

that

the

final

polarization

could

also

depend

on

the

thickness

of

the

barrier

as

illustrated

by

the

calculations

of

MacLaren

et

al.

for

Fe/ZnSe/Fe

junctions

-LRB-

-RRB-

The

influence

of

the

barrier

on

the

spin

polarization

opens

new

ways

to

shape

and

optimize

the

TMR

Interesting

bias

dependencies

can

be

obtained

with

barriers

selecting

the

d

electrons

and

probing

the

fine

structure

of

the

d-DOS

as

in

Fig.

The

DOS

of

a

d-band

can

also

be

easily

tailored

by

alloying

-LRB-

for

example

by

introduction

of

virtual

bound

states

-RRB-

to

produce

specific

bias

dependencies

Although

here

we

concentrated

on

the

problem

of

the

spin

polarization

of

the

Co

electrode

and

regarded

the

strongly

spin-polarized

LSMO

only

as

a

useful

spin

analyzer

the

large

TMR

ratios

obtained

by

combining

Co

and

LSMO

electrodes

-LRB-

with

a

STO

barrier

-RRB-

are

also

an

interesting

result

The

drawback

arising

from

the

low

Curie

temperature

of

LSMO

-LRB-

K

-RRB-

is

the

reduction

of

the

TMR

at

room

temperature

down

to

about

at

K

in

Co/STO

LSMO

-LRB-

-RRB-

However

other

types

of

oxides

of

the

double-perovskite

family

-LRB-

for

example

-RRB-

combine

electronic

properties

similar

to

those

of

manganites

with

a

definitely

higher

Curie

temperature

-LRB-

-RRB-

Their

use

in

magnetic

tunnel

junctions

is

promising

for

a

new

generation

of

tunnel

junctions

with

very

high

magnetoresistance

for

room-temperature

applications

J.

S.

Moodera

L.

R.

Kinder

T.

M.

Wong

R.

Meservey

Phys

Rev.

Lett

-LRB-

-RRB-

J.

De

Boeck

Science

-LRB-

-RRB-

M.

Sharma

S.

X.

Wang

J.

H.

Nickel

Phys

Rev.

Lett

-LRB-

-RRB-

J.

M.

De

Teresa

et

al.

ibid.

p.

P.

M.

Tedrow

and

R.

Meservey

Phys

Rev.

B

-LRB-

-RRB-

R.

Meservey

and

P.

M.

Tedrow

Phys

Rep.

-LRB-

-RRB-

M.

B.

Stearns

J.

Magn

Magn

Mater

-LRB-

-RRB-

J.

A.

Hertz

and

K.

Aoi

Phys

Rev.

B

-LRB-

-RRB-

D.

Nguyen-Mahn

et

al.

Mat

Res

Soc

Symp

Proc

-LRB-

-RRB-

J.

H.

Park

et

al.

Nature

-LRB-

-RRB-

J.

Nassar

M.

Hehn

A.

Vaures

F.

Petroff

A.

Fert

Appl

Phys

Lett

-LRB-

-RRB-

J.

S.

Moodera

et

al.

ibid

-LRB-

-RRB-

K.

Wang

thesis

New

York

University

-LRB-

-RRB-

S.

Zhang

P.

M.

Levy

A.

C.

Marley

S.

S.

P.

Parkin

Phys

Rev.

Lett

-LRB-

-RRB-

J.

M.

MacLaren

X.

G.

Zhang

W.

H.

Butler

X.

Wang

Phys

Rev.

B

-LRB-

-RRB-

K.

I.

Kobayashi

T.

Kimura

H.

Sawada

K.

Terakura

Y.

Tokura

Nature

-LRB-

-RRB-

We

thank

the

group

of

A.

Revcoleschi

for

providing

LSMO

targets

R.

Lyonnet

and

A.

Vaures

for

their

experimental

help

and

J.

Nassar

for

fruitful

discussions

Supported

by

the

European

Union

and

the

New

Energy

and

Industrial

Technology

Development

Organization

of

Japan

July

accepted

September

References

and

Notes

Fig.

Bias

dependence

of

the

TMR

ratio

in

-LRB-

A

-RRB-

Co/STO/LSMO

and

-LRB-

B

-RRB-

Co/ALO/STO

LSMO

tunnel

junctions

level

of

LSMO

is

situated

above

the

Fermi

level

of

Co

and

a

maximum

of

inverse

TMR

is

expected

when

the

Fermi

level

of

LSMO

is

approximately

at

the

maximum

of

the

spin

DOS

of

Co.

This

is

consistent

with

the

maximum

of

inverse

TMR

observed

at

V

for

Co/STO/LSMO

junctions

-LRB-

Fig.

-RRB-

For

a

positive

bias

the

TMR

is

expected

to

change

sign

and

become

normal

above

V

when

the

Fermi

level

of

LSMO

goes

down

into

the

energy

range

of

the

majority

spin

d-band

of

Co.

This

is

also

observed

in

Fig.

For

ALO

and

ALO/STO

barriers

a

predominant

tunneling

of

s-character

electrons

-LRB-

see

arrow

in

Fig.

-RRB-

is

the

usual

explanation

of

the

positive

polarization

-LRB-

-RRB-

The

rapid

drop

with

bias

-LRB-

Fig.

-RRB-

is

similar

to

what

has

been

observed

in

most

junctions

with

ALO

barriers

and

completely

different

from

what

is

obtained

when

the

tunneling

is

predominantly

by

d-character

electrons

-LRB-

Fig.

-RRB-

The

origin

of

this

rapid

decrease

of

the

TMR

at

relatively

small

bias

has

never

been

clearly

explained

This

is

roughly

consistent

with

the

energy

dependence

of

the

DOS

induced

by

sp-d

bonding

effects

on

the

first

atomic

layer

of

ALO

in

the

calculation

of

Nguyen-Mahn

et

al.

-LRB-

-RRB-

for

the

Co-ALO

interface

But

Zhang

et

al.

-LRB-

-RRB-

have

also

shown

that

a

large

part

of

the

TMR

drop

can

be

attributed

to

the

excitation

of

spin

waves

The

experiments

reported

here

and

in

several

recent

publications

-LRB-

-RRB-

demonstrate

the

important

role

of

the

electronic

structure

of

the

metal-oxide

interface

in

determining

the

spin

polarization

of

the

tunneling

electrons

The

negative

polarization

for

the

Co-STO

interface

has

been

ascribed

to

d-d

bonding

effects

between

Al

and

Ti

-LRB-

-RRB-

This

interpretation

is

similar

to

Emergence

of

Scaling

in

Random

Networks

Albert-Laszlo

Barabasi

and

Reka

Albert

Systems

as

diverse

as

genetic

networks

or

the

World

Wide

Web

are

best

described

as

networks

with

complex

topology

A

common

property

of

many

large

networks

is

that

the

vertex

connectivities

follow

a

scale-free

power-law

distribution

This

feature

was

found

to

be

a

consequence

of

two

generic

mechanisms

-LRB-

i

-RRB-

networks

expand

continuously

by

the

addition

of

new

vertices

and

-LRB-

ii

-RRB-

new

vertices

attach

preferentially

to

sites

that

are

already

well

connected

A

model

based

on

these

two

ingredients

reproduces

the

observed

stationary

scale-free

distributions

which

indicates

that

the

development

of

large

networks

is

governed

by

robust

self-organizing

phenomena

that

go

beyond

the

particulars

of

the

individual

systems

The

inability

of

contemporary

science

to

describe

systems

composed

of

nonidentical

elements

that

have

diverse

and

nonlocal

interDepartment

of

Physics

University

of

Notre

Dame

Notre

Dame

IN

USA

To

whom

correspondence

should

be

addressed

Email

alb@nd.edu

actions

currently

limits

advances

in

many

disciplines

ranging

from

molecular

biology

to

computer

science

-LRB-

-RRB-

The

difficulty

of

describing

these

systems

lies

partly

in

their

topology

Many

of

them

form

rather

complex

networks

whose

vertices

are

the

elements

of

the

system

and

whose

edges

represent

the

interactions

between

them

For

example

liv

SCIENCE

VOL

OCTOBER

Downloaded

from

on

May

REPORTS

ing

systems

form

a

huge

genetic

network

whose

vertices

are

proteins

and

genes

the

chemical

interactions

between

them

representing

edges

-LRB-

-RRB-

At

a

different

organizational

level

a

large

network

is

formed

by

the

nervous

system

whose

vertices

are

the

nerve

cells

connected

by

axons

-LRB-

-RRB-

But

equally

complex

networks

occur

in

social

science

where

vertices

are

individuals

or

organizations

and

the

edges

are

the

social

interactions

between

them

-LRB-

-RRB-

or

in

the

World

Wide

Web

-LRB-

WWW

-RRB-

whose

vertices

are

HTML

documents

connected

by

links

pointing

from

one

page

to

another

-LRB-

-RRB-

Because

of

their

large

size

and

the

complexity

of

their

interactions

the

topology

of

these

networks

is

largely

unknown

Traditionally

networks

of

complex

topology

have

been

described

with

the

random

graph

theory

of

Erdos

and

Renyi

-LRB-

ER

-RRB-

-LRB-

-RRB-

but

in

the

absence

of

data

on

large

networks

the

predictions

of

the

ER

theory

were

rarely

tested

in

the

real

world

However

driven

by

the

computerization

of

data

acquisition

such

topological

information

is

increasingly

available

raising

the

possibility

of

understanding

the

dynamical

and

topological

stability

of

large

networks

Here

we

report

on

the

existence

of

a

high

degree

of

self-organization

characterizing

the

large-scale

properties

of

complex

networks

Exploring

several

large

databases

describing

the

topology

of

large

networks

that

span

fields

as

diverse

as

the

WWW

or

citation

patterns

in

science

we

show

that

independent

of

the

system

and

the

identity

of

its

constituents

the

probability

P

-LRB-

k

-RRB-

that

a

vertex

in

the

network

interacts

with

k

other

vertices

decays

as

a

power

law

following

P

-LRB-

k

-RRB-

k

This

result

indicates

that

large

networks

self-organize

into

a

scale-free

state

a

feature

unpredicted

by

all

existing

random

network

models

To

explain

the

origin

of

this

scale

invariance

we

show

that

existing

network

models

fail

to

incorporate

growth

and

preferential

attachment

two

key

features

of

real

networks

Using

a

model

incorporating

these

two

ingredients

we

show

that

they

are

responsible

for

the

power-law

scaling

observed

in

real

networks

Finally

we

argue

that

these

ingredients

play

an

easily

identifiable

and

important

role

in

the

formation

of

many

complex

systems

which

implies

that

our

results

are

relevant

to

a

large

class

of

networks

observed

in

nature

Although

there

are

many

systems

that

form

complex

networks

detailed

topological

data

is

available

for

only

a

few

The

collaboration

graph

of

movie

actors

represents

a

well-documented

example

of

a

social

network

Each

actor

is

represented

by

a

vertex

two

actors

being

connected

if

they

were

cast

together

in

the

same

movie

The

probability

that

an

actor

has

k

links

-LRB-

characterizing

his

or

her

popularity

-RRB-

has

a

power-law

tail

for

large

k

following

P

-LRB-

k

-RRB-

k

actor

where

actor

-LRB-

Fig.

-RRB-

A

more

complex

network

with

over

million

vertices

-LRB-

-RRB-

is

the

WWW

where

a

vertex

is

a

document

and

the

edges

are

the

links

pointing

from

one

document

to

another

The

topology

of

this

graph

determines

the

Web

's

connectivity

and

consequently

our

effectiveness

in

locating

information

on

the

WWW

-LRB-

-RRB-

Information

about

P

-LRB-

k

-RRB-

can

be

obtained

using

robots

-LRB-

-RRB-

indicating

that

the

probability

that

k

documents

point

to

a

certain

Web

page

follows

a

power

law

with

www

-LRB-

Fig.

-RRB-

-LRB-

-RRB-

A

network

whose

topology

reflects

the

historical

patterns

of

urban

and

industrial

development

is

the

electrical

power

grid

of

the

western

United

States

the

vertices

being

generators

transformers

and

substations

and

the

edges

being

to

the

high-voltage

transmission

lines

between

them

-LRB-

-RRB-

Because

of

the

relatively

modest

size

of

the

network

containing

only

vertices

the

scaling

region

is

less

prominent

but

is

nevertheless

approximated

by

a

power

law

with

an

exponent

-LRB-

Fig.

-RRB-

Finally

a

rather

large

power

complex

network

is

formed

by

the

citation

patterns

of

the

scientific

publications

the

vertices

being

papers

published

in

refereed

journals

and

the

edges

being

links

to

the

articles

cited

in

a

paper

Recently

Redner

-LRB-

-RRB-

has

shown

that

the

probability

that

a

paper

is

cited

k

times

-LRB-

representing

the

connectivity

of

a

paper

within

the

network

-RRB-

follows

a

power

law

with

exponent

cite

The

above

examples

-LRB-

-RRB-

demonstrate

that

many

large

random

networks

share

the

common

feature

that

the

distribution

of

their

local

connectivity

is

free

of

scale

following

a

power

law

for

large

k

with

an

exponent

between

and

which

is

unexpected

within

the

framework

of

the

existing

network

models

The

random

graph

model

of

ER

-LRB-

-RRB-

assumes

that

we

start

with

N

vertices

and

connect

each

pair

of

vertices

with

probability

p

In

the

model

the

probability

that

a

vertex

has

k

edges

follows

a

Poisson

distribution

P

-LRB-

k

-RRB-

k

k

where

e

N

k

In

the

small-world

model

recently

introduced

by

Watts

and

Strogatz

-LRB-

WS

-RRB-

-LRB-

-RRB-

N

vertices

form

a

one-dimensional

lattice

each

vertex

being

connected

to

its

two

nearest

and

next-nearest

neighbors

With

probability

p

each

edge

is

reconnected

to

a

vertex

chosen

at

random

The

long-range

connections

generated

by

this

process

decrease

the

distance

between

the

vertices

leading

to

a

small-world

phenomenon

-LRB-

-RRB-

often

referred

to

as

six

degrees

of

separation

-LRB-

-RRB-

For

p

the

probability

distribution

of

the

connectivities

is

P

-LRB-

k

-RRB-

-LRB-

k

z

-RRB-

where

z

is

the

coordination

number

in

the

lattice

whereas

for

finite

p

P

-LRB-

k

-RRB-

still

peaks

around

z

but

it

gets

broader

-LRB-

-RRB-

A

common

feature

of

the

ER

and

WS

models

is

that

the

probability

of

finding

a

highly

connected

vertex

-LRB-

that

is

a

large

k

-RRB-

decreases

exponentially

with

k

thus

vertices

with

large

connectivity

are

practically

absent

In

contrast

the

power-law

tail

characterizing

P

-LRB-

k

-RRB-

for

the

networks

studied

indicates

that

highly

connected

-LRB-

large

k

-RRB-

vertices

have

a

large

chance

of

occurring

dominating

the

connectivity

There

are

two

generic

aspects

of

real

networks

that

are

not

incorporated

in

these

models

First

both

models

assume

that

we

start

with

a

fixed

number

-LRB-

N

-RRB-

of

vertices

that

are

then

randomly

connected

-LRB-

ER

model

-RRB-

or

reconnected

-LRB-

WS

model

-RRB-

without

modifying

N

In

contrast

most

real

world

networks

are

open

and

they

form

by

the

continuous

addition

of

new

vertices

to

the

system

thus

the

number

of

vertices

N

increases

throughout

the

lifetime

of

the

network

For

example

the

actor

network

grows

by

the

addition

of

new

actors

to

the

system

the

WWW

grows

exponentially

over

time

by

the

addition

of

new

Web

pages

-LRB-

-RRB-

and

the

research

literature

constantly

grows

by

the

publication

of

new

papers

Consequently

a

common

feature

of

pk

p

N

k

Fig.

The

distribution

function

of

connectivities

for

various

large

networks

-LRB-

A

-RRB-

Actor

collaboration

graph

with

N

vertices

and

average

connectivity

k

-LRB-

B

-RRB-

WWW

N

k

-LRB-

-RRB-

-LRB-

C

-RRB-

Power

grid

data

N

k

The

dashed

lines

have

slopes

-LRB-

A

-RRB-

actor

-LRB-

B

-RRB-

www

and

-LRB-

C

-RRB-

power

OCTOBER

VOL

SCIENCE

Downloaded

from

on

May

N

REPORTS

these

systems

is

that

the

network

continuously

expands

by

the

addition

of

new

vertices

that

are

connected

to

the

vertices

already

present

in

the

system

Second

the

random

network

models

assume

that

the

probability

that

two

vertices

are

connected

is

random

and

uniform

In

contrast

most

real

networks

exhibit

preferential

connectivity

For

example

a

new

actor

is

most

likely

to

be

cast

in

a

supporting

role

with

more

established

and

better-known

actors

Consequently

the

probability

that

a

new

actor

will

be

cast

with

an

established

one

is

much

higher

than

that

the

new

actor

will

be

cast

with

other

less-known

actors

Similarly

a

newly

created

Web

page

will

be

more

likely

to

include

links

to

well-known

popular

documents

with

already-high

connectivity

and

a

new

manuscript

is

more

likely

to

cite

a

wellknown

and

thus

much-cited

paper

than

its

less-cited

and

consequently

less-known

peer

These

examples

indicate

that

the

probability

with

which

a

new

vertex

connects

to

the

existing

vertices

is

not

uniform

there

is

a

higher

probability

that

it

will

be

linked

to

a

vertex

that

already

has

a

large

number

of

connections

We

next

show

that

a

model

based

on

these

two

ingredients

naturally

leads

to

the

observed

scale-invariant

distribution

To

incorporate

the

growing

character

of

the

network

starting

with

a

small

number

-LRB-

m

-RRB-

of

vertices

at

every

time

step

we

add

a

new

vertex

with

m

-LRB-

m

-RRB-

edges

that

link

the

new

vertex

to

m

different

vertices

already

present

in

the

system

To

incorporate

preferential

attachment

we

assume

that

the

probability

that

a

new

vertex

will

be

connected

to

vertex

i

depends

on

the

connectivity

k

i

of

that

vertex

so

that

k

i

j

k

j

After

t

time

steps

the

-LRB-

k

i

-RRB-

model

leads

to

a

random

network

with

t

m

vertices

and

mt

edges

This

network

evolves

into

a

scale-invariant

state

with

the

probability

that

a

vertex

has

k

edges

following

a

power

law

with

an

exponent

model

-LRB-

Fig.

-RRB-

Because

the

power

law

observed

for

real

networks

describes

systems

of

rather

different

sizes

at

different

stages

of

their

development

it

is

expected

that

a

correct

model

should

provide

a

distribution

whose

main

features

are

independent

of

time

Indeed

as

Fig.

demonstrates

P

-LRB-

k

-RRB-

is

independent

of

time

-LRB-

and

subsequently

indet

-RRB-

indicatpendent

of

the

system

size

m

ing

that

despite

its

continuous

growth

the

system

organizes

itself

into

a

scale-free

stationary

state

The

development

of

the

power-law

scaling

in

the

model

indicates

that

growth

and

preferential

attachment

play

an

important

role

in

network

development

To

verify

that

both

ingredients

are

necessary

we

investigated

two

variants

of

the

model

Model

A

keeps

the

growing

character

of

the

network

but

preferential

attachment

is

eliminated

by

assuming

that

a

new

vertex

is

connected

with

equal

probability

to

any

vertex

in

the

system

-LSB-

that

t

-RRB-

-RSB-

Such

is

-LRB-

k

-RRB-

const

-LRB-

m

a

model

-LRB-

Fig.

-RRB-

leads

to

P

-LRB-

k

-RRB-

exp

-LRB-

k

-RRB-

indicating

that

the

absence

of

preferential

attachment

eliminates

the

scalefree

feature

of

the

distribution

In

model

B

we

start

with

N

vertices

and

no

edges

At

each

time

step

we

randomly

select

a

vertex

ki

and

connect

it

with

probability

-LRB-

k

i

-RRB-

j

k

j

to

vertex

i

in

the

system

Although

at

early

times

the

model

exhibits

power-law

scaling

P

-LRB-

k

-RRB-

is

not

stationary

because

N

is

constant

and

the

number

of

edges

increases

with

time

after

T

N

time

steps

the

system

reaches

a

state

in

which

all

vertices

are

connected

The

failure

of

models

A

and

B

indicates

that

both

ingredients

growth

and

preferential

attachment

are

needed

for

the

development

of

the

stationary

power-law

distribution

observed

in

Fig.

Because

of

the

preferential

attachment

a

vertex

that

acquires

more

connections

than

another

one

will

increase

its

connectivity

at

a

higher

rate

thus

an

initial

difference

in

the

connectivity

between

two

vertices

will

increase

further

as

the

network

grows

The

rate

at

which

a

vertex

acquires

edges

is

k

i

t

m

-LRB-

t/t

i

-RRB-

where

k

i

which

gives

k

i

-LRB-

t

-RRB-

t

i

is

the

time

at

which

vertex

i

was

added

to

the

system

-LRB-

see

Fig.

-RRB-

a

scaling

property

that

could

be

directly

tested

once

time-resolved

data

on

network

connectivity

becomes

available

Thus

older

-LRB-

with

smaller

t

i

-RRB-

vertices

increase

their

connectivity

at

the

expense

of

the

younger

-LRB-

with

larger

t

i

-RRB-

ones

leading

over

time

to

some

vertices

that

are

highly

connected

a

rich-get-richer

phenomenon

that

can

be

easily

detected

in

real

networks

Furthermore

this

property

can

be

used

to

calculate

analytically

The

probability

that

a

vertex

i

has

a

connectivity

smaller

than

k

k

-RSB-

can

be

written

as

P

-LRB-

t

i

P

-LSB-

k

i

-LRB-

t

-RRB-

m

t/k

-RRB-

Assuming

that

we

add

the

vertices

to

the

system

at

equal

time

intervals

we

m

t/k

-RRB-

P

-LRB-

t

i

obtain

P

-LRB-

t

i

m

t/k

-RRB-

m

t/k

-LRB-

t

m

-RRB-

The

probability

density

P

-LRB-

k

-RRB-

can

be

obtained

from

k

-RSB-

k

which

over

long

P

-LRB-

k

-RRB-

P

-LSB-

k

i

-LRB-

t

-RRB-

time

periods

leads

to

the

stationary

solution

P

k

giving

independent

of

m.

Although

it

reproduces

the

observed

scale-free

distribution

the

proposed

model

can

not

be

expected

to

account

for

all

aspects

of

the

studied

networks

For

that

we

need

to

model

these

systems

in

more

detail

For

example

in

the

model

we

assumed

linear

preferential

attachment

that

is

-LRB-

k

-RRB-

k.

However

although

in

general

-LRB-

k

-RRB-

could

have

an

arbitrary

nonlinear

form

-LRB-

k

-RRB-

k

simulations

indicate

that

scaling

is

present

only

for

Furthermore

the

exponents

obtained

for

the

different

networks

are

scattered

between

and

However

it

is

easy

to

modify

our

model

to

account

for

exponents

different

from

For

example

if

we

assume

that

a

fraction

p

of

the

links

is

directed

we

obtain

-LRB-

p

-RRB-

p

which

is

supported

by

numerical

simulations

-LRB-

-RRB-

Finally

some

networks

evolve

not

only

by

adding

new

vertices

but

by

adding

-LRB-

and

sometimes

removing

-RRB-

connections

between

established

vertices

Although

these

and

other

system-specific

features

could

modify

the

exponent

our

model

offers

the

first

successful

mechanism

accounting

for

the

scale-invariant

nature

of

real

networks

Growth

and

preferential

attachment

are

mechanisms

common

to

a

number

of

complex

systems

including

business

networks

-LRB-

-RRB-

social

networks

-LRB-

describing

individuals

or

organizations

-RRB-

transportation

networks

-LRB-

-RRB-

and

so

on

Consequently

we

expect

that

the

scale-invariant

state

observed

in

all

systems

for

which

detailed

data

has

been

available

to

us

is

a

generic

property

of

many

complex

networks

with

applicability

reaching

far

beyond

the

quoted

examples

A

better

description

of

these

systems

would

help

in

understanding

other

complex

systems

Fig.

-LRB-

A

-RRB-

The

power-law

connectivity

distribution

at

t

-LRB-

E

-RRB-

and

t

-LRB-

-RRB-

as

m

The

slope

of

the

dashed

line

is

-LRB-

B

-RRB-

The

obtained

from

the

model

using

m

m

-LRB-

E

-RRB-

m

m

exponential

connectivity

distribution

for

model

A

in

the

case

of

m

m

-LRB-

-LCB-

-RRB-

and

m

m

-LRB-

-RRB-

-LRB-

C

-RRB-

Time

evolution

of

the

connectivity

for

two

-LRB-

-RRB-

m

and

t

The

dashed

line

has

slope

vertices

added

to

the

system

at

t

SCIENCE

VOL

OCTOBER

Downloaded

from

on

May

REPORTS

as

well

for

which

less

topological

information

is

currently

available

including

such

important

examples

as

genetic

or

signaling

networks

in

biological

systems

We

often

do

not

think

of

biological

systems

as

open

or

growing

because

their

features

are

genetically

coded

However

possible

scale-free

features

of

genetic

and

signaling

networks

could

reflect

the

networks

evolutionary

history

dominated

by

growth

and

aggregation

of

different

constituents

leading

from

simple

molecules

to

complex

organisms

With

the

fast

advances

being

made

in

mapping

out

genetic

networks

answers

to

these

questions

might

not

be

too

far

away

Similar

mechanisms

could

explain

the

origin

of

the

social

and

economic

disparities

governing

competitive

systems

because

the

scale-free

inhomogeneities

are

the

inevitable

consequence

of

selforganization

due

to

the

local

decisions

made

by

the

individual

vertices

based

on

information

that

is

biased

toward

the

more

visible

-LRB-

richer

-RRB-

vertices

irrespective

of

the

nature

and

origin

of

this

visibility

R.

Gallagher

and

T.

Appenzeller

Science

-LRB-

-RRB-

R.

F.

Service

ibid.

p.

G.

Weng

U.

S.

Bhalla

R.

Iyengar

ibid.

p.

C.

Koch

and

G.

Laurent

ibid.

p.

S.

Wasserman

and

K.

Faust

Social

Network

Analysis

-LRB-

Cambridge

Univ.

Press

Cambridge

-RRB-

Members

of

the

Clever

project

Sci

Am

-LRB-

June

-RRB-

R.

Albert

H.

Jeong

A.-L

Barabasi

Nature

-LRB-

-RRB-

A.-L

Barabasi

R.

Albert

H.

Jeong

Physica

A

-LRB-

-RRB-

see

also

P.

Erdos

and

A.

Renyi

Publ

Math

Inst

Hung

Acad

Sci

-LRB-

-RRB-

B.

Bollobas

Random

Graphs

-LRB-

Aca

demic

Press

London

-RRB-

S.

Lawrence

and

C.

L.

Giles

Science

-LRB-

-RRB-

Nature

-LRB-

-RRB-

In

addition

to

the

distribution

of

incoming

links

the

WWW

displays

a

number

of

other

scale-free

features

characterizing

the

organization

of

the

Web

pages

within

a

domain

-LSB-

B.

A.

Huberman

and

L.

A.

Adamic

Nature

-LRB-

-RRB-

-RSB-

the

distribution

of

searches

-LSB-

B.

A.

Huberman

P.

L.

T.

Pirolli

J.

E.

Pitkow

R.

J.

Lukose

Science

-LRB-

-RRB-

-RSB-

or

the

number

of

links

per

Web

page

-LRB-

-RRB-

D.

J.

Watts

and

S.

H.

Strogatz

Nature

-LRB-

-RRB-

S.

Redner

Eur

Phys

J.

B

-LRB-

-RRB-

We

also

studied

the

neural

network

of

the

worm

Caenorhabditis

elegans

-LRB-

-RRB-

and

the

benchmark

References

and

Notes

June

accepted

September

Osmium

Isotope

Constraints

on

Ore

Metal

Recycling

in

Subduction

Zones

Brent

I.

A.

McInnes

Jannene

S.

McBride

Noreen

J.

Evans

David

D.

Lambert

Anita

S.

Andrew

Veined

peridotite

xenoliths

from

the

mantle

beneath

the

giant

Ladolam

gold

deposit

on

Lihir

Island

Papua

New

Guinea

are

to

times

more

enriched

in

copper

gold

platinum

and

palladium

than

surrounding

depleted

arc

mantle

Gold

ores

have

osmium

isotope

compositions

similar

to

those

of

the

underlying

subduction-modified

mantle

peridotite

source

region

indicating

that

the

primary

origin

of

the

metals

was

the

mantle

Because

the

mantle

is

relatively

depleted

in

gold

copper

and

palladium

tectonic

processes

that

enhance

the

advective

transport

and

concentration

of

these

fluid

soluble

metals

may

be

a

prerequisite

for

generating

porphyry-epithermal

copper-gold

deposits

The

tectonic

relationship

between

subductionrelated

magmatism

at

convergent

margins

and

porphyry

copper-gold

-LRB-

Cu-Au

-RRB-

ore

formation

has

long

been

recognized

-LRB-

-RRB-

However

the

physical

and

chemical

processes

that

govern

Cu-Au

metallogeny

and

the

ultimate

source

-LRB-

s

-RRB-

of

the

metals

in

these

ore

deposits

are

poorly

understood

The

rhenium-osmium

-LRB-

Re-Os

-RRB-

isodecay

of

topic

system

-LRB-

based

on

the

Commonwealth

Scientific

and

Industrial

Research

Organization

Exploration

and

Mining

Post

Office

Box

North

Ryde

New

South

Wales

Australia

Department

of

Earth

Sciences

Monash

University

Clayton

Victoria

Australia

Scientific

and

Industrial

Research

Organization

Petroleum

Resources

Post

Office

Box

North

Ryde

New

South

Wales

Australia

To

whom

correspondence

should

be

addressed

Email

Brent.McInnes@dem.csiro.au

to

-RRB-

is

a

tracer

of

metallogenic

processes

at

convergent

margins

because

both

elements

have

geochemical

properties

similar

to

metals

that

occur

in

porphyry

ore

deposits

-LRB-

-RRB-

Because

Re

is

highly

concentrated

in

crustal

rocks

and

Os

is

concentrated

in

the

mantle

-LRB-

-RRB-

this

isotopic

system

is

particularly

useful

for

quantifying

the

flux

of

ore

elements

in

island

arc

settings

where

the

two

principal

reservoirs

for

metals

are

subducted

crust

and

mantle

wedge

peridotite

Os

isotope

studies

in

subduction

zones

are

currently

limited

because

of

the

rarity

of

deep-seated

suprasubduction

samples

-LRB-

rocks

overlying

a

subducted

slab

-RRB-

Previous

studies

-LRB-

-RRB-

have

demonstrated

that

radiogenic

Os

is

introduced

into

the

subarc

mantle

by

hydrous

oxidizing

fluids

derived

during

slab

dehydration

We

report

on

results

from

a

suprasubduction

xenolith

locality

the

Tubaf

seamount

in

the

Lihir

island

group

of

the

Tabar-Lihir-Tanga-Feni

island

arc

in

Papua

New

Guinea

-LRB-

Fig.

-RRB-

This

xenolith

locality

is

important

for

the

following

reasons

-LRB-

i

-RRB-

it

contains

samples

that

represent

a

complete

section

of

oceanic

lithosphere

at

an

intraoceanic

convergent

margin

-LRB-

ii

-RRB-

it

is

located

adjacent

to

one

of

the

world

's

largest

and

youngest

volcano-hosted

Au

deposits

and

-LRB-

iii

-RRB-

it

contains

metasomatized

mantle

peridotite

xenoliths

with

Au-enriched

vein

minerals

that

crystallized

in

the

mantle

from

oxidizing

alkali

and

sulfur-rich

hydrous

fluids

During

the

oceanographic

investigation

of

submarine

hydrothermal

systems

in

Papua

New

Guinea

-LRB-

-RRB-

a

submarine

cinder

cone

-LRB-

Tubaf

volcano

m

below

sea

level

S

E

-RRB-

was

discovered

km

southwest

of

the

giant

Ladolam

gold

mine

-LRB-

million

oz

contained

Au

-RRB-

on

Lihir

Island

Dredge

and

video-grab

sampling

of

the

volcanic

cone

returned

ultramafic

mafic

and

sedimentary

xenoliths

The

study

of

these

samples

has

provided

an

unprecedented

view

of

the

source

region

of

an

island

arc

magmatic

system

with

a

propensity

to

produce

giant

porphyry-epithermal

ore

deposits

The

xenolith

assemblage

includes

spinel

lherzolite

harzburgite

websterite

orthopyroxenite

clinopyroxenite

syenite

serpentinite

gabbro

hornblende

gabbro

diabase

basalt

pelagic

deep-sea

sediment

and

shallow-water

volcaniclastic

sediment

as

well

as

coralline

and

coralgal

limestone

These

lithologies

represent

a

cross

section

of

the

suprasubduction

assemblage

and

can

be

reassembled

into

an

ophiolite-type

model

of

oceanic

lithosphere

-LRB-

Fig.

-RRB-

-LRB-

-RRB-

OCTOBER

VOL

SCIENCE

Downloaded

from

on

May

diagram

of

a

computer

chip

-LRB-

see

ucla.edu

-RRB-

We

found

that

P

-LRB-

k

-RRB-

for

both

was

consistent

with

power-law

tails

despite

the

fact

that

for

C.

elegans

the

relatively

small

size

of

the

system

-LRB-

vertices

-RRB-

severely

limits

the

data

quality

whereas

for

the

wiring

diagram

of

the

chips

vertices

with

over

edges

have

been

eliminated

from

the

database

S.

Milgram

Psychol

Today

-LRB-

-RRB-

M.

Kochen

ed.

The

Small

World

-LRB-

Ablex

Norwood

NJ

-RRB-

J.

Guare

Six

Degrees

of

Separation

A

Play

-LRB-

Vintage

Books

New

York

-RRB-

M.

Barthelemy

and

L.

A.

N.

Amaral

Phys

Rev.

Lett

-LRB-

-RRB-

For

most

networks

the

connectivity

m

of

the

newly

added

vertices

is

not

constant

However

choosing

m

randomly

will

not

change

the

exponent

-LRB-

Y.

Tu

personal

communication

-RRB-

W.

B.

Arthur

Science

-LRB-

-RRB-

Preferential

attachment

was

also

used

to

model

evolving

networks

-LRB-

L.

A.

N.

Amaral

and

M.

Barthelemy

personal

communication

-RRB-

J.

R.

Banavar

A.

Maritan

A.

Rinaldo

Nature

-LRB-

-RRB-

We

thank

D.

J.

Watts

for

providing

the

C.

elegans

and

power

grid

data

B.

C.

Tjaden

for

supplying

the

actor

data

H.

Jeong

for

collecting

the

data

on

the

WWW

and

L.

A.

N.

Amaral

for

helpful

discussions

This

work

was

partially

supported

by

NSF

Career

Award

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B B O O O O O O O O O O O O O O B B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O