mapping change in large integrated systems
TRANSCRIPT
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
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
Molecular &cell biology
Medicine
Oncology
Psychiatry
NeurologyPsychologyNephrology
Molecular &cell biology
Medicine
Neuroscience
Oncology
Psychiatry
Urology
Infectiousdiseases
PsychologyNephrology
2001 2003 2005 2007
Rosvall & Bergstrom (2010)
Barabasi, A.L.
Jeong, H.
Albert, R.
Oltvai, Z.N.
Ravasz, E.
Yook, S.H.
Neda, Z.
Schuberts, A.
Tombor, B.
Dezso, Z.
Kovacs, B.
Mongru, D.A.
Podani, J.
Ravasz, R.
Somera, A.L.
Szathmary, E.
Wuchty, S.
Beg, Q.K.
Dobrin, R.
Mason, S.P.
Tu, Y.
Arenas, A.
Diaz-Guilera, A.
Guimera, R.
Danon, L.
Duch, J.
Gomez, S.
Fernandez, A.
Giralt, F.
Cabrales, A.
Gleiser, P.M.
Vega-Redondo, F.
Diaz-Aguilera, A.
Jensen, P.
Rubi, M.
Pastor-Satorras, R.Vespignani, A.
Vazquez, A.
Boguna, M.
Barrat, A.
Weigt, M.
Rubi, J.
Soffer, S.N.
Flammini, A.
Zecchina, R.
Latora, V.
Marchiori, M.
Crucitti, P.
Rapisarda, A.
Pluchino, A.
Porta, S.
Castellani, G.C.
Franceschi, C.
Remondini, D.
Tieri, P.
Valensin, S.
Ivanchenko, M.
Caruso, F.
Kinney, R.
Amaral, L.A.N.
Barthelemy, M.
Stanley, H.E.
Sales-Pardo, M.
Edling, C.R.
Liljeros, F.
Moreira, A.A.
Mossa, S.
Provero, P.
Scala, A.
Aberg, Y.
Andrade J.S., Jr.
Camacho, J.
Gondran, B.
Guichard, E.
Herrmann, C.
Sawardecker, E.N.
Turtschi, A.
Newman, M.E.J.
Moore, C.
Clauset, A.
Girvan, M.
Gastner, M.T.
Leicht, E.A.
Lusseau, D.
Barkema, T.
De Montjoye, Y.
Good, B.H.
Jin, E.M.
Kalapala, V.K.
Karrer, B.
Levina, E.
Sanwalani, V.
Sole, R.V.
Valverde, S.
Ferrer I Cancho, R.
Kuntz, P.
Montoya, J.M.
Smith, E.
Buhl, J.
Deneubourg, J.L.
Gautrais, J.
Theraulaz, G.
Cancho, R.F.
De Fraysseix, H.
Garcia-Fernandez, J.
Janssen, C.
Kohler, R.
Salazar-Ciudad, I.
Kepler, T.B.
Boccaletti, S.
Hwang, D.U.
Valladares, D.L.
Bragard, J.
Chavez, M.
Mancini, H.
Hentschel, H.G.E.
Amann, A.
Mancini, H.L.
Maza, D.
Mendoza, C.
Vannucchi, F.S.
Lopez-Ruiz, R.
Vallone, A.
Moreno, Y.
Pacheco, A.F.
Gomez-Gardenes, J.
Gomez, J.B.
Echenique, P.
Nekovee, M.
Floria, L.M.
Vazquez-Prada, M.
Kertesz, J.
Kaski, K.
Onnela, J.P.
Kumpula, J.M.
Saramaki, J.
Alava, M.
Lahtinen, J.
Szabo, G.
Chakraborti, A.
Kanto, A.
Tibely, G.
Heimo, T.
Jari Saramaki, J.
Kivela, M.
Kahng, B.
Goh, K.I.
Kim, D.
Oh, E.S.
Oh, E.
Rho, K.
Jung, S.
Kim, S.
Park, Y.
Kim, J.H.
Lee, D.S.
Kurths, J.
Pikovsky, A.S.
Rosenblum, M.G.
Zhou, C.S.
Osipov, G.V.
Abel, H.H.
Park, E.H.
Schafer, C.
Zaks, M.A.
Freund, H.J.
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.
Roux, S.
Sousa, A.O.
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.
White, D.R.
Reitz, K.P.
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.
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
Two-level partitions with themap equation
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
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Citation flow from B to ACitation flow within field
Citation flow from A to BCitation flow out of field
B
Molecular & Cell Biology
Medicine
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Tribology
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Orthopedics Veterinary
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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
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Ecology & Earth 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
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
Mapping change in science
Molecular &cell biology
Medicine
Oncology
Psychiatry
NeurologyPsychologyNephrology
Molecular &cell biology
Medicine
Neuroscience
Oncology
Psychiatry
Urology
Infectiousdiseases
PsychologyNephrology
2001 2003 2005 2007
Rosvall & Bergstrom (2010)
Significance clustering
Clustering
Significanceclustering
Clustering
Bootstrap world
Resampling
Real world
Significance clusteringReal world
Resampling
Clustering
Bootstrap world
Significanceclustering
Clustering
Significance clusteringReal world
Resampling
Clustering Clustering
Bootstrap world
Significanceclustering
Significance clusteringReal world
Resampling
Clustering Clustering
clusteringSignificance
Bootstrap world
Mapping change
Significanceclustering
Time 1 Time 2
Mapping change with alluvial diagram
Real world 2 Bootstrap world 2
Resampling
ClusteringClustering
Significanceclustering
Real world 1 Bootstrap world 1
Resampling
ClusteringClustering
Time 2Time 1
Mapping changeSignificanceclustering
Time 1 Time 2
Mapping change with alluvial diagram
Real world 2 Bootstrap world 2
Resampling
ClusteringClustering
Significanceclustering
Real world 1 Bootstrap world 1
Resampling
ClusteringClustering
Time 2Time 1
Mapping change in science
Molecular &cell biology
Medicine
Oncology
Psychiatry
NeurologyPsychologyNephrology
Molecular &cell biology
Medicine
Neuroscience
Oncology
Psychiatry
Urology
Infectiousdiseases
PsychologyNephrology
2001 2003 2005 2007
Rosvall & Bergstrom (2010)
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
Nuclear & particle physics
1989 1990
Article resampling
1991 1992 19931992
A
General physicsGeneral physics
Nuclear & particle physics
1989 1990
Multinomial resampling
1991 1992 1993
B
C