mapping change in large integrated systems

Mapping changein large integrated systems

Martin RosvallAlcides V. Esquivel, AtiehMirshahvalad,Daniel Edler, and Carl Bergstrom

Aquatic Ecology

Ecology & EvolutionGeology

Geophysics & Climatology

Entomology

Environmental Science

Soil Science

Hydrology

Aquaculture Fisheries Science

Molecular & Cell Biology

Medicine

Neuroscience

Immunology & Hematology

Psychiatry & Psychology

Oncology

Infectious Disease

Applied Microbiology

Gastroenterology & Hepatology

Botany

Respiratory Medicine

Radiology

Nephrology

Reproductive Medicine

Orthopedics

Ophthalmology

Toxicology

Food scienceRheumatology

Veterinary

Urology

Dentistry

Dermatology

Sports Medicine

Otolaryngology

Anesthesiology

Bone

Neurosurgery

Pharmacokinetics & Pharmacodynamics

Gene Therapy

Ethnopharmacology

Physics

Chemistry

Materials Science

High-energy Physics

Polymer Science

Optics

Chemical Engineering

Analytical Chemistry

Medicinal Chemistry

ElectrochemistryAstrophysics & Astronomy

Drug Delivery

Fluid Mechanics

Applied Mechanics

Radiation

Plasma Physics

AcousticsRenewable Energy

Mathematics

Applied Math

Discrete Math & Computer Science

Microwaves & Circuits

Communications Engineering

Software Engineering

Computer Vision

Probability Theory

Applied Computation

Operations Research

Robotics

Cybernetics

Power Engineering

EconomicsFinance

Political Science

Business & Management

Statistics

Sociology

Geography

Law

Education

Physical Sciences

Life Sciences

Ecology & Earth Sciences

Social Sciences

Rosvall & Bergstrom (2011)

Aquatic Ecology



Entomology


Soil Science

Hydrology



Medicine

Neuroscience



Oncology

Infectious Disease



Botany


Radiology

Nephrology


Orthopedics

Ophthalmology

Toxicology


Veterinary

Urology

Dentistry

Dermatology

Sports Medicine

Otolaryngology

Anesthesiology

Bone

Neurosurgery


Gene Therapy

Ethnopharmacology

Physics

Chemistry

Materials Science

High-energy Physics

Polymer Science

Optics



Medicinal Chemistry


Drug Delivery

Fluid Mechanics

Applied Mechanics

Radiation

Plasma Physics


Mathematics

Applied Math





Computer Vision

Probability Theory

Applied Computation

Operations Research

Robotics

Cybernetics

Power Engineering

EconomicsFinance

Political Science


Statistics

Sociology

Geography

Law

Education

Physical Sciences

Life Sciences


Social Sciences

Molecular &cell biology

Medicine

Oncology

Psychiatry

NeurologyPsychologyNephrology


Medicine

Neuroscience

Oncology

Psychiatry

Urology

Infectiousdiseases

PsychologyNephrology

2001 2003 2005 2007


Courtesy of Aaron Koblin

How are these systems organized?with respect to the flow?

Networksdescribe flow

beyond nearest neighbors

Barabasi, A.L.

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Pacheco, A.F.

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Nekovee, M.

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Vazquez-Prada, M.

Kertesz, J.

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Tass, P.

Volkmann, J.

Weule, M.G.

Kim, B.J.

Holme, P.

Choi, M.Y.

Trusina, A.

Hong, H.

Minnhagen, P.

Han, S.K.

Huss, M.

Yoon, C.N.

Chung, J.S.

Park, H.

Vicsek, T.

Palla, G.

Farkas, I.J.

Derenyi, I.

Adamcsek, B.

D'Ovidio, F.

Marodi, M.

Pollner, P.

Abel, D.

Schurbert, A.

Tomkins, A.S.

Kumar, R.S.

Raghavan, P.

Rajagopalan, S.

Novak, J.

Chakrabarti, D.

Kumar, S.R.

Sivakumar, D.

Upfal, E.

Backstrom, L.

Broder, A.

Maghoul, F.

Stata, R.

Wiener, J.

Fortunato, S.

Castellano, C.

Lancichinetti, A.

Radicchi, F.

Ramasco, J.J.

Cecconi, F.

Loreto, V.

Parisi, D.

Vilone, D.

Mungan, M.

Porter, M.A.

Mucha, P.J.

Jones, N.S.

Traud, A.L.

Fenn, D.J.

Fowler, J.H.

Friend, A.J.

Johnson, N.F.

Mcdonald, M.

Richardson, T.

Williams, S.

Deane, C.M.

Kelsic, E.D.

Lewis, A.C.F.

Warbrand, C.M.

Warmbrand, C.M.

Havlin, S.

Cohen, R.

Ben-Avraham, D.

Erez, K.

Makse, H.A.

Schwartz, N.

Song, C.

Rozenfeld, A.F.

Dorogovtsev, S.N.

Mendes, J.F.F.

Samukhin, A.N.

Goltsev, A.V.

Di, Z.

Fan, Y.

Zhang, P.

Li, M.

Wu, J.

Hu, Y.

Chen, H.

Gan, G.

Ma, C.

Bornholdt, S.

Ebel, H.

Reichardt, J.

Leone, M.

Davidsen, J.

Mielsch, L.I.

Rohlf, T.

Stauffer, D.

Da Fontoura Costa, L.

Aharony, A.

Adler, J.

Bernardes, A.T.

Diambra, L.

Meyer-Ortmanns, H.

Rodrigues, F.A.

Travieso, G.

Araujo, A.D.

Costa, U.M.S.

Motter, A.E.

Lai, Y.C.

Liu, Z.

De Moura, A.P.S.

Hoppensteadt, F.C.

Nishikawa, T.

Grebogi, C.

Ye, N.

Zhou, C.

Dasgupta, P.

Lawrence, S.R.

Giles, C.L.

Flake, G.W.

Coetzee, F.M.

Bollacker, K.

Glover, E.J.

Pennock, D.M.

Lee Giles, C.

Sneppen, K.

Maslov, S.

Eriksen, K.A.

Simonsen, I.

Bak, P.

Zaliznyak, A.

Rinaldo, A.

Maritan, A.

Rodriguez-Iturbe, I.

Banavar, J.R.

Rigon, R.

Cieplak, M.

Fedroff, N.V.

Giacometti, A.

Holter, N.S.

Mitra, M.

Watts, D.J.

Dodds, P.S.

Muhamad, R.

Rothman, D.H.

Sabel, C.F.

Caldarelli, G.

Capocci, A.

Servedio, V.D.P.

Castri, M.

Rios, P.D.L.

Colaiori, F.

Krapivsky, P.L.

Redner, S.

Antal, T.

Leyvraz, F.

Vazquez, F.

Kim, J.

Chen, G.

Li, C.

Li, X.

Wang, X.F.

Lu, J.

Yu, X.

Xu, J.

Chen, D.

Tadic, B.

Rodgers, G.J.

Thurner, S.

Darby-Dowman, K.

Ergun, G.

Mitrovic, M.

Sokolov, I.M.

Blumen, A.

Jespersen, S.

Sander, L.M.

Warren, C.P.

Koopman, J.

Simon, C.

Xulvi-Brunet, R.

Strogatz, S.H.

Mirollo, R.E.

Callaway, D.S.

Hopcroft, J.E.

Matthews, P.C.

Yeung, M.K.S.

Hu, G.

Hu, B.

Yang, J.

Gao, Z. Liu, W.

Zhan, M.

Zheng, Z.

Bianconi, G.

Perez Vicente, C.J.

Acebron, J.A.

Bonilla, L.L.

Coolen, A.C.C.

Marsili, M. Pin, P.

Ritort, F.

Spigler, R.

Schnitzler, A.

Gross, J.

Hamalainen, M.Hari, R.

Ilmoniemi, R.

Knuutila, J.

Kujala, J.

Lounasmaa, O.V.

Salmelin, R.

Timmermann, L.

Pecora, L.M.

Carroll, T.L.

Heagy, J.F.

Pelaez, A.

Fink, K.S.

Johnson, G.

De Los Rios, P.

Gfeller, D.

Chappelier, J.C.

Coccetti, F.

Brunet, R.

Petermannn, T.

Fortuna, L.

La Rosa, M.

Frasca, M.

Cosenza, S.

Stagni, C.

Usai, L.

Bucolo, M.

Spata, A.

Mantegna, R.N.

Bonanno, G.

Lillo, F.

Tumminello, M.

Vandewalle, N.

Almaas, E.

Kulkarni, R.V.

Stroud, D.

Macdonald, P.J.

Williams, R.J.

Martinez, N.D.

Dunne, J.A.

Berlow, E.L.

Koch, C.

Schuster, H.G.

Crick, F.

Kreiman, G.

Laurent, G.

Ress, G.

Kammen, D.M.

Niebur, E.

Moukarzel, C.F.

Argollo De Menezes, M.

De Menezes, M.A.

Penna, T.J.P.

Zhou, T.

Wang, B.H.

Chen, E.H.

Ding, J.F.

Jiang, X.Y.

Liu, J.G.

Xiang, B.

Xie, Y.B.

Faloutsos, C.

Yu, P.S.

Faloutsos, M.

Faloutsos, P.

Palmer, C.R.

Papadimitriou, S.

Sun, J.

Ben-Jacob, E.

Ayali, A.

Cohen, I.

Czirok, A.

Golding, I.

Segev, R.

Shefi, O.

Shochet, O.

Herrmann, H.J.

De Arcangelis, L.

Gonzales, M.C.

Hong, D.C.

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Barahona, M.

Delvenne, J.C.

Lambiotte, R.

Evans, T.S.

Yaliraki, S.N.

Kleinberg, J.M.

Gorman, S.P.

Kulkarni, R.

Liben-Nowell, D.

Rosvall, M.

Bergstrom, C.T.

Axelsson, D.

Holyst, J.A.

Fronczak, A.

Aleksiejuk, A.

Fronczak, P.

Jedynak, M.

Sienkiewicz, J.

Munoz, M.A.

Marro, J.

Dickman, R.

Donetti, L.

Garrido, P.L.

Torres, J.J.

Noh, J.D.

Rieger, H.

Son, S.W.

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Koput, K.W.

Owen-Smith, J.

Powell, W.W.

Park, J.

Teichmann, S.A.

Apic, G.

Gough, J.

Lappe, M.

Wang, J.

De Wilde, P.

Fu, Z.

Krishnamurthy, B.

Zhang, Z.

Coppersmith, S.N.

Kadanoff, L.P.

Zhang, Y.

Braun, T.

Cerdeira, H.A.

Chen, S.

Yao, Y.

Bearman, P.S.

Moody, J.

Stovel, K.

Leskovec, J.

Dasgupta, A.

Lang, K.J.

Mahoney, M.W.

Xu, X.

Feng, Z.

Long, B.

Schweiger, T.

Yuruk, N.

Almendral, J.A.

Buldu, J.M.

Leyva, I.

Li, D.

Sendina-Nadal, I.

Wang, B.

Du, N.

Wang, Y.

Wu, B.

Yan, G.

Liao, X.

Ren, W.

Xiao, L.

Pietronero, L.

Bottaccio, M.

De Lucia, M.

Montuori, M.

Arecchi, F.T.

Allaria, E.

Di Garbo, A.

Meucci, R.

Louis, E.

Vragovic, I.

Chate, H.

Gregoire, G.

Rudzick, O.

Forrest, S.

Balthrop, J.

Kaplan, T.D.

Garlaschelli, D.

Loffredo, M.I.

Battiston, S.

Catanzaro, M.

Guardiola, X.

Llas, M.

Perez, C.J.

Angel, M.L.

Martin, M.

Schrag, S.

Albert, I.

Nakarado, G.L.

GoogleMaps for networks

Mapsdepict regularities

in the dynamics on networksusing less information

Finding regularities⇐⇒Compression5.8MB (tiff)→ 0.91MB (tiff + LZW)

5.8MB (tiff)→ 2.8MB (tiff + LZW)

If we can find a good codefor describing flow on a network,

wewill have solved the dual problemof finding the important structures

with respect to that flow

Weuse amodular code structurethat can exploit regions in the networkin which units of flow tend to stay

for a relatively long time

Two-level partitions

Howmanymodules are present? Andwhichnodes aremembers of whichmodules?Maximal compression of flowwith constraints:1. Modular code structure2. Nomore than two levels3. Each node can only belong to onemodule

Two-level partitions with themap equation

Howmanymodules are present? Andwhichnodes aremembers of whichmodules?Maximal compression of flowwith constraints:1. Modular code structure2. Nomore than two levels3. Each node can only belong to onemodule


L(M) = qyH(Q) +m∑i=1

pi�H(P i)


BA

2

1

3

L(M) = H(P) = 4.75 bits.

L(M) = qxH(Q) +

p1�H(P1)

p2�H(P2)

p3�H(P3)

= 3.68 bits.

︸︷︷︸0.12 bits

︸︷︷︸3.56 bits

Two-level partitions with themap equationA B

2

7

89

4

56

1

3

23

1

L(M) = qxH(Q) +

p1

�H(P1)p2

�H(P2)p3

�H(P3)= 3.68 bits.

︸︷︷︸0.12 bits

︸︷︷︸3.56 bits

L(M) = qxH(Q) +

p1�H(P1)

p2�H(P2)

p3�H(P3)

p4�H(P4)

p5�H(P5)

p6�H(P6)

p7�H(P7)

p8�H(P8)

p9�H(P9)

= 3.57 bits.

︸︷︷︸0.97 bits

︸︷︷︸2.60 bits

Science 201010,000 journals1,000,000 articles10,000,000 citations

Thomson Scientific Journal Citation Reports2010


Medicine

Physics

Ecology & Evolution

Economics

Geosciences

Psychology

Chemistry

Psychiatry

Environmental Chemistry & Microbiology

Mathematics

Computer Science

Analytic ChemistryBusiness & Marketing

Political Science

Fluid Mechanics

Medical Imaging

Material Engineering

Sociology

Probability & Statistics

Astronomy & Astrophysics

Gastroenterology

Law


Education

Telecommunication

Control Theory

Operations Research

Ophthalmology

Crop Science

Geography

Anthropology

Computer Imaging

Agriculture

Parasitology

Dentistry

Dermatology

Urology

Rheumatology

Applied Acoustics

Pharmacology

Pathology

Otolaryngology

Electromagnetic Engineering

Circuits

Power Systems

Tribology

Neuroscience

Orthopedics Veterinary

Environmental Health

A

Citation flow from B to ACitation flow within field

Citation flow from A to BCitation flow out of field

B

Multilevel partitionsInto howmany hierarchical levels is a givennetwork organized? Howmanymodules arepresent at each level? Andwhich nodes aremembers of whichmodules?Maximal compression of flowwith constraints:1. Modular code structure2. Nomore than two levels3. Each node can only belong to onemodule

Multilevel partitions with themap equationInto howmany hierarchical levels is a givennetwork organized? Howmanymodules arepresent at each level? Andwhich nodes aremembers of whichmodules?Maximal compression of flowwith constraints:1. Modular code structure2. Nomore than two levels3. Each node can only belong to onemodule

Multilevel partitions with themap equationA B1

3 2

4

56

7

89

3 2

1

3

1

1

23

1

23

2

L(M) = qxH(Q) +

p1�H(P1)

p2�H(P2)

p3�H(P3)

p4�H(P4)

p5�H(P5)

p6�H(P6)

p7�H(P7)

p8�H(P8)

p9�H(P9)

= 3.57 bits.

︸︷︷︸0.97 bits

︸︷︷︸2.60 bits

L(M) = qxH(Q) +

q1�H(Q1) +

p11

� H(P11)p12

� H(P12)p13

� H(P13)

q2�H(Q2) +

p21

� H(P21)p22

� H(P22)p23

� H(P23)

q3�H(Q3) +

p31

� H(P31)p32

� H(P32)p33

� H(P33)

= 3.48 bits.

︸︷︷︸0.12 bits

︸︷︷︸0.76 bits

︸︷︷︸2.60 bits

Aquatic Ecology



Entomology


Soil Science

Hydrology



Medicine

Neuroscience



Oncology

Infectious Disease



Botany


Radiology

Nephrology


Orthopedics

Ophthalmology

Toxicology


Veterinary

Urology

Dentistry

Dermatology

Sports Medicine

Otolaryngology

Anesthesiology

Bone

Neurosurgery


Gene Therapy

Ethnopharmacology

Physics

Chemistry

Materials Science

High-energy Physics

Polymer Science

Optics



Medicinal Chemistry


Drug Delivery

Fluid Mechanics

Applied Mechanics

Radiation

Plasma Physics


Mathematics

Applied Math





Computer Vision

Probability Theory

Applied Computation

Operations Research

Robotics

Cybernetics

Power Engineering

EconomicsFinance

Political Science


Statistics

Sociology

Geography

Law

Education

Physical Sciences

Life Sciences


Social Sciences

Overlapping partitionsHowmanymodules are present? Which nodesaremembers of whichmodules? Andwhichnodes should belong tomultiple modules and towhat degree?Maximal compression of flowwith constraints:1. Modular code structure2. Nomore than two levels3. Each node can only belong to onemodule

Overlapping partitions with themap equationHowmanymodules are present? Which nodesaremembers of whichmodules? Andwhichnodes should belong tomultiple modules and towhat degree?Maximal compression of flowwith constraints:1. Modular code structure2. Nomore than two levels3. Each node can only belong to onemodule

Overlapping partitions with themap equation

BA

2211

L(M) = qxH(Q) +

{p1�H(P1)

p2�H(P2)= 2.86 bits.

︸︷︷︸0.13 bits

︸︷︷︸2.73 bits

L(M) = qxH(Q) +

{p1�H(P1)

p2�H(P2)= 2.67 bits.

︸︷︷︸0.063 bits

︸︷︷︸2.61 bits

Memory networks capture real organization

Helsinski

Trondheim

Oslo StockholmBergen

StavangerCopenhagen

FrankfurtBerlin

Paris

Barcelona

Orlando

New York

BostonToronto

Minneapolis

Seattle

Reykjavik

Madrid

0.2 0.4 0.6 0.8 1.00.0

One big module

Two nonoverlapping modules

Two overlapping modules

Code

leng

th

Return rate

Multilevel and overlapping partitionsInto howmany hierarchical levels is a givennetwork organized? Howmanymodules arepresent at each level? Which nodes aremembersof whichmodules? Andwhich nodes shouldbelong tomultiple modules and to what degree?Maximal compression of flowwith constraints:1. Modular code structure2. Nomore than two levels3. Each node can only belong to onemodule

Multilevel and overlapping partitions with themap. . .Into howmany hierarchical levels is a givennetwork organized? Howmanymodules arepresent at each level? Which nodes aremembersof whichmodules? Andwhich nodes shouldbelong tomultiple modules and to what degree?Maximal compression of flowwith constraints:1. Modular code structure2. Nomore than two levels3. Each node can only belong to onemodule

CHANGE

Mapping change in science


Medicine

Oncology

Psychiatry



Medicine

Neuroscience

Oncology

Psychiatry

Urology

Infectiousdiseases


2001 2003 2005 2007


What is real changeandwhat is mere noise?

Significance clustering

Clustering

Significanceclustering

Clustering

Bootstrap world

Resampling

Real world

Significance clusteringReal world

Resampling

Clustering

Bootstrap world


Clustering


Resampling

Clustering Clustering

Bootstrap world



Resampling

Clustering Clustering

clusteringSignificance

Bootstrap world

Mapping change


Time 1 Time 2

Mapping change with alluvial diagram

Real world 2 Bootstrap world 2

Resampling

ClusteringClustering



Resampling


Time 2Time 1

Mapping changeSignificanceclustering

Time 1 Time 2

Mapping change with alluvial diagram


Resampling




Resampling


Time 2Time 1

Mapping change in science


Medicine

Oncology

Psychiatry



Medicine

Neuroscience

Oncology

Psychiatry

Urology

Infectiousdiseases


2001 2003 2005 2007


Mapping change in the federal fundsmarket

Jul 16

KT2,...

XM3,...

KA8,...

DC8,...IB6,...

HQ3,...

Jul 30 Aug 13 Aug 27 Sep 10 Sep 24 Oct 8 Oct 22 Nov 5 Nov 19 Dec 3 Dec 17 Dec 31

UC5,...

KA8,...

UT1,...

CV1,...KE5,...

HQ3,...

IB6,...

ZL1,...

JL1,...

Mapping change in the federal fundsmarket

UC5,... KA8,...

DC8,...

HQ3,...

XG9,...

UC5,...

IB6,...ZT4

RA1

ZL1,...

HE3

YS3,...

GT8,...

QG4,...

A B C

Oct 8

UC5,...

KA8,...

IB6,...DC8,...HQ3,...XG9,...

Oct 22

UC5,...

IB6,...

ZT4RA1ZL1,...

HE3

YS3,...

GT8,...

QG4,...

IB6,...

Resampling effects

J1

J2 J3

J5J4

J1

J2 J3

J5J4

J1

J2 J3

J5J4

Article resampling

Multinomial resampling

Poisson resampling

A

B

C

Resampling effects on ranking

0.020.0150.010.0050

Page rank

ArticleMultinomialPoisson

tteLveRsyhP ecneicS

New Engl J Med

J Neurosci

Environ Sci Technol

(8) (7)

(8) (7)

(7)(8)

(6) (5)

(5)(6)

(5)(6)

(14)

(14)

(14)(14)

(14)

(14)

(71)

(69)

(68)

(76)

(78)

(78)

(1)

(2)(2)

(1)

(1)(1)

Resampling effects on clustering

1991 1992 1993

Poisson resampling

General physics

Nuclear & particle physics

General physics

1989 1990

General physicsGeneral physics


1989 1990

Article resampling

1991 1992 19931992

A

General physicsGeneral physics


1989 1990

Multinomial resampling

1991 1992 1993

B

C

mapping change in large integrated systems

Documents