community structures of multilayer political cosponsorship networks

Sang Hoon Lee Department of Energy Science, Sungkyunkwan University

http://sites.google.com/site/lshlj82

Community Structures of Multilayer Political Cosponsorship Networks

The 2nd Daegu Gyeongbuk International Social Network Conference (DISC 2014)

Community structures in networks

modularity (the objective function to be maximized)

M. A. Porter, J.-P. Onnela, and P. J. Mucha, Not. Am. Math. Soc. 56, 1082 (2009); S. Fortunato, Phys. Rep. 486, 75 (2010).

Q =1

2m

Xij

Aij kikj

2m

(gi, gj)

where the adjacency matrixAij 6= 0 if nodes i and j are connected and Aij = 0 otherwise,ki is the degree (number of neighboring nodes of i)or strength (sum of weights around i),gi is the community to which i belongs,and m is the total number of edges or sum of weights in the network

resolution parameter: controlling the characteristic size of communities

importing network dataidentifying community structure

visualizing

smal

ler

com

mun

ities

TA

XO

NO

MIE

SO

FN

ETW

ORK

SFR

OM

COM

MU

NIT

YST

RUCT

URE

PHY

SICA

LR

EVIE

WE

86,

0361

04(20

12)

that

alle

dges

are

antif

erro

mag

netic

atre

solu

tion="

max

and

ther

eby

forc

esea

chnode

into

itsow

nco

mm

unity

.

III.

MES

OSC

OPI

CR

ESPO

NSE

FUN

CTI

ON

S(M

RFS)

Tode

scrib

eho

wa

net

work

disin

tegr

ates

into

com

mu

niti

esas

thev

alue

of

isin

crea

sed

from"

min

to"

max

[see

Fig.

1(a)

fora

sche

mat

ic],o

ne

nee

dsto

sele

ctsu

mm

ary

stat

istic

s.Th

ere

are

man

ypo

ssib

lew

ays

tosu

mm

ariz

esu

cha

disin

tegr

atio

npr

oces

s,an

dw

efo

cus

on

thre

edi

agno

stics

that

char

acte

rize

fund

amen

talp

rope

rties

ofn

etw

ork

com

muniti

es.

Firs

t,w

euse

the

val

ueoft

heH

amilt

onia

nH(

)(

1),w

hich

isa

scal

arqu

antit

ycl

osel

yre

late

dto

net

work

modu

larit

yan

dqu

antifi

esth

een

ergy

of

the

syste

m[1

3,14

].Se

cond

,w

eca

lcul

ate

apa

rtitio

nen

trop

yS

()

toch

arac

teriz

eth

eco

mm

unity

size

distr

ibutio

n.To

doth

is,le

tnk

deno

teth

enum

ber

of

node

sin

com

munity

kan

dde

finepk=nk/N

tobe

the

prob

abili

tyto

choo

sea

node

from

com

munity

kunifo

rmly

atra

ndo

m.T

hisy

ield

sa(S

hann

on)p

artit

ione

ntr

opy

ofS

()=

(

)k=1pk

logp

k,

whi

chqu

antifi

esth

edi

sord

erin

the

asso

ciat

edco

mm

unity

size

distr

ibutio

n.Th

ird,w

euse

the

num

bero

fcom

muniti

es

().

=1,

=34

=0,

=1

=0.2

, =

8=0

.4,

=12

=0.6

, =

17=0

.8,

=24

= 0.

2 =

0.

4 =

0.

6 =

0.

8 =

0

= 1

00.2

0.4

0.6

0.81

ferro

mag

netic

link

snonlin

ksantif

erro

mag

netic

link

s

(a)

(c)

(b)

Heff

S eff

eff

FIG

.1.

(Colo

ronlin

e)(a)

Sche

mat

icofs

om

eoft

hew

ays

that

a

net

work

can

brea

kup

into

com

muniti

esas

the

val

ue

of

(or

)is

incr

ease

d.(b)

Zach

ary

Kar

ate

Club

net

work

[23]

for

diffe

ren

tval

ues

oft

heef

fect

ive

fract

ion

ofa

ntif

erro

mag

netic

edge

s.A

llin

tera

ctio

ns

are

eith

erfe

rrom

agnet

icor

antif

erro

mag

net

ic;i

.e.,

for

the

val

ues

of

th

atw

euse

d,th

ere

are

no

neu

tral

inte

ract

ions

.We

colo

red

ges

inbl

ueif

the

corr

espo

ndin

gin

tera

ctio

nsar

efe

rrom

agne

tic,a

nd

we

colo

rth

emin

red

ifth

ein

tera

ctio

ns

are

antif

erro

mag

net

ic.

We

colo

r

the

node

sbas

edon

com

munity

affil

iatio

n.(c)

TheH e

ff,S

eff,

and

eff

MR

Fs,a

nd

the

inte

ract

ion

mat

rixJ

for

diffe

rent

val

ues

of

.W

e

colo

rel

emen

tsof

the

inte

ract

ion

mat

rixby

depi

ctin

gth

eab

sence

of

aned

gein

whi

te,

ferr

om

agnet

iced

ges

inbl

ue

(dark

gray

),an

dan

tifer

rom

agne

ticed

gesi

nre

d(li

ghtg

ray).

Bec

ause

we

nee

dto

norm

aliz

eH,S

,an

dto

com

pare

them

acro

ssnet

work

s,w

ede

fine

aneff

ectiv

eener

gy

H eff

()=

H(

)H m

in

H maxH m

in=

1H(

)

H min,

(4)

whe

reH m

in=H(

"m

in)a

ndH m

ax=H(

"m

ax);

aneff

ectiv

een

tropy

Sef

f()=

S(

)S

min

Sm

axS

min=

S(

)lo

gN,

(5)

whe

reS

min=S

("m

in)a

ndS

max=S

("m

ax);

and

aneff

ectiv

enum

bero

fcom

muniti

es

ef

f()=

(

)

min

m

ax

min=

(

)1

N

1,

(6)

whe

re

min=

("m

in)=

1an

d

max=

("m

ax)=

N.

Som

enet

work

sco

nta

ina

smal

lnum

ber

of

entr

ies"

ij

that

are

ord

ers

of

mag

nitu

dela

rger

than

most

oth

eren

trie

s.Fo

rex

ampl

e,in

the

net

work

of

Face

boo

kfri

ends

hips

atCa

ltech

[21,

22],

98%

of

the"

ijen

trie

sar

ele

ssth

an10

0,bu

t0.

02%

of

them

are

larg

erth

an80

00.

Thes

ela

rge"

ij

val

ues

arise

whe

ntw

olo

w-st

ren

gth

no

des

beco

me

con

nec

ted.

Usin

gthe

null

mode

lPij=k ik j/(2

m),t

hein

tera

ctio

nbet

wee

ntw

onode

si

andj

beco

mes

antif

erro

mag

netic

whe

n>

Aij/P

ij=

2mAij/(k

ikj).

Ifa

net

work

has

ala

rge

tota

ledg

ew

eigh

tbu

tbo

thi

andj

have

smal

lst

reng

ths

com

pare

dto

oth

ernode

sin

the

net

work

,th

en

nee

dsto

bela

rge

tom

ake

the

inte

ract

ion

antif

erro

mag

netic

.In

prio

rst

udie

s,net

work

com

munity

stru

ctur

eha

sbee

ninv

estig

ated

atdi

ffere

nt

mes

osc

opi

csc

ales

byco

nsid

erin

gpl

otso

fvar

ious

diag

nosti

csas

afu

nct

ion

of

the

reso

lutio

npa

ram

eter

[1

3,14

,17

].In

the

pres

ent

exam

ple,

such

plot

sw

ould

bedo

min

ated

byin

tera

ctio

nsth

atre

quire

larg

ere

solu

tion-

para

met

erval

ues

tobe

com

ean

tifer

rom

agne

tic.T

oover

com

eth

isiss

ue,w

ede

fine

the

effec

tivef

ractio

nofa

ntife

rrom

agn

etic

edge

s

=

()=

A(

)A

("m

in)

A("

max

)A

("m

in)

[0,1

],(7)

whe

reA

()i

sth

eto

tal

nu

mbe

ro

fan

tifer

rom

agn

etic

in-

tera

ctio

nsfo

rth

egi

ven

val

ueof

.In

oth

erw

ord

s,it

isth

enum

ber

of"

ijel

emen

tsth

atar

esm

alle

rth

an

.Th

us,

A("

min

)is

the

larg

estn

um

bero

fantif

erro

mag

netic

inte

rac-

tions

forw

hich

anet

work

still

form

sasin

gle

com

munity

,an

dth

eef

fect

ive

num

ber

of

antif

erro

mag

netic

inte

ract

ions

(

)is

the

num

ber

of

antif

erro

mag

netic

inte

ract

ions

(norm

alize

dto

the

unit

inte

rval

)in

exce

ssof

A("

min

).Th

efu

nct

ion

()

incr

ease

smonoto

nica

llyin

.

Swee

ping

fro

m"

min

to"

max

corr

espo

nds

tosw

eepi

ngth

eval

ueof

from

0to

1.(O

neca

nth

ink

of

asa

contin

uous

var

iabl

ean

d

asa

disc

rete

var

iabl

etha

tcha

nges

with

even

ts.)

Asw

epe

rform

such

swee

ping

fora

given

net

work

,the

num

ber

ofc

om

muniti

esin

crea

sesf

rom

(=

0)=

1to

(=

1)=N

andy

ield

savec

tor[H e

ff(

),Sef

f(),

eff(

)]who

seco

mpo

nent

sw

eca

llth

em

esosc

opi

cre

spon

sefun

ction

s(M

RFs)

of

that

net

work

.(W

eal

soso

met

imes

refe

rto

the

vec

tor

itsel

fas

anM

RF.

)Bec

auseH e

ff

[0,1

],S

eff

[0,1

],

eff

[0,1

],an

d

[0,1

]for

ever

ynet

work

,we

can

com

pare

theM

RFs

acro

ssnet

work

san

duse

them

toid

entif

ygr

oups

of

net

work

sw

ithsim

ilar

mes

osc

opi

cst

ruct

ures

.In

Fig.

1(b),

we

show

the

Zach

ary

Kar

ate

Club

net

work

[23]

for

diffe

rent

valu

esof

0361

04-3

J.-P. Onnela et al., Phys. Rev. E 86, 036104 (2012).

Community structures in networks

modularity (the objective function to be maximized)

M. A. Porter, J.-P. Onnela, and P. J. Mucha, Not. Am. Math. Soc. 56, 1082 (2009); S. Fortunato, Phys. Rep. 486, 75 (2010).

Q =1

2m

Xij

Aij kikj

2m

(gi, gj)

where the adjacency matrixAij 6= 0 if nodes i and j are connected and Aij = 0 otherwise,ki is the degree (number of neighboring nodes of i)or strength (sum of weights around i),gi is the community to which i belongs,and m is the total number of edges or sum of weights in the network

resolution parameter: controlling the characteristic size of communities

importing network dataidentifying community structure

visualizing

smal

ler

com

mun

ities

TA

XO

NO

MIE

SO

FN

ETW

ORK

SFR

OM

COM

MU

NIT

YST

RUCT

URE

PHY

SICA

LR

EVIE

WE

86,

0361

04(20

12)

that

alle

dges

are

antif

erro

mag

netic

atre

solu

tion="

max

and

ther

eby

forc

esea

chnode

into

itsow

nco

mm

unity

.

III.

MES

OSC

OPI

CR

ESPO

NSE

FUN

CTI

ON

S(M

RFS)

Tode

scrib

eho

wa

net

work

disin

tegr

ates

into

com

mu

niti

esas

thev

alue

of

isin

crea

sed

from"

min

to"

max

[see

Fig.

1(a)

fora

sche

mat

ic],o

ne

nee

dsto

sele

ctsu

mm

ary

stat

istic

s.Th

ere

are

man

ypo

ssib

lew

ays

tosu

mm

ariz

esu

cha

disin

tegr

atio

npr

oces

s,an

dw

efo

cus

on

thre

edi

agno

stics

that

char

acte

rize

fund

amen

talp

rope

rties

ofn

etw

ork

com

muniti

es.

Firs

t,w

euse

the

val

ueoft

heH

amilt

onia

nH(

)(

1),w

hich

isa

scal

arqu

antit

ycl

osel

yre

late

dto

net

work

modu

larit

yan

dqu

antifi

esth

een

ergy

of

the

syste

m[1

3,14

].Se

cond

,w

eca

lcul

ate

apa

rtitio

nen

trop

yS

()

toch

arac

teriz

eth

eco

mm

unity

size

distr

ibutio

n.To

doth

is,le

tnk

deno

teth

enum

ber

of

node

sin

com

munity

kan

dde

finepk=nk/N

tobe

the

prob

abili

tyto

choo

sea

node

from

com

munity

kunifo

rmly

atra

ndo

m.T

hisy

ield

sa(S

hann

on)p

artit

ione

ntr

opy

ofS

()=

(

)k=1pk

logp

k,

whi

chqu

antifi

esth

edi

sord

erin

the

asso

ciat

edco

mm

unity

size

distr

ibutio

n.Th

ird,w

euse

the

num

bero

fcom

muniti

es

().

=1,

=34

=0,

=1

=0.2

, =

8=0

.4,

=12

=0.6

, =

17=0

.8,

=24

= 0.

2 =

0.

4 =

0.

6 =

0.

8 =

0

= 1

00.2

0.4

0.6

0.81

ferro

mag

netic

link

snonlin

ksantif

erro

mag

netic

link

s

(a)

(c)

(b)

Heff

S eff

eff

FIG

.1.

(Colo

ronlin

e)(a)

Sche

mat

icofs

om

eoft

hew

ays

that

a

net

work

can

brea

kup

into

com

muniti

esas

the

val

ue

of

(or

)is

incr

ease

d.(b)

Zach

ary

Kar

ate

Club

net

work

[23]

for

diffe

ren

tval

ues

oft

heef

fect

ive

fract

ion

ofa

ntif

erro

mag

netic

edge

s.A

llin

tera

ctio

ns

are

eith

erfe

rrom

agnet

icor

antif

erro

mag

net

ic;i

.e.,

for

the

val

ues

of

th

atw

euse

d,th

ere

are

no

neu

tral

inte

ract

ions

.We

colo

red

ges

inbl

ueif

the

corr

espo

ndin

gin

tera

ctio

nsar

efe

rrom

agne

tic,a

nd

we

colo

rth

emin

red

ifth

ein

tera

ctio

ns

are

antif

erro

mag

net

ic.

We

colo

r

the

node

sbas

edon

com

munity

affil

iatio

n.(c)

TheH e

ff,S

eff,

and

eff

MR

Fs,a

nd

the

inte

ract

ion

mat

rixJ

for

diffe

rent

val

ues

of

.W

e

colo

rel

emen

tsof

the

inte

ract

ion

mat

rixby

depi

ctin

gth

eab

sence

of

aned

gein

whi

te,

ferr

om

agnet

iced

ges

inbl

ue

(dark

gray

),an

dan

tifer

rom

agne

ticed

gesi

nre

d(li

ghtg

ray).

Bec

ause

we

nee

dto

norm

aliz

eH,S

,an

dto

com

pare

them

acro

ssnet

work

s,w

ede

fine

aneff

ectiv

eener

gy

H eff

()=

H(

)H m

in

H maxH m

in=

1H(

)

H min,

(4)

whe

reH m

in=H(

"m

in)a

ndH m

ax=H(

"m

ax);

aneff

ectiv

een

tropy

Sef

f()=

S(

)S

min

Sm

axS

min=

S(

)lo

gN,

(5)

whe

reS

min=S

("m

in)a

ndS

max=S

("m

ax);

and

aneff

ectiv

enum

bero

fcom

muniti

es

ef

f()=

(

)

min

m

ax

min=

(

)1

N

1,

(6)

whe

re

min=

("m

in)=

1an

d

max=

("m

ax)=

N.

Som

enet

work

sco

nta

ina

smal

lnum

ber

of

entr

ies"

ij

that

are

ord

ers

of

mag

nitu

dela

rger

than

most

oth

eren

trie

s.Fo

rex

ampl

e,in

the

net

work

of

Face

boo

kfri

ends

hips

atCa

ltech

[21,

22],

98%

of

the"

ijen

trie

sar

ele

ssth

an10

0,bu

t0.

02%

of

them

are

larg

erth

an80

00.

Thes

ela

rge"

ij

val

ues

arise

whe

ntw

olo

w-st

ren

gth

no

des

beco

me

con

nec

ted.

Usin

gthe

null

mode

lPij=k ik j/(2

m),t

hein

tera

ctio

nbet

wee

ntw

onode

si

andj

beco

mes

antif

erro

mag

netic

whe

n>

Aij/P

ij=

2mAij/(k

ikj).

Ifa

net

work

has

ala

rge

tota

ledg

ew

eigh

tbu

tbo

thi

andj

have

smal

lst

reng

ths

com

pare

dto

oth

ernode

sin

the

net

work

,th

en

nee

dsto

bela

rge

tom

ake

the

inte

ract

ion

antif

erro

mag

netic

.In

prio

rst

udie

s,net

work

com

munity

stru

ctur

eha

sbee

ninv

estig

ated

atdi

ffere

nt

mes

osc

opi

csc

ales

byco

nsid

erin

gpl

otso

fvar

ious

diag

nosti

csas

afu

nct

ion

of

the

reso

lutio

npa

ram

eter

[1

3,14

,17

].In

the

pres

ent

exam

ple,

such

plot

sw

ould

bedo

min

ated

byin

tera

ctio

nsth

atre

quire

larg

ere

solu

tion-

para

met

erval

ues

tobe

com

ean

tifer

rom

agne

tic.T

oover

com

eth

isiss

ue,w

ede

fine

the

effec

tivef

ractio

nofa

ntife

rrom

agn

etic

edge

s

=

()=

A(

)A

("m

in)

A("

max

)A

("m

in)

[0,1

],(7)

whe

reA

()i

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04-3

J.-P. Onnela et al., Phys. Rev. E 86, 036104 (2012).

6www.nature.com/nature

doi: 10.1038/nature09182 SUPPLEMENTARY INFORMATION

Gavroche

ValjeanBossuet

Mabeuf

Bahorel

Grantaire

Gervais

Fauchelevent

Gribier

Fameuil

Listolier

Thenardier

Bamatabois

Champmathieu

MmeHucheloup

Montparnasse

Courfeyrac

Enjolras

Gillenormand

Fantine

Tholomyes

MariusJoly

Brujon

GueulemerFavourite

Zephine

Eponine

MmeMagloireMyriel

MmeThenardier

Cosette

LtGillenormand

MlleGillenormand

Feuilly

MlleBaptistine

Blacheville

Claquesous

CombeferreJavert

Woman1

Dahlia

Child1

Child2

Perpetue

Simplice

Babet

Pontmercy

Chenildieu

Napoleon

CravatteChamptercier

Scaufflaire

Boulatruelle

Labarre

Judge

BaronessTCountessDeLo

Isabeau

Marguerite Brevet

Cochepaille

MmePontmercy

MlleVaubois

Magnon

Woman2

Prouvaire

MmeDeR

Toussaint

Count

MotherPlutarch

MmeBurgon

MotherInnocent

Anzelma

OldMan

Jondrette

Geborand

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 3: Link communities for the coappearance network of characters in the novel Les Miserables [9]. (Top) the networkwith link colors indicating the clustering, with grey indicating single-link clusters. Each node is depicted as a pie-chartrepresenting its membership distribution. The main characters have more diverse community membership. (Bottom) thefull link dendrogram (left) and partition density (right). Note the internal blue community in the large blue and red cliquecontaining Valjean. Link clustering is able to unveil hierarchical structure even inside of cliques.

2.3.1 Clique percolation

Clique percolation [11, 15] provides an elegant and highly useful method to uncover overlapping com-munity structure [16]. It is currently the most popular and most successful tool available for this task.A particularly interesting feature of this method is that it presents the experimenter with a knob k, theclique size, which can be used to tune the result between high coverage, low community quality (sparsecommunities) and low coverage, high community quality (dense communities). For some networks,such as the mobile phone network, a precedent exists for the choice of k, which we follow. Wheneverthat is not the case, we have computed the composite performance for a range of ks and chosen the kwhich results in the optimum overall performance2. This weighs coverage and quality equally, however,and it remains at the discretion of the researcher to decide if this is optimal for his or her application.See Appendix A.2.

2For some of the very large or very dense networks, we were not able to run clique percolation for large values of k with thefastest existing software (even on a machine with 32 Gb of RAM), using the fast algorithm developed by Kumpala et al. [17].

6

Y.-Y. Ahn, J. P. Bagrow, and S. Lehmann, Nature 466, 761 (2010).

note: i and j are node indices, and s and r are layer indices.The adjacency tensor Aijs 6= 0 if nodes i and j are connectedin layer s, and Aijs = 0 otherwise.kis is the degree (or strength) of node i in layer s,ms is the number of edges (or sum of weights) in layer s,and s = is the resolution parameter in layer s.Cjsr = ! 6= 0 if layers s and r are connected via node j,and Cjsr = 0 otherwise.The normalization factor 2 =

PijsAijs +

Pjsr Cjsr for Qmultilayer 2 [1, 1].

Qmultilayer =1

2

Xijsr

Aijs s kiskjs

2ms

sr + ijCjsr

(gis, gjr)

Community Structure inTime-Dependent, Multiscale,and Multiplex NetworksPeter J. Mucha,1,2* Thomas Richardson,1,3 Kevin Macon,1 Mason A. Porter,4,5 Jukka-Pekka Onnela6,7

Network science is an interdisciplinary endeavor, with methods and applications drawn from acrossthe natural, social, and information sciences. A prominent problem in network science is thealgorithmic detection of tightly connected groups of nodes known as communities. We developed ageneralized framework of network quality functions that allowed us to study the communitystructure of arbitrary multislice networks, which are combinations of individual networks coupledthrough links that connect each node in one network slice to itself in other slices. This frameworkallows studies of community structure in a general setting encompassing networks that evolve overtime, have multiple types of links (multiplexity), and have multiple scales.

Thestudy of graphs, or networks, has a longtradition in fields such as sociology andmathematics, and it is now ubiquitous inacademic and everyday settings. An importanttool in network analysis is the detection ofmesoscopic structures known as communities (orcohesive groups), which are defined intuitively asgroups of nodes that are more tightly connected toeach other than they are to the rest of the network(13). One way to quantify communities is by aquality function that compares the number ofintracommunity edges to what one would expectat random.Given the network adjacencymatrixA,where the element Aij details a direct connectionbetween nodes i and j, one can construct a qual-ity functionQ (4, 5) for the partitioning of nodesinto communities as Q = ij (Aij Pij)d(gi, gj),where d(gi, gj) = 1 if the community assignmentsgi and gj of nodes i and j are the same and 0otherwise, and Pij is the expected weight of theedge between i and j under a specified null model.

The choice of null model is a crucial con-sideration in studying network community struc-ture (2). After selecting a null model appropriateto the network and application at hand, one canuse a variety of computational heuristics to assignnodes to communities to optimize the quality Q(2, 3). However, such null models have not beenavailable for time-dependent networks; analyseshave instead depended on ad hoc methods to

piece together the structures obtained at differenttimes (69) or have abandoned quality functionsin favor of such alternatives as the MinimumDescriptionLength principle (10). Although tensordecompositions (11) have been used to clusternetwork data with different types of connections,no quality-function method has been developedfor such multiplex networks.

We developed a methodology to remove theselimits, generalizing the determination of commu-nity structure via quality functions to multislicenetworks that are defined by coupling multipleadjacency matrices (Fig. 1). The connectionsencoded by the network slices are flexible; theycan represent variations across time, variationsacross different types of connections, or evencommunity detection of the same network atdifferent scales. However, the usual procedure forestablishing a quality function as a direct count ofthe intracommunity edge weight minus that

expected at random fails to provide any contribu-tion from these interslice couplings. Because theyare specified by common identifications of nodesacross slices, interslice couplings are either presentor absent by definition, so when they do fall insidecommunities, their contribution in the count of intra-community edges exactly cancels that expected atrandom. In contrast, by formulating a null model interms of stability of communities under Laplaciandynamics, we have derived a principled generaliza-tion of community detection to multislice networks,

REPORTS

1Carolina Center for Interdisciplinary Applied Mathematics,Department of Mathematics, University of North Carolina,Chapel Hill, NC 27599, USA. 2Institute for Advanced Materials,Nanoscience and Technology, University of North Carolina,Chapel Hill, NC 27599, USA. 3Operations Research, NorthCarolina State University, Raleigh, NC 27695, USA. 4OxfordCentre for Industrial and Applied Mathematics, MathematicalInstitute, University of Oxford, Oxford OX1 3LB, UK. 5CABDyNComplexity Centre, University of Oxford, Oxford OX1 1HP, UK.6Department of Health Care Policy, Harvard Medical School,Boston, MA 02115, USA. 7Harvard Kennedy School, HarvardUniversity, Cambridge, MA 02138, USA.

*To whom correspondence should be addressed. E-mail:[email protected]

1

2

3

4

Fig. 1. Schematic of amultislice network. Four slicess= {1, 2, 3, 4} represented by adjacencies Aijs encodeintraslice connections (solid lines). Interslice con-nections (dashed lines) are encoded byCjrs, specifyingthe coupling of node j to itself between slices r and s.For clarity, interslice couplings are shown for only twonodes and depict two different types of couplings: (i)coupling between neighboring slices, appropriate forordered slices; and (ii) all-to-all interslice coupling,appropriate for categorical slices.

no

des

resolution parameters

coupling = 0

1 2 3 4

5

10

15

20

25

30

no

des


coupling = 0.1

1 2 3 4

5

10

15

20

25

30

no

des


coupling = 1

1 2 3 4

5

10

15

20

25

30

Fig. 2. Multislice community detection of theZachary Karate Club network (22) across multipleresolutions. Colors depict community assignments ofthe 34 nodes (renumbered vertically to groupsimilarly assigned nodes) in each of the 16 slices(with resolution parameters gs = {0.25, 0.5,, 4}),for w = 0 (top), w = 0.1 (middle), and w =1 (bottom). Dashed lines bound the communitiesobtained using the default resolution (g = 1).

14 MAY 2010 VOL 328 SCIENCE www.sciencemag.org876

CORRECTED 16 JULY 2010; SEE LAST PAGE

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Multilayer community detection

P. J. Mucha, T. Richardson, K. Macon, M. A. Porter, and J.-P. Onnela, Science 328, 876 (2010).

different slices: time series or categories

nodes in individual slices

(weighted) edges

with a single parameter controlling the interslicecorrespondence of communities.

Important to our method is the equivalencebetween themodularity quality function (12) [witha resolution parameter (5)] and stability of com-munities under Laplacian dynamics (13), whichwe have generalized to recover the null models forbipartite, directed, and signed networks (14). First,we obtained the resolution-parameter generaliza-

tion of Barbers null model for bipartite networks(15) by requiring the independent joint probabilitycontribution to stability in (13) to be conditionalon the type of connection necessary to stepbetween two nodes. Second, we recovered thestandard null model for directed networks (16, 17)(again with a resolution parameter) by generaliz-ing the Laplacian dynamics to include motionalong different kinds of connectionsin this case,

both with and against the direction of a link. Bythis generalization, we similarly recovered a nullmodel for signed networks (18). Third, weinterpreted the stability under Laplacian dynamicsflexibly to permit different spreading weights onthe different types of links, giving multiple reso-lution parameters to recover a general null modelfor signed networks (19).

We applied these generalizations to derive nullmodels for multislice networks that extend theexisting quality-function methodology, includingan additional parameter w to control the couplingbetween slices. Representing each network slice sby adjacencies Aijs between nodes i and j, withinterslice couplingsCjrs that connect node j in slicer to itself in slice s (Fig. 1), we have restricted ourattention to unipartite, undirected network slices(Aijs = Ajis) and couplings (Cjrs = Cjsr), but we canincorporate additional structure in the slices andcouplings in the same manner as demonstrated forsingle-slice null models. Notating the strengths ofeach node individually in each slice by kjs =iAijsand across slices by cjs = rCjsr, we define themultislice strength by kjs = kjs + cjs. The continuous-time Laplacian dynamics given by

pis jrAijsdsr dijCjsrpjr

kjr pis 1

respects the intraslice nature of Aijs and theinterslice couplings of Cjsr. Using the steady-stateprobability distribution pjr kjr=2m, where 2m = jrkjr, we obtained the multislice null model interms of the probability ris| jr of sampling node i inslice s conditional on whether the multislice struc-ture allowsone to step from ( j, r) to (i, s), accountingfor intra- and interslice steps separately as

risj jrpjr

kis2ms

kjrkjr

dsr Cjsrcjrcjrkjr

dij

! "kjr2m

2

where ms = jkjs. The second term in parentheses,which describes the conditional probability ofmotion between two slices, leverages the definitionof the Cjsr coupling. That is, the conditionalprobability of stepping from ( j, r) to (i, s) alongan interslice coupling is nonzero if and only if i = j,and it is proportional to the probability Cjsr/kjr ofselecting the precise interslice link that connects toslice s. Subtracting this conditional joint probabilityfrom the linear (in time) approximation of theexponential describing the Laplacian dynamics,weobtained a multislice generalization of modularity(14):

Qmultislice 12m ijsrh#

Aijs gskiskjs2ms

dsr$

dijCjsridgis,gjr 3

where we have used reweighting of the conditionalprobabilities, which allows a different resolution gsin each slice. We have absorbed the resolution pa-rameter for the interslice couplings into the mag-nitude of the elements ofCjsr, which, for simplicity,we presume to take binary values {0,w} indicatingthe absence (0) or presence (w) of interslice links.

1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 2000

40PA, 24F, 8AA

151DR, 30AA, 14PA, 5F141F, 43DR

44D, 2R

1784R, 276D, 149DR, 162J, 53W, 84other

176W, 97AJ, 61DR, 49A,24D, 19F, 13J, 37other

3168D, 252R, 73other

222D, 6W, 11other

1490R, 247D, 19other

Year

Sena

tor

10 20 30 40 50 60 70 80 90 100 110CTMEMANHRI VTDE NJNY PAIL INMI OHWI IAKSMNMONENDSDVA ALAR FLGA LAMSNCSC TXKYMDOK TNWVAZCO IDMTNVNMUTWYCAORWAAK HI

Congress #

A

B

Fig. 3. Multislice community detection of U.S. Senate roll call vote similarities (23) withw = 0.5 couplingof 110 slices (i.e., the number of 2-year Congresses from 1789 to 2008) across time. (A) Colors indicateassignments to nine communities of the 1884 unique senators (sorted vertically and connected acrossCongresses by dashed lines) in each Congress in which they appear. The dark blue and red communitiescorrespond closely to the modern Democratic and Republican parties, respectively. Horizontal barsindicate the historical period of each community, with accompanying text enumerating nominal partyaffiliations of the single-slice nodes (each representing a senator in a Congress): PA, pro-administration;AA, anti-administration; F, Federalist; DR, Democratic-Republican; W, Whig; AJ, anti-Jackson; A, Adams; J,Jackson; D, Democratic; R, Republican. Vertical gray bars indicate Congresses in which three communitiesappeared simultaneously. (B) The same assignments according to state affiliations.

www.sciencemag.org SCIENCE VOL 328 14 MAY 2010 877

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Social Facebook Political: voting

Political: cosponsorship Political: committee Protein interaction

Metabolic Brain Fungal

Financial

Language Collaboration

effeffeff

FIG. 7. (Color online) MRFs for all of the network categoriescontaining at least eight networks (see Table I). At each value of , theupper curve shows the maximum value ofHeff (magenta, left panel ineach category), Seff (blue, center panels), and eff (black, right panels)for all networks in the category and the lower curve shows the mini-mum value. The dashed curves show the corresponding mean MRFs.

A. Voting in the United States SenateOur first example deals with roll-call voting in the US

Senate [3134,48]. Establishing a taxonomy of networksdetailing the voting similarities of individual legislators com-plements previous studies of these data, and it facilitatesthe comparison of voting similarity networks across time.We consider Congresses 1110, which cover the period17892008. As in Ref. [34], we construct networks from theroll-call data [31,32] for each two-year Congress such that theadjacency matrix element Aij [0,1] represents the numberof times Senators i and j voted the same way on a bill (eitherboth in favor of it or both against it) divided by the total numberof bills on which both of them voted. Following the approachof Ref. [32], we consider only nonunanimous roll-call votes,which are defined as votes in which at least 3% of the Senatorswere in the minority.

Much research on the US Congress has been devoted tothe ebb and flow of partisan polarization over time and theinfluence of parties on roll-call voting [33,34]. In highlypolarized legislatures, representatives tend to vote alongparty lines, so there are strong similarities in the votingpatterns of members of the same party and strong differencesbetween members of different parties. In contrast, duringperiods of low polarization, the party lines become blurred.The notion of partisan polarization can be used to helpunderstand the taxonomy of Senates in Fig. 8, in which weconsider two measures of polarization. The first measure usesDW-Nominate scores (a multidimensional scaling techniquecommonly used in political science [32,33]), where the extentof polarization is given by the absolute value of the differencebetween the mean first-dimension DW-Nominate scores formembers of one party and the same mean for members ofthe other party [3133]. In particular, we use the simplestsuch measure of polarization, called MPR polarization, whichassumes a competitive two-party system and hence cannot becalculated prior to the 46th Senate. The second measure thatwe consider is the maximum modularity Q over partitions of

0

0.11

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10 20 30 40 50 60 70 80 90 100 1100

0.2

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Mod

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ity (Q

)D

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ate

pola

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Mod

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ity (Q

)D

W-N

omin

ate

pola

rizat

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(a)

(b)

FIG. 8. (Color) (a) Dendrogram for Senate roll-call voting net-works for the 1st110th Congresses. Each leaf in the dendrogramrepresents a single Senate. The two horizontal color bars below thedendrograms indicate polarization measured in terms of optimizedmodularity (upper bar) and DW-Nominate scores (lower bar). Wecolor the branches in the dendrogram corresponding to periods ofsimilar polarization. (b) Polarization of the US Senate as a function oftime, which we label using the Congress number. The height of eachstem indicates the level of polarization measured using optimizedmodularity, and the color of each stem gives the cluster membershipof each Senate in (a). The black curve shows the DW-Nominatepolarization. Note that we have normalized both measures to lie inthe interval [0,1].

a network. It was shown recently that Q is a good measure ofpolarization even for Congresses without clear party divisions[34]. Modularity is given in terms of the energy H in Eq. (1)by Q = H( = 1)/(2m).

In Fig. 8(a), we include bars under the dendrogramsto represent the two polarization measures, both of whichhave been normalized to lie in the interval [0,1]. The barsdemonstrate that Senates with similar levels of polarization(measured in terms of both DW-Nominate scores and opti-mized modularity values) are usually assigned to the samegroup, suggesting that our MRF clustering technique groupsSenates based on the polarization of roll-call votes. We havealso colored dendrogram groups according to their mean levelsof polarization using optimized modularity, where the browngroup in the dendrogram corresponds to the most polarizedSenates and the blue group corresponds to the least polarizedSenates. We chose the specific number of groups by inspectionof the dendrogram. Although one ought to expect similarity inthe results from the modularity-based measure of polarizationand the MRF clustering, it is important to stress that theMRF clustering method is based on different principles;modularity attempts to quantify the extent to which a givennetwork is modular, whereas the MRF clustering explicitly

036104-8

data: US senators


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(weighted) edges







kjr pis 1


risj jrpjr

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dij

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Aijs gskiskjs2ms

dsr$

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1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 2000

40PA, 24F, 8AA

151DR, 30AA, 14PA, 5F141F, 43DR

44D, 2R

1784R, 276D, 149DR, 162J, 53W, 84other

176W, 97AJ, 61DR, 49A,24D, 19F, 13J, 37other

3168D, 252R, 73other

222D, 6W, 11other

1490R, 247D, 19other

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multilayer community index:for node i on layer s


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all the layers connected to each other: categorical multilayer communities

only the adjacent layers connected to each other: ordered multilayer communities



(weighted) edges







kjr pis 1


risj jrpjr

kis2ms

kjrkjr

dsr Cjsrcjrcjrkjr

dij

! "kjr2m

2



Aijs gskiskjs2ms

dsr$

dijCjsridgis,gjr 3


1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 2000

40PA, 24F, 8AA

151DR, 30AA, 14PA, 5F141F, 43DR

44D, 2R

1784R, 276D, 149DR, 162J, 53W, 84other

176W, 97AJ, 61DR, 49A,24D, 19F, 13J, 37other

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parameter space = [ (intralayer resolution),! (interlayer coupling strength)]

congressional cosponsorship networks

1

1

2

1

1

2

1 1

congressperson a congressperson d

congressperson b congressperson c congressperson e


congressperson b congressperson e

congressperson c

bill A bill B bill C

(a) bipartite network

(b) congresspersonmode projection (c) billmode projection

bill A

bill B bill C

FIG. 1: The construction procedure for the weighted projected networks from the bipartite network.

Figure 1 illustrates the method to project the bipartite cosponsorship network to the weighted

bill and congressperson networks similar to the one used in the relationship between protein com-

plexes and component proteins [1], where Tables I and II show the detailed statistics divided by

eight periods. There are 15 dierent status assigned for the bills as listed in Table III, but we do not

distinguish those status in the following analysis for now. For the bipartite network, Fig. 2 shows

the degree distributions of bills and congresspersons, indicating that the number of congressper-

sons cosponsoring bills is much more heterogeneously distributed compared to the number of bills

in which individual congresspersons participates.

The bills are clearly partitioned into ten periods (2006 I, 2006 II, 2007 I, 2007 II, 2008 I,

2008 II, 2009 I, 2009 II, 2010 I, and 2010 II), but the weighted congressperson-mode projection

networks for dierent years (composed of 130 members in total) shares most congresspersons (ex-

cept for the congresspersons invisible for a specific year due to their absence in the cosponsoring

activities), which allows the temporally changing or multiplex relations over the eight periods. In

2

Congress of the Republic of Peru (20062011)

Senate of the United States (1973-2009)


1

1

2

1

1

2

1 1


congressperson b congressperson c congressperson e


congressperson b congressperson e

congressperson c

bill A bill B bill C

(a) bipartite network

(b) congresspersonmode projection (c) billmode projection

bill A

bill B bill C

FIG. 1: The construction procedure for the weighted projected networks from the bipartite network.

Figure 1 illustrates the method to project the bipartite cosponsorship network to the weighted

bill and congressperson networks similar to the one used in the relationship between protein com-

plexes and component proteins [1], where Tables I and II show the detailed statistics divided by

eight periods. There are 15 dierent status assigned for the bills as listed in Table III, but we do not

distinguish those status in the following analysis for now. For the bipartite network, Fig. 2 shows

the degree distributions of bills and congresspersons, indicating that the number of congressper-

sons cosponsoring bills is much more heterogeneously distributed compared to the number of bills

in which individual congresspersons participates.

The bills are clearly partitioned into ten periods (2006 I, 2006 II, 2007 I, 2007 II, 2008 I,

2008 II, 2009 I, 2009 II, 2010 I, and 2010 II), but the weighted congressperson-mode projection

networks for dierent years (composed of 130 members in total) shares most congresspersons (ex-

cept for the congresspersons invisible for a specific year due to their absence in the cosponsoring

activities), which allows the temporally changing or multiplex relations over the eight periods. In

2

Congress of the Republic of Peru (20062011)


temporally ordered multilayer


93 97

101 105 109

50 100 150 200 250 300c on g

r es s

i nd e

x

senator index

19 time points, a=1.0, t=20.0

93 97

101 105 109

50 100 150 200 250 300c on g

r es s

i nd e

x

senator index


93 97

101 105 109

50 100 150 200 250 300c on g

r es s

i nd e

x

senator index


93 97

101 105 109

50 100 150 200 250 300c on g

r es s

i nd e

x

senator index


93 97

101 105 109

50 100 150 200 250 300c on g

r es s

i nd e

x

senator index


log(modularity Q) log[flexibility (average number of community switching normalized by the number of time points)]

paint drip plots

US senate (93rd-110th): 18 time slices

0 20 40 60 80 100t

0

0.5

1

1.5

2

a

-9-8-7-6-5-4-3-2-1 0

US Senate (93rd-110th): 18 time slices

0 20 40 60 80 100t

0

0.5

1

1.5

2

a

-10-9-8-7-6-5-4-3-2-1 0

19 time points: = 1, ! = 20

19 time points: = 1, ! = 40

19 time points: = 1, ! = 60

19 time points: = 1, ! = 80

19 time points: = 1, ! = 100


(a) (b)

10-4

10-3

10-2

10-1

100

100 1

community structures of multilayer political cosponsorship networks

Science

nodes i

hh minh maxh

total number of edges

sungkyunkwan universityhttp

adjacency matrixaij

mxijaij kikj2mgi

hminandh max

networkresolution parameter