Mapping change
in large integrated systems
Martin Rosvall
Alcides V. Esquivel, Atieh Mirshahvalad,
Daniel Edler, and Carl Bergstrom
Aquatic Ecology
Ecology & Evolution Geology
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 science Rheumatology 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
Electrochemistry Astrophysics & Astronomy
Drug Delivery Fluid Mechanics
Applied Mechanics
Radiation Plasma Physics Acoustics
Renewable 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
Economics Finance
Political Science Business & Management
Statistics Sociology
Geography
Law Education
Physical Sciences
Life Sciences
Ecology & Earth Sciences Social Sciences
Rosvall & Bergstrom (2011)
Aquatic Ecology
Ecology & Evolution Geology
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 science Rheumatology
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
Electrochemistry
Astrophysics & Astronomy
Drug Delivery Fluid Mechanics
Applied Mechanics
Radiation
Plasma Physics Acoustics
Renewable 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
Economics Finance
Political Science Business & Management
Statistics
Sociology Geography
Law
Education
Physical Sciences
Life Sciences
Ecology & Earth Sciences
Social Sciences
Molecular &
cell biology Medicine Oncology Psychiatry Neurology Psychology Nephrology
Molecular &
cell biology Medicine Neuroscience Oncology Psychiatry Urology
Infectious diseases Psychology Nephrology
2001 2003 2005 2007
Rosvall & Bergstrom (2010)
Courtesy of Aaron Koblin
How are these systems organized?
with respect to the flow?
How are these systems organized?
with respect to the flow?
Networks describe flow
beyond nearest neighbors
Networks describe flow
beyond nearest neighbors
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.
Google Maps for networks
depict regularities Maps
in the dynamics on networks
using less information
depict regularities Maps
in the dynamics on networks
using less information
Finding regularities ⇐⇒ Compression
5.8MB (tiff) → 0.91MB (tiff + LZW)
5.8MB (tiff) → 2.8MB (tiff + LZW)
If we can find a good code for describing flow on a network, we will have solved the dual problem
of finding the important structures
with respect to that flow
We use a modular code structure that can exploit regions in the network
in which units of flow tend to stay
for a relatively long time
Two-level partitions
How many modules are present? And which nodes are members of which modules?
Maximal compression of flow with constraints:
1. Modular code structure 2. No more than two levels
3. Each node can only belong to one module
Two-level partitions with the map equation
How many modules are present? And which nodes are members of which modules?
Maximal compression of flow with constraints:
1. Modular code structure 2. No more than two levels
3. Each node can only belong to one module
Two-level partitions with the map equation
L ( M ) = q y H (Q) +
m
X
i = 1
p i H (P i )
Two-level partitions with the map equation
B A
2 1
3
L(M) = H(P) = 4.75 bits.
L(M) = q x H(Q) +
p 1 H(P 1 ) p 2 H(P 2 ) p 3 H(P 3 )
= 3.68 bits.
| {z } 0.12 bits
| {z }
3.56 bits
Two-level partitions with the map equation
A B
2
7
8 9
4
5 6 1
3
2 3
1
L(M) = q x H(Q) +
p 1 H(P 1 ) p 2 H(P 2 ) p 3 H(P 3 )
= 3.68 bits.
| {z } 0.12 bits
| {z } 3.56 bits
L(M) = q x H (Q) +
p 1 H(P 1 ) p 2 H(P 2 ) p 3 H(P 3 ) p 4 H(P 4 ) p 5 H(P 5 ) p 6 H(P 6 ) p 7 H(P 7 ) p 8 H(P 8 ) p 9 H(P 9 )
= 3.57 bits.
| {z } 0.97 bits
| {z }
2.60 bits
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A
Citation flow from B to A Citation flow within field
Citation flow from A to B
Citation flow out of field
B
Molecular & Cell Biology
Medicine
Physics
Ecology & Evolution Economics
Geosciences
Psychology
Chemistry
Psychiatry
Environmental Chemistry & Microbiology Mathematics
Computer Science
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Control Theory Operations Research
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Parasitology
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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 A
Citation flow within field
Citation flow from A to B
Citation flow out of field
B
Multilevel partitions
Into how many hierarchical levels is a given network organized? How many modules are present at each level? And which nodes are members of which modules?
Maximal compression of flow with constraints:
1. Modular code structure 2. No more than two levels
3. Each node can only belong to one module
Multilevel partitions with the map equation Into how many hierarchical levels is a given network organized? How many modules are present at each level? And which nodes are members of which modules?
Maximal compression of flow with constraints:
1. Modular code structure 2. No more than two levels
3. Each node can only belong to one module
Multilevel partitions with the map equation
A 1 B
3 2
4
5 6 7
8 9
3 2
1
3
1
1
2 3 1
2 3
2
L(M) = q x H(Q) +
p 1 H(P 1 ) p 2 H(P 2 ) p 3 H(P 3 ) p 4 H(P 4 ) p 5 H(P 5 ) p 6 H(P 6 ) p 7 H(P 7 ) p 8 H(P 8 ) p 9 H(P 9 )
= 3.57 bits.
| {z } 0.97 bits
| {z } 2.60 bits
L(M) = q x H(Q) +
q 1 H(Q 1 ) +
p 11 H(P 11 ) p 12 H(P 12 ) p 13 H(P 13 ) q 2 H(Q 2 ) +
p 21 H(P 21 ) p 22 H(P 22 ) p 23 H(P 23 ) q 3 H(Q 3 ) +
p 31 H(P 31 ) p 32 H(P 32 ) p 33 H(P 33 )
= 3.48 bits.
| {z } 0.12 bits
| {z } 0.76 bits
| {z }
2.60 bits
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Food science Rheumatology
Veterinary
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Dermatology Sports Medicine
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Neurosurgery
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Overlapping partitions
How many modules are present? Which nodes are members of which modules? And which nodes should belong to multiple modules and to what degree?
Maximal compression of flow with constraints:
1. Modular code structure 2. No more than two levels
3. Each node can only belong to one module
Overlapping partitions with the map equation How many modules are present? Which nodes are members of which modules? And which nodes should belong to multiple modules and to what degree?
Maximal compression of flow with constraints:
1. Modular code structure 2. No more than two levels
3. Each node can only belong to one module
Overlapping partitions with the map equation
B A
2 1 2
1
L(M) = q x H (Q) + ( p 1 H (P 1 )
p 2 H (P 2 ) = 2.86 bits.
| {z } 0.13 bits
| {z } 2.73 bits
L(M) = q x H(Q) + ( p 1 H(P 1 )
p 2 H(P 2 ) = 2.67 bits.
| {z } 0.063 bits
| {z }
2.61 bits
Memory networks capture real organization
Helsinski Trondheim Oslo Stockholm Bergen Stavanger
Copenhagen Frankfurt Berlin Paris
Barcelona
Orlando New York
Boston Toronto Minneapolis Seattle
Reykjavik
Madrid
0.2 0.4 0.6 0.8 1.0
0.0
One big module
Two nonoverlapping modules Two o verlapp
ing mo dules
Codelength
Return rate
Memory networks capture real organization
Helsinski Trondheim Oslo Stockholm Bergen Stavanger
Copenhagen Frankfurt Berlin Paris
Barcelona
Orlando New York
Boston Toronto Minneapolis Seattle
Reykjavik
Madrid
0.2 0.4 0.6 0.8 1.0
0.0
One big module
Two nonoverlapping modules Two o verlapp
ing mo dules
Codelength
Return rate
Multilevel and overlapping partitions
Into how many hierarchical levels is a given network organized? How many modules are present at each level? Which nodes are members of which modules? And which nodes should
belong to multiple modules and to what degree?
Maximal compression of flow with constraints:
1. Modular code structure 2. No more than two levels
3. Each node can only belong to one module
Multilevel and overlapping partitions with the map...
Into how many hierarchical levels is a given network organized? How many modules are present at each level? Which nodes are members of which modules? And which nodes should
belong to multiple modules and to what degree?
Maximal compression of flow with constraints:
1. Modular code structure 2. No more than two levels
3. Each node can only belong to one module
CHANGE
Mapping change in science
Molecular &
cell biology Medicine Oncology Psychiatry Neurology Psychology Nephrology
Molecular &
cell biology Medicine Neuroscience Oncology Psychiatry Urology
Infectious diseases Psychology Nephrology
2001 2003 2005 2007
Rosvall & Bergstrom (2010)
What is real change
and what is mere noise?
Significance clustering
Clustering
Significance clustering
Clustering
Bootstrap world
Resampling
Real world
Significance clustering
Real world
Resampling
Clustering
Bootstrap world
Significance clustering
Clustering
Significance clustering
Real world
Resampling
Clustering Clustering
Bootstrap world
Significance
clustering
Significance clustering
Real world
Resampling
Clustering Clustering
clustering Significance
Bootstrap world
Mapping change
Significance clustering
Time 1 Time 2
Mapping change with alluvial diagram
Real world 2 Bootstrap world 2
Resampling
Clustering Clustering
Significance clustering
Real world 1 Bootstrap world 1
Resampling
Clustering Clustering
Time 2
Time 1
Mapping change Significance clustering
Time 1 Time 2
Mapping change with alluvial diagram
Real world 2 Bootstrap world 2
Resampling
Clustering Clustering
Significance clustering
Real world 1 Bootstrap world 1
Resampling
Clustering Clustering
Time 2
Time 1
Mapping change in science
Molecular &
cell biology Medicine Oncology Psychiatry Neurology Psychology Nephrology
Molecular &
cell biology Medicine Neuroscience Oncology Psychiatry Urology
Infectious diseases Psychology Nephrology
2001 2003 2005 2007
Rosvall & Bergstrom (2010)
Mapping change in the federal funds market
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 funds market
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,...
ZT4 RA1
ZL1,...HE3 YS3,...
GT8,...
QG4,...
