How to Tame a Very Large ER Diagram (using Link Analysis and Force-Directed Drawing Algorithms)

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1 Proceedings of 24th Int. Conf. on Conceptual Modeling, ER 2005 Klagenfurt, Austria, Oct How to Tame a Very Large ER Diagram (using Link Analysis and Force-Directed Drawing Algorithms) Yannis Tzitzikas 1 and Jean-Luc Hainaut 2 1 University of Crete and FORTH-ICS, Heraklion, Greece 2 Institut d Informatique, University of Namur (F.U.N.D.P.), Belgium Abstract. Understanding a large schema without the assistance of persons already familiar with it (and its associated applications), is a hard and very time consuming task that occurs very frequently in reverse engineering and in information integration. In this paper we describe a novel method that can aid the understanding and the visualization of very large ER diagrams that is inspired by the link analysis techniques that are used in Web Searching. Specifically, this method takes as input an ER diagram and returns a smaller (top-k) diagram that consists of the major entity and relationship types of the initial diagram. Concerning the drawing of the resulting top-k graphs in the 2D space, we propose a force-directed placement algorithm especially adapted for ER diagrams. Specifically, we describe and analyze experimentally two different force models and various configurations. The experimental evaluation on large diagrams of real world applications proved the effectiveness of this technique. 1 Prologue It has been recognized long ago that the usefulness of conceptual diagrams (e.g. ER/UML diagrams) degrades rapidly as they grow in size. Understanding a large schema without the assistance of persons already familiar with it, is often a nightmare. Unfortunately, large conceptual schemas are becoming more and more frequent. The integration of information systems, the development or reverse engineering of large systems, the usage of ERP (the SAP database includes tables) and the development of the Semantic Web (structured into ontologies potentially including dozens of thousands of classes) naturally lead to the building of very large schemas. Although a good drawing of a conceptual schema could aid its understanding, and several approaches for automatic placement have been already proposed (e.g. see [33, 23, 8, 28, 10]), it is a widely accepted opinion that the automatic layout facilities offered by current UMLbased CASE tools are not satisfactory even for very small diagrams (for more 1

2 see [14]) 1. Consequently, the vast majority of layouts created today are done by hand ; a human designer makes most, if not all, of the decisions about the position of the objects to be presented [26]. The visualization and drawing of large conceptual graphs is even less explored. The classical hierarchical decomposition techniques that are used for visualizing large plain graphs (for a survey see Chapter 3 of [29]), have not been applied or tested on conceptual graphs. Consequently, only manual collapsing mechanisms (like those described in [22]) are currently available for decreasing the visual clutter and for aiding the understanding of big conceptual graphs. In addition, the techniques that have have been proposed for reducing the size of a conceptual graph (in order to aid its comprehension), specifically ER clustering, either require human input [15, 34, 18, 7], or they are automated but not tested on large conceptual schemas [30, 1]. We decided to devise an automatic technique for identifying the major entity and relationship types of a very large conceptual graph as a means for facilitating its understanding. As Link Analysis has been proved very successful in Web Searching [6, 25] and recently in several other application domains [19, 20], we decided to design a similar in spirit technique for one very common and important kind of conceptual schemas, namely Entity-Relationship (ER) diagrams [9]. Concerning the drawing in the 2D space of the resulting top-k graphs, we describe a force-directed placement algorithm especially adapted for ER diagrams. Specifically, we describe and analyze experimentally two different force models and various configurations. Both of the techniques that are presented in this paper can be applied not only to ER diagrams, but also on other kinds of conceptual graphs. The Semantic Web is one interesting application area because it is founded on ontologies (potentially including dozens of thousands of classes) that are exchanged in a layout-missing format. In this context, the provision of top-k diagrams and automatic layout services is very important as they can aid understanding that is very important for accomplishing tasks like semantic annotation, creation of ontology mappings, ontology specialization, etc. This paper is structured as follows: Section 2 describes Link Analysis for ER diagrams, Section 3 introduces force-directed placement algorithms for ER diagrams and finally, Section 4 concludes this paper. 2 Link Analysis for ER Diagrams Our objective here is to identify the major entity and relationship types of a very large ER diagram in order to facilitate its understanding. We designed a PageRank [6] style scoring method because PageRank is described in terms on the entire Web, while HITS [25] is mainly applied on small collections of pages (say those retrieved in response to a query). We view an ER diagram as a triple (E, R, I) where E = {e 1,..., e N } denotes the entity types, R = {r 1,..., r m } denotes the relationship types, and I the isa relationships over E (i.e. I 1 General graph drawings algorithms (e.g. see [2]) usually make some assumptions that are not always valid in conceptual graphs. 2

3 E E). For any given e E, we shall use conn R (e) to denote those entity types that are connected with e through relationship types, conn sb (e) denote the direct subtypes of e, and conn sp (e) the direct supertypes of e. We shall also use the following shorthands: conn I (e) = conn sb (e) conn sp (e) and conn(e) = conn R (e) conn I (e). Since two entity types may be connected with more than one relationship types we consider conn R (e) as a bag for being able to record duplicates. In addition, we shall use attrs(e) to denote the attributes of an entity type e. Now the score (or EntityRank) of an entity type e in E, denoted Sc(e), can be defined as follows: Sc(e) = q/n + (1 q) e conn R (e) Sc(e ) conn R (e ) where q stands for a constant less than 1 (e.g as in the case of Google) 2. One can easily see that the above formula simulates a random walk in the schema. Under this view each relationship type is viewed as a bidirectional transition and the probability of randomly jumping to an entity type is the same for all entity types (i.e. q/n). The resulting scores of the entity types correspond to the stationary probabilities of the Markov chain. A rising question here is how we can incorporate n-ary (n > 2) relationship types into the aforementioned model. This can be achieved by replacing each n- ary relationship type (n > 2) over n entity types e 1,..., e n by n(n 1)/2 binary relationship types that form a complete graph 3 over e 1,..., e n. Consequently, an n-ary relationship type is viewed as n(n 1)/2 binary relationships. An alternative approach is to assume that the probability of jumping to a random entity is not the same for all entities, but it depends on the number of its attributes. In this case we define the score (or BEntityRank) of an entity type e in E as follows: Sc(e) = q attrs(e) Attr + (1 q) e conn R (e) Sc(e ) conn R (e ) where Attr denotes the set of all attributes of all entity types (i.e. Attr = { attrs(e) e E}). This particular formula simulates a user navigating randomly in the schema who jumps to a random entity e with probability q attrs(e) Attr or follows a random relationship type (on the current entity). The probability attrs(e) Attr corresponds to the probability of selecting e by clicking randomly on a list that enumerates the attributes of all entity types of the schema. The linear algebra version of EntityRank and BEntityRank is given in Appendix A. 2 If we set q = 0.15 or below then an iterative method for computing the scores (e.g. the Jacobi method) requires at most 100 iterations to convergence. 3 A complete graph is a graph in which each pair of graph vertices is connected by an edge. (1) (2) 3

4 Let s now discuss the differences between link analysis for ER diagrams and link analysis for the Web. Firstly, Web links are directed, while relationship types are not directed thus the latter are considered as bidirectional transitions. Secondly, we do not collapse all relationships types between two entity types into one (as it is done with Web links), and this is the reason why we consider conn R as a bag. Thirdly, in ER diagrams we should count self hyperlinks (i.e. cyclic relationship types), although the Web techniques ignore them. For example, consider a schema consisting of two entity types {Person, City} and two relationship types {Person lives City, Person f atherof Person}. If we ignore the relationship type fatherof, then both Person and City would be equally scored, a not so good choice. At last, although in the Web link analysis is exploited mainly for ranking the results of retrieval queries, in our case we don t need just a ranked list of entity types, but rather another diagram that consists of the major entity and relationship types. We have implemented and evaluated the above scoring schemes into the DB MAIN CASE tool (for more see [17, 21]). The designer provides a threshold per between 0 and 100. Subsequently, all entity types with score lower than per% ScM ax, where ScM ax denotes the highest score, disappear. Controlling the visibility of entity types according their score, and not according to their rank, is preferred as it better handles ties. Concerning relationship types, only those that connect the visible entity types are displayed. The computation of the scores takes taeg nombre valeur maximum réel acompte mensualités termes Vente devise autorisé à crédit inter exter type top tpbsta/toptyp D'INTERVENTION PAR DOSIE top/action only some seconds on a conventional PC. Specifically, to compute the scores répartition/tva toptyp/tpcsuivre toptyp/tpbcli Heure Texte suivi LE DOSIER SAV tpbmtf/tpbsta tpb/tpbsta we use the Jacobi iterative algorithm. We have noticed that 50 iterations give quite stable orderings and their application on schemas with 1000 entity and relationship types takes less than 2 seconds a conventional PC. suivi prxpaycat/prxpgecat prxpaycat/prxraycat prxcat/prxpaycat bnb prxpaycat/prxrefcat REFERENCE CONTIENTservices prxpgecat/prxrefcat quantité Top modification proposés logistiqueprxposcat/prxposexe prxpos/prxposcat prxref/prxpos prxraycat/prxrefcat prxref/prxrefcat prxray/prxref prxray/prxraycatsuivi Nombre Clients TOPS CLIENT TRANSPORTEUR PAR CANAL code Code Cumul operatrice type commande ligne temps valeur relais SUIVI délai suivi catalogue nette d'encodage de opératrice OPERATRICE livraiso date Numéro Etat pour de systeme séquence SUIVI de suivi COMMANDES mesure comma tpd/tpdliv Suivi des jets application SUIVI ACTIVITE support du logement Code Cumul mouvement type commande colis article mouvements PRODUIT/CATEGORIE Nombre Top information d'articlesstat Code Nombre Chiffre d'affaire d'articles FOURNISEUR 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type état catalogue francais d'activité vente cn#c facturation dossier du de lien fournisseur d'adresse dossier page client réparation colis de SAV article si DOSSIER base du (interne) DETAIL Numéro Texte SAV absent document courrier livraison article de de SAAV la DOSIER ligne dossier 3 SUISSES COMPLEMENT Nom Localité Code Rue Date Numéro SAAV facture pays postal l'adresse du téléphone l'adresse réparation dossier colis numero ligne nbre prix commentaire etat DOSIER facturation article unitaire de FACTURATION Date Quantitée Nombre Numéro facturé SAAV initiale de minimum ligne réele proposée LIGNE l'offre proposition proposée livraison PROPOSITION à d'ach D'ACHATcode MARQUE marquenumero quantité prix date taux etat numéro Date fiche tissu bois cde livraison rendez-vous demande butoir achat initiale ristourne demandée technique commande pour lettre problable FB devise lay-out COMMANDE livraisonnuméro JET jetnuméro type date nombre texte top code compte marquage reconditionnement fin article état secteur responsable libre réception ligne articles avancement livré rendu d'activité à LIGNE receptionnés réceptionnés contrôle ligne RECEPTION à théo quali réc cndate top Numéro Motif réponse circuit état type user saisie subsidiairelibellé code FILIALE filiale demande commande[] saisie clientnom groupe code 10 nombre numéro taux Période périodes pays acheteur achat couverture d'échelonnement[10] personne jour de provenance fabrication cadencement sécurité ACHETEUR par Code Libelé Cout d Libellé Code TYPE type de frais FRAIS d'expédition D'EXPEDITIONcode Texte Début Fin Top Action promotion Sans de validité RESERVATION prévue la commande Cadeau promotion ETAT type code libellé Libellé Nombre Code d'état[] refus D'AVANCEMENT de jours 310 de caractères rétention (LIGNE Libellé UNITE d'unité budgétaire BUDGETAIRE Secteur activite code libelé libele Top code pour le LANGUE client libellé Libellé Délaide Code Libellé prévu type de TYPE de PROMOTION de reser stock STOCK TOMBOLA numéro Numéro Date Stock Quantité Prix Nouveau du Code billetsuport TYPE CHANCE avant fournisseur revalorisation type de prix CENTRE livrée revalorisation centre REVALORISATION ARTICLE moyen / fournisseur DISPATCHING dispatching pondérélibellé du jour JOURCode Libelé Longueur Largeur Hauteur Poids Nombre support à francais vide du logements suport support de du stockage suport DE contigus opposés STOCKAGELIGNE Indice code Référence origine de DE la article COLIS lignesans ADRESEREGROUPEMENT Code Libellé regroupement STATISTIQUE code Date canal début fin fermeture RELAIScode libellé CO Poids Unité Quantité quantité Longueur Largeur Hauteur Nombre Type Volume CONDITIONNEMENT Prix Code conditionement unitaire Top top Date article achat vente technique abandon modif initialisation coloris édition exception pli taille fournisseur frais type logement report francais de néerlandais livraison réservée emballée montée de minimum vente message composé[] standard hors solde l'article sur-stock colis prix VPC campagne remplacement pour 12cession solde base pour LDF l'article 24 interne externe statistiqu ARTICLE h transport bordereau l'article VPC suivant stock pickivpc Code CENTRE Code DISPATCHING DISPONIBILITE disponibilitécontrole Libellé circuitcontrole CIRCUITDate opératrice Pays Code Numéro Créateur Indice Commentaire de postal supplémentaire de retour commentaire d'ouverture localité colis retrouvé top code secteur mode pays pour de déviation activité réception goulotte CALCUL goulotte tournée GOULOTTELibellé Commentaire Dernier Date Top Action Valeur Taux Code circuit dernier d'envoi Nombres Prix Type Numéro Code TYPE bon minimale à type composition ENVOIS ristourne jet éditer retardé d'achat de d'envois etatenvoi probable jet paiement PAIEMENT COMPOSITION vente de d'articles commande EXEDENCE à prélèvements sélection externe éditer d'envoi excedence a réception code Valeur Numéro RECEPTION canal excédence exter facture D'ENVOI Libelle Donnée centre entreprise dispatching centre libellé délai type top nom numéro code Top Type Origine Numéro Code Nombre dispacanal saisie scission attribution blocage[] canal circuit[] langue[] postal[] du type depot responsable maximum type du livraison[] bordereau canal[] des centre téléphone mouvement code poste commande(o/n)[] livraison(o/n)[] retour[] du déroutage[] commandes[] colis magasin[] no postal/localité dispatching du travail/heu colis du PAR paiement[] canal[] PREPARATION facture FACTURE interne externe CONTROLE Numéro Nombre Code Date Identification User Top Valeur archivage traitement colis totale paiement réception contrôle records Nombre Type code numéro MOUVEMENTS paiement bande anomalie mouvement état bande d'envoi sur paiement d'articles bande interne externe prélevements sélection excédence EXCEDENCE PAIEMENTS code Type Date Top Code protection logement blocage pseudoniveau état type groupe dernier inventaire du d'inventaire Libellé Code Top prélèvement LOGEMENT bon ristourne à logement incendie - catalogue/cadeau payer 1234TYPE type d'achat circulaire CIRCULAIRENuméro Libellé francais néerlandais commentaire libelé top commentaire canal saisie COMMENTAIRE du type commandes[] commande(o/n)[] livraison(o/n)[] Type Code Date Libellé canal[] PARTICULARITE début fin particularité mouvement validité mouvement MOUVEMENT Taux de répartition de la tva tva/echnombre de pilesprxposcat/prxrefcattraduction codes/type emballag Code TVATaux change Cumul valeur catalogue mode livraison/pays ech lan/tpbcli est commandé individu/pays cn/circuit ComposteurCOLIS SANS ADRESSES code type de canaltype DE CANAL Fig. 1. Excerpt (< 10%) of a large ER diagram drawn using a force-directed placement algorithm Figure 1 shows a very small part of the ER diagram of a Belgian distribution company. Though the schema comprises about 450 nodes and 800 edges only, the 4

5 layout is definitely useless for understanding the schema and the corresponding application. CANAL LIVRAISON COLIS COLIS cn/suicol réexpédition colis colcom/suicol SUIVI LIVRAISON COLIS col/eta ETAT D'AVANCEMENT (LIGNE OU CO STAT FOURNISSEUR artvpc/frnmensuelles décomposition d'un ensemble SUIVRE LE DOSSIER article vpc pièce article vpc ensemble POS-DTE ARTICLE VPC description décrire Notice FOURNISSEUR remise fin année fournisseur date de fin de validité date début de validité DATE achfrn/lan fournisseur/langue est remplace remplace Remplacement HISTORIQUE cd/suicolcolis COMMANDE cd/eta artref/artvpc Canal commande livraison POSITIONS/ACTIVITE/MOIS Mouvement de stock SUIVI COMMANDES cumul mouvement systeme mesure atvsec/cd Poids max/canal Document par langue SUIVI OPERATRICE artref/atvsec SECTEUR ACTIVITE REFERENCE ARTICLE ALIMENTER pondération calcul SUIVI canalactivite calcul goulotte FRAIS PAR DEFAUT Cadeau associé pays de la devise EARLY BIRD cn/pay pays du fournisseur artref/tva simulation prix artref/douref de vente Pays d'origine PAYS ENSEIGNE LANGUE dte/pay fournisseur/pays Date de depot LOT FORMULE PAR PAYS/LANGUE SUIVI DES TOPS CLIENT PAYS LANGUE Composition PAYS LANGUE pays d'origine pays destination FRAIS DE DOUANE Fig. 2. The diagram of the top-11 entity types of the schema of Figure 1 Now Figure 2 shows in micrography the diagram of the top-11 entity types of the schema of Figure 1 according to EntityRank (for reasons of space, the attributes are not displayed in this figure). Although this schema has 54 relationship types it is extremely more easy to visualize, and thus to understand, than the original schema. Of course, one user could start from even smaller diagrams. For instance, Figure 3 shows the graph of the top-5 entity types of the same schema. It indeed contains the major entity types of this application and a user can immediately understand the application domain of this schema. In case of diagrams with big isa hierarchies, some entity types, although major, may not receive high scores because their relationship types are scattered in several subentity types. To handle this case, we introduced a (optional) preprocessing step in which each isa hierarchy of the schema is collapsed into one entity type that collects all the attributes and relationship types of its subentity types. Concerning evaluation, at first we have to note that the evaluation of the effectiveness of link analysis techniques for ER diagrams (for conceptual graphs in general), is more difficult than in the case of Web. In the latter case, it is not so hard to judge whether the top ranked pages are indeed relevant to the submitted query. However, in the case of conceptual graphs, one has to know well in advance the conceptual graph in order to judge whether the resulting small graph indeed contains the major concepts of the conceptual graph and of its underlying domain. Inevitably, the most reliable evaluation of such techniques can be done only in already known conceptual graphs. For this reason we applied this method to almost every conceptual schema that the DBMAIN group has produced the last 3 years. This was a quite representative test bed as it includes 5

6 description decomposition Item describes Notice partof Bonus replaces EARLY BIRD replacedby replacement cn/pay channelcalculation follow-up activity supplies origin Channel Country destination Operator POSITIONS/ACTIVITIES/MONTHS Stock Movement Max Weight/Channel path Calculation country language State of Progress accumulated movements SectorOfActivity Customs costs DefaultCost Fig. 3. The diagram of the top-5 entity types of the schema of Figure 1 big schemas of existing (and non artificial) real world applications. We always obtained surprisingly good results. For reasons of space we cannot report here the exact results of the evaluation of EntityRank (and BEntityRank) using metrics coming from the area of IR (for more [35]). In addition, it is an advantage that the proposed formulas for link analysis are mathematically founded and that the underlying model (the random walk model) is quite relevant to browsing, i.e. to the most widely used method for understanding a conceptual graph. At last, another evidence that link analysis is indeed appropriate for ER diagrams is that all large schemas that we have tested have a small set of elements, usually less than 5% of the total ones, whose scores are significantly higher than the rest. This at least indicates that big ER diagrams tend to have a well connected kernel which, at least in our experiments, always comprised the more important concepts of the application domain. 3 Automatic ER Drawing For drawing automatically the top-k ER diagrams that are derived by the previous technique, we shall view them as mechanical systems. Below we present two force models that combine the spring-model (proposed and developed in [13, 24, 16]) with the magnetic-spring model (proposed in [32, 31]) in a way that is appropriate for ER diagrams. 3.1 Force Model A Here entity types are viewed as equally charged particles which repel each other. Relationship types and isa relationships are viewed as springs that pull their adjacent entity types. Moreover, we assume that the springs that correspond to isa links are all magnetized and that there is a global magnetic field that acts on these springs. Specifically, this magnetic field is parallel (i.e. all magnetic forces operate in the same direction) and the isa springs are magnetized unidirectionally, so they tend to align with the direction of the magnetic field, here upwards. Figure 4 illustrates this metaphor. 6

7 Person worksat Company Manager Secretary magnet magnetic field electrical repulsion Fig. 4. Viewing an ER diagram as a mechanical system Under the above force model, the force on a entity type e i is given by: F (e i ) = f(e j, e i ) + g(e j, e i ) + h(e j, e i ) (3) e j E,e j e i e j conn(e i ) e j conn I (e i ) where: f(e j, e i ) is the force exerted on e i by the spring between e j and e i (note that e i and e j are connected by a relationship type or an isa link), g(e j, e i ) is the electrical repulsion exerted on e i by the entity type e j, and h(e j, e i ) is the rotational force exerted on e i by the entity type e j (here e i and e j are connected by an isa link). Figure 5 gives some indicative examples that explain the role of the forces f, g and h. Specifically, figure (a) justifies the spring force, figure (b) justifies the electrical repulsion and shows that high electrical repulsion (high K e ) results in symmetrical drawings, and figure (c) illustrates how the magnetic field can be used in order to obtain the classical top-down drawings for isa hierarchies. no force with f (a) with f with f and g (low Ke) with f and g (high Ke) (b) no force with f and g (high Ke) with f and g (high Ke) and h (c) Fig. 5. Forces and ER Drawings The spring force f(e j, e i ) follows Hooke s law, i.e. it is proportional to the difference between the distance between e j and e i and the zero-energy length of 7

8 the spring. Let d(p, p ) denote the Euclidean distance between two points p and p and let p i = (x i, y i ) denote the position of an entity type e i. The x component of the force f(e i ) is given by: f x (e i ) = e j conn(e i ) K s i,j(d(p i, p j ) L i,j ) x j x i d(p i, p j ) where L i,j denotes the natural (zero energy) length of the spring between e i and e j. This means that if d(p i, p j ) = L i,j then no force is exerted by the spring between e i and e j. Now K s i,j denotes the stiffness of the spring between e i and e j. The larger the value of K s i,j, the more tendency for the distance d(p i, p j ) to be close to L i,j. The y component of the force f(e i ) is defined analogously. The electrical force g(e j, e i ) follows an inverse square law. The x component of the force g(e i ) is given by: g x (e i ) = e j E,e j e i Ke i,j d(p i, p j ) 2 x i x j d(p i, p j ) where Ki,j e is used to control the repulsion strength between e i and e j. The y component of the force g(e i ) is defined analogously. The magnetic force h(e j, e i ) depends on the angle between the isa spring (that connects e j and e i ) and the direction of the magnetic field and it induces a rotational force on that spring. For example, Figure 6 shows an isa link between e i and e j and the exerted forces on e i and e j due to the magnetic field. The x and y components of the magnetic force h(e i ) are given by: h x (e i ) = h y (e i ) = e j conn sp(e i) e j conn sp (e i ) K m x j x i L i,j + K m L i,j + y j y i L i,j e j conn sb (e i) K m x j x i L i,j e j conn sb (e i ) where K m is used to control the strength of the magnetic field. K m L i,j + y i y j L i,j l y j y i e i h x(e j) l h x(e i) h y(e j) e j h y(e i) x j xi Magnetic Field Fig. 6. Magnetic forces and isa links The x and y components of the composed force F (e i ) on an entity type e i are obtained by summing up, i.e.: F x (e i ) = f x (e i ) + g x (e i ) + h x (e i ) and F y (e i ) = f y (e i ) + g y (e i ) + h y (e i ). 8

9 As in the link analysis technique, we view an n-ary relationship type as n(n 1)/2 springs. 3.2 Force Model B One weakness of the above model is that the resulting drawings can have several overlaps. The reason is that: (a) there is no repulsion among relationship types, and (b) there is no repulsion between entity and relationship types. Figure 7 illustrates this problem. This drove us to introduce a different force model where each relationship type is viewed as a particle too. Clearly, the resulting electrical repulsion discourages the creation of overlaps (between entity and relationship types, or between relationship types themselves). Notice that according to this view, a relationship type does no longer correspond to one spring. Specifically, the particle of a relationship type over k entity types, is connected with one spring with each one of them. The forces on entity types and relationship types are computed analogously to the force model A. Without rel rel repulsion With rel rel repulsion (a) Without ent rel. repulsion With ent rel repulsion (b) Fig. 7. Forces and ER Drawings 3.3 The Drawing Algorithm We can reach a drawing by an algorithm that simulates the mechanical system. Such a algorithm would seek for a configuration with locally minimal energy, i.e. a drawing in which the forces on each node is zero. A variety of numerical techniques can be used to find an equilibrium configuration, and thus the final drawing. We have adopted the iterative method based on the method proposed in [13]. At first the nodes are placed at random positions. At each iteration, the force on each node is computed and then the node is moved towards the corresponding direction by a small amount proportional to the magnitude of the force. This can be continued until convergence, but we can also limit the number of iterations. Note that if we would like to find a drawing that corresponds to a state with globally minimal energy, then we would have to resort to very general optimization methods. For instance, a method based on simulated annealing is proposed in [11], while an approach based on genetic algorithms is described in [4]. However the computational complexity of these techniques turns them 9

10 not very appropriate for interactive design systems. In addition, and according to the results of the extensive empirical analysis of several force-directed algorithms (including globally minimal energy algorithms) upon plain graphs that are reported in [3], there is no universal winner and the general approach is to try several methods and choose the best. 3.4 Experimental Evaluation We have investigated and evaluated all these issues in the context of the CASE tool DB-MAIN. The specification of the parameters L, K s, K e and K m is not a trivial task as these parameters determine in a high degree how the final drawing will look like. One flexibility of the proposed approach is that we can adjust the spring length (L i,j ), spring stiffness(ki,j s ) and electrical repulsion(ke i,j ), in order to customize the appearance of the drawing according to the semantics of the ER diagram constructs. For instance, as it is desirable to keep the nodes of an isa hierarchy close enough and since between any two isa-related entity types we only have to draw a line (and not any hexagon-enclosed string), we can use a smaller length for isa-springs than that of relationship-springs. In any case, the user can change their value at run-time. Figure 8 shows one drawing obtained by the algorithm using low electrical repulsion. Although the isa hierarchy is drawn as a top-down drawing and we have no overlaps, this drawing is not satisfying because a designer would hardly manually place into the space occupied by an isa hierarchy an entity type that does not belong to that hierarchy. After we increased the repulsion and the magnetic field we never faced again such a drawing. The lesson learned is that high repulsion not only results in symmetrical drawings but its combination with a strong magnetic field results in clear isa drawings. Another drawing of a diagram with 4 isa hierarchies that is derived by the algorithm according to force model A, is shown in Figure 9. low Ke high Ke, high Km Fig. 8. How to obtain clean isa drawings A more complex case is shown in Figure 10. Figure 10.(a) shows a manually placed diagram where all subentity types have been placed at the outer part of the drawing. Figure (b) shows the drawing obtained according to force model B. Notice that every isa hierarchy now corresponds to a top-down drawing and that the entire drawing is symmetrical and satisfying. 10

11 ENTITY_1 R ENTITY ENTITY_3 R_1 ENTITY_4 ENTITY_5 R_2 ENTITY_2 ENTITY_10 ENTITY_11 ENTITY_6 ENTITY_7 ENTITY_8 R_3 ENTITY_9 Fig. 9. A drawing of a diagram with 4 isa hierarchies according to force model A ENTITY_5 R_2 ENTITY R_6 ENTITY_9 R_10 ENTITY_1 R_3 R_2 ENTITY R_6 R ENTITY_4 R_5 ENTITY_6 ENTITY_1 R_10 ENTITY_9 R_3 ENTITY_5 ENTITY_4 R ENTITY_6 R_5 R_9 ENTITY_2 R_4 ENTITY_3 R_7 R_9 ENTITY_2 R_4 ENTITY_3 R_7 ENTITY_8 R_8 ENTITY_7 ENTITY_8 R_8 ENTITY_7 (a) (b) Fig. 10. Drawing of a diagram with several IsA hierarchies (a): manual drawing where isa links are not vertical. (b): drawing obtained according to force model B The experimental evaluation showed that the drawings according to force model A suffer from overlaps, while those according to force model B have a few (or none) overlaps. The difference between force model A and force model B is even more evident in dense diagrams. Figure 11 shows the drawings obtained by these two models when applied on the top-5 ( E = 5, R = 17) diagram of Figure 3. Again, the second drawing is evidently better. A noteworthy remark here is that the second diagram is more clear and intuitive than the manually specified layout that is shown in Figure 3. This indicates that in certain cases (at least when the diagram is very dense) the automatically-derived drawings can be better than the manually drawn. However, we have to note that force model B has two weaknesses comparing to force model A: (i) it is computational more expensive, and (ii) in the resulting drawings the tentacles of binary relationship types are in many cases unnecessarily not aligned. This is evident in Figure 12. Although this is not a major problem it is an issue for further research. Figure 13 shows the automatic layout obtained for the top-11 diagram (that was presented in Figure 2). The high relative number of relationships makes the drawing almost unreadable. This example suggests that we should take into account the density of a diagram, in order to reach readable and clear drawings. Roughly, we could handle dense diagrams by considering: (i) larger springs, (ii) higher electrical repulsion, (iii) less stiff springs. For example, and assuming force model A, Figure 14.(a) shows the drawing obtained with spring length 11

12 ENTITY_7 R_10 R_9 ENTITY_6 R_6 ENTITY_8 R_1 R_12 R_13 ENTITY_3 R_5 R_2 ENTITY_4 R_4 R ENTITY_5 R_15 R_14 R_8 ENTITY R_3 ENTITY_2 R_11 R_7 ENTITY_1 R_10 ENTITY_7 R_9 ENTITY_6 R_1 R_6 ENTITY_8 R_12 R_5 R_13 ENTITY_3 R_2 ENTITY_4 R_4 ENTITY_5 R_15 R R_14 ENTITY R_3 R_8 ENTITY_2 R_11 ENTITY_1 R_7 ENTITY_14 ENTITY_15 R_27 R_28 R_29 ENTITY_7 R_10 R_9 R_26 ENTITY_16 R_6 R_30 ENTITY_8 R_25 ENTITY_6 ENTITY_13 R_31 R_1 R_12 R_13 R_5 R_24 ENTITY_3 R_2 ENTITY_4 R_4 ENTITY_5 R_16 R R_15 R_3 R_14 R_8 R_23 ENTITY_9 R_17 ENTITY ENTITY_2 R_22 ENTITY_12 R_18 R_11 R_7 ENTITY_1 R_19 R_20 R_21 ENTITY_10 ENTITY_11 ENTITY_15 ENTITY_14 R_28 R_27 R_29 ENTITY_7 R_10 R_9 R_26 R_30 R_6 R_25 ENTITY_6 ENTITY_8 ENTITY_16 ENTITY_13 R_1 R_5 R_12 R_13 R_31 R_24 ENTITY_3 R_2 ENTITY_4 R_4 ENTITY_5 R_16 R_23 R_15 R_3 R_14 R R_8 ENTITY_9 ENTITY_2 ENTITY_12 ENTITY R_17 R_22 R_11 R_7 R_18 ENTITY_1 R_21 R_19 R_20 ENTITY_10 ENTITY_11 Channel replacement replacedby describes Bonus EARLY BIRD cn/pay Item decomposition replacement Notice description replaces partof Max Weight/Channel Operator channelcalculation follow-up path activity destination accumulated movements supplies POSITIONS/ACTIVITIES/MONTHS Stock Movement Country Customs costs origin country DefaultCost language State of Progress replacedby Item replaces partof description Bonus describes decomposition POSITIONS/ACTIVITIES/MONTHS Stock Movement Notice cn/pay EARLY BIRD country language path Calculation channelcalculation Country Max Weight/Channel SectorOfActivity destination origin Channel supplies Customs costs follow-up activity Operator State of Progress accumulated movements SectorOfActivity DefaultCost (a) (b) Fig. 11. Force model A vs force model B on a diagram with E = 5 and R = 17 (a) (b) (c) Fig. 12. Force model A vs force model B (a): force model A. (b): force model B. (c): force model A. (d): force model B. (d) FOURNISSEUR remise fin année fournisseur date de fin de validité fournisseur/langue achfrn/lan REFERENCE ARTICLE date début de validité COLIS STAT FOURNISSEUR artvpc/frn LANGUE MENSUELLES Notice DATE Date de depot fournisseur/pays du de la devise LIVRAISON cn/suicol COLIS artref/artvpc SUIVI réexpédition colcom/suicol LIVRAISON col/eta colis COLIS SUIVRE LE DOSSIER simulation Pays POS-DTE LOT Document par langue d'origine prix de vente description décrire CANAL artref/douref artref/tva FORMULE SUIVI EARLY PAYS DES Composition PAR TOPS PAYS/LANGUE CLIENT remplace HISTORIQUE COLIS Cadeau BIRD article vpc ensemble dte/pay associé cd/suicol artref/atvsec SUIVI COMMANDES pondération PAYS ENSEIGNE cn/pay LANGUE décomposition Remplacement d'un ensemble ARTICLE VPC ETAT D'AVANCEMENT (LIGNE OU pays d'origine Canal article est vpc remplace pièce commande livraison CO ALIMENTER SUIVI Poids OPERATRICE SUIVI max/canal ACTIVITE cd/eta calcul goulotte canal POSITIONS/ACTIVITE/MOIS Mouvement PAYS de stock FRAIS DE DOUANE cumul mouvement systeme mesure FRAIS pays destination COMMANDE PAR DEFAUT atvsec/cd SECTEUR ACTIVITE Fig. 13. Dense diagram drawing 12

13 COLIS HISTORIQUE cd/suicol COLIS SUIVI réexpédition colcom/suicol LIVRAISON col/eta colis COLIS COMMANDE LIVRAISON cn/suicol COLIS cd/eta ETAT D'AVANCEMENT (LIGNE OU CO SUIVI COMMANDES Canal commande livraison SUIVRE LE DOSSIER atvsec/cd REFERENCE ARTICLE CANAL cumul mouvement systeme mesure DATE dte/pay simulation Pays artref/douref artref/tva d'origine prix de cn/pay vente ALIMENTER pondération artref/atvsec SUIVI Poids OPERATRICE max/canal SUIVI calcul ACTIVITE goulotte canal Date de depot artref/artvpc POS-DTE FRAIS PAR DEFAUT SECTEUR ACTIVITE FOURNISSEUR date de fin de validité fournisseur/pays du de la devise LOT STAT FOURNISSEUR artvpc/frn MENSUELLES date début de validité POSITIONS/ACTIVITE/MOIS Mouvement de stock EARLY BIRD PAYS PAYS ENSEIGNE LANGUE remise fin année fournisseur fournisseur/langue achfrn/lan ARTICLE VPC Cadeau associé pays destination FORMULE SUIVI PAYS DES Composition PAR TOPS PAYS/LANGUE CLIENT pays d'origine Document par langue LANGUE article vpc remplace ensemble article est vpc remplace pièce FRAIS DE DOUANE description décrire décomposition Remplacement d'un ensemble Notice COLIS REFERENCE ARTICLE ETAT D'AVANCEMENT (LIGNE OU CO artref/atvsec simulation Pays artref/douref artref/tva d'origine prix de vente SUIVRE LE DOSSIER DATE cumul mouvement systeme mesure ALIMENTER date de fin de validité pondération artref/artvpc SUIVI réexpédition LIVRAISON colis COLIS SUIVI COMMANDES Date dte/pay colcom/suicol col/eta de depot date début de validité cd/eta SECTEUR ACTIVITE pays d'origine FRAIS PAR DEFAUT PAYS FRAIS fournisseur/pays du DE de DOUANE la devise pays destination FOURNISSEUR remise fin année fournisseur LOT POS-DTE atvsec/cd PAYS ENSEIGNE LANGUE POSITIONS/ACTIVITE/MOIS Mouvement de stock SUIVI calcul ACTIVITE goulotte canal FORMULE SUIVI PAYS DES Composition PAR TOPS PAYS/LANGUE CLIENT HISTORIQUE COLIS STAT FOURNISSEUR fournisseur/langue achfrn/lan cd/suicol artvpc/frnmensuelles COMMANDE SUIVI Poids OPERATRICE max/canal cn/pay EARLY BIRD Cadeau article associé vpc remplace ensemble Notice LIVRAISON COLIS LANGUE cn/suicol Document par langue ARTICLE VPC décomposition Remplacement d'un description ensemble décrire Canal commande livraison article est vpc remplace pièce CANAL FOURNISSEUR remise fin année fournisseur date de fin de validité fournisseur/langue achfrn/lan REFERENCE ARTICLE date début de validité COLIS LANGUE Notice DATE SUIVI réexpédition colcom/suicol LIVRAISON col/eta colis COLIS Date de depot SUIVRE LE DOSSIER STAT FOURNISSEUR artvpc/frnmensuelles LIVRAISON cn/suicol COLIS artref/artvpc LOT Document par langue ETAT D'AVANCEMENT (LIGNE OU CO POS-DTE description décrire fournisseur/pays du de la devise simulation Pays artref/douref artref/tva d'origine prix de vente FORMULE SUIVI PAYS DES Composition PAR TOPS PAYS/LANGUE CLIENT SUIVI COMMANDES dte/pay EARLY BIRD HISTORIQUE cd/suicol COLIS CANAL Cadeau associé article vpc remplace ensemble ARTICLE VPC décomposition Remplacement d'un ensemble article est vpc remplace pièce cd/eta ALIMENTER cn/pay Canal commande livraison artref/atvsec pondération PAYS ENSEIGNE LANGUE pays d'origine PAYS FRAIS DE DOUANE pays destination cumul mouvement systeme mesure SUIVI calcul ACTIVITE goulotte canal COMMANDE SUIVI Poids OPERATRICE max/canal POSITIONS/ACTIVITE/MOIS Mouvement de stock FRAIS PAR DEFAUT atvsec/cd SECTEUR ACTIVITE L = 5L, Figure 14.(b) shows the drawing obtained with K e = 100K e, and Figure 14.(c) shows the drawing obtained with K s = K s / Indeed, all are better than the original drawing shown in Figure 13. Another simple method that is both effective and efficient is to scale up the entire drawing (i.e. multiply each coordinate by a constant c > 1). Nevertheless, an issue that is worth further research is to investigate the effectiveness of local-density adaptations, e.g. to adapt the spring lengths according to the local density of the graph. EntityRank and BEntityRank scores could be exploited for this purpose. (a) (b) (c) Fig. 14. (a): larger springs; (b): higher repulsion; (c) less stiff springs As a final remark note that the above drawing techniques can be applied for drawing the structural part of ontologies expressed in RDFS [5] and OWL [12]. The only difference is that RDFS supports property specialization which however will be handled correctly due to the magnetic field that is applied on specialization/generalization links (also indicated by Figure 9). 4 Conclusion We described a novel method for identifying the major elements of an ER diagram that is based on link analysis. This method can significantly aid (a) the understanding, (b) the visualization, and (c) the drawing of very large schemas. The proposed technique can elevate automatically the major elements and allows exploring the schema gradually: from the more important elements to the less. Consequently, it can be very useful in reverse engineering and in information integration. Moreover, the scores can be exploited for ordering the schema elements that match a keyword query of the user. In addition, and given the inability to produce automatically aesthetically satisfying layouts for large schemas, the small (top-k graphs) that can be derived by this technique can be visualized effectively and this is very useful during communication (e.g. between designers and application programmers or in requirements engineering and training). For this purpose we investigated a force-directed drawing algorithm and evaluated two different force models upon several conceptual schemas of real applications. For small and medium sized diagrams the results were satisfying in most of the cases. In the rest cases, human intervention (moving, nailing) and rerun of the drawing algorithm could rectify the problems. 13

14 Acknowledgements The first author wants to thank Tonia Dellaporta for the several fruitful and really enjoyable discussions on this issue. Also many thanks to Jean-Rock Maurisse, Anne- France Brogneaux and Jean Herald for their help on using the DB-MAIN toolkit. References 1. Jacky Akoka and Isabelle Comyn-Wattiau. Entity-Relationship and Object- Oriented Model Automatic Clustering. Data and Knowledge Engineering, 20(2):87 117, Giuseppe Di Battista, Peter Eades, Roberto Tamassia, and Ioannis Tollis. Graph drawing: algorithms for the visualization of graphs. Prentice Hall Englewood Cliffs (N.J.), ISBN/ISSN : F. J. Braedenburg, M. Himsolt, and C. Rohrer. An Experimental Comparison of Force-Direceted and Randomized Graph Drawing Algorithms. In Procs of Graph Drawing, GD 95, pages 76 87, J. Branke, F. Bucher, and H. Schmeck. Using Genetic Algorithms for Drawing Undirected Graphs. In Procs of the 3rd Nordic Workshop on Genetic Algorithms and Their Applications, 3NWGA, pages , Dan Brickley and R. V. Guha. Resource Description Framework (RDF) Schema specification: Proposed Recommendation, W3C, March Sergey Brin and Lawrence Page. The Anatomy of a Large-scale Hypertextual Web Search Engine. In Proceedings of the 7th International WWW Conference, Brisbane, Australia, April L. J. Campbell, Terry A. Halpin, and Henderik Alex Proper. Conceptual Schemas with Abstractions: Making Flat Conceptual Schemas More Comprehensible. Data and Knowledge Engineering, 20(1):39 85, Rodolfo Castello, Rym Mili, and I. Tollis. A Framework for the Static and Interactive Visualization of Statecharts. Journal of Graph Algorithms and Applications, 6(3): , P. Chen. The Entity-Relationship Model - Toward a Unified View of Data. ACM Transactions on Database Systems, 1(1):9 36, March Richard Cole. Automatic Layout of Concept Lattices using Force Directed Placement and Genetic Algorithms. In Proc. of the 23th Australiasian Computer Science Conference, pages Australian Computer Science Communications 1, IEEE Computer Society, R. Davidson and D. Harel. Drawing Graphics Nicely Using Simulated Annealing. ACM Trans. Graph., 15, M. Dean, D. Connolly, F. van Harmelen, J. Hendler, I. Horrocks, D. L. McGuinness, P. F. Patel-Schneider, and L.A. Stein. OWL Web Ontology Language 1.0 Reference, (http://www.w3c.org/tr/owl-ref). 13. P. Eades. A Heuristic for Graph Drawing. Congressus Numerantium, 42, Holger Eichelberger and Jurgen Wolff von Gudenberg. UML Class Diagrams - State of the Art in Layout Techniques. In Proceeding of Vissoft 2003, International Workshop on Visualizing Software for Understanding and Analysis, pages 30 34, P. Feldman and D. Miller. Entity Model Clustering: Structuring a Data Model by Abstraction. The Computer Journal, 29(4): ,

15 16. T. Fruchterman and E. Reingold. Graph Drawing by Force-directed Placement. Software - Practice and Experience, 21(11): , F.U.N.D.P. DB-MAIN. (http://www.info.fundp.ac.be/ dbm/). 18. Munish Gandhi, EdwardL Robertson, and Dirk Van Gucht. Levelled Entity Relationship Model. In Procs of the 13rd Intern. Conf. on the Entity Relationship Approach, ER 94, pages , Manchester, U.K., December Floris Geerts, Heikki Mannila, and Evimaria Terzi. Relational Link-based ranking. In Procs of the 30th Intern. Conference on Verly Large Data Bases, VLDB 2004, Toronto, Canada, August Zoltan Gyongyi, Hector Garcia-Molina, and Jan Pedersen. Combating Web Spam with TrustRank. In Procs of the 30th Intern. Conference on Verly Large Data Bases, VLDB 2004, Toronto, Canada, August Jean-Luc Hainaut. Transformation-based Database Engineering. In Transformation of Knowledge, Information and Data: Theory and Applications. IDEA Group Pub., Jouni Huotari, Kalle Lyytinen, and Marketta Niemela. Improving Graphical Information System Model Use with Elision and Connecting Lines. ACM Transactions on Computer-Human Interaction, 10(4), Yannis E. Ioannidis, Miron Livny, Jian Bao, and Eben M. Haber. User-Oriented Visual Layout at Multiple Granularities. In Proc. of the 3rd International Workshop on Advanced Visual Interfaces, pages , Gubbio, Italy, May T. Kamada. On Visualization of Abstract Objects and Relations. PhD thesis, Dept. of Information Science, Univ. of Tokyo, Dec Jon Kleinberg. Authoritative Sources in a Hyperlinked Environment. In Proceedings of 9th ACM-SIAM Symposium on Discrete Algorithms, San Francisco, USA, Simon Lok and Steven Feiner. A Survey of Automated Layout Techniques for Information Presentations. In Procs of the 1st. Int. Symp. on Smart Graphics, Hawthorne, NY, Rajeev Motwani and Prabhakar Raghavan. Randomized algorithms. Cambridge University Press, Arthur Ouwerkerk and Heiner Stuckenschmidt. Visualizing RDF Data for P2P Information Sharing. In Procs of the workshop on Visualizing Information in Knowledge Engineering, VIKE 03, Sanibel Island, FL, Aaron J. Quigley. Large Scale Relational Information Visualization, Clustering, and Abstraction. PhD thesis, University of Newcastle, Australia, August (http://www.it.usyd.edu.au/ aquigley/thesis/aquigley-thesis-mar-02.pdf). 30. O. Rauh and E. Stickel. Entity Tree Clustering: A Method for Simplifying ER Design. In Procs of the 11th Int. Conf. on Entity-Relationship Approach, ER K. Sugiyama and K. Misue. A Simple and Unified Method for Drawing Graphs: Magnetic-Spring Algorithm. In Procs of Graph Drawing Conference, GD 94, pages , K. Sugiyama and K. Misue. Graph Drawing by Magnetic-Spring Model. Journal on Visual Lang. Comput., 6(3), R. Tamassia, C. Batini, and M. Talamo. An algorithm for automatic layout of entity-relationship diagrams. In Procs of the 3rd International Conference on Entity-relationship approach to software engineering, pages Elsevier North-Holland, Inc., T. J. Teory, W. Guangping, D. L. Bolton, and J. A. Koenig. ER Model Clustering as an Aid for User Communication and Documentation in Database Design. Communications of the ACM, 32(8): ,

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