Fuzzy Meta Node Fuzzy Metagraph and its Cluster Analysis

Transcription

1 Journal of Computer Siene 4 (): 9-97, 008 ISSN Siene Publiations Fuzzy Meta Node Fuzzy Metagraph and its Cluster Analysis Deepti Gaur, Aditya Shastri and Ranjit Biswas Department of Computer Engineering, Institute of Tehnology and Management, Setor-3/A, Gurgaon-07, India Department of Computer Siene and Eletronis, Rajasthan-3040, India Abstrat: Problem statement: In this study researhers propose a new fuzzy graph theoreti onstrut alled fuzzy metagraph and a new method of lustering finding the similar fuzzy nodes in a fuzzy metagraph. Approah: We adopted T-norms (Triangular Norms) funtions and join two or more T- norms to luster the fuzzy nodes. Fuzzy metagraph is the fuzzyfiation of the risp Metagraphs using fuzzy Generating sets and the fuzzy edge set. We ould effiiently analyze the inexat information and investigate the fuzzy relation by applying the fuzzy graph theory. Results: In this study researhers suggesting a new method of lustering of a new graph theoreti struture i.e., fuzzy metagraph and investigated fuzzy metanode and fuzzy metagraph struture. Conlusion/Reommendations: Our future researh will be to explore all its useful operations on fuzzy metagraph. We will give the more appliation based implementation of fuzzy metagraph. Key words: Fuzzy metagraph, fuzzy metanode, luster, fuzzy generating set INTRODUCTION An important onept in design of many information proessing systems suh as transation proessing systems, deision support systems, projet management systems and worflow systems is that a graph struture. In this simplest form a graph onsists of set a of elements (or nodes) and a set of ordered and unordered pairs of nodes (or edges). We would extend the fuzzy graph theory and propose a fuzzy node fuzzy graph. Sine a fuzzy node fuzzy graph is ompliated to analyze, we would transform it to a simple fuzzy graph by using T-norm family A Graph is defined by a pair G = {X, E} where X = {x 3 x n } is a finite set of verties and E a olletion of edges that happen to onnet these verties. The edge set E graphially represented as X X. The onept of fuzzy graph is the fuzzyfiation of the risp graphs using fuzzy sets. Definition : Fuzzy node fuzzy graph and T-norm: Crisp node fuzzy graph a risp node fuzzy graph G is defined by: G = (V, F): V = {v i }, F = (f ), 0 f Where: V = The set of the nodes F = An n n matrix whose (i, j) omponent f is a fuzziness of the ar from the node v i to the node v j Definition : Fuzzy node fuzzy graph a fuzzy node fuzzy graph G is defined by: G = (V, F): V = {v i /u i }, Y = (y ), 0 u i,0 y Where: V = The set of the nodes and the fuzziness ui = A fuzziness of the node v i Y = A n n matrix whose (i, j) omponent y = A fuzziness of the ar from the node v i to the node v j [] Over the past years, a number of fuzzy graphs have been proposed to represent unertain relationship between fuzzy elements and sets of fuzzy elements however, as mentioned before; existing fuzzy graphs are not apable of effetively modeling and direted relationships between sets of fuzzy elements. A fuzzy node fuzzy graph is haraterized by the fuzziness of the nodes and the fuzziness of the ars. Therefore, the struture of a fuzzy node fuzzy graph is usually very ompliated. Then, it should be interesting to transform a fuzzy node fuzzy graph to a risp node fuzzy graph and we present a method to transform a fuzzy node fuzzy graph to a risp node fuzzy graph. MATERIALS AND METHODS Fuzzy node fuzzy graph would be applied to the T- norm family. As Shown in Fig. every node has some Corresponding Author: Deepti Gaur, Department of Computer Engineering, Institute of Tehnology and Management, Setor-3/A, Gurgaon-07, India 9

2 J. Computer Si., 4 (): 9-97, 008 fuzzyness fator assoiated with it as well as every edge has some fuzzyness fator [0, ] and this fuzzyness fator satisfies the properties of T-norm binary operations. By applying the f = T (u i, y ) we get the risp node fuzzy fuzzy edge graph. Definition 3: Transformation from fuzzy node fuzzy graph to risp node fuzzy graph: A fuzzy node fuzzy graph G = (V, Y) an be transformed to a risp node fuzzy graph G = (V, F) by the following method. Let: G = (V, F): V = {v i }, F = (f ), f = T (u i, y ) where, the fuzziness f of the ar from the node v i to the node v j ould be defined by applying T-norms. T -norm and its family: T-norm (Triangular Norm) is a binary operation that is defined by the following: Definition 5: T-norm is a binary operation: p, q [0,] T(p, q) [0,] Satisfying the following properties: Commutative: T (p, q) = T(q, p) Assoiative: T(p, T(q, r )) = T(T(p,q), r) Monotoniity: p q, r st (p, r) T (q, s) Boundary onditions: T(p, 0) = 0, T(p,) = p The typial T -norm are following: Logial produt: T L (p, q) = p q Algebrai produt: T A (p, q) = pq Multi-valued produt (Luaszewiz produt): T M (p, q) = (p+q-) 0 Drasti produt: T D (p, q) = 0 if p q< or T D (p, q) = p q if p q = Here, a b means min (a, b), a b means max (a, b). Definition 6: Order of t-norms: For two T -norms T a and T b if a relation. T a (p, q) T b (p, q), (p, q) [0,] holds, we denote it by T a T b. For any T-norm T, a relation T D T T L always holds. Definition 7: T -norm family: For any [a, b], when T is T-norm, then we say that {T } is T -norm family that onnets T a with T b. Typial T-norm families are: Dubois produt: T (p, q) = pq/ (p q ), [0, ] Weber produt: T (p, q) = 0 {(+) (p+q-)- pq}, - λ Shweizer produt: T (p, q) = λ λ (0 p + q ), >0 Quasi-logial produt: T (p, q) = 0 if p q < - or T (p, q) = p q if p q - Here [0, ] and so on We ould understand that by viewing the above rules that Dubois produt onnets logial produt with algebrai produt, Weber produt does algebrai produt with mulh-valued produt, Shweizer produt does multi-valued produt with drasti produt and quasi-logial produt does logial produt with drasti produt [-4]. Fuzzy graph and luster analysis: In order to analyze the similarity struture of nodes for a fuzzy graph, we use the symmetri relation matrix: S = (s) A symmetri relation matrix S ould be defined by using the arithmeti mean, the geometri mean, the harmoni mean and so on. Here, we define the symmetri relation matrix S by using the harmoni mean [4]. A symmetri relation matrix S is defined by: S = (s ), = + s f f ji where, s = 0 if f f = 0. In order to analyze the lustering struture among nodes, we have its max-min transitive losure: Ŝ = ( s ^ ) whih is omputed by Ŝ = Sn After that we define the -ut iruit matrix S of Fig. : G fuzzy node fuzzy edge graph G 93 Ŝ = ( s ^ ) as follows:

3 J. Computer Si., 4 (): 9-97, 008 Fig. : Fuzzy matrix F of fuzzy graph Fig. 3: Symmetri matrix S S = (s ) s = { if (sˆ ) 0 if (sˆ < ) where, (0 ). From the matrix S we define the luster CLs (i): J (i) = { j s =, j n } CLs(i) = {v j j J (i)} The luster CLs(i) gives an equivalent relationship among nodes. Then we an onstrut the partition tree by hanging the level of -ut matrix whih represents the lustering situation of nodes in a fuzzy graph. Suppose a Fuzzy Matrix of a fuzzy graph is shown in Fig. then we obtain the symmetri matrix S in Fig. 3 and then transitive losure in Fig. 4 Conlusion is that The transitive losure matrix shows that luster and 4 have been merged at the value = 0.83 we say that and 4 are merged at luster level R Conerning the luster analysis of the fuzzy graph by viewing the above matrix we an analyzed that node and 4 are equivalent so we luster them as a single node. Similarly the node, 7, 8 are equivalent so we luster them and form a single node. Following are the limitations of fuzzy graph with onventional graphs: A fuzzy undireted graph an represent relationships existing between two Variables; it annot provide the diretion of relationships 94 Fig. 4: Transitive Closure Ŝ = ( ŝ ) The input output relationships between the element pairs an be desribed by the fuzzy direted graph. However, it annot represent the relationships where there is more than one variable in the input and/or in the output Fuzzy hyper graph desribes any fuzzy relationship as a set of fuzzy elements, but it is not able to distinguish the input variables from the outputs Using ars to ombine edges, a fuzzy AND/OR graph attempts to represents relationships even where there are more than one input and output variables. When desribing the relationships between the sets of variables however, their are too many ombined edges for fuzzy AND/OR graph to distinguish them [5-7]

4 J. Computer Si., 4 (): 9-97, 008 Fig. 5: Metagraph It is ritial that all the graphs la powerful algebrai analysis methods for manipulating the direted relationships between sets of elements. This motivates the development of FM i.e., fuzzy metagraphs. The major distintion between fuzzy metagraph and traditional graph-theoreti onstruts is that an FM desribes the direted relationships between sets of elements inside of single elements. Eah edge in the fuzzy metagraph is an ordered pair of sets of elements. So, the edge is neither an ordered pair of elements in a fuzzy direted graph nor a disordered set of elements in a fuzzy hyper graph [8-0]. Metagraphs: In 99 in the lassial paper [], Basu and Blanning introdued the onept of metagraph. In this study we reprodue the metagraph. We begin with few definitions: Definition: Generating set: The generating set of a metagraph is the set of elements X = {x,x...x n }, whih represent variables of interest. Definition: Edge: An edge e in a metagraph is a pair e = V e, W e E (where E is the set of edges) onsisting of an invertex V e X and an outvertex W e X, eah of whih is a set and may ontain any number of elements. The different elements in the invertex (outvertex) are o-inputs (o-outputs) of eah other. Definition: Metagraph: A metagraph S = X, E is a graphial representation onsisting of two tuples X and E. Here X is its generating set and E is the set of edges defined on generating sets. The generating set X of the metagraph S i.e. the set of elements X = {x, x 3,,x n } represents variables and ours in the edges of the metagraphs [9]. Example: A metagraph an be understood by the following example (Fig. 5): S = X, E is a metagraph when X = {x 3 4 5, x 6 7 } is the generating set and E = {e, e, e 3, e 4 } is the set of edges. 95 Fig. 6: Fuzzy metagraph The edge set an be speified as: E = {{x }, {x 4 }, {x 3 }, {x 5 }, {x 4 5 }, {x 6, x 7 }, {x 5 }, {x 7 }} In-vertex is an operation (funtion) having one argument whih an find out the internal verties from a given set. For example: In-vertex ({x 4 5 }, {x 6 7 }) = {x 4 5 } Out-vertex is another operation (funtion) having one argument whih an find out what are the out verties from the given set. For example: Out-vertex ({x 4 5 }, {x 6 7 }) = {x 6 7 } Two more operations of metagraph are the o-input and o-output operations eah having two arguments. For example: Co-input {x 4, {x 4 5 }, {x 6 7 }} = {x 5 } Co-output{x 6, {x 4 5 }, {x 6 7 }} = {x 7 } The edges of the metagraph are labeled as: e = {x }, {x 4 } e = {x 3 }, {x 5 } e 3 = {x 4 5 }, {x 6 7 } e 4 = {x 5 }, {x 7 } Fuzzy metagraphs: The fuzzy metagraph is the onept of fuzzyfiation of the risp Metagraph using fuzzy generating set. Fuzzy generating set is the node set of all the elements of fuzzy metagraph. Definition of fuzzy metanode fuzzy metagraph: Consider a finite set X = {x i, =,,...,I}. A fuzzy metagraphs is a triplet shown in Fig. 6. RESULTS AND DISCUSSION Cluster in fuzzy metagraph: The Desription of members of the generating sets of fuzzy metagraph of

5 J. Computer Si., 4 (): 9-97, 008 Fig. 7 is shown in Table. Fuzzy matrix as well as the Symmetri matrix of Fig. 7 is shown in Fig. 8 and 9. S = {X, X, E } where X is a fuzzy set on X is nown Fuzzy MetaNode and E is a fuzzy edge set { e, =,, 3,..., K} nown as fuzzy edge set. Eah Component of e in E is haraterized by an ordered pair V,W in pair V X is the invertex of e and the W X is the outvertex. The o-input of any x V isv {x} and the o-output of any x W is W {x}. Figure 6 fuzzy metagraph whose element set is X = { x,... 6 } is nown as fuzzy MetaNode and whose edge set onsists of: e = { x }, { x 3 } and e = { x 3 4 }, { x 5 6 } Fig. 7: A fuzzy Metagraph The in-vertex and out-vertex of e are { x } and { x 3 } respetively. An important property of graph is its onnetivity. In Fig. 6 there is a sequene of edges e, e that onnets x to x 5 whih means there is a path from x to x 5 exists. Fig. 8: Fuzzy matrix of fuzzy metagraph Fig. 9: Symmetri matrix S of fuzzy metagraph Table : Members of generating Sets their proposition Sr. No Member of generating sets Proposition x Altitude is high x 3 x 3 4 x 4 5 x 5 6 x 6 7 x 7 8 x 8 9 x 9 0 x 0 x Altitude is low ontrol system breadown Cabin breadown Net breadown System breadown Output large Relay is abnormal Relay is normal Voltage is high Voltage is low 96 Fig. 0: Transitive losure of fuzzy metagraph S ˆ = (s ˆ )

6 J. Computer Si., 4 (): 9-97, 008 Fuzzy metagraphs and luster analysis of fuzzy metagraph: The tabular representation of the Fuzzy generating set of the fuzzy Metagraph with there meaning is shown in Table. In order to analyze the similarity struture of nodes for a fuzzy Metagraph, we use the symmetri relation matrix S = (s). Before that we have to onstrut the fuzzy matrix of Fuzzy Metagraph. Fuzzy matrix of above fuzzy Metagraph will be shown in Fig. 7. The Symmetri matrix will be shown in Fig. 8. Transitive losure matrix of above fuzzy matrix in Fig. 7 will be shown in Fig. 0. By viewing this transitive losure matrix we an analyze that some nodes are equivalent we will mae a luster of those nodes. In spite of putting them separately. In the above Fig. 0 node,, 6 will be lustered as a single node. CONCLUSION Metagraph is a graphial model that not only visualized the proess of any system but also their formal analysis where the analysis will be aomplished by means of an algebrai representation of the graphial struture. The graphial struture an be represented by the adjaeny and inidene matrix of a metagraph. Metagraphs have lot of appliations in the field of information proessing systems; deision support systems, models Management of database and rule base management of wor flow systems in whih the wor onsists of information proessing tass to be performed by the human or the mahines metagraphs provide a useful and omprehensive funtion for modeling. In our example of metagraph the entities will orrespond to the elements of the generating sets and the tass will orrespond to the edges in the metagraph. This onstrut extends the features offered by the traditional graph strutures i.e., digraphs, hyper graphs. Metagraph allows different omponents of the proess to be represented both graphially and analytially. Our future wor will be to explore all its useful operations on fuzzy metagraph. We will give the more appliation based implementation of fuzzy metagraph. REFERENCES. Zadeh, L.A., 965. Fuzzy Sets*. Inform. Control, 8: Zadesh, L.A., 975. Fuzzy Sets and Their Appliation to Cognitive and Deision Proesses. st Edn., Aademi Press In., New Yor, USA., ISBN: 0: , pp: Nishhida, T. and E. Taeda, 978. Fuzzy Set and its Appliations. st Edn., Moriita Shuppan, (In Japanese). 4. Uesu, H., 004. Approximated analysis of fuzzy node fuzzy graph and its appliations. Proeedings of the IEEE International Conferene on Fuzzy Systems, July 5-9, IEEE Computer Soiety, Washington DC., USA., pp: DOI: 0.09/FUZZY Basu, A. and R.W. Blanning, 99. Metagraphs and petri nets in model management. Proeedings of the Seond Annual Worshop on Information Tehnologies and Systems, De. -5, Dallas, pp: Banerjee, S. and A. Basu, 993. Model type seletion in an integrated DSS environment. De. Support Syst., 9: DOI: 0.06/ (93)9004-W 7. Barua, A., S.C.H. Lee and A.B. Whinston, 005. The alulus of reengineering. Inform. Syst. Res., 7: DOI: 0.87/isre Basu, A. and R.W. Blanning, 99. Enterprise modeling using metagraphs. Proeedings of the IFIP TC8/WG8.3 Woring Conferene on Deision Support Systems: Experienes and Expetations, June 30-July 3, North-Holland Publishing Co., Amsterdam, The Netherlands, pp: Basu, A. and R.W. Blanning, 997. Synthesis and deomposition of proesses in organizations. Inform. Syst. Res., 4: DOI: 0.87/isre Basu, A., R.W. Blanning and A. Shtub, 997. Metagraphs in hierarhial modeling. Manage. Si., 43: Basu, A. and R.W. Blanning, 995. Meta-graphs. Omega, 3: Basu, A. and R.W. Blanning, 005. A formal approah to worflow analysis. Inform. Syst. Res., : DOI: 0.87/isre