Finding Lagrangian Coherent Structures Using Community Detection

Finding Lagrangian Coherent Structures Using Community Detection

Sang Hoon Lee Department of Energy Science, Sungkyunkwan University, South Korea

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

in collaboration with Mohammad Farazmand (Georgia Tech), George Haller (ETH Zürich), and Mason A. Porter (Univ. of Oxford)

2015년 한국물리학회 봄학술논문발표회 통계물리학분과회 [B10.11], 2015년 4월 23일

Lagrangian Coherent Structures (LCSs)“time-evolving surfaces that shape trajectory patterns in non-autonomous dynamical systems, such as turbulent fluid flows”from Mohammad Farazmand’s Ph.D. thesis


www.physicstoday.org February 2013 Physics Today 41

In April 2010, fine, airborne ash from a volcaniceruption in Iceland caused chaos throughout European airspace. The same month, the explo-sion at the Deepwater Horizon drilling rig in theGulf of Mexico left a gushing oil well on the sea

floor that caused the largest offshore oil spill in UShistory. A year later the Tohoku tsunami hit the coastof Japan, causing great loss of life, the Fukushimanuclear-reactor disaster, and the release of substan-tial amounts of debris and radioactive contamina-tion into the Pacific Ocean.

Those three globally significant events, de-picted in figure 1, share a common theme. In eachcase, material was released into the environmentfrom what was essentially a point source, and pre-dicting where that material would be transportedby the surrounding oceanic or atmospheric flowwas of paramount importance.

To predict the outcomes of such events, thestandard approach is to run numerical simulationsof the atmosphere or the sea and use the resultingvelocity-field data sets to forecast pollutant trajec-tories. Although that approach does predict the fu-ture of individual fluid parcels, the predictions arehighly sensitive to small changes in the time and lo-cation of release. Attempts to address the excessivesensitivity to initial conditions include running sev-eral different models for the same scenario. But thattypically produces even larger distributions of ad-vected particles—those transported by the fluid

flow—and thus hides key organizing structures ofthat flow.

Furthermore, traditional trajectory analysis fo-cuses on full trajectory histories that yield convoluted“spaghetti plots” that are hard to interpret. Improvedunderstanding and forecasting therefore requiresnew concepts and methods that provide more insightinto why fluid flows behave as they do.

Lagrangian coherent structuresRecently, ideas that lie at the interface between non-linear dynamics—the mathematical discipline thatunderlies chaos theory—and fluid dynamics havegiven rise to the concept of Lagrangian coherentstructures (LCSs), which provides a new way of un-derstanding transport in complex fluid flows.

Although advances have been made in the de-tection of LCSs in fully three- dimensional flows,this article focuses primarily on the many advancesthat have been made for 2D flows. There, LCSs takethe form of material lines—continuous, smoothcurves of fluid elements advected by the flow. Theyare conceptually simpler than the 2D material sur-faces required for LCSs in 3D flows. Furthermore,2D flows are particularly relevant for studies of

New techniques promise better forecastingof where damaging contaminants in the

ocean or atmosphere will end up.

Thomas Peacock and George Haller

Thomas Peacock is a professor of mechanical engineering at the MassachusettsInstitute of Technology in Cambridge. George Haller is a professor of nonlineardynamics at ETH Zürich in Switzerland.

Lagrangiancoherent structures

The hidden skeleton of fluid flows

Downloaded 01 Feb 2013 to 195.176.113.187. Redistribution subject to AIP license or copyright; see http://www.physicstoday.org/about_us/terms

T. Peacock and G. Haller, Physics Today (February, 2013), pp. 41-47.


www.physicstoday.org February 2013 Physics Today 41

In April 2010, fine, airborne ash from a volcaniceruption in Iceland caused chaos throughout European airspace. The same month, the explo-sion at the Deepwater Horizon drilling rig in theGulf of Mexico left a gushing oil well on the sea

floor that caused the largest offshore oil spill in UShistory. A year later the Tohoku tsunami hit the coastof Japan, causing great loss of life, the Fukushimanuclear-reactor disaster, and the release of substan-tial amounts of debris and radioactive contamina-tion into the Pacific Ocean.

Those three globally significant events, de-picted in figure 1, share a common theme. In eachcase, material was released into the environmentfrom what was essentially a point source, and pre-dicting where that material would be transportedby the surrounding oceanic or atmospheric flowwas of paramount importance.

To predict the outcomes of such events, thestandard approach is to run numerical simulationsof the atmosphere or the sea and use the resultingvelocity-field data sets to forecast pollutant trajec-tories. Although that approach does predict the fu-ture of individual fluid parcels, the predictions arehighly sensitive to small changes in the time and lo-cation of release. Attempts to address the excessivesensitivity to initial conditions include running sev-eral different models for the same scenario. But thattypically produces even larger distributions of ad-vected particles—those transported by the fluid

flow—and thus hides key organizing structures ofthat flow.

Furthermore, traditional trajectory analysis fo-cuses on full trajectory histories that yield convoluted“spaghetti plots” that are hard to interpret. Improvedunderstanding and forecasting therefore requiresnew concepts and methods that provide more insightinto why fluid flows behave as they do.

Lagrangian coherent structuresRecently, ideas that lie at the interface between non-linear dynamics—the mathematical discipline thatunderlies chaos theory—and fluid dynamics havegiven rise to the concept of Lagrangian coherentstructures (LCSs), which provides a new way of un-derstanding transport in complex fluid flows.

Although advances have been made in the de-tection of LCSs in fully three- dimensional flows,this article focuses primarily on the many advancesthat have been made for 2D flows. There, LCSs takethe form of material lines—continuous, smoothcurves of fluid elements advected by the flow. Theyare conceptually simpler than the 2D material sur-faces required for LCSs in 3D flows. Furthermore,2D flows are particularly relevant for studies of

New techniques promise better forecastingof where damaging contaminants in the

ocean or atmosphere will end up.

Thomas Peacock and George Haller

Thomas Peacock is a professor of mechanical engineering at the MassachusettsInstitute of Technology in Cambridge. George Haller is a professor of nonlineardynamics at ETH Zürich in Switzerland.

Lagrangiancoherent structures

The hidden skeleton of fluid flows



https://vimeo.com/68802165“Density field and 3D Lagrangian coherent structures obtained from 7 million particle releases in a transitional multi-scale flow in which surface buoyancy driven frontal instabilities trigger deeper baroclinic instabilities. Dispersion characteristics of pollutants in such oceanic flows has been explored.”

The importance of

pollution transport on the ocean surface and on sur-faces of constant density in the atmosphere.

Generally speaking, the LCS approach pro-vides a means of identifying key material lines thatorganize fluid-flow transport. Such material linesaccount for the linear shape of the ash cloud in figure 1a, the structure of the oil spill in 1b, and thetendrils in the spread of radioactive contaminationin 1c. More specifically, the LCS approach is basedon the identification of material lines that play thedominant role in attracting and repelling neighbor-ing fluid elements over a selected period of time.Those key lines are the LCSs of the fluid flow. To de-velop an understanding of them, we must first con-sider several ideas.

Lagrange versus EulerThere are two different perspectives one can take indescribing fluid flow. The Eulerian point of viewconsiders the properties of a flow field at each fixedpoint in space and time. The velocity field is a primeexample of an Eulerian description. It gives the in-stantaneous velocity of fluid elements throughoutthe domain under consideration. The identity andprovenance of fluid elements are not important; atany given point and instant, the velocity field sim-ply refers to the motion of whatever fluid elementhappens to be passing.

By contrast, the Lagrangian perspective is con-cerned with the identity of individual fluid ele-ments. It tracks the changing velocity of individualparticles along their paths as they are advected bythe flow. It’s the natural perspective to use when

considering flow transport because patterns such asthose in figure 1 arise from material advection.

Another driving force behind the developmentof the LCS approach is the concept of objectivity, orframe invariance. Characterizations of flow struc-tures in terms of the properties of Eulerian fieldssuch as the velocity field tend not to be objective;they don’t remain invariant under time- dependentrotations and translations of the reference frame.For instance, a common way to visualize flow fieldsis to use streamlines, which are Eulerian entities thatfollow the local direction of the velocity field at agiven instant.

Traditionally, vortices in fluid flows have beenidentified as regions filled with closed streamlines.But velocity fields, and hence their streamlines,change when viewed from different referenceframes. So what looks like a domain full of closedstreamlines in one frame can appear completely dif-ferent when viewed from another frame. For exam-ple, an unsteady vortex flow may look like a steadysaddle-point flow in an appropriate rotating frame.

For unsteady flows, which are the rule ratherthan the exception in nature, there is no obvious pre-ferred frame of reference. So any conclusion abouttransport-guiding dynamic structures should holdfor any choice of reference frame. With regard to an

42 February 2013 Physics Today www.physicstoday.org

Lagrangian structures

a b

c

Figure 1. Large-scale contaminant flows. (a) A 150-km-wide view of theash cloud from the 2010 Icelandic volcano eruption. (b) A 300-km-wideview of the 2010 Deepwater Horizon oil spill in the Gulf of Mexico. (c) A prediction of the eastward spread of radioactive contaminationinto the Pacific Ocean from the 2011 Fukushima reactor disaster in Japan.

NA

SA

NA

SA

AS

R


Lagrangian Coherent Structures (LCSs)


The importance of

pollution transport on the ocean surface and on sur-faces of constant density in the atmosphere.

Generally speaking, the LCS approach pro-vides a means of identifying key material lines thatorganize fluid-flow transport. Such material linesaccount for the linear shape of the ash cloud in figure 1a, the structure of the oil spill in 1b, and thetendrils in the spread of radioactive contaminationin 1c. More specifically, the LCS approach is basedon the identification of material lines that play thedominant role in attracting and repelling neighbor-ing fluid elements over a selected period of time.Those key lines are the LCSs of the fluid flow. To de-velop an understanding of them, we must first con-sider several ideas.

Lagrange versus EulerThere are two different perspectives one can take indescribing fluid flow. The Eulerian point of viewconsiders the properties of a flow field at each fixedpoint in space and time. The velocity field is a primeexample of an Eulerian description. It gives the in-stantaneous velocity of fluid elements throughoutthe domain under consideration. The identity andprovenance of fluid elements are not important; atany given point and instant, the velocity field sim-ply refers to the motion of whatever fluid elementhappens to be passing.

By contrast, the Lagrangian perspective is con-cerned with the identity of individual fluid ele-ments. It tracks the changing velocity of individualparticles along their paths as they are advected bythe flow. It’s the natural perspective to use when

considering flow transport because patterns such asthose in figure 1 arise from material advection.

Another driving force behind the developmentof the LCS approach is the concept of objectivity, orframe invariance. Characterizations of flow struc-tures in terms of the properties of Eulerian fieldssuch as the velocity field tend not to be objective;they don’t remain invariant under time- dependentrotations and translations of the reference frame.For instance, a common way to visualize flow fieldsis to use streamlines, which are Eulerian entities thatfollow the local direction of the velocity field at agiven instant.

Traditionally, vortices in fluid flows have beenidentified as regions filled with closed streamlines.But velocity fields, and hence their streamlines,change when viewed from different referenceframes. So what looks like a domain full of closedstreamlines in one frame can appear completely dif-ferent when viewed from another frame. For exam-ple, an unsteady vortex flow may look like a steadysaddle-point flow in an appropriate rotating frame.

For unsteady flows, which are the rule ratherthan the exception in nature, there is no obvious pre-ferred frame of reference. So any conclusion abouttransport-guiding dynamic structures should holdfor any choice of reference frame. With regard to an

42 February 2013 Physics Today www.physicstoday.org

Lagrangian structures

a b

c

Figure 1. Large-scale contaminant flows. (a) A 150-km-wide view of theash cloud from the 2010 Icelandic volcano eruption. (b) A 300-km-wideview of the 2010 Deepwater Horizon oil spill in the Gulf of Mexico. (c) A prediction of the eastward spread of radioactive contaminationinto the Pacific Ocean from the 2011 Fukushima reactor disaster in Japan.

NA

SA

NA

SA

AS

R


Lagrangian Coherent Structures (LCSs)


Maria Antonova

@mashant

RETWEETS

2,635

FAVORITES

1,468

What happens when you flush a bunch of GPS trackers down a St. Petersburg toilet "@leprasorium: "

2:51 AM - 17 Nov 2014

Follow

Related headlines

Süddeutsche Zeitung @SZ18. November 2014

Reply to @mashant @leprasorium

Humble Gateau @kirschly · Nov 17@RobLawrence Well, I thought plants generally filter out solid objects before further processing. The one I visited years ago certainly did.

Home Notifications Messages Discover

Search Twitter

Lagrangian vs Eulerian viewpoint on fluid

Lagrangian Eulerian

from S. Takagi, K. Sugiyama, S. Ii, and Y. Matsumoto, J. Appl. Mech. 79, 010911 (2011).

Joseph-Louis Lagrange (1736-1813) Leonhard Euler (1707-1783)

“fluid-element network”

Community structure in network

“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

X

ij

✓Aij � �

kikj2m

◆�(gi, gj)

where the adjacency matrix

Aij 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

importing network dataidentifying community structure

visualizing

resolution parameter: controlling the characteristic size of communities

smal

ler

com

mun

ities

�TA

XO

NO

MIE

SO

FN

ET

WO

RK

SFR

OM

CO

MM

UN

ITY

STR

UC

TU

RE

PHY

SIC

AL

RE

VIE

WE

86,0

3610

4(2

012)

that

alle

dges

are

antif

erro

mag

netic

atre

solu

tion

λ=

"m

axan

dth

ereb

yfo

rces

each

node

into

itsow

nco

mm

unity

.

III.

ME

SOSC

OPI

CR

ESP

ON

SEFU

NC

TIO

NS

(MR

FS)

Tode

scri

beho

wa

netw

ork

disi

nteg

rate

sin

toco

mm

uniti

esas

the

valu

eof

λis

incr

ease

dfr

om"

min

to"

max

[see

Fig.

1(a)

fora

sche

mat

ic],

one

need

sto

sele

ctsu

mm

ary

stat

istic

s.T

here

are

man

ypo

ssib

lew

ays

tosu

mm

ariz

esu

cha

disi

nteg

ratio

npr

oces

s,an

dw

efo

cus

onth

ree

diag

nost

ics

that

char

acte

rize

fund

amen

talp

rope

rtie

sof

netw

ork

com

mun

ities

.Fi

rst,

we

use

the

valu

eof

the

Ham

ilton

ianH

(λ)(

1),w

hich

isa

scal

arqu

antit

ycl

osel

yre

late

dto

netw

ork

mod

ular

ityan

dqu

antifi

esth

een

ergy

ofth

esy

stem

[13,

14].

Seco

nd,

we

calc

ulat

ea

part

ition

entr

opy

S(λ

)to

char

acte

rize

the

com

mun

itysi

zedi

stri

butio

n.To

doth

is,

let

nk

deno

teth

enu

mbe

rof

node

sin

com

mun

ityk

and

defin

ep

k=

nk/N

tobe

the

prob

abili

tyto

choo

sea

node

from

com

mun

ityk

unif

orm

lyat

rand

om.T

hisy

ield

sa(S

hann

on)p

artit

ion

entr

opy

ofS

(λ)=

−!

η(λ

)k=

1p

klo

gp

k,w

hich

quan

tifies

the

diso

rder

inth

eas

soci

ated

com

mun

itysi

zedi

stri

butio

n.T

hird

,we

use

the

num

bero

fcom

mun

ities

η(λ

).

ξ=1,

η=3

4ξ=

0,η =

1ξ=

0.2,

η=8

ξ=0.

4, η

=12

ξ=0.

6, η

=17

ξ=0.

8, η

=24

ξ = 0

.2ξ =

0.4

ξ = 0

.6ξ =

0.8

ξ = 0

ξ = 1

0

0.2

0.4

0.6

0.81

ξ

ferr

omag

netic

link

sno

nlin

ksan

tifer

rom

agne

tic li

nks

(a)

(c)

(b)

Hef

f

Sef

fη ef

f

FIG

.1.

(Col

oron

line)

(a)

Sche

mat

icof

som

eof

the

way

sth

ata

netw

ork

can

brea

kup

into

com

mun

ities

asth

eva

lue

ofλ

(or

ξ)

isin

crea

sed.

(b)Z

acha

ryK

arat

eC

lub

netw

ork

[23]

ford

iffer

entv

alue

sof

the

effe

ctiv

efr

actio

nof

antif

erro

mag

netic

edge

sξ.A

llin

tera

ctio

nsar

eei

ther

ferr

omag

netic

oran

tifer

rom

agne

tic;i

.e.,

for

the

valu

esof

ξth

atw

eus

ed,

ther

ear

eno

neut

ral

inte

ract

ions

.W

eco

lor

edge

sin

blue

ifth

eco

rres

pond

ing

inte

ract

ions

are

ferr

omag

netic

,and

we

colo

rth

emin

red

ifth

ein

tera

ctio

nsar

ean

tifer

rom

agne

tic.W

eco

lor

the

node

sba

sed

onco

mm

unity

affil

iatio

n.(c

)T

heH

eff,

Sef

f,an

dη

eff

MR

Fs,

and

the

inte

ract

ion

mat

rix

Jfo

rdi

ffer

ent

valu

esof

ξ.

We

colo

rel

emen

tsof

the

inte

ract

ion

mat

rix

byde

pict

ing

the

abse

nce

ofan

edge

inw

hite

,fe

rrom

agne

ticed

ges

inbl

ue(d

ark

gray

),an

dan

tifer

rom

agne

ticed

ges

inre

d(l

ight

gray

).

Bec

ause

we

need

tono

rmal

izeH

,S,a

ndη

toco

mpa

reth

emac

ross

netw

orks

,we

defin

ean

effe

ctiv

een

ergy

Hef

f(λ

)=

H(λ

)−H

min

Hm

ax−

Hm

in=

1−

H(λ

)H

min

,(4

)

whe

reH

min

=H

("m

in)

andH

max

=H

("m

ax);

anef

fect

ive

entr

opy

Sef

f(λ

)=

S(λ

)−S

min

Sm

ax−

Sm

in=

S(λ

)lo

gN

,(5

)

whe

reS

min

=S

("m

in)

and

Sm

ax=

S("

max

);an

dan

effe

ctiv

enu

mbe

rof

com

mun

ities

ηef

f(λ

)=

η(λ

)−η

min

ηm

ax−

ηm

in=

η(λ

)−1

N−

1,

(6)

whe

reη

min

=η

("m

in)=

1an

dη

max

=η

("m

ax)=

N.

Som

ene

twor

ksco

ntai

na

smal

lnu

mbe

rof

entr

ies

"ij

that

are

orde

rsof

mag

nitu

dela

rger

than

mos

tot

her

entr

ies.

For

exam

ple,

inth

ene

twor

kof

Face

book

frie

ndsh

ips

atC

alte

ch[2

1,22

],98

%of

the

"ij

entr

ies

are

less

than

100,

but

0.02

%of

them

are

larg

erth

an80

00.

The

sela

rge

"ij

valu

esar

ise

whe

ntw

olo

w-s

tren

gth

node

sbe

com

eco

nnec

ted.

Usi

ngth

enu

llm

odel

Pij

=k i

k j/(

2m),

the

inte

ract

ion

betw

een

two

node

si

and

jbe

com

esan

tifer

rom

agne

ticw

hen

λ>

Aij/P

ij=

2mA

ij/(

k ik j

).If

ane

twor

kha

sa

larg

eto

tal

edge

wei

ght

but

both

ian

dj

have

smal

lst

reng

ths

com

pare

dto

othe

rno

des

inth

ene

twor

k,th

enλ

need

sto

bela

rge

tom

ake

the

inte

ract

ion

antif

erro

mag

netic

.In

prio

rst

udie

s,ne

twor

kco

mm

unity

stru

ctur

eha

sbee

nin

vest

igat

edat

diff

eren

tm

esos

copi

csc

ales

byco

nsid

erin

gpl

ots

ofva

riou

sdi

agno

stic

sas

afu

nctio

nof

the

reso

lutio

npa

ram

eter

λ[1

3,14

,17]

.In

the

pres

ent

exam

ple,

such

plot

sw

ould

bedo

min

ated

byin

tera

ctio

nsth

atre

quir

ela

rge

reso

lutio

n-pa

ram

eter

valu

esto

beco

me

antif

erro

mag

netic

.To

over

com

eth

isis

sue,

we

defin

eth

eef

fect

ive

frac

tion

ofan

tifer

rom

agne

ticed

ges

ξ=

ξ(λ

)=

ℓA(λ

)−ℓA

("m

in)

ℓA("

max

)−ℓA

("m

in)

∈[0

,1],

(7)

whe

reℓA

(λ)

isth

eto

tal

num

ber

ofan

tifer

rom

agne

ticin

-te

ract

ions

for

the

give

nva

lue

ofλ

.In

othe

rw

ords

,it

isth

enu

mbe

rof

"ij

elem

ents

that

are

smal

ler

than

λ.

Thu

s,ℓA

("m

in)

isth

ela

rges

tnu

mbe

rof

antif

erro

mag

netic

inte

rac-

tions

forw

hich

ane

twor

kst

illfo

rms

asi

ngle

com

mun

ity,a

ndth

eef

fect

ive

num

ber

ofan

tifer

rom

agne

ticin

tera

ctio

nsξ

(λ)

isth

enu

mbe

rof

antif

erro

mag

netic

inte

ract

ions

(nor

mal

ized

toth

eun

itin

terv

al)

inex

cess

ofℓA

("m

in).

The

func

tion

ξ(λ

)in

crea

ses

mon

oton

ical

lyin

λ.

Swee

ping

λfr

om"

min

to"

max

corr

espo

nds

tosw

eepi

ngth

eva

lue

ofξ

from

0to

1.(O

neca

nth

ink

ofλ

asa

cont

inuo

usva

riab

lean

dξ

asa

disc

rete

vari

able

that

chan

ges

with

even

ts.)

Asw

epe

rfor

msu

chsw

eepi

ngfo

ragi

ven

netw

ork,

the

num

ber

ofco

mm

uniti

esin

crea

sesf

rom

η(ξ

=0)

=1

toη

(ξ=

1)=

Nan

dyi

elds

ave

ctor

[Hef

f(ξ

),S

eff(ξ

),η

eff(ξ

)]w

hose

com

pone

nts

we

call

the

mes

osco

pic

resp

onse

func

tions

(MR

Fs)

ofth

atne

twor

k.(W

eal

soso

met

imes

refe

rto

the

vect

orits

elf

asan

MR

F.)

Bec

ause

Hef

f∈

[0,1

],S

eff∈

[0,1

],η

eff∈

[0,1

],an

dξ

∈[0

,1]f

orev

ery

netw

ork,

we

can

com

pare

the

MR

Fsac

ross

netw

orks

and

use

them

toid

entif

ygr

oups

ofne

twor

ksw

ithsi

mila

rm

esos

copi

cst

ruct

ures

.In

Fig.

1(b)

,w

esh

owth

eZ

acha

ryK

arat

eC

lub

netw

ork

[23]

for

diff

eren

tva

lues

of

0361

04-3

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

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 connected

in 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 2µ =

Pijs Aijs +

Pjsr Cjsr for Qmultilayer 2 [�1, 1].

Qmultilayer =1

2µ

X

ijsr

✓Aijs � �s

kiskjs2ms

◆�sr + �ijCjsr

��(gis, gjr)

Community structure in time-dependent or “multilayer” network

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 in

academic 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(1–3). 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 (6–9) 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.

node

s

resolution parameters

coupling = 0

1 2 3 4

5

10

15

20

25

30

node

s


coupling = 0.1

1 2 3 4

5

10

15

20

25

30

node

s


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

on

Nov

embe

r 8, 2

011

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fro

m

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

different slices: “time series”

nodes in individual slices

(weighted) edges

multilayer community index: for node i on layer s

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 Barber’s 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 connections—in 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 ¼ ∑jr

ðAijsdsr þ dijCjsrÞpjrkjr

− pis ð1Þ

respects the intraslice nature of Aijs and theinterslice couplings of Cjsr. Using the steady-stateprobability distribution p∗jr ¼ 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 jrp∗jr ¼

kis2ms

kjrkjr

dsr þCjsr

cjr

cjrkjr

dij

! "kjr2m

ð2Þ

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 ¼12m

∑ijsr

h#Aijs − gs

kiskjs2ms

dsr$þ

dijCjsr

idðgis,gjrÞ ð3Þ

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

Sen

ator

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

REPORTS

on

Nov

embe

r 8, 2

011

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fro

m

An Example of LCS: simulated flow

Preliminary Results of Community Detection in Flow Maps (last updated: January 1, 2014)

I. FLOW MAP DATA

FIG. 1. Original flow map.

• Original flow map: Fig. 1 [512 ⇥ 512 uniform grid points corresponding to [0, 2⇡) ⇥ [0, 2⇡)

with the periodic boundary condition (PBC)—all the metrics such as distance between two

points consider the PBC, as presented in Sec. II].

• Sampled flow map: sampling every fourth (n = 4) element for x and y axes, which yields

(128 ⇥ 128 grid points = 16 384 nodes and their interactions)

II. DEFINITION OF WEIGHTS

W

(1)AB

=|r

i

(A, B)||r

f

(A, B)| , (1)

1

M. Farazmand and G. Haller, e-print arXiv:1402.4835.


Sang Hoon Lee,1, 2, ⇤ Mohammad Farazmand,3 George Haller,4 and Mason A. Porter2, 5

1Integrated Energy Center for Fostering Global Creative Researcher (BK 21 plus)and Department of Energy Science, Sungkyunkwan University, Suwon 440–746, Korea

2Oxford Centre for Industrial and Applied Mathematics (OCIAM),Mathematical Institute, University of Oxford, Oxford, OX2 6GG, United Kingdom3School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA

4Institute of Mechanical Systems, ETH Zurich, Switzerland5CABDyN Complexity Centre, University of Oxford, Oxford, OX1 1HP, United Kingdom

Lagrangian coherent structures (LCSs) refer to dynamically distinct groups of fluid elements in fluid flowsand provide valuable mesoscale geographical information to identify the most essential elements of such flows.Using relative dispersion, we define the pairwise correlation between fluid elements and use the community de-tection for the systematic identification of LCSs as the community or modular structures underlying interactionsystems e↵ectively provide substructures behind the system of interest. We detect communities using modular-ity maximization with a tunable resolution for simulation data and two of real satellite-tracked drifter data inthe ocean to examine LCSs in various scales. In particular, to obtain more detailed spatiotemporal LCSs, wemaximize a multilayer version of modularity on a multilayer network in which temporal slices of interactionsare properly considered. We believe that our approach illustrates a new way to e�ciently detect LCSs for givendynamical systems and opens new possibilities of applications in dynamical systems in general.

PACS numbers: 45.20.Jj, 47.54.-r, 89.75.Fb, 89.75.Hc

Introduction.—Recent developments in the theory of dy-namical systems have given rise to new concepts of coherentstructures in fluid flow [1–4]. These methods seek exceptionalmaterial surfaces (or curves, in the case of two-dimensionalflow) that play a key role in mixing and transport over a giventime interval [5–8]. Their approach is Lagrangian in nature, incontrast to the Eulerian point of view that studies the instanta-neous velocity field [9].

The Lagrangian methods each rely on specific mathemati-cal tools such as probability, di↵erential geometry and calcu-lus of variations. Here, we develop a new approach to coher-ent structure analysis using recent advances in network the-ory [10]. After laying down the theoretical frame work, weshow on two examples how our approach can complementearlier methods, as well as, providing new insight that is onlyaccessible to our network theory-based method.

Before turning to network theory, we present typical typesof Lagrangian coherent structures (LCSs) on an example: aturbulent flow. Figure 1 shows LCSs from a direct numericalsimulation of the forced Navier–Stokes equation

@u@t+ u · ru = �rp + ⌫r2u + f, r · u = 0, (1)

over the domain [0, 2⇡] ⇥ [0, 2⇡] with doubly periodic bound-ary conditions. The Lagrangian analysis is carried out over afew eddy turn-over times after the flow has reached its fullyturbulent state (see Ref. [11] for a detailed analysis).

The repelling and attracting LCSs (red and blue curves, re-spectively, in figure 1) are the main drivers of mixing throughextensive stretching and folding of nearby material elements.The green islands, in contrast, represent elliptic LCSs thatinhibit mixing by preserving their shape over relatively longtime scales.

Network Representation.—A fresh way to look at those

FIG. 1. (color online). Repelling (red), attracting (blue) and elliptic(green) Lagrangian coherent structures (LCSs).

systems for that purpose is to consider such systems as dis-crete interacting objects, as in the force-chain networks de-scribing granular material systems [12, 13] or the plumedetection problem in fluid [14]. General relationships be-tween community finding, transport, and partition are dis-cussed in Refs. [15, 16]. Another example of using thenetwork-theory tools to analyze the flow network is presentedin Refs. [17, 18], where the mass transport is represented asthe directed edges between geographical sub-areas (nodes),which in fact is rather in line with the spirit of the Eulerianpoint of view. We, by contrast, consider the fluid elementsthemselves as the nodes, so we can highlight more fundamen-tal structural properties of LCSs.

simulated flow from the forced Navier-Stokes equation

repelling LCSs attracting LCSs elliptic LCSs

pressure

external forceviscosity

u(x, t) is the velocity field

defined on the two-dimensional

domain U as x 2 U = [0, 2⇡]⇥ [0, 2⇡]at time t with doubly periodic

boundary conditions

An Example of LCS: simulated flow


I. FLOW MAP DATA








W

(1)AB

=|r

i

(A, B)||r

f

(A, B)| , (1)

1

M. Farazmand and G. Haller, e-print arXiv:1402.4835.











@u@t+ u · ru = �rp + ⌫r2u + f, r · u = 0, (1)






simulated flow from the forced Navier-Stokes equation

repelling LCSs attracting LCSs elliptic LCSs

pressure

external forceviscosity

u(x, t) is the velocity field

defined on the two-dimensional

domain U as x 2 U = [0, 2⇡]⇥ [0, 2⇡]at time t with doubly periodic

boundary conditions

Network community analysis of LCSPreliminary Results of Community Detection in Flow Maps (last updated: January 1, 2014)

I. FLOW MAP DATA








W

(1)AB

=|r

i

(A, B)||r

f

(A, B)| , (1)

1











@u@t+ u · ru = �rp + ⌫r2u + f, r · u = 0, (1)







I. FLOW MAP DATA








W

(1)AB

=|r

i

(A, B)||r

f

(A, B)| , (1)

1











@u@t+ u · ru = �rp + ⌫r2u + f, r · u = 0, (1)







I. FLOW MAP DATA








W

(1)AB

=|r

i

(A, B)||r

f

(A, B)| , (1)

1

where we can define the relative dispersion for each grid element as

maxB2nnhd(A)

|rf

(A, B)||r

i

(A, B)| , (2)

where nnhd(A) is the set of nodes in the nearest neighbors of A (four for each grid point) and Fig. 2

shows the relative dispersion map for the 512⇥ 512 original grid elements. The data including the

relative dispersion for the original grid elements is original dispersion matrix.txt.

W

(2)AB

=|r

i

(A, B)||F(A)r

i

(A, B)| , (3)

where ri

(A, B) = ri

(B) � ri

(A) [the vector from ri

(A) to ri

(B)], rf

(A, B) = rf

(B) � rf

(A) [the

vector from rf

(A) to rf

(B)], ri

(A) = [x

i

(A), yi

(A)] which is the initial point (t = 0) of the element

A], rf

(A) = [x

f

(A), yf

(A)] which is the final point (t = 50) of the element A], and F(A) is the

deformation gradient tensor at A, i.e., |F(A)ri

(A, B)| =p{F

xx

(A)[x

i

(B) � x

i

(A)] + F

xy

(A)[yi

(B) � y

i

(A)]}2 + {Fyx

(A)[x

i

(B) � x

i

(A)] + F

yy

(A)[yi

(B) � y

i

(A)]}2.

The distance measures such as ri

(A, B) and the coordinates such as ri

(A) take the shortest distance

among all the possible distances considering the PBC: x itself and x ± 2⇡ for x, and y itself and

y ± 2⇡ for y (9 combinations in total).

III. COMMUNITY DETECTION METHODS

• W

(1)AB

(= W

(1)BA

) in Eq. (1): the Louvainmethod [1, 2] with the Girvan-Newman null model [3]

and the resolution parameter � [4] is used for Fig. 3, where the number of communities and

the values of quality measure QGN are specified for four di↵erent � values. The communities

here describe the (mutually exclusive for now—we can extend this to take the “overlapping”

communities into account by using other methods) groups of nodes where the intra-group

interactions are significantly stronger than the inter-group interactions. For the resolution

parameter �, roughly speaking, the larger the value of � is, the smaller (in terms of typical

number of nodes in a community, thus larger number of communities) communities are

identified.

The modularity for the Girvan-Newman null model, which is the objective function QGN

where the purpose is to find the set of communities {gA

} that maximizes QGN, is given by

2

3

Supplemental Figure S1. The relative dispersion lnhmaxB2⌫(A) |r f (A, B)|/|ri(A, B)|

ifor the same grid elements as in Fig. 1 of the main text. The

set ⌫(A) is the set of nodes that are adjacent to node A.

3

Supp

lem

enta

lFig

ure

S1.T

here

lativ

edi

sper

sion

lnh m

axB2⌫(

A)|r f

(A,B

)|/|r i

(A,B

)|ifo

rthe

sam

egr

idel

emen

tsas

inFi

g.1

ofth

em

ain

text

.The

set⌫

(A)i

sth

ese

tofn

odes

that

are

adja

cent

tono

deA.

A, B: discretized 512✕512 grid cell indices


I. FLOW MAP DATA








W

(1)AB

=|r

i

(A, B)||r

f

(A, B)| , (1)

1











@u@t+ u · ru = �rp + ⌫r2u + f, r · u = 0, (1)







I. FLOW MAP DATA








W

(1)AB

=|r

i

(A, B)||r

f

(A, B)| , (1)

1


maxB2nnhd(A)

|rf

(A, B)||r

i

(A, B)| , (2)




W

(2)AB

=|r

i

(A, B)||F(A)r

i

(A, B)| , (3)

where ri

(A, B) = ri

(B) � ri


(A) to ri

(B)], rf

(A, B) = rf

(B) � rf

(A) [the

vector from rf

(A) to rf

(B)], ri

(A) = [x

i

(A), yi


A], rf

(A) = [x

f

(A), yf



(A, B)| =p{F

xx

(A)[x

i

(B) � x

i

(A)] + F

xy

(A)[yi

(B) � y

i

(A)]}2 + {Fyx

(A)[x

i

(B) � x

i

(A)] + F

yy

(A)[yi

(B) � y

i

(A)]}2.







• W

(1)AB

(= W

(1)BA









identified.




2

3




3

Supp

lem

enta

lFig

ure

S1.T

here

lativ

edi

sper

sion

lnh m

axB2⌫(

A)|r f

(A,B

)|/|r i

(A,B

)|ifo

rthe

sam

egr

idel

emen

tsas

inFi

g.1

ofth

em

ain

text

.The

set⌫

(A)i

sth

ese

tofn

odes

that

are

adja

cent

tono

deA.


A

B

C

D


I. FLOW MAP DATA








W

(1)AB

=|r

i

(A, B)||r

f

(A, B)| , (1)

1











@u@t+ u · ru = �rp + ⌫r2u + f, r · u = 0, (1)







I. FLOW MAP DATA








W

(1)AB

=|r

i

(A, B)||r

f

(A, B)| , (1)

1


maxB2nnhd(A)

|rf

(A, B)||r

i

(A, B)| , (2)




W

(2)AB

=|r

i

(A, B)||F(A)r

i

(A, B)| , (3)

where ri

(A, B) = ri

(B) � ri


(A) to ri

(B)], rf

(A, B) = rf

(B) � rf

(A) [the

vector from rf

(A) to rf

(B)], ri

(A) = [x

i

(A), yi


A], rf

(A) = [x

f

(A), yf



(A, B)| =p{F

xx

(A)[x

i

(B) � x

i

(A)] + F

xy

(A)[yi

(B) � y

i

(A)]}2 + {Fyx

(A)[x

i

(B) � x

i

(A)] + F

yy

(A)[yi

(B) � y

i

(A)]}2.







• W

(1)AB

(= W

(1)BA









identified.




2

3




3

Supp

lem

enta

lFig

ure

S1.T

here

lativ

edi

sper

sion

lnh m

axB2⌫(

A)|r f

(A,B

)|/|r i

(A,B

)|ifo

rthe

sam

egr

idel

emen

tsas

inFi

g.1

ofth

em

ain

text

.The

set⌫

(A)i

sth

ese

tofn

odes

that

are

adja

cent

tono

deA.


|ri(A,B)|A

B

C

D|ri(C,D)|


I. FLOW MAP DATA








W

(1)AB

=|r

i

(A, B)||r

f

(A, B)| , (1)

1











@u@t+ u · ru = �rp + ⌫r2u + f, r · u = 0, (1)







I. FLOW MAP DATA








W

(1)AB

=|r

i

(A, B)||r

f

(A, B)| , (1)

1


maxB2nnhd(A)

|rf

(A, B)||r

i

(A, B)| , (2)




W

(2)AB

=|r

i

(A, B)||F(A)r

i

(A, B)| , (3)

where ri

(A, B) = ri

(B) � ri


(A) to ri

(B)], rf

(A, B) = rf

(B) � rf

(A) [the

vector from rf

(A) to rf

(B)], ri

(A) = [x

i

(A), yi


A], rf

(A) = [x

f

(A), yf



(A, B)| =p{F

xx

(A)[x

i

(B) � x

i

(A)] + F

xy

(A)[yi

(B) � y

i

(A)]}2 + {Fyx

(A)[x

i

(B) � x

i

(A)] + F

yy

(A)[yi

(B) � y

i

(A)]}2.







• W

(1)AB

(= W

(1)BA









identified.




2

3




3

Supp

lem

enta

lFig

ure

S1.T

here

lativ

edi

sper

sion

lnh m

axB2⌫(

A)|r f

(A,B

)|/|r i

(A,B

)|ifo

rthe

sam

egr

idel

emen

tsas

inFi

g.1

ofth

em

ain

text

.The

set⌫

(A)i

sth

ese

tofn

odes

that

are

adja

cent

tono

deA.


|rf (A,B)|

|ri(A,B)|

A

BA

B

C

D|ri(C,D)|


I. FLOW MAP DATA








W

(1)AB

=|r

i

(A, B)||r

f

(A, B)| , (1)

1











@u@t+ u · ru = �rp + ⌫r2u + f, r · u = 0, (1)







I. FLOW MAP DATA








W

(1)AB

=|r

i

(A, B)||r

f

(A, B)| , (1)

1


maxB2nnhd(A)

|rf

(A, B)||r

i

(A, B)| , (2)




W

(2)AB

=|r

i

(A, B)||F(A)r

i

(A, B)| , (3)

where ri

(A, B) = ri

(B) � ri


(A) to ri

(B)], rf

(A, B) = rf

(B) � rf

(A) [the

vector from rf

(A) to rf

(B)], ri

(A) = [x

i

(A), yi


A], rf

(A) = [x

f

(A), yf



(A, B)| =p{F

xx

(A)[x

i

(B) � x

i

(A)] + F

xy

(A)[yi

(B) � y

i

(A)]}2 + {Fyx

(A)[x

i

(B) � x

i

(A)] + F

yy

(A)[yi

(B) � y

i

(A)]}2.







• W

(1)AB

(= W

(1)BA









identified.




2

3




3

Supp

lem

enta

lFig

ure

S1.T

here

lativ

edi

sper

sion

lnh m

axB2⌫(

A)|r f

(A,B

)|/|r i

(A,B

)|ifo

rthe

sam

egr

idel

emen

tsas

inFi

g.1

ofth

em

ain

text

.The

set⌫

(A)i

sth

ese

tofn

odes

that

are

adja

cent

tono

deA.


|rf (A,B)|

|ri(A,B)|

A

BA

B

C

D

D

C

|ri(C,D)|

|rf (C,D)|

belonging to the different coherent structures (small )W (1)

CD


I. FLOW MAP DATA








W

(1)AB

=|r

i

(A, B)||r

f

(A, B)| , (1)

1











@u@t+ u · ru = �rp + ⌫r2u + f, r · u = 0, (1)







I. FLOW MAP DATA








W

(1)AB

=|r

i

(A, B)||r

f

(A, B)| , (1)

1


maxB2nnhd(A)

|rf

(A, B)||r

i

(A, B)| , (2)




W

(2)AB

=|r

i

(A, B)||F(A)r

i

(A, B)| , (3)

where ri

(A, B) = ri

(B) � ri


(A) to ri

(B)], rf

(A, B) = rf

(B) � rf

(A) [the

vector from rf

(A) to rf

(B)], ri

(A) = [x

i

(A), yi


A], rf

(A) = [x

f

(A), yf



(A, B)| =p{F

xx

(A)[x

i

(B) � x

i

(A)] + F

xy

(A)[yi

(B) � y

i

(A)]}2 + {Fyx

(A)[x

i

(B) � x

i

(A)] + F

yy

(A)[yi

(B) � y

i

(A)]}2.







• W

(1)AB

(= W

(1)BA









identified.




2

3




3

Supp

lem

enta

lFig

ure

S1.T

here

lativ

edi

sper

sion

lnh m

axB2⌫(

A)|r f

(A,B

)|/|r i

(A,B

)|ifo

rthe

sam

egr

idel

emen

tsas

inFi

g.1

ofth

em

ain

text

.The

set⌫

(A)i

sth

ese

tofn

odes

that

are

adja

cent

tono

deA.


belonging to the same coherent structure (large )W (1)

AB

|rf (A,B)|

|ri(A,B)|

A

BA

B

C

D

D

C

|ri(C,D)|

|rf (C,D)|


I. FLOW MAP DATA








W

(1)AB

=|r

i

(A, B)||r

f

(A, B)| , (1)

1











@u@t+ u · ru = �rp + ⌫r2u + f, r · u = 0, (1)







I. FLOW MAP DATA








W

(1)AB

=|r

i

(A, B)||r

f

(A, B)| , (1)

1


maxB2nnhd(A)

|rf

(A, B)||r

i

(A, B)| , (2)




W

(2)AB

=|r

i

(A, B)||F(A)r

i

(A, B)| , (3)

where ri

(A, B) = ri

(B) � ri


(A) to ri

(B)], rf

(A, B) = rf

(B) � rf

(A) [the

vector from rf

(A) to rf

(B)], ri

(A) = [x

i

(A), yi


A], rf

(A) = [x

f

(A), yf



(A, B)| =p{F

xx

(A)[x

i

(B) � x

i

(A)] + F

xy

(A)[yi

(B) � y

i

(A)]}2 + {Fyx

(A)[x

i

(B) � x

i

(A)] + F

yy

(A)[yi

(B) � y

i

(A)]}2.







• W

(1)AB

(= W

(1)BA









identified.




2

3




3

Supp

lem

enta

lFig

ure

S1.T

here

lativ

edi

sper

sion

lnh m

axB2⌫(

A)|r f

(A,B

)|/|r i

(A,B

)|ifo

rthe

sam

egr

idel

emen

tsas

inFi

g.1

ofth

em

ain

text

.The

set⌫

(A)i

sth

ese

tofn

odes

that

are

adja

cent

tono

deA.


without involving differential terms such as the deformation-gradient tensor

2

FIG. 1. (Color online) Repelling (red), attracting (blue), and elliptic(green filled regions) LCSs in a direct numerical simulation of two-dimensional turbulence. MAP:do we need to worry about using thecolor green with the red with respect to people who are color-blind? SHL:I added “filled regions” after “green” that I hope willcircumvent that problem at least for some degree.

General relationships between community-finding, transport,and partitioning are discussed in Refs. [9, 11]. The Jacobianmatrix in dynamical systems has also been analyzed explicitlyusing a network-based approach to examine mesoscale struc-tures [12].

Community detection has been useful in flow problemssuch as the detection of plumes in fluids [13] and the analysisof force networks in granular materials [14, 15]. Network-theoretic tools, including community detection, were used toanalyze fluid flow networks in Refs. [16–18]. Note, how-ever, that Refs. [16, 17] represented mass transport GH:Howis mass transport Eulerian? via directed edges between ge-ographical areas (nodes), and they thus took what is essen-tially an Eulerian perspective. A particular type of vortexdynamics governed by the Biot-Savart law is investigated inRef. [18], where the spectral sparsification technique in thespectral graph theory is used to reduce the number of vortex-to-vertex interactions, rather than general type of coherentstructures in fluid. In our computations, by contrast, we ex-plicitly take a Lagrangian approach for general types of dy-namics in fluid.

MAP:the above paragraph needs some more work; also,should it be above or in the introduction section? I madechanges, and the accuracy of the various statements needto be double-checked; we should also distinguish what wedo from what is in [18]; GH:! Important! additionally,people like Gary Froyland have used spectra of Laplacianmatrices for LCS; GH:! These are almost invariant sets,not LCS. that is a very old form of community detection, sowe need to deepen our contributions with respect to ”clas-sical” things; I feel like this stu↵ should probably be in the

introductory section for the most part rather than here;anyway, the point is that I think that we need to makethings stronger in establishing the context of what we dovs the rest of the literature

SHL:I added a sentence describing Ref. [18], although Iam not 100% sure about the accuracy of the sentence de-scribing the paper. Please take a look at it for double-checking. I also moved the texts up to this point to theintroduction section.

Network Representation.—A pair of fluid elements A and Bhas a relative dispersion of

W (1)AB =

|ri(A, B)||r f (A, B)| , (1)

where ri = [xi, yi] and r f = [x f , y f ] denote the initial and finalpositions of the fluid element. The initial and final distancesbetween the fluid elements A and B are given by |ri(A, B)| =|ri(B) � ri(A)| and |r f (A, B)| = |r f (B) � r f (A)|. We also definethe relative dispersion for each grid element as

maxB2⌫(A)

|r f (A, B)||ri(A, B)| , (2)

where ⌫(A) is the set of nodes in a prescribed neighborhoodof A. If the fluid elements are initially given on a rectangulargrid, then ⌫(A) is given by the four closest neighbors consti-tute. In Fig. S1 of the SM [5], we show the relative dispersionsfor the same simulation as in Fig. 1.

An alternative way to define the relative dispersion is touse the deformation-gradient tensor [19] to obtain the instan-taneous displacement instead of the final position. This yields

W (2)AB =

|ri(A, B)||F(A)ri(A, B)| , (3)

where F(A) is the deformation-gradient tensor applied to thelocation of node A defined as the Jacobian @(x f , y f )/@(xi, yi)of the flow map. The relative dispersions W (1)

AB and W (2)AB in

Eqs. (1) and (3) give the weighted adjacency-matrix elementsbetween nodes A and B [7]. Larger values of these matrix ele-ments indicate stronger connections between the two fluid el-ements A and B. For each of the measures, distance measuressuch as ri(A, B) and coordinates such as ri(A) use the shortestdistance among all of the possible distances. (Note that we useperiodic boundary conditions for the simulation data, in con-trast to the real drifter data, GH:This does not appear to givea generally usable approach to ocean data SHL:I added abit of comments to avoid the possible confusion. so x itselfand x ± 2⇡ both yield x; and y and y ± 2⇡ both yield y.)

Community Detection.—There are several methods to de-tect communities in networks [8]. In our paper, we will ap-ply methods based on the maximization of a quality functioncalled “modularity,” the most popular approach to communitydetection. We partition a network’s node set V into sets called“communities” (with each node assigned to one community),such that intra-community interactions between nodes are sig-nificantly stronger than the inter-community interactions. We

2








W (1)AB =

|ri(A, B)||r f (A, B)| , (1)


maxB2⌫(A)

|r f (A, B)||ri(A, B)| , (2)



W (2)AB =

|ri(A, B)||F(A)ri(A, B)| , (3)


AB and W (2)AB in



: the Jacobian of the flow map, so that

2








W (1)AB =

|ri(A, B)||r f (A, B)| , (1)


maxB2⌫(A)

|r f (A, B)||ri(A, B)| , (2)



W (2)AB =

|ri(A, B)||F(A)ri(A, B)| , (3)


AB and W (2)AB in




I. FLOW MAP DATA








W

(1)AB

=|r

i

(A, B)||r

f

(A, B)| , (1)

1











@u@t+ u · ru = �rp + ⌫r2u + f, r · u = 0, (1)







I. FLOW MAP DATA








W

(1)AB

=|r

i

(A, B)||r

f

(A, B)| , (1)

1


maxB2nnhd(A)

|rf

(A, B)||r

i

(A, B)| , (2)




W

(2)AB

=|r

i

(A, B)||F(A)r

i

(A, B)| , (3)

where ri

(A, B) = ri

(B) � ri


(A) to ri

(B)], rf

(A, B) = rf

(B) � rf

(A) [the

vector from rf

(A) to rf

(B)], ri

(A) = [x

i

(A), yi


A], rf

(A) = [x

f

(A), yf



(A, B)| =p{F

xx

(A)[x

i

(B) � x

i

(A)] + F

xy

(A)[yi

(B) � y

i

(A)]}2 + {Fyx

(A)[x

i

(B) � x

i

(A)] + F

yy

(A)[yi

(B) � y

i

(A)]}2.







• W

(1)AB

(= W

(1)BA









identified.




2

3




3

Supp

lem

enta

lFig

ure

S1.T

here

lativ

edi

sper

sion

lnh m

axB2⌫(

A)|r f

(A,B

)|/|r i

(A,B

)|ifo

rthe

sam

egr

idel

emen

tsas

inFi

g.1

ofth

em

ain

text

.The

set⌫

(A)i

sth

ese

tofn

odes

that

are

adja

cent

tono

deA.



2








W (1)AB =

|ri(A, B)||r f (A, B)| , (1)


maxB2⌫(A)

|r f (A, B)||ri(A, B)| , (2)



W (2)AB =

|ri(A, B)||F(A)ri(A, B)| , (3)


AB and W (2)AB in



2








W (1)AB =

|ri(A, B)||r f (A, B)| , (1)


maxB2⌫(A)

|r f (A, B)||ri(A, B)| , (2)



W (2)AB =

|ri(A, B)||F(A)ri(A, B)| , (3)


AB and W (2)AB in




2








W (1)AB =

|ri(A, B)||r f (A, B)| , (1)


maxB2⌫(A)

|r f (A, B)||ri(A, B)| , (2)



W (2)AB =

|ri(A, B)||F(A)ri(A, B)| , (3)


AB and W (2)AB in




I. FLOW MAP DATA








W

(1)AB

=|r

i

(A, B)||r

f

(A, B)| , (1)

1











@u@t+ u · ru = �rp + ⌫r2u + f, r · u = 0, (1)







I. FLOW MAP DATA








W

(1)AB

=|r

i

(A, B)||r

f

(A, B)| , (1)

1


maxB2nnhd(A)

|rf

(A, B)||r

i

(A, B)| , (2)




W

(2)AB

=|r

i

(A, B)||F(A)r

i

(A, B)| , (3)

where ri

(A, B) = ri

(B) � ri


(A) to ri

(B)], rf

(A, B) = rf

(B) � rf

(A) [the

vector from rf

(A) to rf

(B)], ri

(A) = [x

i

(A), yi


A], rf

(A) = [x

f

(A), yf



(A, B)| =p{F

xx

(A)[x

i

(B) � x

i

(A)] + F

xy

(A)[yi

(B) � y

i

(A)]}2 + {Fyx

(A)[x

i

(B) � x

i

(A)] + F

yy

(A)[yi

(B) � y

i

(A)]}2.







• W

(1)AB

(= W

(1)BA









identified.




2

3




3

Supp

lem

enta

lFig

ure

S1.T

here

lativ

edi

sper

sion

lnh m

axB2⌫(

A)|r f

(A,B

)|/|r i

(A,B

)|ifo

rthe

sam

egr

idel

emen

tsas

inFi

g.1

ofth

em

ain

text

.The

set⌫

(A)i

sth

ese

tofn

odes

that

are

adja

cent

tono

deA.



2








W (1)AB =

|ri(A, B)||r f (A, B)| , (1)


maxB2⌫(A)

|r f (A, B)||ri(A, B)| , (2)



W (2)AB =

|ri(A, B)||F(A)ri(A, B)| , (3)


AB and W (2)AB in



2








W (1)AB =

|ri(A, B)||r f (A, B)| , (1)


maxB2⌫(A)

|r f (A, B)||ri(A, B)| , (2)



W (2)AB =

|ri(A, B)||F(A)ri(A, B)| , (3)


AB and W (2)AB in




2








W (1)AB =

|ri(A, B)||r f (A, B)| , (1)


maxB2⌫(A)

|r f (A, B)||ri(A, B)| , (2)



W (2)AB =

|ri(A, B)||F(A)ri(A, B)| , (3)


AB and W (2)AB in



applicable for most real data such assparsely released drifters or data on non-uniform grid


I. FLOW MAP DATA








W

(1)AB

=|r

i

(A, B)||r

f

(A, B)| , (1)

1











@u@t+ u · ru = �rp + ⌫r2u + f, r · u = 0, (1)






community detection (considering only nearest neighbor interactions

on the 2D square lattice)

initial positions final positions

community detection method: GenLouvain[Generalized Louvain: http://netwiki.amath.unc.edu/GenLouvain/GenLouvain original Louvain: V. D. Blondel et al., J. Stat. Mech. 2008, P10008 (2008)]

3

(a) (b)

FIG. 2. (Color online) Ten communities (each of a di↵erent color), which we detect algorithmically from a network constructed from nearest-neighbor interactions, from the simulated data that we show in Fig. 1. Panels (a) and (b), respectively, show the fluid elements at the initial andfinal times. We detect the communities using the relative dispersion W (1)

AB in Eq. (1) and the modularity QNG in Eq. (4). The resolution-parametervalue is � = 0.005. See Figs. S2–S5 in SM [5] for similar results using the modularity QLN and various resolution-parameter values.

that intra-community interactions between nodes are signif-icantly stronger than the inter-community interactions. Wedetect the set {gA | A 2 V} of communities, where node Ais assigned to community gA, such that modularity is maxi-mized. We use di↵erent null models for the relative disper-sion W (1)

AB (which is symmetric) and the deformation-gradienttensor W (2)

AB (which is not).For the relative dispersion between nodes A and B, we use

the modularity QNG for the Newman–Girvan (NG) null model[8, 20]:

QNG =1

2m

X

AB

W (1)

AB � �kAkB

2m

!� (gA, gB) , (4)

where kA =P

B W (1)AB =

PB W (1)

BA is the sum of weights corre-sponding to the interactions of A, the quantity 2m =

PA kA is

the total sum of weights in all of the interactions, � is a reso-lution parameter, and �(gA, gB) = 1 if A and B are in the samecommunity and 0 if they are not. The normalization constant1/(2m) enforces Q 2 [�1, 1]. Larger resolution-parameter val-ues � tend to result in smaller communities (with respect to thetypical number of nodes in a community).

For the deformation-gradient tensor between nodes A andB, we use the modularity QLN with the Leicht–Newman (LN)null model [21]:

QLN =1m

X

AB

0BBBB@W (2)

AB � �kin

A koutB

m

1CCCCA � (gA, gB) , (5)

where kinA =

PB W (2)

BA (respectively, koutA =

PB W (2)

AB) is the sumof incoming (respectively, outgoing) weights corresponding

to the interactions of A, and m =P

A kinA =

PA kout

A is the totalsum of weights for all of the interactions. (This sum is nec-essarily the same for both incoming and outgoing weights.)To detect the communities for both Eqs. (4) and (5), we usethe GenLouvain code [22], which is an implementation of amethod that is similar to the locally-greedy Louvain methodof Ref. [23].

Results for Simulation Data.—The turbulence-simulationdata consists of 512 ⇥ 512 grid points = 262144 nodes andtheir interactions given by Eqs. (1) and (3). For computationaltractability, we only consider the four nearest-neighbor inter-actions in the 2D grid system. This corresponds to setting allof the interactions between non-nearest-neighboring nodes inEqs. (1) and (3) to zero. The initial time is tinit = 0, and thefinal time is tfinal = 50 in the simulation time units. In Fig. 2,we show the ten detected communities using W (1)

AB [see Eq. (1)]with � = 0.005 [24]. For small values of �, one detects a smallnumber of large communities. They have blob-like structuresthat do not undergo substantial filamentation. Larger valuesof �, however, yield a larger number of communities, whichexhibit spiral patterns that correspond to vortex filaments. InFigs. S2–S5 of the SM [5], we show the dependence of thedetected communities for a set of values of � and using theweight function W (2)

AB. We find that the weight functions W (1)AB

and W (2)AB yield similar communities for the same value of �.

In Figs. S6–S9 of the SM [5], we present results for sam-pled grid points in which we use every fourth element (whichyields a 128⇥128 grid points and thus 16384 nodes) along thex and y axes. In this case, we also include all of their global

maximizing


I. FLOW MAP DATA








W

(1)AB

=|r

i

(A, B)||r

f

(A, B)| , (1)

1


maxB2nnhd(A)

|rf

(A, B)||r

i

(A, B)| , (2)




W

(2)AB

=|r

i

(A, B)||F(A)r

i

(A, B)| , (3)

where ri

(A, B) = ri

(B) � ri


(A) to ri

(B)], rf

(A, B) = rf

(B) � rf

(A) [the

vector from rf

(A) to rf

(B)], ri

(A) = [x

i

(A), yi


A], rf

(A) = [x

f

(A), yf



(A, B)| =p{F

xx

(A)[x

i

(B) � x

i

(A)] + F

xy

(A)[yi

(B) � y

i

(A)]}2 + {Fyx

(A)[x

i

(B) � x

i

(A)] + F

yy

(A)[yi

(B) � y

i

(A)]}2.







• W

(1)AB

(= W

(1)BA









identified.




2

3




3

Supp

lem

enta

lFig

ure

S1.T

here

lativ

edi

sper

sion

lnh m

axB2⌫(

A)|r f

(A,B

)|/|r i

(A,B

)|ifo

rthe

sam

egr

idel

emen

tsas

inFi

g.1

ofth

em

ain

text

.The

set⌫

(A)i

sth

ese

tofn

odes

that

are

adja

cent

tono

deA.


http://netwiki.amath.unc.edu/GenLouvain/GenLouvain



I. FLOW MAP DATA








W

(1)AB

=|r

i

(A, B)||r

f

(A, B)| , (1)

1

using relative-dispersion-based weights

using deformation-gradient-tensor-based weights

2








W (1)AB =

|ri(A, B)||r f (A, B)| , (1)


maxB2⌫(A)

|r f (A, B)||ri(A, B)| , (2)



W (2)AB =

|ri(A, B)||F(A)ri(A, B)| , (3)


AB and W (2)AB in




6

(a) (b)

(c) (d)

Supplemental Figure S4. In this figure, we detect communities, which we label by color, using the simulated data from Fig. 1 of the maintext. We keep only near-neighbor interactions between fluid elements. We measure the interaction strength between nodes using the relativedispersion W (2)

AB in Eq. (S2) and LN modularity in Eq. (S3). We consider di↵erent resolution-parameter values in the di↵erent panels. (a)� = 0.001 (3 communities and QLN ⇡ 0.99931), (b) � = 0.002 (8 communities and QLN ⇡ 0.99904), (c) � = 0.005 (16 communities andQLN ⇡ 0.99817), and (d) � = 0.01 (263 communities and QLN ⇡ 0.97818). MAP:for the methods with near neighbors, the values of � usedare really tiny; it seems like we at least need some sort of comment about this in the SM text SHL:I added Sec. S5 to address that point.

7

(a) (b)

(c) (d)

Supplemental Figure S5. Communities from the final positions of the fluid elements using the same data and community-detection method fromFig. S4. Corresponding panels in the figures use the same value of the resolution parameter �. MAP:similar question as in past final-positionfigs SHL:addressed in Fig. S7.



I. FLOW MAP DATA








W

(1)AB

=|r

i

(A, B)||r

f

(A, B)| , (1)

1

using relative-dispersion-based weights

using deformation-gradient-tensor-based weights

2








W (1)AB =

|ri(A, B)||r f (A, B)| , (1)


maxB2⌫(A)

|r f (A, B)||ri(A, B)| , (2)



W (2)AB =

|ri(A, B)||F(A)ri(A, B)| , (3)


AB and W (2)AB in




6

(a) (b)

(c) (d)

Supplemental Figure S4. In this figure, we detect communities, which we label by color, using the simulated data from Fig. 1 of the maintext. We keep only near-neighbor interactions between fluid elements. We measure the interaction strength between nodes using the relativedispersion W (2)

AB in Eq. (S2) and LN modularity in Eq. (S3). We consider di↵erent resolution-parameter values in the di↵erent panels. (a)� = 0.001 (3 communities and QLN ⇡ 0.99931), (b) � = 0.002 (8 communities and QLN ⇡ 0.99904), (c) � = 0.005 (16 communities andQLN ⇡ 0.99817), and (d) � = 0.01 (263 communities and QLN ⇡ 0.97818). MAP:for the methods with near neighbors, the values of � usedare really tiny; it seems like we at least need some sort of comment about this in the SM text SHL:I added Sec. S5 to address that point.

7

(a) (b)

(c) (d)

Supplemental Figure S5. Communities from the final positions of the fluid elements using the same data and community-detection method fromFig. S4. Corresponding panels in the figures use the same value of the resolution parameter �. MAP:similar question as in past final-positionfigs SHL:addressed in Fig. S7.

capturing the similar coherent structures without the differential term F(A) involved!









REPORTS



1

2

3

4


node

s


coupling = 0

1 2 3 4

5

10

15

20

25

30

node

s


coupling = 0.1

1 2 3 4

5

10

15

20

25

30

node

s


coupling = 1

1 2 3 4

5

10

15

20

25

30




on

Nov

embe

r 8, 2

011

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fro

m

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

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3 3.5

drite

r ID

time (day)

Time interval of drifters

Supplemental Figure S11. The time intervals of drifters recorded are shown with the blue vertical lines

corresponding to the spacing for the multilayer network analysis. tinit = 0.1 (day), tfinal = 3.1 (day), and

tres = 0.1 (day).

13

Preliminary Results of Community Detection of the 72 Drifters (last updated: February 12, 2014)

I. DRIFTER DATA

40.7 40.8 40.9 41 41.1 41.2 41.3 41.4 41.5 41.6 41.7

−71.1

−71

−70.9

−70.8

−70.7

−70.6

−70.5

−70.4

lat

lon

FIG. 1. The figure of drifters’ trajectories by Hosein Amini.

• Figure 1: the figure of drifters’ trajectories.

A. Drifters’ Time Interval

• As shown in Fig. 2, the initial and final time points are all di↵erent for di↵erent drifters.

Therefore, we first set the initial time t = tinit = 0.2 (day) and consider several di↵erent final

time points as t = tfinal = 0.3, 1.5, and 3.0 (day) for single-layer networks in Sec. II. For

multilayer networks in Sec. III, we set t = tinit = 0.1 (day) and t = tfinal = 3.1 (day) and

divide the time interval into pieces as time windows (see Fig. 4).

• Only the drifters whose time interval entirely contains [tinit, tfinal] are considered for each

case of tfinal, so the numbers of nodes are smaller for larger tfinal (as noted in the caption of

Fig. 6).

1

latitude

long

itude

real ocean flow: Northern Atlantic region (global interactions)

4

interactions (i.e., all of the nonzero elements in the weightfunctions W (l)

AB, where l 2 {1, 2}). Because we use all of theglobal interactions, we obtain much more spiral-like commu-nities (see Figs. S6–S9 of the SM [5]), compared to blob-likecommunities shown in Fig. 2.

Results for Snapshots of Drifter Data.—GH:Where is thisfrom? Irina Rypina, maybe. MAP:we also need to brieflydescribe what is in the data, so that readers know whatwe’re talking about; the first sentence below comes acrossas a non sequitur SHL:I think we need more informationfrom Irina Rypina (starting this section with such infor-mation would make the transition much smoother), as wellas the permission to use it in our paper. In Fig. S10 of theSM [5], we show the drifter trajectories. The drifters are re-leased at di↵erent times, and their trajectories are recordedfor di↵erent time intervals. (See Fig. S11 of the SM [5].) Forconsistency, we need a common time interval [tinit, tfinal] overwhich the trajectories are known for all drifters. The earliestinitial time is tinit = 0.2 (days). For the final time tfinal, weconsider several values: tfinal = 0.3, tfinal = 1.5, and tfinal = 3.0(days). For each final time, we only include drifters whosetime interval of recorded trajectory contains the entire interval[tinit, tfinal] are as nodes in a network. Consequently, the num-ber of network nodes is smaller for later tfinal. (See Fig. S12of the SM [5].)

For all of the drifters, we take the common initial time tinitand the final time tfinal as we described in the previous para-graph. The weight between two drifters A and B is based onrelative dispersion W (1)

AB defined in Eq. (1), where |ri(A, B)| and|r f (A, B)|, respectively, give the initial and final Euclidean dis-tances in the 2D (longitude, latitude) plane [25]. Note that thedeformation-gradient tensor F(A) in Eq. (3) is unknown for thedrifter data, so we cannot use W (2)

AB in Eq. (3) in those cases.The modularity for the NG null model is given by Eq. (4). InFig. S12 of the SM [5], we show the detected communities forfour di↵erent values of the resolution parameter �.

MAP:the discussion of what a community is above seemsrather redundant with earlier discussions? I suspect spacemay be an issue, in which case it should be shortened toonly what we need, but it also seems like we may wish toshorten it to minimize redundancy anyway; I am trying tofigure out what it’s contributing

SHL:I commented out that part, and I moved the Gen-Louvain part to the methodology section because GenLou-vain is used throughout the work.

MAP:should time-dependent community structure start anew section in order to emphasize it?

SHL:I separated the sections.Results for Time-Dependent Drifter Data.—To gain in-

sights into the dynamics of LCSs, it is also helpful to ex-amine time-dependent community structure. To do this,we will split up the full temporal dynamics into a se-ries of intervals. The simplest approach is to divide thetime interval [tinit, tfinal] uniformly into S “layers” usinga temporal resolution of tres. This yields the time in-tervals {[t0 = tinit, t1 = t0 + tres], [t1, t2], · · · , [tS�1, tS = tfinal]},

and similar to W (1)AB in Eq. (2), we define the weight between

nodes A and B in layer s 2 {1, . . . , S } as

W (1)ABs =

|rt=ts�1 (A, B)||rt=ts (A, B)| , (6)

where rt(A, B) is the Euclidean distance between thenodes A and B at time t [see Eq. (1)]. First, we treateach layer separately and detect communities as be-fore. We set tinit = 0.1 days and tfinal = 3.1 days. InFig. S13 of the SM [5], we show results for � = 1.We divide the time interval [tinit = 0.1, tfinal = 3.1]is uniformly into S = 30 non-overlapping intervals{[t0 = tinit = 0.1, 0.2), [0.2, 0.3), · · · , [3.0, tS = tfinal = 3.1]}with a temporal resolution of tres = 0.1.

We then apply multilayer community detection [26] to ourtime-dependent networks by using a generalized version ofthe modularity function in Eq. (4) for multilayer networks:

Qmulti =1

2µ

X

ABsr

" W (1)

ABs � �skAskBs

2ms

!�sr + �ABTBsr

#� (gAs, gBr) ,

(7)where A and B index nodes (i.e., fluid elements) as in the orig-inal modularity function in Eq. (4), and s and r index the timelayers in the multilayer network. (See Refs. [27, 28] for a re-view of multilayer networks.) That is, for each layer s, thereexists a separate network described by the adjacency-matrixelements W (1)

ABs. The quantity W (1)ABs, which describes the

strength of the interaction between fluid elements A and B attime s, is thus an element of an adjacency tensor [26, 27, 29].The intralayer interactions W (1)

ABs , 0 if nodes A and B areconnected in layer s, and W (1)

ABs = 0 otherwise. Additionally,kAs =

PB W (1)

ABs, we normalize in each layer s separately usingms =

PAB W (1)

ABs, and �s is the resolution parameter in layer s.To connect fluid elements to themselves when they are present

0 0.5

1 1.5

2 2.5

3

0 10 20 30 40 50 60 70initi

al p

oint

of t

ime

slic

e (d

ay)

node index

FIG. 3. (Color online) Multilayer (time-dependent) communitystructure of drifters for tinit = 0.1 day, tfinal = 3.1 day, and tres = 0.1.The resolution-parameter value is � = 1, and the interlayer couplingstrength is ! = 25. The vertical axis gives the initial point of timelayers ts in [ts, ts+1) where s = 0, 1, . . . , 29. MAP:I was unable todiscern what was meant by the description, and I am not entirelysure of the axis label either, so we need to adjust phrasing SHL:Ihope it’s clear now, and the horizontal axis indicates the node index.We indicate di↵erent communities by using di↵erent colors and sym-bols (also used in Fig. 4). The solid horizontal lines in the interiorcorrespond to the snapshots in Fig. 4.

4














W (1)ABs =

|rt=ts�1 (A, B)||rt=ts (A, B)| , (6)



Qmulti =1

2µ

X

ABsr

" W (1)

ABs � �skAskBs

2ms

!�sr + �ABTBsr

#� (gAs, gBr) ,






PB W (1)


PAB W (1)


0 0.5

1 1.5

2 2.5

3

0 10 20 30 40 50 60 70initi

al p

oint

of t

ime

slic

e (d

ay)

node index


maximizing









REPORTS



1

2

3

4


node

s


coupling = 0

1 2 3 4

5

10

15

20

25

30

node

s


coupling = 0.1

1 2 3 4

5

10

15

20

25

30

node

s


coupling = 1

1 2 3 4

5

10

15

20

25

30




on

Nov

embe

r 8, 2

011

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fro

m


0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3 3.5

drite

r ID

time (day)




tres = 0.1 (day).

13


I. DRIFTER DATA

40.7 40.8 40.9 41 41.1 41.2 41.3 41.4 41.5 41.6 41.7

−71.1

−71

−70.9

−70.8

−70.7

−70.6

−70.5

−70.4

lat

lon











Fig. 6).

1

latitude

long

itude


4














W (1)ABs =

|rt=ts�1 (A, B)||rt=ts (A, B)| , (6)



Qmulti =1

2µ

X

ABsr

" W (1)

ABs � �skAskBs

2ms

!�sr + �ABTBsr

#� (gAs, gBr) ,






PB W (1)


PAB W (1)


0 0.5

1 1.5

2 2.5

3

0 10 20 30 40 50 60 70initi

al p

oint

of t

ime

slic

e (d

ay)

node index


4














W (1)ABs =

|rt=ts�1 (A, B)||rt=ts (A, B)| , (6)



Qmulti =1

2µ

X

ABsr

" W (1)

ABs � �skAskBs

2ms

!�sr + �ABTBsr

#� (gAs, gBr) ,






PB W (1)


PAB W (1)


0 0.5

1 1.5

2 2.5

3

0 10 20 30 40 50 60 70initi

al p

oint

of t

ime

slic

e (d

ay)

node index


maximizing

16

(a) (b) (c)

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3

node

inde

xinitial point of time slice (day)

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3

node

inde

x

initial point of time slice (day)

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3

node

inde

x


(d) (e) (f)

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3

node

inde

x


0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3

node

inde

x


0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3

node

inde

x


Supplemental Figure S14. Multilayer (i.e., time-dependent) community structure of drifters for tinit = 0.1 day, tfinal = 3.1 day, and tres = 0.1. Weuse a resolution-parameter value of � = 1 and various values of the interlayer coupling !. We measure the interaction strength between nodesusing the relative dispersion W (1)

ABs in Eq. (6) of the main text and multilayer modularity in Eq. (7) of the main text. (a) ! = 2 (Qmulti ⇡ 0.89478),(b) ! = 4 (Qmulti ⇡ 0.80002), (c) ! = 20 (Qmulti ⇡ 0.55415), (d) ! = 25 (Qmulti ⇡ 0.55697), (e) ! = 30 (Qmulti ⇡ 0.57146), and (f) ! = 40(Q ⇡ 0.60159). The horizontal axes gives the initial point of time layers ts in [ts, ts+1) where s = 0, 1, . . . , 29, and the vertical axes gives thenode index. MAP:why did you transpose this from the main text? we need to be consistent; note that I am confused about the ”timeorder” label, just as in the main text We represent the communities using di↵erent colors and symbols (also used in Fig. S15). The solidvertical lines in the interior of (d) correspond to the snapshots in Fig. S15. SHL:I transposed it in the main text due to the space issue. Irealized that the limited palette does not a↵ect this plot as I used the di↵erent symbols as well, so I removed that part.









REPORTS



1

2

3

4


node

s


coupling = 0

1 2 3 4

5

10

15

20

25

30

node

s


coupling = 0.1

1 2 3 4

5

10

15

20

25

30

node

s


coupling = 1

1 2 3 4

5

10

15

20

25

30




on

Nov

embe

r 8, 2

011

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fro

m


0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3 3.5

drite

r ID

time (day)




tres = 0.1 (day).

13


I. DRIFTER DATA

40.7 40.8 40.9 41 41.1 41.2 41.3 41.4 41.5 41.6 41.7

−71.1

−71

−70.9

−70.8

−70.7

−70.6

−70.5

−70.4

lat

lon











Fig. 6).

1

latitude

long

itude


4














W (1)ABs =

|rt=ts�1 (A, B)||rt=ts (A, B)| , (6)



Qmulti =1

2µ

X

ABsr

" W (1)

ABs � �skAskBs

2ms

!�sr + �ABTBsr

#� (gAs, gBr) ,






PB W (1)


PAB W (1)


0 0.5

1 1.5

2 2.5

3

0 10 20 30 40 50 60 70initi

al p

oint

of t

ime

slic

e (d

ay)

node index


4














W (1)ABs =

|rt=ts�1 (A, B)||rt=ts (A, B)| , (6)



Qmulti =1

2µ

X

ABsr

" W (1)

ABs � �skAskBs

2ms

!�sr + �ABTBsr

#� (gAs, gBr) ,






PB W (1)


PAB W (1)


0 0.5

1 1.5

2 2.5

3

0 10 20 30 40 50 60 70initi

al p

oint

of t

ime

slic

e (d

ay)

node index


maximizing

16

(a) (b) (c)

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3

node

inde

xinitial point of time slice (day)

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3

node

inde

x


0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3

node

inde

x


(d) (e) (f)

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3

node

inde

x


0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3

node

inde

x


0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3

node

inde

x


Supplemental Figure S14. Multilayer (i.e., time-dependent) community structure of drifters for tinit = 0.1 day, tfinal = 3.1 day, and tres = 0.1. Weuse a resolution-parameter value of � = 1 and various values of the interlayer coupling !. We measure the interaction strength between nodesusing the relative dispersion W (1)

ABs in Eq. (6) of the main text and multilayer modularity in Eq. (7) of the main text. (a) ! = 2 (Qmulti ⇡ 0.89478),(b) ! = 4 (Qmulti ⇡ 0.80002), (c) ! = 20 (Qmulti ⇡ 0.55415), (d) ! = 25 (Qmulti ⇡ 0.55697), (e) ! = 30 (Qmulti ⇡ 0.57146), and (f) ! = 40(Q ⇡ 0.60159). The horizontal axes gives the initial point of time layers ts in [ts, ts+1) where s = 0, 1, . . . , 29, and the vertical axes gives thenode index. MAP:why did you transpose this from the main text? we need to be consistent; note that I am confused about the ”timeorder” label, just as in the main text We represent the communities using di↵erent colors and symbols (also used in Fig. S15). The solidvertical lines in the interior of (d) correspond to the snapshots in Fig. S15. SHL:I transposed it in the main text due to the space issue. Irealized that the limited palette does not a↵ect this plot as I used the di↵erent symbols as well, so I removed that part.

40.9

41

41.1

41.2

41.3

41.4

41.5

-71.1 -71 -70.9 -70.8 -70.7 -70.6 -70.5 -70.4

latit

ude

longitude

ω=25.0, γ=1.0, 1 community (Q=0.55697)

final positions at t=tfinal=0.2: (61 nodes), time slice of tinit=0.1 and tfinal=0.2)

40.9

41

41.1

41.2

41.3

41.4

41.5

-71.1 -71 -70.9 -70.8 -70.7 -70.6 -70.5 -70.4la

titud

elongitude

ω=25.0, γ=1.0, 9 communities (Q=0.55697)


40.9

41

41.1

41.2

41.3

41.4

41.5

-71.1 -71 -70.9 -70.8 -70.7 -70.6 -70.5 -70.4

latit

ude

longitude

ω=25.0, γ=1.0, 1 community (Q=0.55697)


40.9

41

41.1

41.2

41.3

41.4

41.5

-71.1 -71 -70.9 -70.8 -70.7 -70.6 -70.5 -70.4

latit

ude

longitude

ω=25.0, γ=1.0, 7 communities (Q=0.55697)


40.9

41

41.1

41.2

41.3

41.4

41.5

-71.1 -71 -70.9 -70.8 -70.7 -70.6 -70.5 -70.4

latit

ude

longitude

ω=25.0, γ=1.0, 9 communities (Q=0.55697)


40.9

41

41.1

41.2

41.3

41.4

41.5

-71.1 -71 -70.9 -70.8 -70.7 -70.6 -70.5 -70.4

latit

ude

longitude

ω=25.0, γ=1.0, 2 communities (Q=0.55697)


real ocean flow: The Global Drifter Program (global interactions)

http://www.aoml.noaa.gov/phod/dac/index.php


real ocean flow: The Global Drifter Program (global interactions)


a small sample data in the Northern Atlantic region: Sep. 1-30, 2013

10

20

30

40

50

60

-70 -60 -50 -40 -30 -20

latit

ude

longitude

γ=1.0, 6 communities (Q=0.030906)Sep. 1-30, 2013, wAB = dAB(t=tinit)/dAB(t=tfinal): snapshot on Sep. 1

10

20

30

40

50

60

-70 -60 -50 -40 -30 -20la

titud

elongitude


September 1, 2013 September 30, 2013

3

(a) (b)

FIG. 2. (Color online) Ten communities (each of a di↵erent color), which we detect algorithmically from a network constructed from nearest-neighbor interactions, from the simulated data that we show in Fig. 1. Panels (a) and (b), respectively, show the fluid elements at the initial andfinal times. We detect the communities using the relative dispersion W (1)

AB in Eq. (1) and the modularity QNG in Eq. (4). The resolution-parametervalue is � = 0.005. See Figs. S2–S5 in SM [5] for similar results using the modularity QLN and various resolution-parameter values.

that intra-community interactions between nodes are signif-icantly stronger than the inter-community interactions. Wedetect the set {gA | A 2 V} of communities, where node Ais assigned to community gA, such that modularity is maxi-mized. We use di↵erent null models for the relative disper-sion W (1)

AB (which is symmetric) and the deformation-gradienttensor W (2)

AB (which is not).For the relative dispersion between nodes A and B, we use

the modularity QNG for the Newman–Girvan (NG) null model[8, 20]:

QNG =1

2m

X

AB

W (1)

AB � �kAkB

2m

!� (gA, gB) , (4)

where kA =P

B W (1)AB =

PB W (1)

BA is the sum of weights corre-sponding to the interactions of A, the quantity 2m =

PA kA is

the total sum of weights in all of the interactions, � is a reso-lution parameter, and �(gA, gB) = 1 if A and B are in the samecommunity and 0 if they are not. The normalization constant1/(2m) enforces Q 2 [�1, 1]. Larger resolution-parameter val-ues � tend to result in smaller communities (with respect to thetypical number of nodes in a community).

For the deformation-gradient tensor between nodes A andB, we use the modularity QLN with the Leicht–Newman (LN)null model [21]:

QLN =1m

X

AB

0BBBB@W (2)

AB � �kin

A koutB

m

1CCCCA � (gA, gB) , (5)

where kinA =

PB W (2)

BA (respectively, koutA =

PB W (2)

AB) is the sumof incoming (respectively, outgoing) weights corresponding

to the interactions of A, and m =P

A kinA =

PA kout

A is the totalsum of weights for all of the interactions. (This sum is nec-essarily the same for both incoming and outgoing weights.)To detect the communities for both Eqs. (4) and (5), we usethe GenLouvain code [22], which is an implementation of amethod that is similar to the locally-greedy Louvain methodof Ref. [23].

Results for Simulation Data.—The turbulence-simulationdata consists of 512 ⇥ 512 grid points = 262144 nodes andtheir interactions given by Eqs. (1) and (3). For computationaltractability, we only consider the four nearest-neighbor inter-actions in the 2D grid system. This corresponds to setting allof the interactions between non-nearest-neighboring nodes inEqs. (1) and (3) to zero. The initial time is tinit = 0, and thefinal time is tfinal = 50 in the simulation time units. In Fig. 2,we show the ten detected communities using W (1)

AB [see Eq. (1)]with � = 0.005 [24]. For small values of �, one detects a smallnumber of large communities. They have blob-like structuresthat do not undergo substantial filamentation. Larger valuesof �, however, yield a larger number of communities, whichexhibit spiral patterns that correspond to vortex filaments. InFigs. S2–S5 of the SM [5], we show the dependence of thedetected communities for a set of values of � and using theweight function W (2)

AB. We find that the weight functions W (1)AB

and W (2)AB yield similar communities for the same value of �.

In Figs. S6–S9 of the SM [5], we present results for sam-pled grid points in which we use every fourth element (whichyields a 128⇥128 grid points and thus 16384 nodes) along thex and y axes. In this case, we also include all of their global

maximizing



10

20

30

40

50

60

-70 -60 -50 -40 -30 -20

latit

ude

longitude


10

20

30

40

50

60

-70 -60 -50 -40 -30 -20

latit

ude

longitude




10

20

30

40

50

60

-70 -60 -50 -40 -30 -20

latit

ude

longitude


10

20

30

40

50

60

-70 -60 -50 -40 -30 -20

latit

ude

longitude



-60-40-20

0 20 40 60 80

-150 -100 -50 0 50 100 150

latit

ude

longitude


-60-40-20

0 20 40 60 80

-150 -100 -50 0 50 100 150

latit

ude

longitude


Summary and Outlook

• analysis of Lagrangian coherent structures (LCSs) in terms of interrelated fluid particles or “networks”• advantage: based on the observable relative dispersion

without involving the differential terms in contrast to the conventional variational approach (more practical for real data)

• single- and multilayer (temporal + spatial) community identifications

• future work• more systematic approach by controlling the resolution

parameter, interlayer coupling strength, etc.• any insight from the network approach applicable to the

system that cannot be revealed by conventional fluid dynamics approaches?

these slides in .pdf: http://www.slideshare.net/lshlj82/finding-lagrangian-coherent-structures-using-community-detection-47177457

http://www.slideshare.net/lshlj82/finding-lagrangian-coherent-structures-using-community-detection-47177457