Vol

April

doi

LETTERS

Quantifying

social

group

evolution

Gergely

Albert-Laszlo

Tamas

The

rich

set

of

interactions

between

individuals

in

results

in

complex

community

structure

capturing

highly

connected

circles

of

friends

families

or

professional

cliques

in

a

social

Thanks

to

frequent

changes

in

the

activity

and

communication

patterns

of

individuals

the

associated

social

and

communication

network

is

subject

to

constant

Our

knowledge

of

the

mechanisms

governing

the

underlying

community

dynamics

is

limited

but

is

essential

for

a

deeper

understanding

of

the

development

and

self-optimization

of

society

as

a

We

have

developed

an

algorithm

based

on

clique

that

allows

us

to

investigate

the

time

dependence

of

overlapping

communities

on

a

large

scale

and

thus

uncover

basic

relationships

characterizing

community

evolution

Our

focus

is

on

networks

capturing

the

collaboration

between

scientists

and

the

calls

between

mobile

phone

users

We

find

that

large

groups

persist

for

longer

if

they

are

capable

of

dynamically

altering

their

membership

suggesting

that

an

ability

to

change

the

group

composition

results

in

better

adaptability

The

behaviour

of

small

groups

displays

the

opposite

tendency

the

condition

for

stability

is

that

their

composition

remains

unchanged

We

also

show

that

knowledge

of

the

time

commitment

of

members

to

a

given

community

can

be

used

for

estimating

the

community

's

lifetime

These

findings

offer

insight

into

the

fundamental

differences

between

the

dynamics

of

small

groups

and

large

institutions

The

data

sets

we

consider

are

-LRB-

-RRB-

the

monthly

list

of

articles

in

the

Cornell

University

Library

e-print

condensed

matter

-LRB-

cond-mat

-RRB-

archive

spanning

months

with

over

and

-LRB-

-RRB-

the

record

of

phone

calls

between

the

customers

of

a

mobile

phone

company

spanning

weeks

-LRB-

accumulated

over

two-week-long

periods

-RRB-

and

containing

the

communication

patterns

of

over

million

users

Both

types

of

collaboration

events

-LRB-

a

new

article

or

a

phone

call

-RRB-

document

the

presence

of

social

interaction

between

the

involved

individuals

-LRB-

nodes

-RRB-

and

can

be

represented

as

-LRB-

timedependent

-RRB-

links

The

extraction

of

the

changing

link

weights

from

the

primary

data

is

described

in

Supplementary

Information

In

Fig.

b

we

show

the

local

structure

at

a

given

time

step

in

the

two

networks

in

the

vicinity

of

a

randomly

chosen

individual

-LRB-

marked

by

a

red

frame

-RRB-

The

communities

-LRB-

social

groups

represented

by

more

densely

interconnected

parts

within

a

network

of

social

links

-RRB-

are

colour

coded

so

that

black

nodes/edges

do

not

belong

to

any

community

and

those

that

simultaneously

belong

to

two

or

more

communities

are

shown

in

red

The

two

networks

have

rather

different

local

structure

the

collaboration

network

of

scientists

emerges

as

a

one-mode

projection

of

the

bipartite

graph

between

authors

and

papers

so

it

is

quite

dense

and

the

overlap

between

communities

is

very

significant

In

contrast

in

the

phone-call

network

the

communities

are

less

interconnected

and

are

often

separated

by

one

or

more

inter-community

nodes/edges

Indeed

whereas

the

phone

record

captures

the

communication

between

two

people

the

publication

record

assigns

to

all

individuals

that

contribute

to

a

paper

a

fully

connected

clique

As

a

result

the

phone

data

are

dominated

by

single

links

whereas

the

co-authorship

data

have

many

dense

highly

connected

neighbourhoods

Furthermore

the

links

in

the

phone

network

correspond

to

instant

communication

events

capturing

a

relationship

as

it

happens

In

contrast

the

co-authorship

data

a

Co-authorship

b

Phone

call

c

d

60 80 100 120

Zip-code

Age

Zip-code

Age

nreal

nrand

nreal

s

60 80 100 120

s

s

Contraction

t

t

Splitting

t

e

t

Growth

t

Merging

f

tt

t

t

Birth

t

t

t

t

Death

t

t

t

Figure

Structure

and

schematic

dynamics

of

the

two

networks

considered

a

The

co-authorship

network

The

figure

shows

the

local

community

structure

at

a

given

time

step

in

the

vicinity

of

a

randomly

selected

node

b

As

a

but

for

the

phone-call

network

c

The

filled

black

symbols

correspond

to

the

average

size

of

the

largest

subset

of

members

with

the

same

zip-code

nreal

in

the

phone-call

communities

divided

by

the

same

quantity

found

in

random

sets

nrand

as

a

function

of

the

community

size

s.

Similarly

the

open

symbols

show

the

average

size

of

the

largest

subset

of

community

members

with

an

age

falling

in

a

three-year

time

window

divided

by

the

same

quantity

in

random

sets

The

error

bars

in

both

cases

correspond

to

nreal

-LRB-

nrand

srand

-RRB-

and

nreal

-LRB-

nrand

srand

-RRB-

where

srand

is

the

standard

deviation

in

the

case

of

the

random

sets

d

The

nreal

s

as

a

function

of

s

for

both

the

zip-code

-LRB-

filled

black

symbols

-RRB-

and

the

age

-LRB-

open

symbols

-RRB-

e

Possible

events

in

community

evolution

f

The

identification

of

evolving

communities

The

links

at

t

-LRB-

blue

-RRB-

and

the

links

at

t

-LRB-

yellow

-RRB-

are

merged

into

a

joint

graph

-LRB-

green

-RRB-

Any

CPM

community

at

t

or

t

is

part

of

a

CPM

community

in

the

joined

graph

so

these

can

be

used

to

match

the

two

sets

of

communities

Statistical

and

Biological

Physics

Research

Group

of

the

HAS

Pazmany

P.

stny

Budapest

Hungary

for

Complex

Network

Research

and

Departments

of

Physics

and

Computer

Science

University

of

Notre

Dame

Indiana

USA

of

Biological

Physics

Eotvos

University

Pazmany

P.

stny

Budapest

Hungary

Nature

Publishing

Group

NATURE

Vol

April

LETTERS

record

the

results

of

a

long-term

collaboration

process

These

fundamental

differences

suggest

that

any

common

features

of

the

community

evolution

in

the

two

networks

represent

potentially

generic

characteristics

of

community

formation

rather

than

being

rooted

in

the

details

of

the

network

representation

or

data

collection

process

The

communities

at

each

time

step

were

extracted

using

the

clique

percolation

-LRB-

CPM

-RRB-

The

key

features

of

the

communities

obtained

by

the

CPM

are

that

their

members

can

be

reached

through

well

connected

subsets

of

nodes

and

that

the

communities

may

overlap

-LRB-

share

nodes

with

each

other

-RRB-

This

latter

property

is

essential

as

most

networks

are

characterized

by

overlapping

and

nested

As

a

first

step

it

is

important

to

check

if

the

uncovered

communities

correspond

to

groups

of

individuals

with

a

shared

common

activity

pattern

For

this

purpose

we

compared

the

average

weight

of

the

links

inside

communities

wc

to

the

average

weight

of

the

inter-community

links

wic

For

the

co-authorship

network

wc/wic

is

about

whereas

for

the

phone-call

network

the

difference

is

even

more

significant

as

wc/wic

indicating

that

the

intensity

of

collaboration/communication

within

a

group

is

significantly

higher

than

with

contacts

belonging

to

a

different

Although

for

co-authors

the

quality

of

the

clustering

can

be

directly

tested

by

studying

their

publication

records

in

more

detail

in

the

phone-call

network

personal

information

is

not

available

In

this

case

the

zip-code

and

the

age

of

the

users

provide

additional

information

for

checking

the

homogeneity

of

the

communities

According

to

Fig.

the

nreal

nrand

ratio

is

significantly

larger

than

for

both

the

zip-code

and

the

age

indicating

that

communities

have

a

tendency

to

contain

people

from

the

same

generation

and

living

in

the

same

neighbourhood

-LRB-

nreal

is

the

size

of

the

largest

subset

of

people

having

the

same

zipcode

averaged

over

time

steps

and

the

set

of

available

communities

while

nrand

represents

the

same

average

but

with

randomly

selected

users

-RRB-

It

is

of

specific

interest

that

nreal

nrand

for

the

zip-code

has

a

prominent

peak

at

community

size

s

suggesting

that

communities

of

this

size

are

geographically

the

most

homogeneous

ones

However

as

Fig.

shows

the

situation

is

more

complex

on

average

the

smaller

communities

are

more

homogeneous

in

respect

of

both

the

zip-code

and

the

age

but

there

is

still

a

noticeable

peak

at

s

for

the

zip-code

In

summary

the

phone-call

communities

uncovered

by

the

CPM

tend

to

contain

individuals

living

in

the

same

a

Co-authorship

Phone

call

neighbourhood

and

having

a

comparable

age

a

homogeneity

that

supports

the

validity

of

the

uncovered

community

structure

Further

support

is

given

in

Supplementary

Information

The

basic

events

that

may

occur

in

the

life

of

a

community

are

shown

in

Fig.

a

community

can

grow

or

contract

groups

may

merge

or

split

new

communities

are

born

while

others

may

disappear

We

have

developed

a

method

for

the

appropriate

matching

-LRB-

between

the

subsequent

states

of

the

evolving

communities

-RRB-

from

the

information

available

for

relatively

distant

points

in

time

only

-LRB-

see

Methods

-RRB-

After

determining

the

dynamically

changing

community

structure

we

first

consider

two

basic

quantities

characterizing

a

community

its

size

s

and

its

age

t

representing

the

time

passed

since

its

birth

The

quantities

s

and

t

are

positively

correlated

larger

communities

are

on

average

older

-LRB-

Fig.

-RRB-

Next

we

used

the

auto-correlation

function

C

-LRB-

t

-RRB-

to

quantify

the

relative

overlap

between

two

states

of

the

same

community

A

-LRB-

t

-RRB-

at

t

time

steps

apart

C

-LRB-

t

-RRB-

jA

-LRB-

-RRB-

A

-LRB-

zt

-RRB-

j

jA

-LRB-

-RRB-

A

-LRB-

zt

-RRB-

j

where

jA

-LRB-

-RRB-

A

-LRB-

zt

-RRB-

j

is

the

number

of

common

nodes

-LRB-

members

-RRB-

in

A

-LRB-

-RRB-

and

A

-LRB-

t

-RRB-

and

jA

-LRB-

-RRB-

A

-LRB-

zt

-RRB-

j

is

the

number

of

nodes

in

the

union

of

A

-LRB-

-RRB-

and

A

-LRB-

t

-RRB-

Figure

shows

the

average

time

dependent

auto-correlation

function

for

communities

born

with

different

sizes

The

data

indicate

that

the

collaboration

network

is

more

dynamic

-LRB-

C

-LRB-

t

-RRB-

decays

faster

-RRB-

We

also

find

that

in

both

networks

the

auto-correlation

function

decays

faster

for

the

larger

communities

showing

that

the

membership

of

the

larger

communities

is

changing

at

a

higher

rate

In

contrast

small

communities

change

at

a

smaller

rate

their

composition

being

more

or

less

static

To

quantify

this

aspect

of

community

evolution

we

define

the

stationarity

f

of

a

community

as

the

average

correlation

between

subsequent

states

tmax

-LCB-

P

C

-LRB-

t

-RRB-

f

t

tmax

-LCB-

-LCB-

where

denotes

the

birth

of

the

community

and

tmax

is

the

last

step

before

the

extinction

of

the

community

Thus

f

represents

the

average

ratio

of

members

changed

in

one

step

c

s

60 80 100 120

-LRB-

s

-RRB-

b

s

Phone

call

s

Phone

call

s

Phone

call

s

Co-authorship

s

Co-authorship

s

Co-authorship

s

d

C

-LRB-

t

-RRB-

s

t

Figure

Characteristic

features

of

community

evolution

a

The

age

t

of

communities

with

a

given

size

-LRB-

number

of

people

-RRB-

s

averaged

over

the

set

of

available

communities

and

the

time

steps

divided

by

the

average

age

of

all

communities

t

as

a

function

of

s

The

increasing

nature

of

the

plot

indicates

that

larger

communities

are

on

average

older

b

The

autocorrelation

function

C

-LRB-

t

-RRB-

of

communities

with

different

sizes

averaged

over

the

communities

and

unit

of

time

t

is

two

weeks

thus

for

the

coauthorship

network

where

the

data

samples

were

taken

monthly

the

C

-LRB-

t

-RRB-

values

are

shown

for

every

other

time

step

c

The

life-span

t

averaged

over

the

communities

as

a

function

of

the

stationarity

f

and

the

community

size

s

for

the

co-authorship

network

-LRB-

The

communities

still

living

at

the

last

available

time

step

in

the

data

set

were

excluded

from

this

investigation

-RRB-

The

peak

in

t

is

close

to

f

for

small

sizes

whereas

it

is

shifted

towards

lower

f

values

for

large

sizes

d

Similar

results

found

in

the

phone-call

network

In

c

and

d

the

white

line

corresponds

to

the

optimal

stationarity

Nature

Publishing

Group

LETTERS

NATURE

Vol

April

We

observe

an

interesting

effect

when

investigating

the

relationship

between

the

lifetime

t

-LRB-

the

number

of

steps

between

the

birth

and

the

disintegration

of

a

community

-RRB-

the

stationarity

and

the

community

size

The

lifetime

can

be

viewed

as

a

simple

measure

of

fitness

communities

having

higher

fitness

have

an

extended

life

whereas

the

ones

with

small

fitness

quickly

disintegrate

In

Fig.

d

we

show

the

average

life-span

t

-LRB-

colour

coded

-RRB-

as

a

function

of

the

stationarity

f

and

the

community

size

s

-LRB-

both

s

and

f

were

binned

-RRB-

In

both

networks

for

small

community

sizes

the

highest

average

life-span

is

at

a

stationarity

value

very

close

to

one

indicating

a

s

small

stationary

b

small

non-stationary

that

for

small

communities

it

is

optimal

to

have

static

timeindependent

membership

On

the

other

hand

the

peak

in

t

is

shifted

towards

low

f

values

for

large

communities

suggesting

that

for

these

the

optimal

regime

is

to

be

dynamic

that

is

to

have

a

continually

changing

membership

To

illustrate

the

difference

in

the

optimal

behaviour

-LRB-

a

pattern

of

membership

dynamics

leading

to

extended

lifetime

-RRB-

of

small

and

large

communities

in

Fig.

we

show

the

time

evolution

of

four

communities

from

the

co-authorship

network

As

Fig.

indicates

a

typical

small

and

stationary

community

undergoes

minor

changes

but

lives

for

a

long

time

This

is

well

illustrated

by

the

snapshots

of

the

community

structure

showing

that

the

community

's

stability

is

conferred

by

a

core

of

three

individuals

representing

a

collaborative

group

spanning

over

months

In

contrast

a

small

community

with

high

turnover

of

its

members

has

a

lifetime

of

nine

time

steps

only

-LRB-

Fig.

-RRB-

The

opposite

is

seen

for

large

communities

a

large

stationary

community

disintegrates

after

four

time

steps

-LRB-

Fig.

-RRB-

In

contrast

a

large

non-stationary

community

whose

members

change

dynamically

resulting

in

significant

fluctuations

in

both

size

and

composition

has

a

quite

extended

lifetime

-LRB-

Fig.

-RRB-

The

different

stability

rules

followed

by

the

small

and

large

communities

raise

an

important

question

could

the

inspection

of

a

s

c

large

stationary

New

Old

s

s

n

d

Leaving

in

next

step

large

non-stationary

pl

wout

-LRB-

win

wout

-RRB-

Co-authorship

Phone

call

wout

-LRB-

win

wout

-RRB-

b

e

pd

Wout

-LRB-

Win

Wout

-RRB-

Co-authorship

Phone

call

Figure

Evolution

of

four

types

of

communities

in

the

co-authorship

network

The

height

of

the

columns

corresponds

to

the

actual

community

size

s

and

within

one

column

the

yellow

colour

indicates

the

number

of

old

nodes

-LRB-

that

have

been

present

in

the

community

at

least

in

the

previous

time

step

as

well

-RRB-

while

newcomers

are

shown

with

green

The

members

abandoning

the

community

in

the

next

time

step

are

shown

with

orange

or

purple

colour

depending

on

whether

they

are

old

or

new

-LRB-

This

latter

type

of

member

joins

the

community

for

only

one

time

step

-RRB-

We

show

a

small

and

stationary

community

-LRB-

a

-RRB-

a

small

and

non-stationary

community

-LRB-

b

-RRB-

a

large

and

stationary

community

-LRB-

c

-RRB-

and

finally

a

large

and

non-stationary

community

-LRB-

d

-RRB-

A

mainly

growing

stage

-LRB-

two

time

steps

-RRB-

in

the

evolution

of

the

last

community

is

detailed

in

e.

Wout

-LRB-

Win

Wout

-RRB-

Figure

Effects

of

links

between

communities

a

The

probability

pl

of

a

member

abandoning

the

community

in

the

next

step

as

a

function

of

the

ratio

of

the

member

's

aggregated

link

weights

to

other

parts

of

the

network

-LRB-

wout

-RRB-

and

the

member

's

total

aggregated

link

weight

-LRB-

win

wout

-RRB-

The

inset

shows

the

average

time

spent

in

the

community

by

the

nodes

tn

as

a

function

of

wout

-LRB-

win

wout

-RRB-

b

The

probability

pd

of

a

community

disintegrating

in

the

next

step

as

a

function

of

the

ratio

of

the

aggregated

weights

of

links

from

the

community

to

other

parts

of

the

network

-LRB-

Wout

-RRB-

and

the

aggregated

weights

of

all

links

starting

from

the

community

-LRB-

Win

Wout

-RRB-

The

inset

shows

the

average

life-time

t

of

communities

as

a

function

of

Wout

-LRB-

Win

Wout

-RRB-

Nature

Publishing

Group

NATURE

Vol

April

LETTERS

Received

January

accepted

February

Watts

D.

J.

Strogatz

S.

H.

Collective

dynamics

of

small-world

networks

Nature

-LRB-

-RRB-

Barabasi

A.-L

Albert

R.

Emergence

of

scaling

in

random

networks

Science

-LRB-

-RRB-

Scott

J.

Social

Network

Analysis

A

Handbook

edn

-LRB-

Sage

Publications

London

-RRB-

Dorogovtsev

S.

N.

Mendes

J.

F.

F.

Evolution

of

Networks

From

Biological

Nets

to

the

Internet

and

WWW

-LRB-

Oxford

Univ.

Press

Oxford

-RRB-

Watts

D.

J.

Dodds

P.

S.

Newman

M.

E.

J.

Identity

and

search

in

social

networks

Science

-LRB-

-RRB-

Faust

K.

in

Models

and

Methods

in

Social

Network

Analysis

-LRB-

eds

Carrington

P.

Scott

J.

Wasserman

S.

-RRB-

Ch

-LRB-

Cambridge

Univ.

Press

New

York

-RRB-

Liljeros

F.

Edling

Ch

R.

Amaral

L.

A.

N.

Stanley

H.

E.

Aberg

Y

The

web

of

human

sexual

contacts

Nature

-LRB-

-RRB-

Shiffrin

R.

M.

Borner

K.

Mapping

knowledge

domains

Proc

Natl

Acad

Sci

USA

-LRB-

suppl

-RRB-

-LRB-

-RRB-

Newman

M.

E.

J.

Detecting

community

structure

in

networks

Eur

Phys

J.

B

-LRB-

-RRB-

Radicchi

F.

Castellano

C.

Cecconi

F.

Loreto

V.

Parisi

D.

Defining

and

identifying

communities

in

networks

Proc

Natl

Acad

Sci

USA

-LRB-

-RRB-

Barabasi

A.-L

et

al.

Evolution

of

the

social

network

of

scientific

collaborations

Physica

A

-LRB-

-RRB-

Holme

P.

Edling

Ch

R.

Liljeros

F.

Structure

and

time-evolution

of

an

internet

dating

community

Soc

Networks

-LRB-

-RRB-

Ebel

H.

Davidsen

J.

Bornholdt

S.

Dynamics

of

social

networks

Complexity

-LRB-

-RRB-

Wagner

C.

S.

Leydesdorff

L.

Network

structure

self-organization

and

the

growth

of

international

collaboration

in

science

Res

Policy

-LRB-

-RRB-

Yeung

Y.-Y.

Liu

T.

C.-Y

Ng

P.-H

A

social

network

analysis

of

research

collaboration

in

physics

education

Am

J.

Phys

-LRB-

-RRB-

Newman

M.

E.

J.

Park

J

Why

social

networks

are

different

from

other

types

of

networks

Phys

Rev.

E

-LRB-

-RRB-

Guimera

R.

Danon

L.

Diaz-Guilera

A.

Giralt

F.

Arenas

A.

Self-similar

community

structure

in

organisations

Phys

Rev.

E

-LRB-

-RRB-

Hopcroft

J.

Khan

O.

Kulis

B.

Selman

B.

Tracking

evolving

communities

in

large

linked

networks

Proc

Natl

Acad

Sci

USA

-LRB-

-RRB-

Guimera

R.

Uzzi

B.

Spiro

J.

Amaral

L.

A.

N.

Team

assembly

mechanisms

determine

collaboration

network

structure

and

team

performance

Science

-LRB-

-RRB-

Li

Ch

Maini

Ph.

K

An

evolving

network

model

with

community

structure

J.

Phys

Math

Gen.

-LRB-

-RRB-

Pollner

P.

Palla

G.

Vicsek

T.

Preferential

attachment

of

communities

The

same

principle

but

a

higher

level

Europhys

Lett

-LRB-

-RRB-

Kossinets

G.

Watts

D.

J.

Empirical

analysis

of

an

evolving

social

network

Science

-LRB-

-RRB-

Palla

G.

Derenyi

I.

Farkas

I.

Vicsek

T.

Uncovering

the

overlapping

community

structure

of

complex

networks

in

nature

and

society

Nature

-LRB-

-RRB-

Derenyi

I.

Palla

G.

Vicsek

T.

Clique

percolation

in

random

networks

Phys

Rev.

Lett

-LRB-

-RRB-

Warner

S.

E-prints

and

the

open

archives

initiative

Library

Hi

Tech

-LRB-

-RRB-

Granovetter

M.

S

The

strength

of

weak

ties

Am

J.

Sociol

-LRB-

-RRB-

Csermely

P.

Weak

Links

-LRB-

Springer

Heidelberg

-RRB-

Onnela

J.-P

et

al.

Structure

and

tie

strengths

in

mobile

communication

networks

Preprint

at

-LRB-

-RRB-

Everett

M.

G.

Borgatti

S.

P.

Analyzing

clique

overlap

Connections

-LRB-

-RRB-

a

community

itself

predict

its

future

To

address

this

issue

for

each

member

in

a

community

we

measured

the

total

weight

of

this

member

's

connections

-LRB-

a

measure

of

how

much

a

member

is

committed

-RRB-

to

outside

of

the

community

-LRB-

wout

-RRB-

as

well

as

to

members

belonging

to

the

same

community

-LRB-

win

-RRB-

We

then

calculated

the

probability

that

the

member

will

abandon

the

community

as

a

function

of

the

wout

-LRB-

win

wout

-RRB-

ratio

As

Fig.

shows

for

both

networks

if

the

relative

commitment

of

a

user

to

individuals

outside

a

given

community

is

higher

then

it

is

more

likely

that

he/she

will

leave

the

community

In

parallel

the

average

time

spent

in

the

community

by

the

nodes

tn

is

a

decreasing

function

of

the

above

ratio

-LRB-

Fig.

inset

-RRB-

As

Fig.

shows

those

with

the

least

commitment

have

a

quickly

growing

likelihood

of

leaving

the

community

Taking

this

idea

from

individuals

to

communities

we

measured

for

each

community

the

total

weight

of

links

from

the

members

to

others

outside

of

the

community

-LRB-

Wout

-RRB-

as

well

as

the

aggregated

link

weight

inside

the

community

-LRB-

Win

-RRB-

We

find

that

the

probability

for

a

community

to

disintegrate

in

the

next

step

increases

as

a

function

of

Wout

-LRB-

Win

Wout

-RRB-

-LRB-

Fig.

-RRB-

and

the

lifetime

of

a

community

decreases

for

large

Wout

-LRB-

Win

Wout

-RRB-

ratios

-LRB-

Fig.

inset

-RRB-

However

an

interesting

observation

is

that

whereas

the

lifetime

of

the

phone-call

communities

for

moderate

levels

is

relatively

insensitive

to

outside

commitments

the

lifetime

of

the

collaboration

communities

possesses

a

maximum

at

intermediate

levels

of

inter-collaborations

-LRB-

collaboration

between

colleagues

who

belong

to

different

communities

-RRB-

These

results

suggest

that

a

tracking

of

the

individual

's

as

well

as

the

community

's

relative

commitment

to

the

other

members

of

the

community

provides

a

clue

for

predicting

the

community

's

fate

In

summary

our

results

indicate

the

significant

difference

between

smaller

collaborative

or

friendship

circles

and

institutions

At

the

heart

of

small

cliques

are

a

few

strong

relationships

and

as

long

as

these

persist

the

community

around

them

is

stable

It

appears

to

be

almost

impossible

to

maintain

this

strategy

for

large

communities

Our

calculations

show

that

the

condition

for

stability

of

large

communities

is

continuous

change

so

that

after

some

time

practically

all

members

are

exchanged

Such

loose

rapidly

changing

communities

are

reminiscent

of

institutions

which

can

continue

to

exist

even

after

all

members

have

been

replaced

by

new

members

For

example

in

a

few

years

most

members

of

a

school

or

a

company

could

change

yet

the

school

and

the

company

will

be

detectable

as

a

distinct

community

at

any

time

during

its

existence

METHODS

Locating

communities

In

the

CPM

method

a

community

is

defined

as

a

union

of

all

k-cliques

-LRB-

complete

sub-graphs

of

size

k

-RRB-

that

can

be

reached

from

each

other

through

a

series

of

adjacent

k-cliques

-LRB-

where

adjacency

means

sharing

k

nodes

-RRB-

When

applied

to

weighted

networks

the

CPM

has

two

parameters

the

k-clique

size

k

-LRB-

in

Fig.

b

we

show

the

communities

for

k

-RRB-

and

the

weight

threshold

w

-LRB-

links

weaker

than

w

are

ignored

-RRB-

The

criterion

for

selecting

these

parameters

is

discussed

in

the

Supplementary

Information

Identifying

evolving

communities

The

basic

idea

of

the

algorithm

developed

by

us

to

identify

community

evolution

is

shown

in

Fig.

For

each

pair

of

consecutive

time

steps

t

and

t

we

construct

a

joint

graph

consisting

of

the

union

of

links

from

the

corresponding

two

networks

and

extract

the

CPM

community

structure

of

this

joint

network

-LRB-

I.

Derenyi

personal

communication

-RRB-

Any

community

from

either

the

t

or

the

t

snap-shot

is

contained

in

exactly

one

community

in

the

joint

graph

as

by

adding

links

to

a

network

the

CPM

communities

can

only

grow

merge

or

remain

unchanged

Thus

the

communities

in

the

joint

graph

provide

a

natural

connection

between

the

communities

at

t

and

at

t

If

a

community

in

the

joint

graph

contains

a

single

community

from

t

and

a

single

community

from

t

then

they

are

matched

If

the

joint

group

contains

more

than

one

community

from

either

time

steps

the

communities

are

matched

in

descending

order

of

their

relative

node

overlap

-LRB-

see

Supplementary

Information

-RRB-

Supplementary

Information

is

linked

to

the

online

version

of

the

paper

at

Acknowledgements

We

thank

I.

Derenyi

for

suggestions

and

G.

Szabo

and

I.

Farkas

for

their

assistance

with

the

primary

phone-call

and

co-authorship

data

sets

respectively

G.P

and

T.V

are

supported

by

OTKA

A.-L.B.

is

supported

by

the

James

S.

McDonnell

Foundation

and

the

National

Science

Foundation

within

the

DDDAS

and

ITR

programmes

Author

Information

Reprints

and

permissions

information

is

available

at

The

authors

declare

competing

financial

interests

details

accompany

the

full-text

HTML

version

of

the

paper

at

nature

Correspondence

and

requests

for

materials

should

be

addressed

to

T.V.

-LRB-

vicsek@angel.elte.hu

-RRB-

Nature

Publishing

Group

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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 B 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 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 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 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 B 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 B B 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 B 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 B O O O O O B 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 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 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 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 B O O O O O O O O O O O O O B 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 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O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O B O O O O O O O O B 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 B 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 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 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 B 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 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 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 B 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 B 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 B B 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 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B RIEFINGS IN BIOINF ORMATICS . VOL 8 . NO 2 . 109 ^ 116 Advance Access publication April 12 , 2007 doi : 10.1093 / bib/bbm007 Bayesian methods in bioinformatics and computational systems biology Darren J.Wilkinson Abstract Bayesian methods are valuable , inter alia , whenever there is a need to extract information from data that are uncertain or subject to any kind of error or noise -LRB- including measurement error and experimental error , as well as noise or random variation intrinsic to the process of interest -RRB- . Bayesian methods offer a number of advantages over more conventional statistical techniques that make them particularly appropriate for complex data . It is therefore no surprise that Bayesian methods are becoming more widely used in the fields of genetics , genomics , bioinformatics and computational systems biology , where making sense of complex noisy data is the norm . This review provides an introduction to the growing literature in this area , with particular emphasis on recent developments in Bayesian bioinformatics relevant to computational systems biology . Keywords : Bayesian inference ; computational systems biology ; networks ; graphical models ; quantitative , predictive biology INTRODUCTION Bioinformatics and computational systems biology are undergoing a Bayesian revolution similar to that already seen in genetics -LSB- 1 -RSB- . The reason is the same -- biology is complex , and data are noisy . Traditional statistical techniques struggle to cope with complex non-linear models that are only partially observed . Due to the fact that the Bayesian statistical paradigm is fully probabilistic , there is no fundamental distinction between any of the unknowns in a statistical model -- parameters , hidden variables and observations are all treated together in a consistent manner -- and it is from this that the power of the methodology is derived -LSB- 2 -RSB- . Provided that you can write down a statistical model relating the quantities you are interested in to the data you can observe -LRB- possibly via many unobserved intermediary variables -RRB- , then you can -LRB- in principle -RRB- carry out Bayesian inference to extract the information in the data to give fully probabilistic information on all unobserved model variables . The main limiting factor in applying Bayesian methods is computational . For non-trivial problems , analytic approaches to Bayesian inference are not possible , and their numerical solution is often challenging due to the need to solve high-dimensional integration problems -LRB- which in the discrete case translate to combinatorial summation problems -RRB- . Advances in the speed of commodity computing hardware in recent decades has been parallelled by developments in computationally intensive algorithms for Bayesian inference . Arguably the most important advance has been the development of a range of techniques based on Markov chain Monte Carlo -LRB- MCMC -RRB- . The ideas originate from statistical physics -LSB- 3 -RSB- , but are now widely used for Bayesian inference -LSB- 4 , 5 -RSB- . Although by no means a panacea , carefully crafted MCMC algorithms executed on fast computers are able to solve a phenomenal range of problems that would have been considered completely intractable only a few years ago . In the simplest -LRB- continuous -RRB- setting , we are interested in making inferences about the parameter vector of a probability -LRB- density -RRB- model p yj giving rise to an observed data vector y . If we treat the parameters as uncertain , and allocate to them Corresponding author . D.J. Wilkinson , School of Mathematics & Statistics , Newcastle University , Newcastle upon Tyne , NE1 7RU , UK . Tel : +44-191-2227320 ; E-mail : d.j.wilkinson@ncl.ac.uk DarrenWilkinson is a Senior Lecturer in Statistics within the School of Mathematics & Statistics at Newcastle University . He has a background in computational Bayesian statistics , and in recent years has become increasingly interested in applications to statistical bioinformatics and computational systems biology . At Newcastle , he is a member of the Centre for Integrated Systems Biology of Ageing and Nutrition -LRB- CISBAN -RRB- and the Systems Biology Resource Centre -LRB- SBRC -RRB- . The Author 2007 Published by Oxford University Press . For Permissions , please email : journals.permissions@oxfordjournals.org . 1 10 Wilkinson a ` prior ' probability density , then Bayes theorem gives the ` posterior ' density jy p yj ; p y BIOINFORMATICS Biological sequence analysis One of the first areas to benefit from the application of Bayesian approaches was biological sequence analysis . Here it had already been recognised that working with probabilistic models was extremely useful -LSB- 13 -RSB- . Whilst for some simple hidden Markov models -LRB- HMMs -RRB- it is possible to estimate parameters using conventional statistical techniques -LRB- such as maximum likelihood via the EM algorithm -RRB- -LSB- 14 , 15 -RSB- , there are many interesting problems where a conventional approach would be inconvenient or unsatisfactory in terms of the information provided by the analysis ; see -LSB- 16 -RSB- for a good introduction to the use of Bayesian methods in this area . Good examples of this include simultaneous multiple sequence alignment -LSB- 17 , 18 -RSB- , motif discovery and transcription factor binding site prediction -LSB- 19 , 20 -RSB- and protein secondary structure prediction -LSB- 21 -RSB- . One of the key benefits of the Bayesian approach is that it allows proper propagation of uncertainty across different levels of modelling . So whilst a traditional approach to phylogeny estimation would use a pre-calculated multiple alignment , uncertainty in the alignment will not propagate through to uncertainty in the phylogeny . In fact , the converse is also true : models for alignment depend implicitly on an assumed phylogeny , so uncertainty in phylogeny induces alignment uncertainty . Using a Bayesian approach , simultaneous estimation is possible -LSB- 22 -RSB- . Even in the relatively simple context of HMM-based ab initio DNA sequence segmentation , the Bayesian approach enables the convenient inclusion of prior information , and provides much richer information about the model parameters -LSB- 23 -RSB- . Furthermore , since uncertainty about model structure is treated consistently with parameter uncertainty in the Bayesian context , variable dimension algorithms such as reversible jump MCMC -LRB- RJMCMC -RRB- -LSB- 24 -RSB- can be used to estimate the number of segments and the order of the base dependence along with all other aspects of the model -LSB- 25 -RSB- . Liu and Logvinenko -LSB- 26 -RSB- provide a detailed review of Bayesian methods in sequence analysis . where p -LRB- y -RRB- is the marginal density for y obtained by integrating over the prior . Since jy is regarded as a function of for fixed -LRB- observed -RRB- y , we can re-write this as jy / p yj ; so that the posterior is proportional to the prior times the likelihood . Practical complications arise due to the fact that typically the normalising constant p -LRB- y -RRB- is not known , and either p yj will not be known explicitly or marginalisation over some components of will be required . Whilst analytically intractable , these integration problems are typically amenable to a Monte Carlo or MCMC solution . In the highdimensional context , it is often necessary to decompose the full problem according to the underlying conditional independence structure of the model , and it is in this context that graphical models -LSB- 6 -RSB- -LRB- also known as conditional independence graphs -RRB- are particularly useful . In non-statistical communities , the term Bayes -LRB- ian -RRB- network is often used to describe a discrete graphical model . However , it is important to note that graphical models can be used to describe any probabilistic conditional independence structure , and that many of the techniques that are often used to ` learn ' Bayesian networks are not Bayesian . The simplest example of a MCMC method is the Gibbs sampler -LSB- 7 , 8 -RSB- . Here a Markov chain is constructed with equilibrium distribution jy . Each iteration of the sampler involves cycling through each component of the p-dimensional vector in order and sampling from i j i ; y ; i 1 , ... ; p , where i denotes the vector of all components of except i. Knowledge of the conditional independence graph for the model can simplify the computation of these so-called full-conditional distributions . In many cases , the fullconditionals will be straightforward to sample directly , but in others , a Metropolis Hastings method will be required -LSB- 9 , 10 -RSB- . Here a proposed new value is simulated from a largely arbitrary proposal distribution , q ? ji and accepted with a i probability carefully chosen to preserve the detailed balance of the chain . Many practical details of the method are presented in -LSB- 11 , 12 -RSB- . Microarray data analysis The analysis of gene microarray data -LSB- 27 -RSB- is another area where Bayesian methods have proven to offer many advantages over more conventional approaches -LSB- 28 , 29 -RSB- . Although amenable to simple statistical analyses such as ANOVA , microarray data Bayesian methods in bioinformatics and computational systems biology analysis is often broken down into a collection of distinct steps that fail to correctly propagate uncertainty . For example , a typical analysis may begin with some kind of normalisation process that produces ` corrected ' expression levels . These normalised data will then be subject to a secondary statistical analysis -LRB- such as identification of differentially expressed genes -RRB- that ignores any uncertainty in the normalisation processes . Often , then the differentially expressed genes will be used for a further analysis that ignores the uncertainty in the identification procedure . Using Bayesian techniques it is possible to develop integrated models for the analysis of unnormalised cDNA microarray data that correctly propagate uncertainty across the various levels of analysis -LSB- 30 , 31 -RSB- . Detailed modelling combined with a carefully designed experiment can allow coherent estimation of absolute transcript concentrations from cDNA array data -LSB- 32 , 33 -RSB- . It is also much more convenient to pool information across multiple experiments and studies using a Bayesian approach -LSB- 34 -RSB- . For Affymetrix GeneChip data , developing probabilistic models of the hybridisation process down at the probe level again allows extraction of information likely to be missed using simpler stepwise approaches -LSB- 35 , 36 -RSB- . Bayesian methods also offer advantages when clustering of expression profiles is felt to be relevant -LSB- 37 39 -RSB- . In fact , the initial task of segmentation and raw intensity estimation can also benefit from a Bayesian approach -LSB- 40 -RSB- . Further modelling approaches and applications are discussed in -LSB- 41 46 -RSB- . Some recent developments in the field are described in -LSB- 29 -RSB- , which also covers some proteomic applications . 11 1 classification -LSB- 53 , 54 -RSB- , as well as protein identification -LSB- 55 , 56 -RSB- . COMPUTATIONAL SYSTEMS BIOLOGY The analysis of microarray data is also central to much research in computational systems biology , although here the emphasis is slightly different . A major concern of computational systems biology is the development of dynamic predictive models of biological -LRB- especially genetic and biochemical -RRB- processes -LSB- 57 -RSB- . The first stage in this process is the identification of interacting partners -LRB- used in a loose sense -RRB- . One approach to identifying gene gene interactions is to attempt to use observed correlations in gene microarray data to infer networks of interaction . Network inference A variety of different approaches to network inference are possible , and many widely used techniques are fundamentally Bayesian in nature . Again , it is worth emphasising the apparent confusion between discrete Bayesian networks and more general Bayesian methods . The term ` Bayes net ' is generally used in non-statistical communities to refer to discrete probabilistic graphical models , irrespective of whether the techniques used to analyse them are Bayesian . Despite some suggestions to the contrary in the literature , there is no need to discretise continuous data in order to learn a Bayesian network -- only to learn a discrete Bayes net . As mentioned above , graphical models can be estimated without using Bayesian methods , but there are advantages in doing so . This is particularly true when the number of observations is small compared to the number of variables , which is typically the case in the context of microarray data analysis . An early , influential paper on Bayesian networks for expression data was -LSB- 58 -RSB- ; also see -LSB- 59 -RSB- for a more recent perspective . An approach based on manipulation experiments for inferring directed networks is described in -LSB- 60 -RSB- . An efficient method for inferring undirected Gaussian graphical models is described in -LSB- 61 -RSB- . More recently , a detailed comparison of various methods for static network inference has been carried out in -LSB- 62 -RSB- . Such methods do not have to be based on microarray data . Typically , using more quantitative data on a -LRB- small -RRB- system of interest will lead to more reliable conclusions . Single-cell flow Protein informatics There are many applications of Bayesian techniques to problems in protein informatics . Down at the structure level , Bayesian techniques for site matching and alignment have been shown to be particularly valuable -LSB- 47 49 -RSB- . A Bayesian method for predicting protein protein interactions from genomic data is given in -LSB- 50 -RSB- . Mass spectrometry data are widely used for understanding the peptide/protein composition of a sample , but these data are subject to many sources of variation , making Bayesian approaches to data analysis highly desirable . Some methods for processing ` raw ' spectra are discussed in -LSB- 51 , 52 -RSB- in the volume -LSB- 29 -RSB- . Bayesian methods can also be useful in the context of mass spectrometry clustering and 1 12 Wilkinson to improve parameter estimation using proper prior modelling of parameter uncertainty -LSB- 79 -RSB- . A nice application of Bayesian modelling in the context of quantitative modelling is the Characterizing Loss Of Cell Cycle Synchrony -LRB- CLOCCS -RRB- model -LSB- 80 -RSB- for loss of synchrony in yeast populations . A simple application of this model is in the alignment of data sets collected under different conditions . However , this model can also be combined with population level data -LRB- such as gene expression array data -RRB- in order to recover information about single-cell dynamics from the population averaged data . This detailed modelling of both the process of interest and its relationship with the experimental data is a powerful technique in this context , and similar strategies are likely to lead to many other examples of extracting better information from high-throughput data . There is increasing evidence that stochasticity plays an important role in intracellular processes -LSB- 81 -RSB- , and there is therefore a great deal of interest in developing stochastic kinetic models of biological processes -LSB- 82 85 -RSB- . Furthermore , experimental technology is improving rapidly , so that -LRB- semi - -RRB- quantitative high-resolution single-cell data of the type that is most informative for the building of stochastic models is now realistically attainable -LSB- 86 -RSB- . Typically , data is generated via fluorescence microscopy , then processed to extract gene expression time series -LSB- 87 -RSB- . Although fully Bayesian approaches to this image-analysis step are likely to be extremely useful , such techniques do not yet seem to have been described in the literature . Stochastic kinetic models are particularly difficult to estimate using non-Bayesian methods . A valiant attempt is described in -LSB- 88 -RSB- , but the applicability of the methods described is limited due to the extent to which non-Bayesian methods can cope with hidden data . In particular , the parsimony assumptions that are typically required have the effect of downwardbiasing of parameter estimates . However , whilst a fully Bayesian approach to inference for discrete stochastic models is possible -LSB- 85 , 89 -RSB- , it is computationally problematic for models of realistic size and complexity . Also see -LSB- 90 -RSB- for a related approach . It turns out to be possible to instead work with a continuous -LRB- approximate -RRB- formulation of stochastic kinetics , known as the ` chemical Langevin equation ' -LSB- 91 , 85 -RSB- . This model seems to be quite adequate for inferential purposes , and is advantageous due to the fact that inference for this diffusion approximation is cytometry data is potentially useful in this context , and a strategy for using it for inferring network structure is described in -LSB- 63 -RSB- . It should be pointed out , however , that most of these papers are not especially Bayesian in their approach . More Bayesian approaches to the problem of inferring sparse undirected -LRB- Gaussian -RRB- graphical models are described in -LSB- 64 -RSB- and -LSB- 65 -RSB- , based on earlier work for graphical Gaussian model selection -LSB- 66 -RSB- , and these are likely to provide more robust inferences in high-dimensional settings , particularly since most methods are able to provide marginal posterior probabilities for the presence of individual network edges . Time-course expression data provide some information about system dynamics , and therefore dynamic network models provide a useful starting point for top-down systems biology modelling . Dynamic Bayesian networks -LRB- DBNs -RRB- have been widely used in this context ; see -LSB- 67 , 68 -RSB- for details . For dynamic networks based on linear Gaussian models , a fast ` Bayesian-inspired ' algorithm has recently been proposed -LSB- 69 -RSB- . As for static networks , fully Bayesian approaches to this problem are likely to offer significant advantages , and are currently the subject of ongoing research . Using Bayesian inference for integrating multiple sources of data offers great potential , but currently remains largely unexplored ; see -LSB- 70 72 -RSB- for initial attempts and perspectives . Quantitative network models As has already been stated , a key aim of systems biology is to develop quantitative , dynamic models of biological processes of interest . One approach to this problem is to extend the top-down network models so that they provide some quantitative information regarding dynamics -LSB- 73 -RSB- . However , this approach has some shortcomings due to the fact that the elements of the model do not link directly to physical parameters of interest . There is therefore great interest in a different approach , based on using data to parameterise bottom-up mechanistic models of biological processes . Obviously , nonBayesian approaches to this problem are possible -LSB- 74 76 -RSB- , but are limited in terms of the information they can provide . Even in the context of deterministic models of biochemical networks based on ordinary differential equations -LRB- ODEs -RRB- , there is considerable utility in using a Bayesian approach in order to properly address issues of noise modelling and parameter uncertainty -LSB- 77 , 78 -RSB- . It is also possible Bayesian methods in bioinformatics and computational systems biology more computationally amenable than for the discrete formulation . A basic inferential algorithm for this model is described in -LSB- 92 -RSB- . A better algorithm for models of this type , based on ideas of sequential Monte Carlo -LSB- 93 -RSB- , is developed in -LSB- 94 -RSB- , and applied to a general and flexible class of stochastic kinetic models in -LSB- 95 -RSB- . Finally , an efficient non-sequential MCMC algorithm for stochastic kinetic models is described in -LSB- 96 -RSB- . A recent review of fitting models to data by Jaqaman and Danuser -LSB- 97 -RSB- includes references to both the Bayesian and non-Bayesian literature . There is another area of statistical methodology that has obvious applications to systems biology modelling : Bayesian analysis of computer code outputs -LRB- BACCO -RRB- -LSB- 98 -RSB- . Here , a complex -LRB- but typically , deterministic -RRB- computer simulation model is treated as a ` black-box ' from a statistical perspective , and the relationships between model inputs , outputs and experimental data are studied in a non-parametric way , often utilising Gaussian processes -LSB- 99 -RSB- . Although these techniques do not yet seem to have been applied to systems biology modelling problems , they have been applied to challenging problems in other application areas -LSB- 100 , 101 -RSB- , so it seems inevitable that as systems biology models become larger and more complex , and BACCO techniques become more sophisticated -LRB- better suited to high-dimensional inputs and outputs , and intrinsic stochasticity in the computer models -RRB- , that applications of BACCO methods to problems in computational systems biology will become common-place . 1 13 framework provides a coherent mathematical solution to the problem , but not always an efficient computational algorithm for practical implementation . Even in difficult scenarios , however , probabilistic statistical models -LRB- such as Hidden Markov Models -RRB- are becoming the accepted framework for analysis -LSB- 13 -RSB- , and used in conjunction with point estimation methods -LRB- such as the EM algorithm -RRB- for parameter fitting . However , experience from closely related disciplines suggests that fully Bayesian approaches will turn out to provide the most satisfactory solutions to the complex statistical inference problems which lie at the heart of computational systems biology . Improvements in computing hardware , the widespread availability of parallel computer clusters and the development of computational Bayesian algorithms that are able to exploit them -LSB- 102 -RSB- mean that there is likely to be an increasing tendency to push for fully Bayesian solutions to the challenging inferential problems in this area , in order to maximise the information that can be extracted from expensive experimental data . 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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 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 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 B 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 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 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 B 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 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 B 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 B 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 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 B 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 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 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 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 O O O O O O O O O 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 B 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 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 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 B O O B O O B 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 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 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 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 B 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 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 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 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 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 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 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 O O O O O O 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 B 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 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 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 B O O O O O O O O B 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 B O O O O O B 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 O O O O O O O O O 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 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 B 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 B 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 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 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 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 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 B 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 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 B 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 B 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 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 B O O O O B 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 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 B 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 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 B 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 B O O B 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 B 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 B O O O O O O O O O O O O

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