XYZ decom- positions
V. Batagelj XYZ decompositions of graphs Dagstuhl Seminar 08191
Motivation
XYZ decom- Vladimir Batagelj positions
Results University of Ljubljana, FMF, Dept. of Mathematics; and References IMFM Ljubljana, Dept. of Theoretical Computer Science joint work with Franz J. Brandenburg, Walter Didimo, Giuseppe Liotta, Maurizio Patrignani
23rd Leoben-Ljubljana Graph Theory Seminar Ljubljana, 14-15. November 2008
V. Batagelj XYZ decompositions Outline
XYZ decom- positions
V. Batagelj
Dagstuhl Seminar 08191
Motivation 1 XYZ decom- Dagstuhl Seminar 08191 positions 2 Motivation Results 3 XYZ decompositions References 4 Results 5 References
V. Batagelj XYZ decompositions Dagstuhl Seminar 08191
XYZ decom- positions
V. Batagelj
Dagstuhl Seminar 08191
Motivation
XYZ decom- positions
Results
References
Dagstuhl Seminar 08191: Graph Drawing with Applications to Bioinformatics and Social Sciences. Dagstuhl, May 4-9, 2008
V. Batagelj XYZ decompositions Motivation
XYZ decom- positions For dense graphs the matrix representation is better [3].
Pajek - shadow [0.00,1.00]
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Snyder and Kick World Trade data (SaKtrade.paj; 118 vertices, 515 arcs, 2116 edges) [18].
V. Batagelj XYZ decompositions Motivation
XYZ decom- positions Abello and van Ham (Matrix Zoom, ASK-GraphView) [1] and V. Batagelj Henry and Fekete (MatrixExplorer, NodeTrix) [9, 10] developed systems for mixed representation of large networks. Dagstuhl Seminar 08191
Motivation
XYZ decom- positions
Results
References
(a) Clustered node-link (b) NodeTrix (c) Node duplications
V. Batagelj XYZ decompositions Figure 7.1: NodeTrix and duplications
7.1 Research problem
As previously explained in section 2.2, social networks can vary a lot in structure: from sparse graphs exhibiting a tree structure to very dense ones presenting a table-like structure. Select- ing the most suited representation is strongly correlated to the network density. For example, node-link diagrams are particularly effective for very sparse networks while matrix representa- tion clearly outperform them for very dense networks [GFC05]. The difficulty is to identify the density threshold beyond which matrices are more suited than node-link diagrams. This choice is especially ambiguous for small-world networks, a very common category of social networks. The particularity of small-world networks is their global sparse structure with dense local parts. The major difficulty faced when representing these network is to show first how members of communities are connected (intra-community connectivity), then how communities are connected (inter-community connectivity) and finally who the central actors are. Thus, in this chapter, we attempt to solve the following question:
®How can we design a representation for small-world networks? i.e., improving intra-community and inter-community connectivity readability as well as highlighting central actors?
The second part of this chapter is dedicated to the problem of ambiguous clustering. When an actor is connected to two or more communities, there are three solutions: placing the actor in one or the other, extracting it and placing it between them or generating overlapping commu- nities. While extracting the actor and producing overlapping communities dramatically decrease the representation readability, choosing one or the other community to place the central actor also raises problems as it changes the visual representation, which is potentially misleading. Thus our research question is:
®How can we solve the problem of ambiguous clustering without degrading the representation readability or misleading the user? To solve these problems, we present the NodeTrix [HFM07] representation (Figure 7.1b), merging node-link diagrams and matrices to visualize social networks as well as the technique of node duplication [HBF08] (Figure 7.1c) to solve the ambiguous clustering problem. NodeTrix 149
Supporting the exploration of matrices One weakness of the matrix representation, when ex- ploring a network, is the tedious work required to perform path-related tasks. For example, finding how two communities are connected is tedious as it requires going back and forth alternately read- ing rows and columns. Moreover, if communities are far apart in the matrix, this task requires a scan of the full length of matrix rows or columns, and connections in a large matrix may lie outside the viewport. Obviously, the task is worse when dealing with three matrices as the user needs to check for intersections of rows and columns in each of the three communities. We noticed in a participatory-design session reported in [HF06a] that social network analysts also use the matrix representation for some of their analyses. To help perform community analysis and provide support for path-related tasks in general, we provide users with a couple of interaction techniques that work across separate matrix-NodeTrix windows that might be arranged in a dual- viewport or split-screen fashion. These techniques are still based on drag-and-drop, however this time, the user drags a group of elements from one window to another one.
From Matrix to NodeTrix The interaction is made of two steps: first, the user selects a group of nodes in the window of the pure matrix visualization and then drags this group to the MotivationNodeTrix window (Figure 7.5). To select the group of nodes, we provide lasso selection directly on the pure matrix representation. Alternatively, the selection can be done on an axis (rows and columns). When a group of cells is selected, the corresponding set of vertices transferred is the XYZ decom- union of the edges’ source vertices and sink vertices. Dropping the selected group inside the positions NodeTrix window performs the addition of an aggregated node to the NodeTrix visualization. The
V. Batagelj group is then displayed as a matrix. Selecting and dropping a second group allows the user to see how these groups are connected to each other visualizing the result with links. The process can
Dagstuhl continue to visualize connections between several communities. Seminar 08191
Motivation
XYZ decom- positions
Results
References
(a) From matrix to NodeTrix (b) Inter-community connectivity appears
Figure 7.5: Dragging communities from standard matrix to NodeTrix helps analyzing how they are connected. (a) A community has already been dragged into NodeTrix (rows are colored in white in the matrix to show they are already copied). User is transferring another community (rows selected in red in the matrix), the cursor shows that he can drop the selection into NodeTrix. (b) Three communities have beenV. dragged Batagelj into NodeTrix,XYZ decompositions inter-community relations can be studied. Motivation NodeTrix 155
XYZ decom- positions
V. Batagelj (a) Compact NodeTrix
Dagstuhl Seminar 08191
MotivationNodeTrix 155
XYZ decom- positions
Results
References
(a) Compact NodeTrix
(b) Detailed NodeTrix
Figure 7.11: Two NodeTrix representations of the information visualization field. The top one presents a compact version, which aims at presenting communities and their connectivity patterns (intra-community and inter-community). The second representation shows all details of the exact same dataset, which can be used for exploration. Colors on axes of matrices represent the number V. Batagelj XYZof citations decompositions of each author. Color intensity within the matrices represents the strength of each collaboration.
(b) Detailed NodeTrix
Figure 7.11: Two NodeTrix representations of the information visualization field. The top one presents a compact version, which aims at presenting communities and their connectivity patterns (intra-community and inter-community). The second representation shows all details of the exact same dataset, which can be used for exploration. Colors on axes of matrices represent the number of citations of each author. Color intensity within the matrices represents the strength of each collaboration. Motivation
XYZ decom- positions
V. Batagelj UIST Coauthorship Network 20 Years of Collaboration Sidner Stahovich Rich Kara Newfield Sethi Hindmarsh Benford Fraser Heath Goldberg J. Rich Dahlquist Partridge Foreman Borriello Kara Veiseh
Latulipe Hardock Sidner McMahon
Kaplan Wittenburg Tapia Cain Clarke
Heinrichs Stahovich Esenther MacKay D. McKay Lanning Thevenin Stanton Lecoanet Miyachi Harada Blackwell Newfield York Lemort Chatty Mertz Vinot Ward Churchill Sire Helfman
Yamada Sethi Denoue Murphy Nelson Shingu
Kurtenbach Fitzmaurice Fraser Almeida Burtnyk Meier Buxton Benford Cain Tsang Pieké Khan
Herranz Latulipe Oscarsson Danielsson Nordgard Holmquist Hindmarsh Redstrom Dahlquist McKay Karlgren Franzén Raeder Bolt
Clarke Heinrichs Hauge Bretan Lecoanet Blackwell Bjork Heath York
Veiseh Stonebraker Kaplan Esenther Sire Weiser Dagstuhl Yamada Woodruff Ward McMahon Foreman schraefel Harada Takashima
Mertz Kitamura Murphy Smith MacKay D. Partridge Kishino Sharlin J2 Abdulla G1 Forlines Ginsburg Miyachi Asano Denoue Kramer Shieber Chatty G1 Lanning F2 Ryall Griswold Goldberg J. Stanton Tsang J1 Aboobaker E3 Forlizzi Jeffery Nelson Lemort E2 Larner C5 Räihä Herranz Borriello Wittenburg Kurtenbach F1 Abotel I5 Forsberg Franzén Helfman Oscarsson Thevenin Bolt Lanning Buxton I5 Larson K. J3 S. Zhao H3 Abowd J6 Fox Redstrom Churchill Raeder Tashman Vinot H1 Larson R. H4 Sakai
Wood S. Fitzmaurice Hornof
Abotel Woodruff Kieras A3 Ackerman B6 Fracchia Karlgren Shingu Smith Hauge Rhyne Khan Takashima Wolf H2 Lassen D2 Sakamura D3 Adar D5 Frank Bretan schraefel Nordgard Stonebraker Shieber Pieké Sharlin F1 Latulipe K3 Salesin D6 Adelstein F1 Frankel Griswold Danielsson Hightower
Rogers D. Ginsburg Stewart J. Bederson Almeida Kishino F3 Lau H. H3 Salisbury
Jeffery Hollan D2 Aggarwal B1 Franzén Bjork Druin Proft Ring Burtnyk Asano L2 Lau T. K1 Samminger J3 Agrawala H1 Fraser Holmquist Kitamura Hong K2 Lavine H3 Sannella
Wood S. Larson R. G4 Ahlberg G3 Frederick Wolf I1 Lecoanet J5 Sarkar Kieras Miller Li B2 Akers D5 Fredrickson Blumenthal Dragicevic Rhyne Golovchinsky Hégron
Dumas H5 Lecolinet E2 Saund Fekete
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C5 Aliakseyeu D3 Freeman Boreczky Christensen Doherty Aboobaker Busse I6 Lee G. E1 Sazawal
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Rogers D. Wang A. Chiu Carlsen D5 Allan A2 Freeman-Benson Chen J. Landay Tucker Sinha Stewart J. Schmidt-Nielsen A5 Lee I. C6 Scacchi
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Harsham G2 Lee J. J5 Schee Barnwell Forlines Mühlhäuser Samminger Hightower Wigdor Li
B2 Fujima Dietz Shen J3 Alvarado Gajewska Borchers Proft Manasse Miller A3 Lee K. J2 Schilit B3 Fukuchi Redell A6 Amento Berc Grossman Hollan Chatterjee Larson R. Dragicevic F1 Leigh A3 Schmandt C4 Amon I6 Furnas Busse Bederson Foote Anderson D. Huot I1 Lemort G1 Schmidt-Nielsen F2 Anderson D. F1 Gajewska Tonomura Higashino Wolber Miyaoku Christensen Sazawal Chen J. Sullivan Druin Uchihashi Frankel Yedidia Dumas A2 Letessier E4 Schneider-Hufschmidt Vicente Marks Leigh A2 Anderson R. A3 Gajos Tucker Fisher G. Ring Ryall Fekete Landay Golovchinsky Shen L2 Letondal H6 Schwesig G3 Anjewierden G4 Gandy Carlsen Vogel Hégron Aboobaker Guzdial Doherty Berc Dietz Mühlhäuser 1 L5 Lewis H6 Sciammarella L4 Apitz C2 Ganoe Wang A. 1 Moulder Wilcox Manasse Samminger J1 Li F2 Sears
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Vlissides Cheng Hu Skepner Tang S. Loeckenhoff K4 Cheng G5 Ingram Thomas Linton Kober Smith Feiner Kawachiya Jones H6 Oba K2 Trevor Liao L5 Chew E6 Inkpen Chapuis Schneider-Hufs. Höllerer Igarashi L2 Oblinger I2 Truong Sukaviriya Kovacevic Winograd Johanson L3 Chi A4 Inoue Phillips D. Bell Tanaka H. Kassell Hutchins Szekely Dannenberg K2 Ohara G1 Tsang Stone Foley J3 Chimera C2 Isenhour Roussel Giuse Korfhage Kadobayashi Proffitt A4 Ohsawa B3 Tsujita Stuerzlinger Amon D1 Chiu E4 Ishak Zanden Plaisant Rumbaugh Sakai Lipson 4 4 Rennison A4 Okada B1 Tucker A2 Chok F2 Ishii Linton Hascoët-Zizi Song H. Skepner H3 Olano H4 Tullio Tang S. N. Young Goble Guimbretiere MacKenzie D4 Choudhary H6 Ishizawa DeLine Wiseman Chalmers Bos D5 Olsen J5 Tversky Lee I. Vlissides Hu Smith B1 Christensen I3 Izadi Hsu Sukaviriya Rao Stone K3 Oren D1 Uchihashi Jones I4 Christiansen E5 Jacob Rosenberg Ahlberg Kochhar Friedell Amon Kovacevic Johanson H1 Oscarsson F2 Ullmer
Asente Hill R.D. Kober
F6 Chuah C2 Jacobs Palay Dannenberg Schneider-Hufs. Gossweiler Hutchins
Borenstein J6 Ouhyoung I3 Underwood Weyers Durbin D4 Chung B2 Jain Gosling van Dantzich Foley Winograd Lewis
Czerwinski A5 Oviatt G5 Ungar Robertson Larson K. D1 Churchill D5 Je eries Ishak N. Young Robbins J5 P. Williams I6 Urnes Hsu Thiel F1 Clarke B1 Je ery Kurlander DeLine Starkweather Wiseman Apitz L2 Pai A3 Vallejo Sugiura Koseki
L2 Cockayne G5 Jellinek Koike Lee I. Friedell Seligmann B5 Palay E2 Van Melle Rosenberg Shamash
F4 Coelho E2 Jerding Subramanian
Kochhar Palay Aliakseyeu F2 Pangaro C5 Vandenberg A5 Cohen J. C6 Jin Borenstein Durbin Lewis Asente Lucero K4 Pangoli C5 Vander Zanden
G3 Cohen J.D. G6 Jo Gossweiler Mao Weyers Mackinlay Gosling Kobayashi Robbins Tai I6 Papathomas E1 Veiseh L3 Cohen M. G6 Jo Russell Jellinek Kim W.C. Card Robertson D5 Parslow C5 Venckus K4 Cohen P. L4 Johanson Koseki van Dantzich P. Williams Sugiura Frank Conner M. E1 Partridge J5 Verplank Eisenberg Larson K. Jones W. A3 Cole F3 John Jacob Wrensch K4 Vertegaal Koike Kaye Czerwinski I2 Patel K. J3 Collomb D6 Johnston Subramanian Fredrickson Norman Gross M.H. Mao I3 Patel S. F1 Vicente Jefferies Parslow
Nielsen Thiel Moyes
I5 Conner D. L4 Jones Cotting Olsen
Aliakseyeu Tversky
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Sarkar K4 Paterno J4 Viega Duby J5 Conner M. B2 Jones S. Lucero Card Reiss Zarmer I4 Patten I1 Vinot
Mackinlay Chew I3 Conway J5 Jones W. A3 Patterson B4 Vlissides L3 Corbin I6 Jul Russell Lieberman Jellinek Jones W. Wrensch B6 Pattison G6 Vo Goldberg D. H6 Correia F4 Julier Conner M. Goodisman Eisenberg I4 Pausch H1 Vogel H4 Corso H4 Kadobayashi Cotting Olsen Zellweger P. Williams Chang
Ungar Meyers D5 Peachey H4 Voida I4 Cosgrove B6 Kajler Gross M.H. Parslow Henderson Nielsen Reiss Zarmer D2 Perlin L2 Vronay A5 Cotting H5 Kamada Duby Chew Moyes H3 Petersen K3 Wade B6 Cowperthwaite C3 Kandogan Sarkar K3 Petschnigg J1 Wang A. Young M. Fredrickson Wilson S.
I3 Curtis F1 Kaplan Rasure Hallett Argiro Tversky Neher Teran G5 Pfranger K3 Wang C.
Jefferies Hughes J.F. C3 Cypher H1 Kara Moscovich Goldberg D. Chang Stevens C5 Phillips C. B4 Wang H. I5 Czerwinski A4 Karger Phillips C. Goodisman Nadeau Badler Zellweger
Elvins A4 Phillips D. B6 Wang J. Hetzler Totten Kirsh E1 Dahlquist B1 Karlgren Ungar Cable Pfranger G1 Pieké E2 Want Lahtinen Ingram Verplank MacLean Roderick B1 Danielsson C6 Karrer Rohall Shaw R. Scheeff
Snibbe J3 Pieper K1 Ward Yonezawa Takahashi Matsuoka Nakazawa Allan Miyashita Watanuki
C4 Dannenberg H2 Karsenty Wilson S. Kamada Miyoshi I4 Pierce H5 Watanuki Mukai Vandenberg
L2 Darken D3 Kashiwagi Neher Seki McDonald Hughes J.F. C4 Pignol B3 Webber Kristensson
Smartt Young M. D4 Davidson I4 Kassell Badler Peachey Elvins Stevens Powers G6 Weghorst
Hunter I3 Pinkerton
Conner D. Cable Smith Rasure Forsberg van Dam Herndon Zhai Zeleznik J3 Davis K2 Kataoka McMillan Phillips C. Kirsh Moscovich Dorsey Taufer Totten I4 Pittman K1 Weiser
Tolba Pfranger Fails J2 Davis Donath Lahtinen Hallett Naur H4 Kawachiya Xiong Nadeau Deng Ingram Snibbe Hetzler G4 Plaisant A3 Weld H4 DeLine K2 Kawada Yin Rohall Halversen Argiro Teran Matsuoka Nakazawa Verplank J6 Polito K2 Wellner D5 Deng E5 Kaye Yonezawa Miyoshi MacLean L2 Pothier L5 Weyers Cohen J. McDonald
D1 Denoue Arthur D3 Kembel Oviatt Kristensson Kamada Watanuki Scheeff G6 Poupyrev C6 Wiecha F6 Derthick G1 Khan Smartt Raisamo Roderick Tolba Hunter Räihä Fails Takahashi Mukai Forsberg K5 Powers D6 Wiecha Donath Vandenberg D4 Dewan J3 Kieninger Dorsey Zhai Taufer Miyashita Seki van Dam Shaw R. I2 Prante G3 Wielimaker Xiong Hauptmann 5
H3 Dey Rudnicky F1 Kieras Witbrock Zeleznik 5 Kobayash McMillan Smith Mostow Kim W.S. J2 Price G1 Wigdor
L2 DiFilippo G5 Kijima Kijima Conner D.
Bae I4 Pro tt D1 Wilcox D6 Diakopoulos A2 Kim Ja. Arthur Herndon Vander Zanden Herczeg C1 Proft I4 Williams G./ Ressel
Cohen J. Räihä Hohl
K4 Dickie K3 Kim Ji. Greenberg Roseman Venckus Visualization of the UIST co-authorship network using the NodeTrix A4 Pugmire K3 Wilson A. Oviatt Raisamo Fitchett Baker I6 Dieterich D5 Kim W.C. Boyle A4 Quan F5 Wilson S. Jin
G1 Dietz G5 Kim W.S. Matsushita N. Rudnicky Chawathe J4 R. Williams B4 Wilson T.
Bae Gribble Ayatsuka Ling B. Brewer
Huang representation. The PDF le can be downloaded at www.aviz.fr E6 Dill K3 Kimbrough Polito
Mostow Fox H1 Raeder L4 Winograd I3 Dixon A4 Kimura T. Kim W.S.
Kuzunuki Hohl Hauptmann C5 Raisamo A4 Wiseman Papathomas Guimaraes Lowgren Kobayash Breiteneder
L4 Do G6 Kimura T.D. Machii Montagnese Witbrock Correia Carmo Terveen
Arai Herczeg Amento J3 Ramos F5 Witbrock Kijima de Mey Henigman F3 Doerry Venckus Fitchett Gibbs E5 Kirsh Bartram Hill Hix Vander Zanden Ressel Boyle H2 Ranjan G1 Wittenburg Cowperthwaite D1 Doherty J1 Kishino Ho Large rectangles represent communities of researchers closely G4 Rao B2 Wloka Barrientos Baker Greenberg B5 Donath Carpendale Miller Huang J1 Kitamura Wang J. Ayatsuka Fracchia Pattison Canny Jin Roseman J3 Rappaport D3 Wobbrock Ayers Fox Ligh K3 Dontcheva I4 Kitrick Matsunaga Matsushita N. F2 Raskar C1 Wolber Machii Gribble collaborating, visualized as adjacency matrices. Each researcher is A5 Dorsey J2 Klemmer Kubo Guimaraes Meyer T. F5 Rasure K1 Wolf F3 Douglas A3 Kleyn Amento Kuzunuki Gibbs Dieterich Chawathe Adelstein Ho Correia Hill Arai Halterman Johnston Jo Breiteneder Polito H3 Rattenbury C2 Wong H3 Dourish B2 Kliger Henigman Carmo represented as a named row and column. When two researchers Terveen Ellis de Mey Ling B. H2 Rau I3 Wood A. G4 Dow F3 Knudsen Bartram Hix Wang J. Fracchia Papathomas Brewer F1 Redell F1 Wood S. H1 Dragicevic C4 Ko Barrientos Kolojejchick Derthick Cowperthwaite Swindells co-signed an article, the cell at the intersection of their respective B1 Redstrom J1 Woodru Chuah Bennett
Mattis Jo Inkpen Greene Wiecha Roth K3 Drucker G5 Kobayash Canny Gould Boies Ligh Tory K4 Rehg L5 Wrensch Dill Kubo C1 Druin H5 Kobayashi Pattison J5 Reiss H2 Wu .M Adelstein Matsunaga Smith D. row and columns is colored. Links also join researchers from J5 Duby L4 Kober Carpendale Ellis H6 Rekimoto J6 Wu J.R. H2 Ducheneaut A5 Kochhar Ouhyoung K4 Rennison A2 Xiao
Johnston Wu J.R. H1 Dumas K2 Koh di erent communities who co-signed an article. E5 Ressel K3 Xie Rekimoto
Chuah Schwesig J5 Durbin B5 Koike Greene Tory Ishizawa Derthick B4 Rha B5 Xiong Oba H3 Edwards B4 Kok Boies Swindells Kimura T.D.
Scacchi Roth K1 Rhyne D1 Yamada
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B3 Eick E3 Koller Sohn B. Apte Mattis Lee G. H1 Rich F3 Yap L5 Eisenberg F6 Kolojejchick Wiecha Dill Vo Bennett Berry Kolojejchick Ouhyoung © 2007 Nathalie Henry, Pierre Dragicevic, Jean-Daniel L3 Riedl D3 Yeatts D6 Ellis L3 Konstan 1 Paper C1 Ring F1 Yedidia
Fisher B. Wu J.R. Fekete and INRIA, France. The network may contain Argue E5 Elvins G4 Korfhage Booth Lin C. Ishizawa I5 Robbins G1 Yerazunis G1 Esenther J3 Kortuem Rekimoto Giacalone errors; please report them to the authors. Reference: Karrer Kimura T.D. 2 Papers I5 Robertson B5 Yin Schwesig Graham D6 Essa B5 Koseki Nejabi Wiecha Lee G. Urnes Nathalie Henry, Jean-Daniel Fekete, Michael J. J2 Robinson-Mosher H5 Yonezawa Scacchi Vo Oba Berlage E5 Fails D2 Koshizuka Spenke Beilken Sohn B. Genau Apte 3 Papers McGu n. NodeTrix: an Hybrid Visualization of Social I3 Rodden I1 York I2 Farrell E4 Kovacevic G3 Rodenstein F5 Young M. Diakopoulos Furnas D3 Fass C3 Kovar Fisher B. Zhang Networks. IEEE Transactions on Visualization and
Jul J5 Roderick C4 Zanden C4 Faulring F1 Kramer Argue 4 Papers Computer Graphics, 13(6):8, Nov-Dec 2007. Essa Nagao G3 Rodriguez L5 Zarmer E4 Feiner J3 Kray Booth Sciammarella Urnes Lin C. C1 Rogers D. I5 Zeleznik H1 Fekete B5 Kristensson Spenke Graham Kajler Billinghurst I3 Rogers Y. G5 Zellweger Weghorst B4 Ferrency G6 Kubo Beilken Poupyrev Nejabi D5 Rohall J2 Zeltzer E6 Fisher B. K3 Kuribayashi Berlage Zhang Essa Furnas E6 Roseman B5 Zhai C1 Fisher G. E4 Kurlander Genau B5 Rosenberg I6 Zhang E2 Fishkin G1 Kurtenbach Diakopoulos Jul 6 6 Maruyama C4 Rosenfeld D2 Zhou E5 Fitchett K2 Kuwari C2 Rosson D4 Zorin G1 Fitzmaurice B6 Kuzunuki Weghorst F6 Roth I6 de Mey K4 Flagg H3 LaMarca Billinghurst C3 Rothrock E4 del R. Millán E2 Fleet D5 Lahtinen Poupyrev A4 Roussel E1 schraefel E3 Fogarty A2 Lakshmipathy L3 Rowe F2 van Baar E4 Foley H3 Lamping F5 Rudnicky I5 van Dam D1 Foote J1 Landay G4 Rumbaugh I5 van Dantzich E1 Foreman E2 Lank G5 Russell A B C D E F G Ichikawa H I J K L
V. Batagelj XYZ decompositions XYZ decompositions
XYZ decom- positions Given two graph classes X and Y, a graph G = (V , E) is an
V. Batagelj X -graph of Y-graphs (or( X , Y)-graph, for short) if a family V1, V2,..., Vh of disjoint subsets of V , called clusters, can be Dagstuhl Seminar 08191 identified, such that: Motivation 1 every cluster induces a graph belonging to class Y, and XYZ decom- positions 2 the reduced graphG ∗ obtained from G by collapsing each Results cluster into a single vertex and replacing multiple edges References with a single one is a graph of class X .
If subset V1, V2,... Vh are requested to be a partition of V , that is, if we add the constraint that V = V1 ∪ V2 ∪ · · · ∪ Vh, then we call G a strong (X , Y)-graph, otherwise we call G a weak (X , Y)-graph or, simply, an (X , Y)-graph. The strong model of X -graph of Y-graphs, also known as two level clustered graphs [11, 17, 13], was introduced in [4].
V. Batagelj XYZ decompositions An example: (planar,K5)-graphs
XYZ decompositions
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Dagstuhl Seminar 08191
Motivation
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Results
References
(planar,K5) - decomposition
V. Batagelj XYZ decompositions XYZ decompositions X -graphs
XYZ decom- positions Both for the strong model and for the weak one, by considering
V. Batagelj different families for X - and Y-graphs one obtains different (X , Y)-decomposition problems which have diverse importance Dagstuhl Seminar 08191 with respect to applications or to the insight into Motivation graph-theoretic decomposition problems. XYZ decom- Only for the strong model it makes sense considering the case positions when X -graphs are general graphs, that is, when they are not Results
References constrained, since with such a hypothesis any graph is an (X , Y)-graph in the weak model. More generally, for their impact on applications it is worth exploring cases when X -graphs are “low-density” graphs; planar graphs; connected graphs with bounded or specified number of nodes; and directed acyclic graphs. Also, the cases when X -graphs are trees, paths, cycles, and bounded size graphs are interesting from a more theoretic perspective.
V. Batagelj XYZ decompositions XYZ decompositions Y-graphs
XYZ decom- positions V. Batagelj Similarly, regarding the families of Y-graphs, we signal as Dagstuhl important for applications the cases when Y-graphs are Seminar 08191 “high-density” graphs, as, for example, graphs with high Motivation density; k-cores; cliques; k-connected graphs; strongly XYZ decom- positions connected digraphs; and stars. Results We recall that the density of a graph G = (V , E) is defined as References 2|E| |V |(|V |−1) , and that a (sub)graph has core number k if all its vertices have (internal) degree at least k. Again, from a more theoretic perspective, the cases when Y-graphs are trees, paths, cycles, and bounded size graphs are interesting.
V. Batagelj XYZ decompositions XYZ decompositions Z-graphs
XYZ decom- positions V. Batagelj In generalized blockmodeling
Dagstuhl [6] also the structure of bipar- Seminar 08191 tite subgraphs induced by the Motivation sets (Vi , Vj ) is important. This XYZ decom- positions adds the Z term to the decom- Results positions: References 3 every pair of clusters induces a bipartite graph belonging to class Z.
Most of blockmodeling problems seem NP-hard [16,7].
V. Batagelj XYZ decompositions Known results for paths, cycles and cliques Y-graphs in the strong model.
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Dagstuhl X -graphs Seminar 08191 Y-graphs tree non-trivial path single k-tree single k-star Motivation paths NP-complete [17] NP-complete [13] XYZ decom- cycles NP-complete [17] NP-complete [13] positions bounded paths Polynomial [17] bounded cycles Polynomial [17] Results
References X -graphs Y-graphs general graph planar graph 3-cycle cycle cliques NP-hard [12, 11] NP-hard [4] Polynomial if diameter > 3 [4] 3-cliques NP-complete [8]
V. Batagelj XYZ decompositions Results
XYZ decom- positions For the strong model we showed [2]: V. Batagelj X -graphs Dagstuhl Y-graphs tree bounded size Seminar 08191 cliques NP-hard Motivation bounded size O(n) O(1)
XYZ decom- positions Regarding the weak model, we explored some cases when dense Results Y-graph are involved. In particular, we observe that identifying References cliques in large graphs may lead to a very effective strategy for information visualization. In fact, when the user is able to tell that a subset of nodes is a clique, its internal edges are understood and do not need to be explicitly displayed. Unfortunately, recognizing (planar, K5)-graphs is NP-hard. This result parallels the analogous result for the strong model [12, 11].
V. Batagelj XYZ decompositions Results
XYZ decom- positions The k-core of a graph G(V , E) is the graph obtained by V. Batagelj recursively removing vertices of degree less than k. Let n and Dagstuhl m be the number of vertices and edges of G, respectively. We Seminar 08191 showed [2] that there exists an O(m + n log(n)) algorithm to Motivation
XYZ decom- find the maximum k such that the reduced graph obtained by positions collapsing each connected component of the k-core of G is Results planar. References
X -graphs Y-graphs planar graph 5-cliques NP-hard k-core graphs (max k) O(m + n log(n))
In real problems we are often searching for almost XYZ-decompositions – optimization problems.
V. Batagelj XYZ decompositions ReferencesI
XYZ decom- positions Abello J., van Ham F., Krishnan N.: ASK-GraphView : A Large Scale V. Batagelj Graph Visualization System. IEEE Transactions on Visualization and Computer Graphics, Vol. 12, No. 5, September/October (2006). Dagstuhl Seminar 08191 Batagelj V., Brandenburg F.J., Didimo Walter., Liotta G., Patrignani Motivation M.: Dagstuhl Seminar 08191 Working Group Report – X-graphs of XYZ decom- positions Y-graphs and their Representations. Dagstuhl, 2008. page Results Batagelj, V., Mrvar, A. and Zaverˇsnik,M.: Partitioning approach to References visualization of large graphs, In Kratochv´ıl,J. (ed), Lecture notes in computer science, 1731, Springer, Berlin, 90–97 (1999).
Brandenburg, F.J.: Graph clustering I: Cycles of cliques. In Di Battista, G., ed.: Graph Drawing (Proc. GD ’97). Volume 1353 of Lecture Notes Comput. Sci., Springer-Verlag (1997) 158–168
Dagstuhl Seminar 08191: Graph Drawing with Applications to Bioinformatics and Social Sciences. Dagstuhl, May 4-9, 2008. page
V. Batagelj XYZ decompositions ReferencesII
XYZ decom- positions Doreian P., Batagelj V., Ferligoj A.: Generalized Blockmodeling, V. Batagelj Cambridge University Press (2005).
Dagstuhl Fialaa J., Paulusmaa D.: A complete complexity classification of the Seminar 08191 role assignment problem. Theoretical Computer Science 349(2005) 1, Motivation pp. 67-81 XYZ decom- positions Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to Results the Theory of NP-Completeness. W. H. Freeman, New York, NY References (1979)
Henry, N., Fekete, J.D., McGuffin, M.J.: NodeTrix: A hybrid visualization of social networks. IEEE Trans. Visual. and Comp. Graphics 13(6) (2007) 1302–1309
Henry, N.: Exploring Large Social Networks with Matrix-Based Representations. Ph.D. Thesis, Cotutelle Universit Paris-Sud (France) and University of Sydney (Australia), July 2008. pdf
V. Batagelj XYZ decompositions ReferencesIII
XYZ decom- positions
V. Batagelj Kratochv`ıl,J., Goljan, M., Kuˇcera,P.: String Graphs. Academia, Prague (1986) Dagstuhl Seminar 08191 Kratochv`ıl,J.: String graphs II: Recognizing string graphs is NP-hard. Motivation J. of Combinatorial Theory, Series B 52 (1991) 67–78 XYZ decom- positions Le, H.O., Le, V.B., M¨uller,H.: Splitting a graph into disjoint induced Results paths or cycles. Discr. Appl. Math. 131 (2003) 199–212 References Lichtenstein, D.: Planar formulae and their uses. SIAM J. Comput. 11 (1982) 185–225
Pajek’s Wiki: http://pajek.imfm.si
Roberts F.S., Sheng L.: How hard is it to determine if a graph has a 2-role assignment? Networks 37 (2001) 2, pp. 6773.
V. Batagelj XYZ decompositions ReferencesIV
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Dagstuhl Schreiber, F., Skodinis, K.: NP-completeness of some tree-clustering Seminar 08191 problems. In Whitesides, S.H., ed.: Graph Drawing (Proc. GD ’98). Motivation Volume 1547 of Lecture Notes Comput. Sci., Springer-Verlag (1998) XYZ decom- 288–301 positions
Results Snyder, D., Kick, E.: The World System and World Trade: An
References Empirical Exploration of Conceptual Conflicts, Sociological Quaterly, 20(1979)1, 23-36.
The latest version of these slides is available at: http://vlado.fmf.uni-lj.si/pub/networks/doc/seminar/XYZgraphs.pdf
V. Batagelj XYZ decompositions