insight

review

articles

Exploring

complex

networks

Steven

H.

Strogatz

Department

of

Theoretical

and

Applied

Mechanics

and

Center

for

Applied

Mathematics

Kimball

Hall

Cornell

University

Ithaca

New

York

USA

-LRB-

e-mail

strogatz@cornell.edu

-RRB-

The

study

of

networks

pervades

all

of

science

from

neurobiology

to

statistical

physics

The

most

basic

issues

are

structural

how

does

one

characterize

the

wiring

diagram

of

a

food

web

or

the

Internet

or

the

metabolic

network

of

the

bacterium

Escherichia

coli

Are

there

any

unifying

principles

underlying

their

topology

From

the

perspective

of

nonlinear

dynamics

we

would

also

like

to

understand

how

an

enormous

network

of

interacting

dynamical

systems

be

they

neurons

power

stations

or

lasers

will

behave

collectively

given

their

individual

dynamics

and

coupling

architecture

Researchers

are

only

now

beginning

to

unravel

the

structure

and

dynamics

of

complex

networks

etworks

are

on

our

minds

nowadays

Sometimes

we

fear

their

power

and

with

good

reason

On

August

a

fault

in

two

power

lines

in

Oregon

led

through

a

cascading

series

of

failures

to

blackouts

in

US

states

and

two

Canadian

provinces

leaving

about

million

customers

without

power

for

up

to

The

Love

Bug

worm

the

worst

computer

attack

to

date

spread

over

the

Internet

on

May

and

inflicted

billions

of

dollars

of

damage

worldwide

In

our

lighter

moments

we

play

parlour

games

about

connectivity

Six

degrees

of

Marlon

Brando

broke

out

as

a

nationwide

fad

in

Germany

as

readers

of

Die

Zeit

tried

to

connect

a

falafel

vendor

in

Berlin

with

his

favourite

actor

through

the

shortest

possible

chain

of

And

during

the

height

of

the

Lewinsky

scandal

the

New

York

Times

printed

a

of

the

famous

people

within

six

degrees

of

Monica

Meanwhile

scientists

have

been

thinking

about

networks

too

Empirical

studies

have

shed

light

on

the

topology

of

food

electrical

power

grids

cellular

and

metabolic

the

World-Wide

the

Internet

the

neural

network

of

the

nematode

worm

Caenorhabditis

telephone

call

coauthorship

and

citation

networks

of

and

the

quintessential

old-boy

network

the

overlapping

boards

of

directors

of

the

largest

companies

in

the

United

-LRB-

Fig.

-RRB-

These

databases

are

now

easily

accessible

courtesy

of

the

Internet

Moreover

the

availability

of

powerful

computers

has

made

it

feasible

to

probe

their

structure

until

recently

computations

involving

million-node

networks

would

have

been

impossible

without

specialized

facilities

Why

is

network

anatomy

so

important

to

characterize

Because

structure

always

affects

function

For

instance

the

topology

of

social

networks

affects

the

spread

of

information

and

disease

and

the

topology

of

the

power

grid

affects

the

robustness

and

stability

of

power

transmission

From

this

perspective

the

current

interest

in

networks

is

part

of

a

broader

movement

towards

research

on

complex

systems

In

the

words

of

E.

O.

The

greatest

challenge

today

not

just

in

cell

biology

and

ecology

but

in

all

of

science

is

the

accurate

and

complete

description

of

complex

systems

Scientists

have

broken

down

many

kinds

of

systems

They

think

they

know

most

of

the

elements

and

forces

The

next

task

is

to

reassemble

them

at

least

in

mathematical

models

that

capture

the

key

properties

of

the

entire

ensembles

N

But

networks

are

inherently

difficult

to

understand

as

the

following

list

of

possible

complications

illustrates

Structural

complexity

the

wiring

diagram

could

be

an

intricate

tangle

-LRB-

Fig.

-RRB-

Network

evolution

the

wiring

diagram

could

change

over

time

On

the

World-Wide

Web

pages

and

links

are

created

and

lost

every

minute

Connection

diversity

the

links

between

nodes

could

have

different

weights

directions

and

signs

Synapses

in

Box

Nonlinear

dynamics

terminology

and

Dynamical

systems

can

often

be

modelled

by

differential

equations

dx/dt

v

-LRB-

x

-RRB-

where

x

-LRB-

t

-RRB-

-LRB-

-LRB-

t

-RRB-

xn

-LRB-

t

-RRB-

-RRB-

is

a

vector

of

state

variables

t

is

time

and

v

-LRB-

x

-RRB-

-LRB-

-LRB-

x

-RRB-

vn

-LRB-

x

-RRB-

-RRB-

is

a

vector

of

functions

that

encode

the

dynamics

For

example

in

a

chemical

reaction

the

state

variables

represent

concentrations

The

differential

equations

represent

the

kinetic

rate

laws

which

usually

involve

nonlinear

functions

of

the

concentrations

Such

nonlinear

equations

are

typically

impossible

to

solve

analytically

but

one

can

gain

qualitative

insight

by

imagining

an

abstract

n-dimensional

state

space

with

axes

xn

As

the

system

evolves

x

-LRB-

t

-RRB-

flows

through

state

space

guided

by

the

velocity

field

dx/dt

v

-LRB-

x

-RRB-

like

a

speck

carried

along

in

a

steady

viscous

fluid

Suppose

x

-LRB-

t

-RRB-

eventually

comes

to

rest

at

some

point

x

Then

the

velocity

must

be

zero

there

so

we

call

x

a

fixed

point

It

corresponds

to

an

equilibrium

state

of

the

physical

system

being

modelled

If

all

small

disturbances

away

from

x

damp

out

x

is

called

a

stable

fixed

point

it

acts

as

an

attractor

for

states

in

its

vicinity

Another

long-term

possibility

is

that

x

-LRB-

t

-RRB-

flows

towards

a

closed

loop

and

eventually

circulates

around

it

forever

Such

a

loop

is

called

a

limit

cycle

It

represents

a

self-sustained

oscillation

of

the

physical

system

A

third

possibility

is

that

x

-LRB-

t

-RRB-

might

settle

onto

a

strange

attractor

a

set

of

states

on

which

it

wanders

forever

never

stopping

or

repeating

Such

erratic

aperiodic

motion

is

considered

chaotic

if

two

nearby

states

flow

away

from

each

other

exponentially

fast

Long-term

prediction

is

impossible

in

a

real

chaotic

system

because

of

this

exponential

amplification

of

small

uncertainties

or

measurement

errors

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articles

the

nervous

system

can

be

strong

or

weak

inhibitory

or

excitatory

Dynamical

complexity

the

nodes

could

be

nonlinear

dynamical

systems

In

a

gene

network

or

a

Josephson

junction

array

the

state

of

each

node

can

vary

in

time

in

complicated

ways

Node

diversity

there

could

be

many

different

kinds

of

nodes

The

biochemical

network

that

controls

cell

division

in

mammals

consists

of

a

bewildering

variety

of

substrates

and

only

a

few

of

which

are

shown

in

Fig.

Meta-complication

the

various

complications

can

influence

each

other

For

example

the

present

layout

of

a

power

grid

depends

on

how

it

has

grown

over

the

years

a

case

where

network

evolution

-LRB-

-RRB-

affects

topology

-LRB-

-RRB-

When

coupled

neurons

fire

together

repeatedly

the

connection

between

them

is

strengthened

this

is

the

basis

of

memory

and

learning

Here

nodal

dynamics

-LRB-

-RRB-

affect

connection

weights

-LRB-

-RRB-

To

make

progress

different

fields

have

suppressed

certain

complications

while

highlighting

others

For

instance

in

nonlinear

dynamics

we

have

tended

to

favour

simple

nearly

identical

dynamical

systems

coupled

together

in

simple

geometrically

regular

ways

Furthermore

we

usually

assume

that

the

network

architecture

is

static

These

simplifications

allow

us

to

sidestep

any

issues

of

structural

complexity

and

to

concentrate

instead

on

the

system

's

potentially

formidable

dynamics

Laser

arrays

provide

a

concrete

In

the

single-mode

approximation

each

laser

is

characterized

by

its

time-dependent

gain

polarization

and

the

phase

and

amplitude

of

its

electric

field

These

evolve

according

to

four

coupled

nonlinear

differential

equations

We

usually

hope

the

laser

will

settle

down

to

a

stable

state

corresponding

to

steady

emission

of

light

but

periodic

pulsations

and

even

chaotic

intensity

fluctuations

can

occur

in

some

Now

suppose

that

many

identical

lasers

are

arranged

side

by

side

in

a

regular

or

interacting

with

their

neighbours

by

evanescent

coupling

or

by

overlap

of

their

electric

Will

the

lasers

lock

their

phases

together

spontaneously

or

break

up

into

a

standing

wave

pattern

or

beat

each

other

into

incoherence

From

a

technological

standpoint

self-synchronization

would

be

the

most

desirable

outcome

because

a

perfectly

coherent

array

of

N

lasers

would

Figure

Wiring

diagrams

for

complex

networks

a

Food

web

of

Little

Rock

Lake

Wisconsin

currently

the

largest

food

web

in

the

primary

Nodes

are

functionally

distinct

trophic

species

containing

all

taxa

that

share

the

same

set

of

predators

and

prey

Height

indicates

trophic

level

with

mostly

phytoplankton

at

the

bottom

and

fishes

at

the

top

Cannibalism

is

shown

with

self-loops

and

omnivory

-LRB-

feeding

on

more

than

one

trophic

level

-RRB-

is

shown

by

different

coloured

links

to

consumers

-LRB-

Figure

provided

by

N.

D.

Martinez

-RRB-

b

New

York

State

electric

power

grid

Generators

and

substations

are

shown

as

small

blue

bars

The

lines

connecting

them

are

transmission

lines

and

transformers

Line

thickness

and

colour

indicate

the

voltage

level

red

kV

and

kV

brown

kV

green

kV

grey

kV

and

below

Pink

dashed

lines

are

transformers

-LRB-

Figure

provided

by

J.

Thorp

and

H.

Wang

-RRB-

c

A

portion

of

the

molecular

interaction

map

for

the

regulatory

network

that

controls

the

mammalian

cell

Colours

indicate

different

types

of

interactions

black

binding

interactions

and

stoichiometric

conversions

red

covalent

modifications

and

gene

expression

green

enzyme

actions

blue

stimulations

and

inhibitions

-LRB-

Reproduced

from

Fig.

in

ref

with

permission

Figure

provided

by

K.

Kohn

-RRB-

c

a

b

CAK

Transcription

P

cdk

Cyclin

box

P

P

P

P

Myc

Max

CycH

P

PCNA

cycD

Fos

Replication

PCNA

RPA

P

P

CycE

CycA

CycD

CycA

P

primase

Pol

P

P

CycA

cycE

RPA

P

cycA

box

P

cycB

pRb

APC

P

P

cyclins

pol

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articles

system

its

long-term

behaviour

is

given

by

stable

fixed

points

limit

cycles

or

chaotic

attractors

-LRB-

Box

-RRB-

If

we

now

couple

many

such

systems

together

what

can

be

said

about

their

collective

behaviour

The

answer

is

not

much

the

details

matter

But

I

will

propose

some

rough

generalizations

anyway

If

the

dynamical

system

at

each

node

has

stable

fixed

points

and

no

other

attractors

the

network

tends

to

lock

into

a

static

pattern

Many

such

patterns

may

coexist

especially

if

the

nodes

have

competing

interactions

In

that

case

the

network

may

become

frustrated

and

display

enormous

numbers

of

locally

stable

equilibria

This

kind

of

complex

static

behaviour

is

seen

in

models

of

spin

glasses

associative

memory

neural

networks

and

combinatorial

optimization

At

the

opposite

extreme

suppose

each

node

has

a

chaotic

attractor

Few

rules

have

emerged

about

the

effect

of

coupling

architecture

on

dynamics

in

this

case

It

is

known

that

networks

of

identical

chaotic

systems

can

synchronize

their

erratic

fluctuations

a

curious

phenomenon

with

possible

applications

to

private

For

a

wide

range

of

network

topologies

synchronized

chaos

requires

that

the

coupling

be

neither

too

weak

nor

too

strong

otherwise

spatial

instabilities

are

Related

lines

of

research

deal

with

networks

of

identical

chaotic

maps

lattice

dynamical

systems

and

cellular

automata

These

systems

have

been

used

mainly

as

testbeds

for

exploring

spatiotemporal

chaos

and

pattern

formation

in

the

simplest

mathematical

settings

rather

than

as

models

of

real

physical

systems

Identical

oscillators

Figure

Spontaneous

synchronization

in

a

network

of

limit-cycle

oscillators

with

distributed

natural

frequencies

The

state

of

each

oscillator

is

represented

geometrically

as

a

dot

in

the

complex

plane

The

amplitude

and

phase

of

the

oscillation

correspond

to

the

radius

and

angle

of

the

dot

in

polar

coordinates

Colours

code

the

oscillators

natural

frequencies

running

from

slowest

-LRB-

red

-RRB-

to

fastest

-LRB-

violet

-RRB-

In

the

absence

of

coupling

each

oscillator

would

settle

onto

its

limit

cycle

-LRB-

circle

-RRB-

and

rotate

at

its

natural

frequency

However

here

all

the

oscillators

are

also

pulled

towards

the

mean

field

that

they

generate

collectively

-LRB-

shown

as

an

asterisk

at

the

centre

of

the

population

-RRB-

Time

increases

from

left

to

right

and

from

top

to

bottom

Starting

from

a

random

initial

condition

the

oscillators

self-organize

by

collapsing

their

amplitudes

then

they

sort

their

phases

so

that

the

fastest

oscillators

are

in

the

lead

Ultimately

they

all

rotate

as

a

synchronized

pack

with

locked

amplitudes

and

phases

The

governing

equations

describe

a

mean-field

model

of

a

laser

-LRB-

Simulation

provided

by

R.

Oliva

-RRB-

produce

N

times

as

much

power

as

a

single

one

But

in

practice

semiconductor

laser

arrays

are

notoriously

prone

to

both

spatial

and

temporal

Even

for

a

simple

ring

geometry

this

problem

is

dynamically

complex

The

first

part

of

this

article

reviews

what

is

known

about

dynamical

complexity

in

regular

networks

of

nonlinear

systems

I

offer

a

few

rules

of

thumb

about

the

impact

of

network

structure

on

collective

dynamics

especially

for

arrays

of

coupled

limit-cycle

oscillators

The

logical

next

step

would

be

to

tackle

networks

that

combine

dynamical

and

structural

complexity

such

as

power

grids

or

ecological

webs

Unfortunately

they

lie

beyond

our

mathematical

reach

we

do

not

even

know

how

to

characterize

their

wiring

diagrams

So

we

have

to

begin

with

network

topology

By

a

happy

coincidence

such

architectural

questions

are

being

pursued

in

other

branches

of

science

thanks

to

the

excitement

about

the

Internet

functional

genomics

financial

networks

and

so

on

The

second

part

of

this

article

uses

graph

theory

to

explore

the

structure

of

complex

networks

an

approach

that

has

recently

led

to

some

encouraging

progress

especially

when

combined

with

the

tools

of

statistical

mechanics

and

computer

simulations

Needless

to

say

many

other

topics

within

network

science

deserve

coverage

here

The

subject

is

amazingly

rich

and

apologies

are

offered

to

those

readers

whose

favourite

topics

are

omitted

Regular

networks

of

coupled

dynamical

systems

Networks

of

dynamical

systems

have

been

used

to

model

everything

from

earthquakes

to

ecosystems

neurons

to

To

impose

some

order

on

this

list

consider

the

dynamics

that

each

node

would

exhibit

if

it

were

isolated

Assuming

it

is

a

generic

dynamical

The

intermediate

case

where

each

node

has

a

stable

limit

cycle

has

turned

out

to

be

particularly

fruitful

Much

of

the

research

has

been

inspired

by

biological

examples

ranging

from

the

mutual

synchronization

of

cardiac

pacemaker

cells

to

rhythmically

flashing

fireflies

and

chorusing

crickets

to

wave

propagation

in

the

heart

brain

intestine

and

nervous

Arrays

of

identical

oscillators

often

synchronize

or

else

form

patterns

that

depend

on

the

symmetry

of

the

underlying

Other

common

modes

of

organization

are

travelling

waves

in

one

spatial

dimension

rotating

spirals

in

two

dimensions

and

scroll

waves

in

three

For

fully

connected

networks

where

each

node

is

coupled

equally

to

all

the

others

the

completely

synchronized

state

becomes

likely

These

heuristics

apply

to

systems

coupled

by

smooth

interactions

akin

to

diffusion

But

many

biological

oscillators

communicate

by

sudden

impulses

a

neuron

fires

a

firefly

flashes

a

cricket

chirps

Hence

the

recent

interest

in

pulse-coupled

This

thread

began

with

Peskin

's

model

of

the

sinoatrial

node

the

heart

's

natural

pacemaker

as

a

collection

of

N

identical

integrate-and-fire

For

the

simple

case

where

each

oscillator

is

connected

to

all

the

others

Peskin

conjectured

that

they

would

all

end

up

firing

in

unison

no

matter

how

they

started

He

gave

a

proof

for

N

oscillators

it

was

later

that

the

conjecture

holds

for

all

N.

Peskin

also

conjectured

that

synchronization

would

occur

even

if

the

oscillators

were

not

quite

identical

but

that

problem

remains

unproven

Peskin

's

model

has

been

used

as

a

caricature

of

coupled

by

including

synaptic

delays

refractory

periods

inhibition

and

local

coupling

these

realistic

features

also

remove

some

of

the

undesirable

discontinuities

in

the

mathematics

In

an

example

of

scientific

cross-fertilization

pointed

out

that

the

locally

coupled

version

of

the

model

is

closely

related

to

slider-block

models

of

earthquakes

and

should

therefore

display

self-organized

criticality

That

observation

suggested

intriguing

links

among

neurobiology

geophysics

synchronization

and

self-organized

criticality

and

sparked

a

burst

of

research

activity

as

reviewed

in

ref

Non-identical

oscillators

While

modelling

populations

of

biological

oscillators

Winfree

discovered

a

new

kind

of

cooperative

phenomenon

the

temporal

NATURE

VOL

MARCH

Macmillan

Magazines

Ltd

insight

review

articles

analogue

of

a

phase

He

proposed

a

mean-field

model

of

nearly

identical

weakly

coupled

limit-cycle

oscillators

and

showed

that

when

the

coupling

is

small

compared

to

the

spread

of

natural

frequencies

the

system

behaves

incoherently

with

each

oscillator

running

at

its

natural

frequency

As

the

coupling

is

increased

the

incoherence

persists

until

a

certain

threshold

is

crossed

then

a

small

cluster

of

oscillators

suddenly

freezes

into

synchrony

For

still

greater

coupling

all

the

oscillators

become

locked

in

phase

and

amplitude

-LRB-

Fig.

-RRB-

refined

this

connection

between

nonlinear

dynamics

and

statistical

physics

He

proposed

an

exactly

solvable

model

of

collective

synchronization

given

by

d

i

dt

K

N

sin

-LRB-

Nj

-RRB-

i

N

a

b

i

j

i

where

i

-LRB-

t

-RRB-

is

the

phase

of

the

ith

oscillator

and

i

is

its

natural

frequency

chosen

at

random

from

a

lorentzian

probability

density

c

g

-LRB-

-RRB-

-LSB-

-LRB-

-RRB-

-RSB-

of

width

and

mean

Using

an

ingenious

self-consistency

argument

Kuramoto

solved

for

the

order

parameter

r

-LRB-

t

-RRB-

Nj

N

ei

j

-LRB-

t

-RRB-

-LRB-

a

convenient

measure

of

the

extent

of

synchronization

-RRB-

in

the

limit

N

and

t

He

found

that

r

d

-LRB-

Kc

K

-RRB-

K

Kc

K

Kc

where

Kc

In

other

words

the

oscillators

are

desynchronized

completely

until

the

coupling

strength

K

exceeds

a

critical

value

Kc

After

that

the

population

splits

into

a

partially

synchronized

state

Figure

Schematic

illustration

of

regular

and

random

network

architectures

a

Ring

of

ten

nodes

connected

to

their

nearest

neighbours

b

Fully

connected

network

of

ten

nodes

c

Random

graph

constructed

by

placing

n

nodes

on

a

plane

then

joining

pairs

of

them

together

at

random

until

m

links

are

used

Nodes

may

be

chosen

more

than

once

or

not

at

all

The

resulting

wiring

diagram

-LRB-

not

shown

-RRB-

would

be

a

snarl

of

crisscrossed

lines

to

clarify

it

I

have

segregated

the

different

connected

components

coloured

them

and

eliminated

as

many

spurious

crossings

as

possible

The

main

topological

features

are

the

presence

of

a

single

giant

component

as

for

a

random

graph

with

m

-LRB-

here

n

m

-RRB-

and

the

absence

of

any

dominant

hubs

The

degree

or

number

of

neighbours

is

Poisson

distributed

across

the

nodes

most

nodes

have

between

one

and

four

neighbours

and

all

have

between

zero

and

six

d

Scale-free

graph

grown

by

attaching

new

nodes

at

random

to

previously

existing

nodes

The

probability

of

attachment

is

proportional

to

the

degree

of

the

target

node

thus

richly

connected

nodes

tend

to

get

richer

leading

to

the

formation

of

hubs

and

a

skewed

degree

distribution

with

a

heavy

tail

Colours

indicate

the

three

nodes

with

the

most

links

-LRB-

red

k

links

blue

k

green

k

-RRB-

Here

n

nodes

m

links

Figure

provided

by

D.

Callaway

Network

visualization

was

done

using

the

Pajek

program

for

large

network

analysis

-LRB-

-RRB-

NATURE

VOL

MARCH

Figure

Solvable

model

of

a

small-world

network

The

model

starts

with

a

ring

lattice

of

n

nodes

each

connected

to

its

neighbours

out

to

some

range

k

-LRB-

here

n

and

k

-RRB-

Shortcut

links

are

added

between

random

pairs

of

nodes

with

probability

per

link

on

the

underlying

lattice

In

the

limit

n

the

average

path

length

between

nodes

can

be

approximated

analytically

-LRB-

Adapted

from

ref

-RRB-

Macmillan

Magazines

Ltd

insight

review

articles

everyone

's

delight

that

prediction

was

later

confirmed

by

their

experimental

Complex

network

architectures

l

n

All

the

network

topologies

discussed

so

far

chains

grids

lattices

and

fully-connected

graphs

have

been

completely

regular

-LRB-

Fig.

b

-RRB-

Those

simple

architectures

allowed

us

to

focus

on

the

complexity

caused

by

the

nonlinear

dynamics

of

the

nodes

without

being

burdened

by

any

additional

complexity

in

the

network

structure

itself

Now

I

take

the

complementary

approach

setting

dynamics

aside

and

turning

to

more

complex

architectures

A

natural

place

to

start

is

at

the

opposite

end

of

the

spectrum

from

regular

networks

with

graphs

that

are

completely

random

nk

Random

graphs

Figure

Average

path

length

normalized

by

system

size

is

plotted

as

a

function

of

the

average

number

of

shortcuts

The

circles

are

results

from

simulations

of

the

model

with

range

k

and

size

up

to

n

nodes

The

solid

line

is

given

by

the

formula

in

the

text

-LRB-

Adapted

from

ref

-RRB-

consisting

of

two

groups

of

oscillators

a

synchronized

group

that

contributes

to

the

order

parameter

r

and

a

desynchronized

group

whose

natural

frequencies

lie

in

the

tails

of

the

distribution

g

-LRB-

-RRB-

and

are

too

extreme

to

be

entrained

With

further

increases

in

K

more

and

more

oscillators

are

recruited

into

the

synchronized

group

and

r

grows

accordingly

Twenty-five

years

later

the

Kuramoto

model

continues

to

surprise

us

-LRB-

see

ref

for

a

review

-RRB-

First

the

incoherent

state

with

r

was

found

to

be

neutrally

stable

below

threshold

despite

its

apparent

stability

in

simulations

the

analysis

reveals

a

connection

to

Landau

damping

in

plasmas

Second

the

square-root

critical

behaviour

of

r

almost

a

clich

for

mean-field

models

in

statistical

mechanics

turns

out

to

be

non-generic

if

the

sinusoidal

coupling

is

replaced

by

a

periodic

function

with

second

harmonics

the

scaling

changes

to

r

K

Kc

Third

although

the

model

was

motivated

originally

by

biological

oscillators

it

has

appeared

in

such

far-flung

settings

as

the

flavour

evolution

of

and

arrays

of

Josephson

and

semiconductor

The

main

unsolved

problem

is

the

stability

of

the

partially

synchronized

state

for

K

Kc

Numerical

simulations

indicate

that

it

is

globally

stable

in

the

sense

that

it

attracts

almost

all

solutions

but

even

the

linear

stability

problem

has

yet

to

be

solved

Another

issue

concerns

the

extension

of

the

model

to

nearest-neighbour

coupling

on

a

d-dimensional

cubic

lattice

and

renormalization

indicate

that

the

synchronization

phase

transition

persists

for

d

and

vanishes

for

d

the

results

are

ambiguous

for

d

All

of

this

awaits

a

mathematical

resolution

In

contrast

to

the

mean-field

models

of

Winfree

and

Kuramoto

Ermentrout

and

Kopell

's

classic

work

deals

with

one-dimensional

chains

of

oscillators

first

in

connection

with

neuromuscular

rhythms

in

the

mammalian

and

later

in

their

model

of

the

central

pattern

generator

for

the

lamprey

The

main

phenomena

here

involve

travelling

waves

rather

than

the

synchrony

found

in

mean-field

models

This

is

not

accidental

as

wave

propagation

is

essential

for

the

generation

of

peristalsis

in

the

intestine

and

for

the

creation

of

the

swimming

rhythm

in

lamprey

Ermentrout

and

Kopell

introduced

several

deep

mathematical

innovations

but

perhaps

their

most

impressive

result

is

a

counterintuitive

biological

prediction

Their

lamprey

model

suggested

that

the

tail-to-head

neural

connections

along

the

spinal

cord

would

be

stronger

than

those

running

from

head

to

tail

despite

the

fact

that

the

wave

associated

with

swimming

travels

from

head

to

tail

To

Imagine

n

buttons

strewn

across

the

Pick

two

buttons

at

random

and

tie

them

together

with

thread

Repeat

this

process

m

times

always

choosing

pairs

of

buttons

at

random

-LRB-

If

m

is

large

you

might

eventually

select

buttons

that

already

have

threads

attached

That

is

certainly

allowed

it

merely

creates

clusters

of

connected

buttons

-RRB-

The

result

is

a

physical

example

of

a

random

graph

with

n

nodes

and

m

links

-LRB-

Fig.

-RRB-

Now

slowly

lift

a

random

button

off

the

floor

If

it

is

tied

to

other

buttons

either

directly

or

indirectly

those

are

dragged

up

too

So

what

happens

Are

you

likely

to

pull

up

an

isolated

button

a

small

cluster

or

a

vast

meshwork

Erd

s

and

R

studied

how

the

expected

topology

of

this

random

graph

changes

as

a

function

of

m

When

m

is

small

the

graph

is

likely

to

be

fragmented

into

many

small

clusters

of

nodes

called

components

As

m

increases

the

components

grow

at

first

by

linking

to

isolated

nodes

and

later

by

coalescing

with

other

components

A

phase

transition

occurs

at

m

where

many

clusters

crosslink

spontaneously

to

form

a

single

giant

component

For

m

this

giant

component

contains

on

the

order

of

n

nodes

-LRB-

more

precisely

its

size

scales

linearly

with

n

as

n

-RRB-

while

its

closest

rival

contains

only

about

log

n

nodes

Furthermore

all

nodes

in

the

giant

component

are

connected

to

each

other

by

short

paths

the

maximum

number

of

degrees

of

separation

between

any

two

nodes

grows

slowly

like

log

n

In

the

decades

since

this

pioneering

work

random

graphs

have

been

studied

deeply

within

pure

They

have

also

served

as

idealized

coupling

architectures

for

dynamical

models

of

gene

networks

ecosystems

and

the

spread

of

infectious

diseases

and

computer

Small-world

networks

Although

regular

networks

and

random

graphs

are

both

useful

idealizations

many

real

networks

lie

somewhere

between

the

extremes

of

order

and

randomness

Watts

and

studied

a

simple

model

that

can

be

tuned

through

this

middle

ground

a

regular

lattice

Table

Clustering

for

three

affiliation

networks

Network

Theory

Company

directors

Movie

actors

Biomedical

authors

Clustering

C

Actual

US

corporate

directors

company

directors

linked

by

joint

membership

on

boards

of

the

Fortune

companies

for

Movie

actors

actors

linked

by

mutual

appearances

in

feature

films

as

specified

by

the

Internet

Movie

Database

-LRB-

-RRB-

as

of

May

Biomedical

collaborations

scientists

linked

by

coauthorship

of

at

least

one

of

biomedical

journal

articles

published

between

and

inclusive

as

listed

in

the

MEDLINE

database

The

clustering

coefficient

C

is

defined

as

the

probability

that

a

connected

triple

of

nodes

is

actually

a

triangle

here

nodes

correspond

to

people

as

in

the

unipartite

representation

shown

in

Fig.

Intuitively

C

measures

the

likelihood

that

two

people

who

have

a

mutual

collaborator

are

also

collaborators

of

each

other

The

results

show

that

the

random

model

accurately

predicts

C

for

the

corporate

director

network

given

the

network

's

bipartite

structure

and

its

degree

distributions

no

additional

social

forces

need

to

be

invoked

For

the

networks

of

actors

and

scientists

the

model

accounts

for

about

half

of

the

observed

clustering

The

remaining

portion

depends

on

social

mechanisms

at

work

in

these

communities

-LRB-

see

text

-RRB-

-LRB-

Adapted

from

ref

-RRB-

Macmillan

Magazines

Ltd

NATURE

VOL

MARCH

insight

review

articles

Figure

Degree

distributions

for

real

networks

a

World-Wide

Web

Nodes

are

web

pages

links

are

directed

URL

hyperlinks

from

one

page

to

another

The

log

log

plot

shows

the

number

of

web

pages

that

have

a

given

in-degree

-LRB-

number

of

incoming

links

-RRB-

The

raw

data

have

been

logarithmically

binned

with

bin

ratio

The

tail

of

the

distribution

follows

an

approximate

power

law

with

exponent

-LRB-

Adapted

from

ref

-RRB-

b

Coauthorship

networks

Nodes

represent

scientists

an

undirected

link

between

two

people

means

that

they

have

written

a

paper

together

The

data

are

for

the

years

inclusive

and

come

from

three

databases

arxiv.org

-LRB-

preprints

mainly

in

theoretical

physics

-RRB-

SPIRES

-LRB-

preprints

in

high-energy

physics

-RRB-

and

NCSTRL

-LRB-

preprints

in

computer

science

-RRB-

Symbols

denote

different

scientific

communities

astrophysics

-LRB-

circles

-RRB-

condensed-matter

physics

-LRB-

squares

-RRB-

high-energy

physics

-LRB-

upright

triangles

-RRB-

and

computer

science

-LRB-

inverted

triangles

-RRB-

The

log

log

plot

shows

that

the

probability

of

having

a

total

of

z

coauthors

is

well

approximated

by

a

truncated

power

law

-LRB-

solid

line

-RRB-

of

the

form

pz

z

exp

-LRB-

z/z

c

-RRB-

where

z

c

is

the

cutoff

The

curves

have

been

displaced

vertically

for

clarity

-LRB-

Adapted

from

ref

-RRB-

c

Power

grid

of

the

western

United

States

and

Canada

Nodes

represent

generators

transformers

and

substations

undirected

links

represent

high-voltage

transmission

lines

between

them

The

data

are

plotted

as

a

cumulative

distribution

to

reduce

the

effects

of

noise

on

this

smaller

data

set

The

linear

log

plot

shows

that

the

proportion

of

nodes

with

at

least

k

transmission

lines

decays

exponentially

in

k

The

negative

derivative

of

this

cumulative

distribution

is

the

degree

distribution

also

an

exponential

-LRB-

Adapted

from

ref

-RRB-

d

Social

network

Nodes

are

Mormons

in

Utah

undirected

links

represent

acquaintances

with

other

The

linear

log

plot

of

the

cumulative

distribution

is

well

fit

by

an

error

function

-LRB-

solid

line

-RRB-

so

the

degree

distribution

is

a

gaussian

-LRB-

Adapted

from

ref

-RRB-

a

b

Frequency

c

Cumulative

probability

Indegree

d

Number

of

coauthors

Transmission

lines

Acquaintances

where

the

original

links

are

replaced

by

random

ones

with

some

probability

They

found

that

the

slightest

bit

of

rewiring

transforms

the

network

into

a

small

world

with

short

paths

between

any

two

nodes

just

as

in

the

giant

component

of

a

random

graph

Yet

the

network

is

much

more

highly

clustered

than

a

random

graph

in

the

sense

that

if

A

is

linked

to

B

and

B

is

linked

to

C

there

is

a

greatly

increased

probability

that

A

will

also

be

linked

to

C

-LRB-

a

property

that

call

transitivity

-RRB-

Watts

and

Strogatz

conjectured

that

the

same

two

properties

short

paths

and

high

clustering

would

hold

also

for

many

natural

and

technological

networks

Furthermore

they

conjectured

that

dynamical

systems

coupled

in

this

way

would

display

enhanced

signal

propagation

speed

synchronizability

and

computational

power

as

compared

with

regular

lattices

of

the

same

size

The

intuition

is

that

the

short

paths

could

provide

high-speed

communication

channels

between

distant

parts

of

the

system

thereby

facilitating

any

dynamical

process

-LRB-

like

synchronization

or

computation

-RRB-

that

requires

global

coordination

and

information

flow

Research

has

proceeded

along

several

fronts

Many

empirical

examples

of

small-world

networks

have

been

documented

in

fields

ranging

from

cell

biology

to

On

the

theoretical

side

small-world

networks

are

turning

out

to

be

a

Rorschach

test

different

scientists

see

different

problems

here

depending

on

their

disciplines

Computer

scientists

see

questions

about

algorithms

and

their

complexity

showed

that

graphs

associated

with

many

difficult

search

problems

have

a

small-world

topology

introduced

an

elegant

model

of

the

algorithmic

challenge

posed

by

Milgram

's

original

sociological

how

to

actually

find

a

short

chain

of

acquaintances

linking

yourself

to

a

random

target

person

using

only

local

information

and

he

proved

that

the

problem

is

easily

solvable

for

some

kinds

of

small

worlds

and

essentially

intractable

for

others

Epidemiologists

have

asked

how

local

clustering

and

global

NATURE

VOL

MARCH

contacts

together

influence

the

spread

of

infectious

disease

with

implications

for

vaccination

strategies

and

the

evolution

of

Neurobiologists

have

wondered

about

the

possible

evolutionary

significance

of

small-world

neural

architecture

They

have

argued

that

this

kind

of

topology

combines

fast

signal

processing

with

coherent

unlike

either

regular

or

random

architectures

and

that

it

may

be

selected

by

adaptation

to

rich

sensory

environments

and

motor

Perhaps

the

strongest

response

to

the

Rorschach

test

comes

from

the

statistical

physicists

who

sensed

that

the

toy

model

of

Watts

and

would

yield

to

their

techniques

-LRB-

see

ref

for

a

review

-RRB-

In

its

improved

form

the

model

starts

with

a

ring

of

n

nodes

each

connected

by

undirected

links

to

its

nearest

and

nextnearest

neighbours

out

to

some

range

k.

Shortcut

links

are

then

added

rather

than

rewired

between

randomly

selected

pairs

of

nodes

with

probability

per

link

on

the

underlying

lattice

thus

there

are

typically

nk

shortcuts

in

the

graph

-LRB-

Fig.

-RRB-

The

question

is

on

average

how

many

steps

are

required

to

go

from

one

node

to

another

along

the

shortest

route

If

denotes

that

average

separation

we

find

that

drops

sharply

near

confirming

that

a

few

shortcuts

do

indeed

shrink

the

world

dramatically

One

of

the

most

striking

results

is

the

following

formula

derived

by

Newman

Moore

and

n

f

-LRB-

nk

-RRB-

k

where

f

-LRB-

x

-RRB-

tanh

x

x

x

This

solution

is

asymptotically

exact

in

the

limits

n

-LRB-

large

system

size

-RRB-

and

either

nk

or

nk

-LRB-

large

or

small

number

of

Macmillan

Magazines

Ltd

insight

review

articles

a

M

A

B

C

D

E

F

N

G

H

I

J

K

b

A

B

C

D

E

J

G

F

H

I

K

Figure

Bipartite

versus

unipartite

representations

of

the

corporate

director

network

a

In

the

bipartite

approach

directors

and

boards

are

treated

as

distinct

kinds

of

nodes

The

schematic

example

shows

directors

and

boards

Links

indicate

which

people

sit

on

which

boards

By

definition

there

are

no

links

between

pairs

of

people

or

between

pairs

of

boards

b

The

more

familiar

unipartite

representation

depicts

the

people

as

nodes

with

links

between

those

on

the

same

board

forming

cliques

This

approach

discards

important

information

and

can

conflate

different

structures

For

example

the

triangle

FHI

corresponds

to

board

number

as

seen

in

a

whereas

the

similar-looking

triangle

FGI

does

not

correspond

to

any

single

board

Another

confusing

effect

is

the

large

number

of

cliques

that

occur

automatically

in

this

projection

of

the

full

bipartite

graph

Such

cliques

account

for

much

of

the

high

clustering

observed

in

real

affiliation

The

random

graph

model

teases

out

this

generic

source

of

clustering

from

that

indicative

of

more

interesting

social

interactions

-LRB-

Adapted

from

ref

-RRB-

Nevertheless

the

scale-free

case

has

stimulated

a

great

deal

of

theorizing

The

earliest

work

is

due

to

in

now

independently

rediscovered

by

Barab

si

Albert

and

They

showed

that

a

heavy-tailed

degree

distribution

emerges

automatically

from

a

stochastic

growth

model

in

which

new

nodes

are

added

continuously

and

attach

themselves

preferentially

to

existing

nodes

with

probability

proportional

to

the

degree

of

the

target

node

Richly

connected

nodes

get

richer

and

the

result

is

pk

k

More

sophisticated

include

the

effects

of

adding

or

rewiring

links

allowing

nodes

to

age

so

that

they

can

no

longer

accept

new

links

or

varying

the

form

of

preferential

attachment

These

generalized

models

predict

exponential

and

truncated

power-law

pk

in

some

parameter

regimes

as

well

as

scale-free

distributions

Could

there

be

a

functional

advantage

to

scale-free

architecture

Albert

Jeong

and

Barab

suggested

that

scale-free

networks

are

resistant

to

random

failures

because

a

few

hubs

dominate

their

topology

-LRB-

Fig.

-RRB-

Any

node

that

fails

probably

has

small

degree

-LRB-

like

most

nodes

-RRB-

and

so

is

expendable

The

flip

side

is

that

such

networks

are

vulnerable

to

deliberate

attacks

on

the

hubs

These

intuitive

ideas

have

been

confirmed

and

by

examining

how

the

average

path

length

and

size

of

the

giant

component

depend

on

the

number

and

degree

of

the

nodes

removed

Some

possible

implications

for

the

resilience

of

the

the

design

of

therapeutic

and

the

evolution

of

metabolic

have

been

discussed

Generalized

random

graphs

As

mentioned

above

the

simplest

random

graph

predicts

a

Poisson

degree

distribution

and

so

can

not

accommodate

the

other

types

of

distribution

found

in

real

networks

Molloy

and

introduced

a

more

flexible

class

of

random

graphs

in

which

any

degree

distribution

is

permitted

Given

a

sequence

of

non-negative

integers

-LCB-

dk

-RCB-

where

dk

denotes

the

number

of

nodes

with

degree

k

consider

the

ensemble

of

all

graphs

with

that

prescribed

degree

sequence

and

weight

them

all

equally

when

computing

statistical

averages

of

interest

For

this

class

of

graphs

Molloy

and

Reed

derived

a

simple

condition

for

the

birth

of

the

giant

and

they

also

found

an

implicit

formula

for

its

size

as

a

fraction

of

n

the

total

number

of

Specifically

let

n

and

define

Q

k

shortcuts

-RRB-

Figure

shows

that

it

also

gives

the

correct

qualitative

behaviour

for

nk

Barbour

and

improved

this

result

by

proving

a

rigorous

distributional

approximation

for

along

with

a

bound

on

the

error

Scale-free

networks

pk

k

-LRB-

k

-RRB-

In

any

real

network

some

nodes

are

more

highly

connected

than

others

are

To

quantify

this

effect

let

pk

denote

the

fraction

of

nodes

that

have

k

links

Here

k

is

called

the

degree

and

pk

is

the

degree

distribution

The

simplest

random

graph

predict

a

bell-shaped

Poisson

distribution

for

pk

But

for

many

real

networks

pk

is

highly

skewed

and

decays

much

more

slowly

than

a

Poisson

For

instance

the

distribution

decays

as

a

power

law

pk

k

for

the

Internet

metabolic

reaction

the

telephone

call

and

the

World-Wide

-LRB-

Fig.

-RRB-

Remarkably

the

exponent

for

all

of

these

cases

Taken

literally

this

form

of

heavy-tailed

distribution

would

imply

an

infinite

variance

In

reality

there

are

a

few

nodes

with

many

links

-LRB-

Fig.

-RRB-

For

the

World-Wide

Web

think

Yahoo

for

metabolic

networks

think

ATP

Barab

si

Albert

and

have

dubbed

these

networks

scale-free

by

analogy

with

fractals

phase

transitions

and

other

situations

where

power

laws

arise

and

no

single

characteristic

scale

can

be

defined

The

scale-free

property

is

common

but

not

For

coauthorship

networks

of

scientists

pk

is

fit

better

by

a

power

law

with

an

exponential

-LRB-

Fig.

-RRB-

for

the

power

grid

of

the

western

United

States

pk

is

an

exponential

-LRB-

Fig.

-RRB-

and

for

a

social

network

of

Mormons

in

pk

is

-LRB-

Fig.

-RRB-

where

pk

dk

n

If

Q

the

graph

consists

of

many

small

components

The

average

component

size

diverges

as

from

below

and

a

giant

component

exists

for

Q

-LRB-

In

technical

terms

these

results

hold

almost

surely

that

is

with

probability

tending

to

as

n

-RRB-

Aiello

Chung

and

applied

these

results

to

a

random

graph

model

for

scale-free

networks

For

pk

of

power-law

form

the

condition

on

Q

implies

that

a

giant

component

exists

if

and

only

if

which

holds

for

most

scale-free

networks

measured

so

far

If

there

are

so

many

high-degree

hubs

that

the

network

forms

one

huge

connected

piece

They

also

proved

theorems

about

the

number

and

size

of

small

components

outside

the

giant

component

and

compared

these

to

a

real

graph

of

about

million

telephone

numbers

and

the

calls

between

them

in

one

day

They

found

that

the

data

are

best

fit

by

an

exponent

which

predicts

correctly

that

the

call

graph

is

not

connected

but

has

a

giant

component

The

papers

by

Molloy

and

and

Aiello

et

al.

are

mathematically

rigorous

Newman

Strogatz

and

recently

developed

a

more

heuristic

approach

based

on

generating

functions

By

handling

the

limit

n

in

an

intuitive

way

their

approach

yields

elementary

derivations

of

the

earlier

results

along

with

new

exact

results

for

graphs

with

additional

structure

such

as

directed

or

bipartite

graphs

NATURE

VOL

MARCH

Macmillan

Magazines

Ltd

insight

review

articles

The

bipartite

case

is

especially

interesting

for

By

definition

in

a

bipartite

graph

there

are

two

types

of

nodes

with

links

running

only

between

different

kinds

-LRB-

Fig.

-RRB-

For

example

consider

the

network

of

the

boards

of

directors

of

the

Fortune

companies

the

largest

US

corporations

ranked

according

to

revenues

This

network

is

fascinating

because

the

boards

interlock

some

important

people

sit

on

several

of

them

and

this

overlap

knits

virtually

all

large

US

firms

together

into

a

giant

web

of

corporate

Let

pj

denote

the

probability

that

a

director

sits

on

exactly

j

boards

and

let

qk

denote

the

probability

that

a

board

consists

of

k

directors

Figure

shows

that

pj

is

approximately

exponential

with

most

directors

sitting

on

only

one

board

whereas

qk

is

strongly

peaked

around

k

indicating

that

most

boards

have

about

ten

members

As

a

hypothesis

assume

that

the

Fortune

network

is

a

random

member

of

the

ensemble

of

all

bipartite

graphs

with

the

same

pj

and

qk

Then

generating

functions

yield

predictions

for

various

quantities

of

For

example

suppose

we

want

to

calculate

rz

the

probability

that

a

random

director

works

with

a

total

of

z

other

co-directors

summed

over

all

the

boards

on

which

he

or

she

serves

Let

-LRB-

x

-RRB-

j

a

b

Probability

pj

c

Probability

rz

Number

of

boards

j

Probability

qk

Number

of

members

k

pj

x

j

Number

of

co-directors

z

-LRB-

x

-RRB-

k

qk

x

k

be

the

generating

functions

associated

with

the

empirical

degree

distributions

pj

and

qk

If

we

now

choose

a

random

edge

on

the

bipartite

graph

and

follow

it

to

the

board

at

one

of

its

ends

the

distribution

of

the

number

of

other

edges

leaving

that

board

can

be

shown

to

be

generated

by

-LRB-

x

-RRB-

-LRB-

x

-RRB-

where

-LRB-

-RRB-

Then

for

a

randomly

chosen

director

the

generating

function

for

z

is

given

by

-LRB-

x

-RRB-

-LRB-

-LRB-

x

-RRB-

-RRB-

If

we

expand

in

a

series

as

-LRB-

x

-RRB-

z

rz

x

z

Figure

Structural

properties

of

the

Fortune

network

of

corporate

directors

for

The

data

shown

here

comprise

directors

and

boards

Most

directors

serve

on

the

board

of

only

one

company

but

about

sit

on

two

or

more

boards

thereby

creating

interlocked

directorates

This

has

potentially

important

consequences

for

business

practices

in

the

United

a

Distribution

of

the

number

of

boards

per

director

The

probability

pj

that

a

director

sits

on

exactly

j

boards

decreases

roughly

exponentially

with

j.

b

Distribution

of

the

number

of

directors

per

board

The

probability

qk

that

a

board

has

k

members

is

approximately

lognormally

distributed

with

a

typical

board

size

around

k

members

c

Distribution

of

each

director

's

total

number

of

co-directors

summed

over

all

the

boards

on

which

the

director

sits

The

probability

rz

of

serving

with

z

other

co-directors

is

a

complicated

function

of

z

with

a

long

tail

and

a

secondary

shoulder

near

z

yet

the

theoretical

prediction

-LRB-

solid

line

-RRB-

agrees

almost

perfectly

with

the

actual

distribution

-LRB-

green

circles

-RRB-

-LRB-

Adapted

from

ref

-RRB-

the

coefficients

rz

are

exactly

the

quantities

we

seek

They

can

be

extracted

by

repeated

differentiation

rz

-LRB-

-RRB-

-LRB-

-RRB-

x

Figure

shows

that

the

predicted

rz

agrees

almost

perfectly

with

the

actual

distribution

Similarly

the

clustering

predicted

for

the

directors

lies

within

of

the

observed

value

-LRB-

Table

-RRB-

Clearly

the

random

model

captures

much

of

the

structure

of

the

real

network

However

for

two

other

bipartite

graphs

film

actors

and

the

movies

they

appeared

in

and

biomedical

scientists

and

the

papers

they

coauthored

the

underestimates

the

clustering

coefficients

by

half

-LRB-

Table

-RRB-

The

reason

is

that

the

random

model

quantifies

only

the

generic

portion

of

the

clustering

it

reflects

the

cliques

that

are

formed

automatically

whenever

a

bipartite

collaboration

graph

is

projected

onto

the

space

of

people

as

in

Fig.

For

the

corporate

board

data

those

cliques

account

for

essentially

all

the

clustering

-LRB-

simply

because

most

directors

sit

on

only

one

board

thus

preventing

clustering

across

boards

-RRB-

But

for

the

scientists

and

actors

some

further

mechanisms

must

be

at

work

One

possible

explanation

is

that

scientists

tend

to

introduce

pairs

of

their

collaborators

to

each

other

engendering

new

collaborations

In

this

way

the

random

model

allows

us

to

disentangle

the

generic

features

of

bipartite

graphs

from

those

that

could

reflect

sociological

effects

Beyond

their

benchmarking

role

generalized

random

graphs

provide

a

promising

new

class

of

substrates

on

which

dynamical

processes

can

be

simulated

and

even

approached

analytically

Using

this

approach

has

given

an

intriguing

explanation

of

fads

and

NATURE

VOL

MARCH

normal

accidents

as

the

natural

consequence

of

cascade

dynamics

on

sparse

interaction

networks

Outlook

In

the

short

run

there

are

plenty

of

good

problems

about

the

nonlinear

dynamics

of

systems

coupled

according

to

small-world

scale-free

or

generalized

random

connectivity

The

speculations

that

these

architectures

are

dynamically

advantageous

-LRB-

for

example

more

synchronizable

or

error-tolerant

-RRB-

need

to

be

sharpened

then

confirmed

or

refuted

mathematically

for

specific

examples

Other

ripe

topics

include

the

design

of

self-healing

networks

and

the

relationships

among

optimization

principles

network

growth

rules

and

network

In

the

longer

run

network

thinking

will

become

essential

to

all

branches

of

science

as

we

struggle

to

interpret

the

data

pouring

in

from

neurobiology

genomics

ecology

finance

and

the

World-Wide

Web

Will

theory

be

able

to

keep

up

Time

to

log

back

on

to

the

Internet

s

Western

Systems

Coordinating

Council

-LRB-

WSCC

-RRB-

Disturbance

Report

for

the

Power

System

Outage

that

Occurred

on

the

Western

Interconnection

on

August

at

PAST

-LRB-

October

-RRB-

Anonymous

Media

Six

degrees

from

Hollywood

Newsweek

October

-LRB-

-RRB-

Kirby

D.

Sahre

P.

Six

degrees

of

Monica

New

York

Times

February

op

ed

page

-LRB-

-RRB-

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Acknowledgements

Thanks

to

J.

Ariaratnam

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Barabasi

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Martinez

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E.

J.

Newman

D.

Watts

and

A.

Winfree

for

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comments

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a

draft

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manuscript

and

to

R.

Albert

L.

Amaral

M.

Amin

W.

Blake

A.

Broder

D.

Callaway

J.

Collins

G.

Davis

H.

Ebel

K.

Kohn

N.

Martinez

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Oliva

M.

E.

J.

Newman

J.

Thorp

D.

Watts

J.

Wiener

A.

Winfree

and

H.

Wang

for

providing

data

figures

and

information

Research

supported

in

part

by

the

National

Science

Foundation

Department

of

Defense

and

Electric

Power

Research

Institute

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