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)

Lagrangian vs Eulerian point of view on fluid

Lagrangian Eulerian

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

fluid-element network

photo courtesy of Heetae Kim

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

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






Pijs Aijs +


Qmultilayer =1

2µ

X

ijsr

✓Aijs � �s

kiskjs2ms


��(gis, gjr)










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




(weighted) edges

multilayer community index: for node i on layer s






Pijs Aijs +


Qmultilayer =1

2µ

X

ijsr

✓Aijs � �s

kiskjs2ms


��(gis, gjr)










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


only the adjacent layers areconnected to each other: “ordered” multilayer communities



(weighted) edges

multilayer community index: for node i on layer s

An Example of Lagrangian Coherent Structure (LCS)

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

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


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


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


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


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


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

QGN =1

2m

X

AB

W

(1)AB

� �k

A

k

B

2m

!� (

g

A

, gB

) , (4)

where A and B are node indices, k

A

=P

B

W

(1)AB

=P

B

W

(1)BA

is the sum of weights correspond-

ing to the interactions connected to A, 2m =P

A

k

A

is the total sum of weights in all the

interactions, � is the resolution parameter, �(gA

, gB

) = 1 if A and B are in the same commu-

nity and 0 otherwise, and the overall factor 1/(2m) is used for the normalization condition

Q 2 [�1, 1].

• W

(2)AB

(, W

(2)BA

) in Eq. (3): the Louvain method [1, 2] with the Leicht-Newman null model [5]

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

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

here describe similar structures based on the relative strength di↵erence between intra-group

and inter-group interactions, but since the Leicht-Newman null model [5] considers the

direction of the interactions, the method tends to split the “source” and “sink” groups (but

see Ref. [6]).

The modularity for the Leicht-Newman null model, which is the objective function QLN


} that maximizes QLN, is given by

QLN =1m

X

AB

W

(2)AB

� �k

inA

k

outB

m

!� (

g

A

, gB

) , (5)


inA

=P

B

W

(2)BA

(koutA

=P

B

W

(2)AB

) is the sum of incoming (out-

going) weights corresponding to the interactions connected to A, respectively, m =P

A

k

inA

=P

A

k

outA

is the total sum of weights in all the interactions (same for incoming and outgoing

weights), � is the resolution parameter, �(gA

, gB


nity and 0 otherwise, and the overall factor 1/m is used for the normalization condition

Q 2 [�1, 1].

• Possible extension to time-dependent and/or multilayer networks: see Ref. [7].

3











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






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


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


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


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


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


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

QGN =1

2m

X

AB

W

(1)AB

� �k

A

k

B

2m

!� (

g

A

, gB

) , (4)


A

=P

B

W

(1)AB

=P

B

W

(1)BA

is the sum of weights correspond-

ing to the interactions connected to A, 2m =P

A

k

A

is the total sum of weights in all the

interactions, � is the resolution parameter, �(gA

, gB


nity and 0 otherwise, and the overall factor 1/(2m) is used for the normalization condition

Q 2 [�1, 1].

• W

(2)AB

(, W

(2)BA

) in Eq. (3): the Louvain method [1, 2] with the Leicht-Newman null model [5]

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

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

here describe similar structures based on the relative strength di↵erence between intra-group

and inter-group interactions, but since the Leicht-Newman null model [5] considers the

direction of the interactions, the method tends to split the “source” and “sink” groups (but

see Ref. [6]).

The modularity for the Leicht-Newman null model, which is the objective function QLN


} that maximizes QLN, is given by

QLN =1m

X

AB

W

(2)AB

� �k

inA

k

outB

m

!� (

g

A

, gB

) , (5)


inA

=P

B

W

(2)BA

(koutA

=P

B

W

(2)AB

) is the sum of incoming (out-

going) weights corresponding to the interactions connected to A, respectively, m =P

A

k

inA

=P

A

k

outA

is the total sum of weights in all the interactions (same for incoming and outgoing

weights), � is the resolution parameter, �(gA

, gB


nity and 0 otherwise, and the overall factor 1/m is used for the normalization condition

Q 2 [�1, 1].

• Possible extension to time-dependent and/or multilayer networks: see Ref. [7].

3











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






community detection

initial positions final positions

All right, but so what? Why is it important to detect the LCS?

All right, but so what? Why is it important to detect the LCS?

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

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

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

real flow

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

(the Northern Atlantic region)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

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)


FIG. 2. The time intervals of drifters recorded. The green dashed vertical line shows tinit = 0.2 (day) and

blue dotted vertical lines show tfinal = 0.3, 1.5, and 3.0, used in Sec. II.

• For each drifter, the smallest time point larger than or equal to the (global) tinit is considered

as the drifter’s tfinal, and the largest time point smaller than or equal to the (global) tfinal is

considered as the drifter’s tfinal.

II. SINGLE-LAYER NETWORK ANALYSIS

A. Definition of Weights in the Single-Layer Network

W

i j

=d

i j

(t = tinit)d

i j

(t = tfinal), (1)

where d

i j

is defined as the Euclidean distance in the two-dimensional (longitude,latitude) plane.

B. Single-Layer Community Detection Method

• W

i j

(= W

ji

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



2

real ocean flow

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

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)









W

i j

=d

i j

(t = tinit)d

i j

(t = tfinal), (1)

where d

i j



• W

i j

(= W

ji




2

(a) (b)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


(c) (d)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


(e) (f)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


FIG. 9. Multilayer (time-dependent) community structure of drifters [tinit = 0.1 (day), tfinal = 3.1 (day),

and tres = 0.1] with the interlayer coupling ! = 25 and the community resolution parameter � = 1, where

di↵erent colors correspond to di↵erent communities found [the same colors as in Fig. 8(d) and the vertical

lines in Fig. 8(d) correspond to the snapshots presented in this figure]. The positions of drifters are the ones

at t = tfinal for each case. (a) tinit = 0.1 (day), tfinal = 0.2 (day), 61 nodes, 1 community. (b) tinit = 0.4 (day),

tfinal = 0.5 (day), 55 nodes, 1 community. (c) tinit = 0.9 (day), tfinal = 1.0 (day), 54 nodes, 9 communities.

(d) tinit = 1.2 (day), tfinal = 1.3 (day), 50 nodes, 9 communities. (e) tinit = 1.9 (day), tfinal = 2.0 (day), 45

nodes, 7 communities. (f) tinit = 2.9 (day), tfinal = 3.0 (day), 33 nodes, 2 communities.

12

latitude

long

itude

real ocean flow

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

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)









W

i j

=d

i j

(t = tinit)d

i j

(t = tfinal), (1)

where d

i j



• W

i j

(= W

ji




2

(a) (b)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


(c) (d)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


(e) (f)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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










12

latitude

long

itude

(a) (b) (c)

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)

γ=1.0, ω=2.0: Q = 0.89478

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3

node

inde

x


γ=1.0, ω=4.0: Q = 0.80002

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3

node

inde

x


γ=1.0, ω=20.0: Q = 0.55415

(d) (e) (f)

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3

node

inde

x


γ=1.0, ω=25.0: Q = 0.55697

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3

node

inde

x


γ=1.0, ω=30.0: Q = 0.57146

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3

node

inde

x


γ=1.0, ω=40.0: Q = 0.60159

FIG. 8. Multilayer (time-dependent) community structure of drifters for tinit = 0.1 (day), tfinal = 3.1 (day),

and tres = 0.1. The community resolution parameter � = 1 and various values of interlayer coupling ! are

applied for (a) ! = 2 (Q = 0.89478), (b) ! = 4 (Q = 0.80002), (c) ! = 20 (Q = 0.55415), (d) ! = 25

(Q = 0.55697), (e) ! = 30 (Q = 0.57146), and (f) ! = 40 (Q = 0.60159). The horizontal (vertical) axis

represents the time order (node index), respectively. Di↵erent colors correspond to di↵erent communities

found, with possibly duplicated colors, as the number of color schemes is limited in Gnuplot that I used to

plot them. Check the Matlab data attached for the full result. The additional vertical lines in (d) indicate

the “snapshot” list of tinit used in Fig. 9.

11

real ocean flow

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

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)









W

i j

=d

i j

(t = tinit)d

i j

(t = tfinal), (1)

where d

i j



• W

i j

(= W

ji




2

(a) (b)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


(c) (d)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


(e) (f)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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










12

latitude

long

itude

(a) (b) (c)

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3

node

inde

x


γ=1.0, ω=2.0: Q = 0.89478

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3

node

inde

x


γ=1.0, ω=4.0: Q = 0.80002

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3

node

inde

x


γ=1.0, ω=20.0: Q = 0.55415

(d) (e) (f)

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3

node

inde

x


γ=1.0, ω=25.0: Q = 0.55697

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3

node

inde

x


γ=1.0, ω=30.0: Q = 0.57146

0

10

20

30

40

50

60

70

0 0.5 1 1.5 2 2.5 3

node

inde

x


γ=1.0, ω=40.0: Q = 0.60159

FIG. 8. Multilayer (time-dependent) community structure of drifters for tinit = 0.1 (day), tfinal = 3.1 (day),

and tres = 0.1. The community resolution parameter � = 1 and various values of interlayer coupling ! are

applied for (a) ! = 2 (Q = 0.89478), (b) ! = 4 (Q = 0.80002), (c) ! = 20 (Q = 0.55415), (d) ! = 25

(Q = 0.55697), (e) ! = 30 (Q = 0.57146), and (f) ! = 40 (Q = 0.60159). The horizontal (vertical) axis

represents the time order (node index), respectively. Di↵erent colors correspond to di↵erent communities

found, with possibly duplicated colors, as the number of color schemes is limited in Gnuplot that I used to

plot them. Check the Matlab data attached for the full result. The additional vertical lines in (d) indicate

the “snapshot” list of tinit used in Fig. 9.

11

(a) (b)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


(c) (d)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


(e) (f)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5lo

ngitu

delatitude

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










12

(a) (b)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


(c) (d)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


(e) (f)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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










12

(a) (b)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


(c) (d)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


(e) (f)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5lo

ngitu

de

latitude

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










12

(a) (b)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


(c) (d)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


(e) (f)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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










12

(a) (b)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


(c) (d)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


(e) (f)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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










12

(a) (b)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


(c) (d)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5lo

ngitu

de

latitude

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


-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


(e) (f)

-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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


-71.1

-71

-70.9

-70.8

-70.7

-70.6

-70.5

-70.4

40.9 41 41.1 41.2 41.3 41.4 41.5

long

itude

latitude

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










12

real ocean flow

The Global Drifter Program

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




Maria Antonova

@mashant

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

2:51 AM - 17 Nov 2014

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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.

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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

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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.

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

0

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longitude

γ=1.0, 6 communities (Q=0.030906)

Sep. 1-30, 2013, wAB = dAB(t=tinit)/dAB(t=tfinal): snapshot on Sep. 1

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γ=1.0, 6 communities (Q=0.030906)


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γ=1.0, 6 communities (Q=0.030906)



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γ=1.0, 6 communities (Q=0.030906)


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γ=1.0, 6 communities (Q=0.030906)


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γ=1.0, 5 communities (Q=0.012128)


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γ=1.0, 5 communities (Q=0.012128)



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γ=1.0, 6 communities (Q=0.030906)


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γ=1.0, 6 communities (Q=0.030906)


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γ=1.0, 5 communities (Q=0.012128)


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γ=1.0, 5 communities (Q=0.012128)


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γ=1.0, 5 communities (Q=0.012128)


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γ=1.0, 5 communities (Q=0.012128)


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γ=1.0, 5 communities (Q=0.012128)


Summary and Outlook

Acknowledgments

Mason Porter(University of Oxford)

George Haller (ETH Zürich)

Mohammad Farazmand(Georgia Institute of

Technology)

• analysis of Lagrangian coherent structures (LCSs) in terms of interrelated fluid particles or “networks”• multilayer (temporal + spatial) community identification• simulated & (real) satellite-tracked data

• future work: more systematic approach by controlling the resolution parameter, etc.

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

Thank you for your attention!!!Any question? Ask now, or contact me anytime later.

[email protected]

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

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

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

finding lagrangian coherent structures using community detection

Science

slices s

nodes i

layer s

layers s

slices r

nodes resolution parameters

strength of node i

different slices