A Comparison of the Readability of Graphs Using Node-Link and Matrix-Based Representations

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1 A Comprison of the Redility of Grphs Using Node-Link nd Mtrix-Bsed Representtions Mohmmd Ghoniem 1 Ecole des Mines de Nntes 4 rue Alfred Kstler. B.P NANTES Cedex 3 Jen-Dniel Fekete 2 INRIA Futurs/LRI Bât 490, Université Pris-Sud Orsy Cedex Philippe Cstgliol 3 Ecole des Mines de Nntes & IRCCyN 4 rue Alfred Kstler. B.P NANTES Cedex 3 Figure 1 Two visuliztions of the sme undirected grph contining 50 vertices nd 400 edges. The node-link digrm ) is computed using the neto progrm nd the mtrix representtion ) is computed using our VisAdj progrm. ABSTRACT In this pper, we descrie txonomy of generic grph relted tsks nd n evlution iming t ssessing the redility of two representtions of grphs: mtrix-sed representtions nd nodelink digrms. This evlution ers on seven generic tsks nd leds to importnt recommendtions with regrd to the representtion of grphs ccording to their size nd density. For instnce, we show tht when grphs re igger thn twenty vertices, the mtrix-sed visuliztion performs etter thn node-link digrms on most tsks. Only pth finding is consistently in fvor of nodelink digrms throughout the evlution. Additionl keywords: Visuliztion of grphs, djcency mtrices, node-link representtion, redility, evlution. Ctegories nd Suject Descriptors: H.5 [Informtion Interfces nd Presenttion]: User Interfces Evlution; I3 [Computer Grphics]: Picture/Imge Genertion Disply Algorithms. 1 INTRODUCTION Node-link digrms hve often een used to represent grphs. In 1 Mohmmd.Ghoniem@emn.fr 2 Jen-Dniel.Fekete@inri.fr 3 Philippe.Cstgliol@emn.fr IEEE Symposium on Informtion Visuliztion 2004 Octoer 10-12, Austin, Texs, USA /04/$ IEEE the grph drwing community, mny pulictions del with lyout techniques complying with esthetic rules such s minimizing the numer of edge-crossings, minimizing the rtio etween the longest edge nd the shortest edge, nd reveling symmetries [2]. Most works strive to optimize lgorithms complying with such rules, ut they scrcely try nd vlidte them from cognitive point of view. Recently, Purchse et l. tckled this prolem through on-pper [13] nd online [10, 11] experiments. These works involved smll hndcrfted grphs (grphs with 20 vertices nd 20 to 30 edges), five esthetic criteri nd eight grph lyout lgorithms. They point out tht while some esthetic criteri tken seprtely my improve the perception of the grph t hnd, one cnnot sy tht n lgorithm rings out such n improvement. Moreover, Wre nd Purchse set up study iming t the vlidtion of some esthetic properties put forth in the grph drwing community, such s the influence of good continuity on the perception of pths [13]. In the Informtion Visuliztion (Infovis) community, mny node-link vrints hve een experimented, oth in 2D nd 3D [5, 8]. However, s soon s the size of the grph or the link density increses, ll these techniques re fcing occlusion prolems due to links overlpping (Figure 1). Thus, it ecomes difficult for users to visully explore the grph or interct with its elements. Conversely, mtrix-sed visuliztions of grphs eliminte ltogether occlusion prolems nd provide n outstnding potentil, despite their lck of fmilirity to most users. In this pper, we present n evlution compring the two representtions in order to show their respective dvntges with regrd to set of generic nlysis tsks. 17

2 2 THE MATRIX-BASED VISUALIZATION OF GRAPHS The mtrix-sed visuliztion of grphs relies from forml stndpoint on the fct tht grph my e represented y its connectivity mtrix which is mtrix of Boolens whose rows nd columns represent the vertices of the grph. When deling with directed grphs, columns represent the origin of edges nd the lines represent their endpoint vertices, lthough conventions my vry. When two vertices re connected, the cell t the intersection of the corresponding line nd column contins the vlue true. Otherwise, it contins the vlue flse. Boolen vlues my e replced with vlued ttriutes ssocited with the edges tht cn provide more informtive visuliztion (Figure 1). The mtrix-sed representtion of grphs offers n interesting lterntive to the trditionl node-link digrms. In [4], Bertin shows tht it is possile to revel the underlying structure of network represented y mtrix through successive permuttions of its lines nd columns. In [3], the uthors visulize the lod distriution of telecommuniction network using mtrix ut most of their effort ims t improving the disply of node-link representtion such s displying hlf-links or postponing the disply of importnt links to minimize occlusion prolems. More recently, in [7], the uthors implemented multi-scle mtrix-sed visuliztion representing the cll grph etween softwre components in ig medicl imgery ppliction. In [6], we hve shown tht mtrix-sed representtion cn e used to effectively grsp the structure of co-ctivity grph nd ssess the ctivity tking plce cross time wheres the equivlent node-link representtion ws unusle. This work ws specificlly pplied to monitoring constrint-oriented progrms. 3 COMPARISON OF REPRESENTATIONS The comprison of two visuliztion techniques cn only e crried out for set of tsks nd set of grphs. The list of tsks tht re useful or importnt with regrd to grph explortion is endless. Indeed, one cn relize this fct y choosing concrete exmple, like grph computed from We site nd enumerting ll the tsks tht one cn chieve or wish to chieve on such grph. In order not to venture into this ottomless sink, we tckled the prolem y considering the most generic tsks of informtion visuliztion nd we dpted them to the visuliztion of grphs. We elieve tht the redility of representtion must e relted to the ility of the user to nswer some questions out the overview. As fr s grphs re concerned, some questions my er on their topology while other questions my concern ttriutes relted to tht topology. The most generic questions relted to the topology of grph i.e. the ones independent of the semntics of dt er on its vertices, links, pths nd su-grphs. Bsic chrcteristics of vertices: one my e interested in determining the numer of vertices (their crdinlity), outliers, given vertex (y its lel), nd the most connected or lest connected vertices. Bsic chrcteristics of pths: the numer of links, the existence of common neighor, the existence of pth etween two nodes, the shortest pth, the numer of neighors of given node, loops nd criticl pths. Bsic chrcteristics of sugrphs: one my e interested in given sugrph, ll the vertices rechle from one or severl vertices (connected sets) or group of vertices strongly connected (clusters). Therefore, compring the redility of grph representtions should, in principle, tke ll these chrcteristics into ccount in order to determine the tsks tht re more esily performed with mtrix-sed representtion nd the ones for which it is more pproprite or more resonle to use node-link representtion. This rticle presents comprtive evlution of redility performed on suset of these generic tsks due to time constrints. 3.1 Redility of grph representtion One cn resonly define the redility of grphic representtion s the reltive ese with which the user finds the informtion he is looking for. Put differently, the more redle representtion, the fster the user executes the tsk t hnd nd the less he mkes mistkes. If the user nswers quickly nd correctly, the representtion is very redle for the tsk. If the user needs lot of time or if the nswer he provides is wrong, then the representtion is not well-suited for tht tsk. In our evlution, we selected the following generic tsks: Tsk 1: pproximte estimtion of the numer of nodes in the grph, referred to s nodecount. Tsk 2: pproximte estimtion of the numer of links in the grph, referred to s edgecount. Tsk 3: finding the most connected node, referred to s mostconnected. Tsk 4: finding node given its lel, referred to s find- Node. Tsk 5: finding link etween two specified nodes, referred to s findlink. Tsk 6: finding common neighor etween two specified nodes, referred to s findneighor. Tsk 7: finding pth etween two nodes, referred to s findpth. Redility lso depends on the concrete grphs, their fmilirity to users, their mening nd the lyout used to visulize them. In our evlution, we only compred rndom grphs which re meningless nd never fmilir to user (e.g. eqully unfmilir), focusing only on strct chrcteristics of grphs. We choose populr grph lyout progrm clled neto, prt of the Grph- Viz[1] pckge to compute the node-link digrm. It could e rgued tht nother lyout progrm would provide more redle lyout ccording to our tsks. This is certinly true for ctul figures ut we elieve tht the trends would e similr when incresing the size nd density of grphs. 3.2 Preliminry Hypotheses The trditionl node-link representtion suffers from link overlpping interfering with neighorhood finding nd link counting nd link length interfering with neighorhood finding. Moreover, some tsks involving sequentil serch of grph elements, such s node finding y nme, re incresingly difficult when the numer of nodes ecomes lrge since, in generl, nodes re not lid out in predictive order. Hence, we expect the numer of nodes nd the link density to influence gretly the redility of this representtion. We define the link density d in grph s: d 3.3 Predictions The mtrix-sed representtion hs two min dvntges: it exhiits no overlpping nd is orderle. We therefore expect tsks involving node finding nd link finding to e crried out more esily. Counting nodes should e eqully difficult on oth repre- l n² where l is the numer of links nd n the numer of nodes in the grph. This vlue vries etween 0 for grph without ny edge to 1 for fully connected grph. In grph theory, the density of grph is usully tken s the rtio of the numer of edges y the numer of vertices ut this definition lthough topologiclly meningful is not scle invrint since the numer of potentil edges increses in the squre of the numer of vertices. 18

3 senttions, unless nodes ecome cluttered on node-link digrms. Counting links should e esier on mtrices since no occlusion interferes with the tsk. Finding the most connected node should perform etter on mtrices for dense grphs ecuse, on node-link digrms, links strting or ending t node re hrd to discriminte from links crossing the node. On the other hnd, when it comes to uilding pth etween two nodes, node-link digrms should perform etter; mtrixsed representtions re more complex to pprehend ecuse nodes re represented twice (once on oth xes of the mtrix), which forces the eye to move from the line representing vertex to its ssocited column ck nd forth, unless more pproprite interction is provided. Lstly, we elieve tht node-link digrms re suitle, nd therefore preferle to the less intuitive mtrix representtion, when deling with smll sized grphs. 3.4 Experimentl setup Implementtion nd tuning Node-link digrms were lid out using AT&T s [1] open source grph lyout progrm neto nd the jv drwing lirry grpp. We mde our est effort to tune oth representtions in order to mke the est use of them. We mde the sme interction tech The dt In order to test our hypotheses, we experimented with grphs of three different sizes (20 vertices, 50 vertices nd 100 vertices) with three different link densities (0.2, 0.4 nd 0.6) for ech size, tht is to sy totl of nine different grphs (Tle 1). In order to void ny is introduced y some peculirity of the chosen dt, we opted for rndom undirected grphs generted y the rndom grph server locted t the ISPT [10]. Moreover, in order to eliminte ny miguity with regrd to tsk 3, which consists in finding the most connected node, we dded n extr 10% of links to the most connected node in these grphs. When severl nodes hd initilly the highest degree, one of them ws chosen t rndom nd received n dditionl 10% of links. The distriution of dditionl links ws lso done t rndom. size\density grph 1 grph 2 grph 3 50 grph 4 grph 5 grph grph 7 grph 8 grph 9 Tle 1.The nine types of grphs used for our experiment The rndom grph genertor we used lels the nodes numericlly ccording to the order of their cretion which, s such, mkes tsk 1 mount to finding the gretest numeric lel. Consequently, we decided to mke this tsk more relevnt y renming the nodes lpheticlly (from A to T on the twenty-node grphs, from A1 to F0 on the fifty-node grphs, nd from A1 to K0 on the one-hundred-node grphs) The popultion The popultion tht performed the evlution consisted of postgrdute students nd confirmed reserchers in the fields of computer science. All the sujects knew wht grph ws. No further knowledge of grph theory ws required. The popultion consisted of 36 sujects, ll of whom hd previously seen node-link representtion of grphs. All the sujects prticipted voluntrily to the evlution The evlution progrm We developed n evlution progrm tht represents the selected grphs ccording to oth representtion techniques. It then sks the user to perform the tsks nd records the time to nswer. In terms of interction, our progrm provides picking nd selection mechnisms. On oth visuliztions, when the mouse goes over node, it is highlighted in green s well s its incident links; nodes cn lso e selected through direct pointing, in which cse they re highlighted in red s well s their incident links. Likewise, when the mouse goes over link, it is highlighted in green s well s its endpoints. (Figure 1) These interctive enhncements were dded to help users focus on grph elements fter n initil testing showing high level of frustrtion from users losing focus. A demonstrtion mde on set of two grphs llowed us to explin how the representtions should e red nd how the vrious tsks could e performed. First, the instructor mnipulted the system nd provided guidelines. Then the user mnipulted the system in order to mke sure tht the representtions, the tsks nd the interctions were well understood. At the end of the demonstrtion, we mde sure tht the user ws redy to strt the evlution per se. When necessry, the instructor proposed to repet the demonstrtion gin. At the end, three instructions were given: The user hs to nswer s quickly s possile. The user hs to nswer correctly. The user is llowed to move to the next question without nswering efore the nswer time elpses in cse he felt he ws not le to nswer. To void memoriztion ises, the system selects representtion technique t rndom mtrix or node-link nd represents sequentilly ll nine grphs, sking the user to execute the seven tsks for ech grph. Then, the system moves to the second technique nd does the sme. By interchnging the representtion techniques, we mke sure tht the sujects hd the sme proility to strt the evlution with series of visuliztions elonging to either technique. Ech series ws divided into two prts: the first included three simple grphs (grphs 1, 2 nd 4) nd llowed the user to get fmilir with the system; the second included the six remining grphs. Furthermore, lerning effect ws oserved when user ws confronted to the mtrix representtion. We were le to mesure such n undesirle effect in series of ten preliminry tests where the system selected the grphs from the smllest to the lrgest nd from the sprsest to the most connected. In spite of the incresing complexity of the displyed grphs, users would tend to nswer more quickly s their understnding of mtrix-sed representtions incresed throughout the experiment. To level this effect, during the evlution, our system selects the grphs t rndom within ech hlf-series. In this wy, the grphs hve n equl proility to pper in the eginning, in the middle, or t the end of their respective hlf-series. For tsks involving two nodes, (tsks 5, 6 nd 7), the system selects oth nodes eforehnd in order to void spending time trying to locte them. Therefore, the time we mesure corresponds exctly to the time for executing those tsks. Since ech evlution session contins totl of 126 questions (9 grphs x 2 visuliztion techniques x 7 tsks), we progrmmed three puses: ten-minute puse etween the two series nd five-minute puse etween the two hlves of ech series. Moreover, since the sessions re rther long, ( full hour of mnipultion per user), we chose to limit the nswer time to 45 seconds per question. When the time runs out, the system moves utomticlly to the next question nd produces n udio feedck in order to notify the user. In this cse, we consider tht the representtion is not effective for tht tsk since the user ws not le to provide the nswer in the llotted time. The udio feedck lso incites the user to hurry up for next questions. 19

4 niques ville on oth visuliztions. We pid ttention to the size of nodes nd the redility of their lels on node-link digrms, however lrge or dense they got. We superimposed the lels of picked or selected nodes on semi-trnsprent ckground, which elimintes the occlusion prolems due to links overlpping over these nodes. Given tht we re deling with undirected grphs, we did not disply ny rrows t the extremities of the links, which significntly improves the node-link representtion of dense grphs. Deling with undirected grphs lso simplifies the pth lookup tsk on the mtrix-sed representtion since links pper twice. We stored the prmeters of the tuned node-link digrms in the dot formt nd used those settings long the evlution. We thus gurntee tht the sujects re confronted with exctly the sme representtion for respectively ll nine grphs. Likewise, we exploited the intrinsic orderility of the mtrix representtion nd sorted its lines nd columns lpheticlly. The mtrix eing sorted instntneously, the mtrix-sed visuliztions did not require ny preliminry tuning. The evlution ws crried out on Dell worksttion hving n NVIDIA GeForce2 ccelerted video crd, dul Pentium III 1Ghz processor, 512 Mytes of RAM, under Windows The disply ws performed in full screen mode on 21" monitor. Sujects were seted t sixty centimeters from the monitor nd executed ll tsks using the mouse. the two smples hve the sme medin vlue; otherwise, we conclude tht the two smples hve different medin vlues. On the following ox-plot digrms, lue oxes correspond to the mtrix representtion nd purple oxes correspond to the node-link representtion. On the y-coordintes, we represent the nswer time. For ech tsk, we disply the evolution of time for the three grph sizes on digrms leled (), nd on the ones leled () we disply the evolution of time for the three chosen densities Estimtion of the numer of nodes (nodecount) On Figure 4, on the mtrix-sed representtion (x-coordintes 1, 3 nd 5), medin nswer time nd time distriution vry little when size increses, wheres they grow notly on the node-link representtion (x-coordintes 2, 4 nd 6). We therefore conclude tht with regrd to this tsk, the redility of node-link digrms deteriortes significntly when the size of the grph increses wheres the mtrix-sed representtion is less ffected y size. On Figure 4, on the mtrix-sed representtion (x-coordintes 1, 3 nd 5), medin nswer time nd time distriution increse little when the density increses; they increse slightly on the node-link representtion. 3.5 Results 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0, Figure 2 Percentge of correct nswers split y tsk nd y size. The mtrix representtion ppers in lue nd the node-link in purple. 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0, Figure 3 Percentge of correct nswers split y tsk nd y density. The mtrix representtion ppers in lue nd the node-link in purple. The mesurements were nlyzed using grphic qulittive method (Box-Plot) nd quntittive method (non prmetric test of Wilcoxon). The ltter mkes it possile to compre the centrl position of two smples without prior knowledge of their distriution (normlity for exmple). This test provides p-vlue which is proility. When the p-vlue is less thn 5%, we conclude tht Figure 4 Distriution of nswer time for nodecount () split y size, () split y density Moreover, in Figure 2, we note tht, with regrds to lrge grphs, 96% of the users hve nswered correctly using the mtrix-sed representtion, ginst 81% using the node-link representtion, tht is to sy difference of 15%, deemed sttisticlly significnt ccording to Wilcoxon s test. On the mtrix-sed representtion, the percentge of correct nswers remins stle, round 97%, when density vries (Figure 3), which corresponds to sttisticlly significnt improvement of 7% compred to the node-link representtion for low nd high densities. The redility of the node-link representtion is strongly ffected when the numer of nodes increses, nd is slightly ffected when link density increses, wheres the redility of mtrix-sed representtions is slightly ffected y size vrition nd is not sensitive to density vrition. On top of tht, using the mtrix-sed representtion, users nswer fster when the size nd density re medium or lrge. 20

5 3.5.2 Estimtion of the numer of links (linkcount) Estimting the numer of links in the grph seems reltively independent of size or link density when these prmeters tke medium or lrge vlues. On Figure 5 (x-coordintes 2 nd 4), there is gp in nswer time etween smll nd medium-sized grphs nd, on Figure 5 (x-coordintes 2 nd 4), etween sprse nd modertely dense grphs. However, there seems to e no difference etween the two techniques for ny given size or density. On Figure 2, the mtrix-sed representtion records 57% of correct nswers on lrge grphs. This figure goes s low s 25% using the node-link representtion, tht is to sy significnt discrepncy of 27 % compred to mtrices. The difference recorded with regrd to smll nd medium-sized grphs respectively in fvor of the node-link representtion nd the mtrix-sed representtion is sttisticlly insignificnt. Similr conclusions cn e drwn with regrd to link density (Figure 3). Figure 6 Distriution of nswer time for mostconnected () split y size, () split y density Finding specified node (findnode) Figure 5 Distriution of nswer time for linkcount () split y size, () split y density Finding the most connected node (mostconnected) When chieving this tsk, we note tht, with regrd to nswer time, oth techniques re sensitive when size increses (Figure 6), wheres they re slightly ffected when link density increses (Figure 6). We cnnot differentite these methods with regrd to this tsk sed on nswer time only. Nevertheless, ccording to Figure 2, 85% of the users execute this tsk correctly on smll grphs using mtrix-sed representtion ginst 63% of correct nswers with the node-link representtion. On medium-sized grphs, we hve 73% of correct nswers using mtrix ginst 52% using node-links. Lstly, on lrge grphs, we record 57% of correct nswers using mtrix ginst only 25% of correct nswers with node-link digrms. These differences re deemed sttisticlly significnt using Wilcoxon s test. Similr conclusions cn e reched with regrd to density on Figure 3. Figure 7 Distriution of nswer time for findnode () split y size, () split y density When considering nswer time, we cn see tht the redility of node-link digrms deteriorte quickly when the size of the grph increses (Figure 7) nd re modertely ffected y link density (Figure 7), wheres the nswer time on the mtrix-sed representtion devites little when the size increses nd does not seem to e ffected t ll y link density. The dispersion of nswer time is very smll using mtrix-sed representtion in oth cses. When deling with smll grphs, oth representtions perform eqully well. The percentge of correct nswers (Figure 2) is high, lmost 98%, using the mtrix-sed representtion, irrespective of the size of the grph. The percentge of correct nswers is 21

6 eqully good using node-link digrms, except for lrge grphs whose score flls s low s 67%, with discrepncy of 31% compred to mtrix-sed representtions. Similr conclusions cn e drwn when link density vries (Figure 3) Finding link etween two nodes (findlink) Using the node-link representtion, the lrger the grph the longer it tkes to look up link in the grph (Figure 8); the nswer time does not vry significntly when link density increses (Figure 8). Using mtrices, this tsk is insensitive to size nd density vrition. For lrge grphs, nd for medium nd high link density, significnt gp is mesured in fvor of mtrix-sed representtions. A significnt difference is mesured in fvor of node-link digrms for smll grphs. Both representtions record excellent percentges of correct nswers, out 95%, for smll nd medium-sized grphs (Figure 2). For lrge grphs, the mtrix-sed representtion records 92% of correct nswers ginst 66% with the node-link digrms, tht is discrepncy of 26%. Figure 9 Distriution of nswer time for findneighor () split y size, () split y density Finding pth etween two nodes (findpth) Figure 8 Distriution of nswer time for findlink () split y size, () split y density Finding of common neighor (findneighor) Using the node-link representtion, the lrger the grph the longer it tkes to sy whether common neighor exists (Figure 9); the medin nswer time is mrginlly ffected when link density increses (Figure 9). On the mtrix-sed representtion, size vrition hs no impct on nswer time, while medin nswer time nd vlue dispersion improve slightly when link density is lrge. When deling with smll grphs, node-link digrms record 99% of correct nswers with led of 12% over mtrices. Mtrixsed representtions tke n equivlent led when deling with lrge grphs, towering t 96% of correct nswers. Both techniques record similr percentge of correct nswers on medium-sized grphs (Figure 2). When link density vries (Figure 3), oth representtions score out 90% of correct nswers. Figure 10 Distriution of nswer time for findpth () split y size, () split y density Finding pth etween two nodes proves to e incresingly difficult using node-link digrms when the size increses (Figure 10), wheres the medin nswer time increses slightly when link density increses (Figure 10). The mtrix-sed representtion is much worse for this tsk except for very dense grphs where it outperforms the node-link representtion (Figure 10). This fct is confirmed y the percentge of correct nswers which towers t 95% for the mtrix-sed representtion ginst 85% for node-links when deling with lrge link density (Figure 3). On smll grphs, 99% of the users nswer correctly using node-link digrms ginst 70% only using mtrices. On medium-sized 22

7 grphs, node-link digrms record 93% of correct nswers with led of 10% over mtrices. For lrge grphs, every other user nswers correctly using oth representtions with sttisticlly insignificnt led in fvor of mtrices. Lstly, this tsk is clerly in fvor of node-link digrms when visulizing sprse grphs. 4 DISCUSSION We expected the redility of node-link digrms to deteriorte when the size of the grph nd its link density increse. This hypothesis ws confirmed for the seven tsks we selected. Only for findpth tsk did node-link digrms prove superior to mtrixsed representtions, lthough their performnce deteriortes on lrge nd dense grphs. This conclusion must however e qulified since this tsk is difficult to crry out visully when the distnce etween the extremities is greter thn two or three rcs, s shown in [13]. Another hypothesis concerned the impct of orderility of mtrices on node nd link finding tsks. findnode nd findlink tsks vlidte this hypothesis for lrge grphs nd for dense grphs. As fr s linkcount tsk is concerned, oth visuliztions record lrge shre of erroneous nswers. We ccount for tht ut this hs yet to e proven through experimenttion tht the numer of links is intrinsiclly difficult to estimte on node-link digrms nd tht users filed to compute it correctly using mtrices. Indeed, links re displyed twice ecuse we considered undirected grphs in our study. We my lso question the extensiility of the results otined in this evlution to other node-link lyout progrms thn the one we chose in our experimenttion. However, sed on erlier works [11], we cn sfely ssert tht, with regrd to smll grphs, the lyout progrm hs very little impct on the redility of the displyed output nd would not chnge the trends we oserved. We my further highlight tht ll the users who took prt in the experiment were fmilir with node-link digrms wheres none hd previously herd out the mtrix-sed representtion of grphs. Since they were given little trining (first, users would wtch the instructor perform the tsks on grph, nd then they would trin on one similr grph), we expect users fmilir with oth representtions to perform even etter with mtrices. As first pproch, we hve split the dt y size nd y density in order to mesure the effect of these vriles on the redility of grph representtions. To this end, we hve done our est effort to isolte those fctors nd mke sure tht no other considertions would interfere with the tsks. However, further nlysis of the results is required in order to check for comined effect of size nd density of grphs on the redility of their representtions. It would lso e meningful to rek down the results for ech of the nine grphs we studied for etter control over the prmeters nd finer understnding of the results. For instnce, finding pth on node-link digrm representing sprse grph proves ll the more difficult thn the shortest pth etween the extremities is long. In this cse, the density of the grph my e misleding indictor; the length of the shortest pth my e etter choice nd should e tken into ccount. Conversely, in dense grphs, the shortest pth is likely to e one or two links long, ut the visul clutter produced y links on node-link digrms mkes this tsk unfesile, while mtrices perform very well. In this evlution, we compre two representtions of grphs, mtrix-sed representtion nd node-link representtion produced y force directed lgorithm, ginst nine rndom grphs nd set of seven explortion tsks. In this context, the recommendtions we cn derive from this study re: for smll grphs, node-link digrms re lwys more redle nd more fmilir thn mtrices. For lrger grphs, the performnce of node-link digrms deteriortes quickly while mtrices remin redle with led of 30% of correct nswers, with comprle if not etter nswer time. For more complex tsks such s findpth, we re convinced tht n pproprite interction is lwys preferle, for exmple y selecting node nd displying ll the possile pths strting from it nd ending t pointed node. On the mtrixsed representtion, this pth cn e displyed using curves connecting djcent links, i.e. connecting the cells representing those links. 5 CONCLUSION In this pper, we hve listed generic tsks for the visuliztion of grphs nd hve compred two representtions of grphs on suset of these tsks. We hve proved theses techniques to e complementry: node-link digrms re well suited for smll grphs, nd mtrices re suitle to lrge or dense grphs. Pth relted tsks remin difficult on oth representtions nd require n pproprite interction tht helps perform them. The mtrix-sed representtion seems therefore under exploited nowdys, despite its quick lyout nd its superior redility with regrd to mny tsks. We think tht wider use of this representtion will result in greter fmilirity nd will consequently improve its redility. We currently use the mtrixsed representtion for the rel-time monitoring of constrintoriented progrms where grphs evolve dynmiclly, oth in size nd ctivity. The results we re otining re quite encourging. We re investigting clustering nd ggregtion techniques of mtrices for the visuliztion of very lrge grphs, out tens of thousnds vertices. 6 ACKNOWLEDGEMENTS We would like to thnk Pierre Drgicevic, Véronique Liérti nd Vness Tico for their time nd dvice. We re grteful to the users who volunteered nd took prt in this evlution nd to the memers of the RNTL OADYMPPAC project. REFERENCES [1] AT&T Ls Reserch. Grphviz - open source grph drwing softwre, [2] [3] Bttist, G.D., Edes, P., Tmssi, R. nd Tollis, I.G. Grph Drwing. Prentice Hll, [4] Becker, R.A., Eick, S.G. nd Wills, G.J. Visulizing network dt. IEEE Trnsction on Visuliztions nd Grphics, 1 (1) [5] Bertin, J. 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