Fuzzy Equivalence Relation Based Clustering and Its Use to Restructuring Websites Hyperlinks and Web Pages

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1 Fuzzy Equivaence Reation Based Custering and Its Use to Restructuring Websites Hyperinks and Web Pages Dimitris K. Kardaras,*, Xenia J. Mamakou, and Bi Karakostas 2 Business Informatics Laboratory, Dept. of Business Administration, Athens University of Economics and Business, 76 Patission Street, Athens 434, Greece {kardaras,xenia}@aueb.gr 2 Centre for HCI Design, Schoo of Informatics, City University, Northampton Sq., London ECV HB, UK bik@soi.city.ac.uk Abstract. Quaity design of websites impies that among other factors, hypeinks structure shoud aow the users to reach the information they seek with the minimum number of cicks. This paper utiises the fuzzy equivaence reation based custering in adapting website hyperinks structure so that the redesigned website aows users to meet as effectivey as possibe their informationa and navigationa requirements. The fuzzy toerance reation is cacuated based on the usage rate of hyperinks in a website. The equivaence reation identifies custers of hyperinks. The custers are then used to reaocate hyperinks in webpages and to rearrange webpages into the website structure hierarchy. Keywords: fuzzy equivaence reation, web adaptation, hyperinks custering. Introduction When designing a website, the way that its content is organised and how efficienty users get access to it, infuence the user perceived design quaity. The designers goa is an effective and pain communication of content [2]. Website structure has been identified by many reaserch studies as an important factor that affects web design quaity. The users perception of how different parts of a web site are inked together is a strong indicator of effective design [4], [3]. Thus, the websites hyperinks structure shoud adapt to meet users changing requirements and priorities depending e.g. on the expertise of users in navigating a website, their famiiarity with its content structure, their information needs, etc. Furthermore, hyperinks structure has been extensivey used in web search engines and web mining [2]. The number of inks pointing to a web page is considered as a quaity indicator that refects the authority of the web page the inks point at []. Many agorithms have been deveoped to tacke one of the greatest chaenges for web design and search engines, i.e. how to specify an appropriate website structure, or how to evauate webpages quaity [8]. This paper suggests the use of fuzzy equivaence reation custering to restucturing webpages * Corresponding author. H. Papadopouos et a. (Eds.): AIAI 23, IFIP AICT 42, pp. 52 6, 23. IFIP Internationa Federation for Information Processing 23

2 Fuzzy Equivaence Reation Based Custering and Its Use to Restructuring Websites 53 throught hypeinks popuarity. Unti recenty, the web users browsing behaviour was often overooked in approches that attempted to manage websites structures or determining the quaity of webpages [9]. This paper considers the popuarity of hypeinks as an indicator of users browsing behaviour and cassifies inks into pages and subsequenty pages are aocated to different website eves. An iustrative exampe is provide to expemify the proposed approach. 2 Hyperink Anaysis Athough hyperink anaysis agorithms that produced significant resuts have been deveoped, they sti need to tacke chaenges such as how to incorporate web user behaviour in hyperink anaysis [8]. A website is considered as a graph of nodes and edges representing webpages and hyperinks respectivey. Based on the hyperinks anaysis, the importance of each node can be estimated thus eading to a website structure that refects the reative importance of each hyperink and each web page [7]. Two of the most representative inks anaysis agorithms are the HITS [5] and the Googe s PageRank [], which assume that a user randomy seects a hyperink and then they cacuate the probabiities of seecting other hyperinks and webpages. Many other ink anaysis agorithms stem from these two agorithms. The basic idea behind ink anaysis agorithms is that if a ink points to page (i) from page (j), then it is assumed that there is a semantic reation between the two pages. However, the inks may not represent users browsing behaviour, which is driven by their interests and information needs. Hyperinks are not cicked by users with equa probabiities and they shoud not be treated as equay important [9]. Thus, webpages that are visited and hyperinks that are cicked by users shoud be regarded as more important than those that are not, even if they beong to the same web page. It is therefore reasonabe to use users' preferences to redesign the hyperink graph of a website [8]. Googe Toobar and Live Toobar coect user browsing information. User browsing preferences is an important source of feedback on page reevance and importance and is widey adopted in website usabiity [3], [2], user intent understanding [22] and Web search [9] research studies. By utiising the user browsing behaviour, website structures can be revised by deeting unvisited web pages and hyperinks and reocating web pages and hyperinks according to their importance as perceived by the users, i.e. as the number of cicks show. Liu et a. (28) deveoped a user browsing graph with Web og data []. A website representing graph as derived from user browsing information can ead to a website structure coser to users needs, because inks in the website graph are actuay chosen and cicked by users. Liu et a. (28) aso proposed an agorithm to estimate page quaity, BrowseRank, which is based on continuous-time Markov process mode, which according to their study performs better than PageRank and TrustRank.

3 54 D.K. Kardaras, X.J. Mamakou, and B. Karakostas 3 Fuzzy Reations and Fuzzy Cassification 3. Fuzzy Reations Fuzzy reations are important for they can describe the strength of interactions between variabes []. Fuzzy reations are fuzzy subsets of X Y, that is mapping from X Y. Let X, Y R be universa sets. Then ~ R = {(( x, y), μ R ( x, y)) ( x, y) X Y} () is caed a fuzzy reation on X Y [23]. 3.2 Fuzzy Cassification with Equivaence Reation Numerous cassification methods have been proposed so far incuding custer anaysis, factor anaysis, discriminant anaysis [6], k-means anaysis [4], c-means custering [2]. According to Ross (2) there are two popuar methods of cassification namey the cassification using equivaent reations and the fuzzy c-means. Custer anaysis, factor anaysis and discriminant anaysis are usuay appied in cassic statistica probem where arge sampe or ong term data is avaiabe. When deaing with sma data k-means or c-means methods are preferred [8]. In this paper, we use fuzzy equivaent reations and ambda-cuts (λ-cuts) to cassify inks, according to their importance in a website. Cassification based on λ-cuts of equivaent reations is used in many recent studies [6], [7] and [9]. An important feature of this approach is that for its appication it is not required to assume that the number of custers is known as it is required by other methods such as in the case of k-means and c-means custering [8]. A fuzzy reation on a singe universe X is aso a reation from X to X. It is a fuzzy toerance reation if the two foowing properties define it: Refexivity: μ R (x i, x i ) = Symmetry: μ R (x i, x j ) = μ R (x j, x i ) Moreover, it is a fuzzy equivaence reation if it is a toerance reation and has the foowing property as we: Transitivity: μ R (x i, x j ) = λ and μ R (x i, x k ) = λ 2 μ R (x i, x k ) = λ, where λ min[λ, λ 2 ]. Any fuzzy toerance reation can be reformed into a fuzzy equivaence reation by at most (n ) compositions with itsef. That is: ~ R n ~ ~ ~ ~ = R R... R R (2) = In fuzzy equivaent reations, their λ-cuts are equivaent ordinary reations. The numerica vaues that characterize a fuzzy reation can be deveoped by a number of ways, one of which is simiarity[5]. Min-max method is one of the simiarity methods that can be used when attempting to determine some sort of simiar

4 Fuzzy Equivaence Reation Based Custering and Its Use to Restructuring Websites 55 pattern or structure in data through various metrics. The equation of this method is given beow: r m k= i, j = m k = min( x ik max( x ik, x, x jk jk ) ), (3) where i, j =,2,,n In this paper, we use min-max simiarity method due to its simpicity and common use [5]. 4 Proposed Methodoogy Assume a website (WS) consists of a number (p) of web pages (wp) such as WS={wp}. A website is regarded as a set of partiay ordered web pages, according to their popuarity in terms of users preferences, i.e. visits. Thus, assuming a website has three eves of web pages, the web pages are aocated to a website eve according to the demand users show for the web page. The proposed methodoogy consists of the foowing steps:. Identify the inks of each web page in a website. 2. In order to capture the users browsing behaviour, first measure the cicks made for each ink ( i ), then compute and normaise their demand using the foowing type: D i = n C i= i C i, (4) where D i is the demand for ink i, i=,2,,n and C i are the number of cicks for the ink i. D i [,]. 3. Fuzzify the D i using trianguar fuzzy numbers (TFNs) and specify their corresponding inguistic variabes. 4. Cacuate the membership degree to the corresponding inguistic variabes for each ink, using membership functions. The membership function of a TFN is given by the foowing formua: A, x a ( x a) /( b a), a x b and a < b ( x) = ( c x) /( c b), b x c and b < c, x c μ (5)

5 56 D.K. Kardaras, X.J. Mamakou, and B. Karakostas 5. Cacuate the fuzzy toerance reation ( R ~ t ), using min-max simiarity method, as in Eq. (3). 6. Cacuate the fuzzy equivaent reation using Eq. (2). 7. Decide on the λ-cuts to be used. 8. Cassify web page inks according to λ-cuts. The derived custers constitute the redesigned web pages. 9. Cacuate the new demand for each of the newy formed web pages, according to the type: Dwp j = C i n i=, (6) where Dwp j is the demand for web page j, with (i) and (n) indicating the hyperinks and the number of hyperinks in the webpage respectivey. The demand for each web page is cacuated according to the tota number of the cicks made to a inks in this specific page.. Normaize the demand for the new web pages and then fuzzify the Dwp j. Since a website is a partiay ordered set of web pages, the newy formed we pages are aocated to a eve a website hierarchy eve according to their demand. The higher the demand the higher the eve they are assigned to.. Cacuate the vaidation index as shown in Eq. (7) for different λ-cuts. The λ- cut vaue that maximises the vaidation index indicated the optimum number of custers. λ C (λ)/ m (7) The C(λ) indicates the number of custers and the (m) shows the number of data sequence that are subject to custering. 5 Iustrative Exampe The foowing exampe iustrates the proposed methodoogy. Let us consider of a website that originay has a tota of inks (i=) in a of its 5 web pages as shown in Fig. By the use of cookies we can identify the number of cicks each of these inks has had in a specific time period. Then according to Eq. (4) the demand of each ink can be cacuated. In our exampe, et s say that the D i s are: {.3,.68,.45,.25,.76,.59,.88,.39,.77,.25}. To fuzzify the D i we wi use three TFNs with their corresponding inguistic variabes, which are Low = (,.3,.5), Medium=(.3,.5,.7) and High=(.5,.7, ). Then, we cacuate the membership degree of each ink to the corresponding inguistic variabe using Eq. (5), as seen in Tabe.

6 Fuzzy Equivaence Reation Based Custering and Its Use to Restructuring Websites 57 Fig.. Origina Website with five web pages having inks Tabe. Hyperinks membership degrees to each of the corresponding inguistic variabes 2 Low.95 Medium.5. High Using Eq. (3) we form the fuzzy toerance reation: ~ 5 R t = (8) To produce the fuzzy equivaent reation ( R ~ ) that wi be used to cassify the inks of the website, we use Eq. (2): e

7 58 D.K. Kardaras, X.J. Mamakou, and B. Karakostas ~ R e = R t ~ = The next step is to decide on the λ-cut. Assuming that the λ-cut that maximises the vaidation index shown in formua (7) is.5 we have: ~ R.5 = Eq. () is then used to cassify each of the inks in our exampe. In this case, we find four casses, each of which incudes specific inks: cass ={, 4, }, cass 2 ={ 2, 5, 7, 9 }, cass 3 ={ 3, 8 } and cass 4 ={ 6 }. We then re-create the website, having this time ony four (out of the origina five) web pages. The next step is to cacuate the new demand for each of the four newy created web pages, using Eq. (6). In our case we find: Dwp =395, Dwp 2 =44, Dwp 3 =442 and Dwp 4 =62. Then we normaize each of the Dwp i and we get Dwp =395/943=.42, Dwp 2 =44/943=.5, Dwp 3 =442/943=.47 and Dwp 4 =62/943=.7. To fuzzify the Dwp i we use the same three TFNs we used when fuzzifying the D i. That is Low = (,.3,.5), Medium = (.3,.5,.7) and High = (.5,.7, ). By cacuating the membership functions for each Dwp i to each TFN, we find that μ Low (Dwp ) =.4, μ Medium (Dwp ) =.6, μ High (Dwp ) =, μ Low (Dwp 2 ) =.7, μ Medium (Dwp 2 ) =, μ High (Dwp 2 ) =, μ Low (Dwp 3 ) =.5, μ Medium (Dwp 3 ) =.85, μ High (Dwp 3 ) =, μ Low (Dwp 4 ) =.23, μ Medium (Dwp 4 ) = and μ High (Dwp 4 ) =. We can then easiy understand that wp and wp 3 beong to the Medium category, whereas wp 2 and wp 4 beong to the Low category. To decide on the eve of each web page to the website, we take a web pages found in the High category and put them in Leve, then the web pages found in the Medium category are organized in Leve 2 (9) ()

8 Fuzzy Equivaence Reation Based Custering and Its Use to Restructuring Websites 59 Fig. 2. Restructured website with three web pages having inks in two eves 6 Concusions and web pages that beong to the Low category are eft in Leve 3. In case a catego- next ry does not exist we put in the specific Leve, the web pages that are found in the category. In our exampe, no web page is found in the High category, so Leve wi incude the pages of the Medium category, which are wp and wp 3. Wp 2 and wp 4 are then put in Leve 2 as shown in Fig. 2. This paper suggests an approach to websites structuring. A fuzzy equivaence reation based custering is been adopted, for it does not assumed a known number of custers as other custering techniques do. Further it is a custering technique that has been recenty used in other domains with satisfactory resuts. In order to derive an appro- behaviour. Research studies caim that it is important to take into account users browsing information in determining websites structure and webpages quaity. This priate web structure, this paper considers the data that refect the users browsing paper suggests that custering of hyperinks usage data can be used in order to custer inks of simiar popuarity into the same web page and then simiar demand web page to be grouped into the same web page hierarchy eve. An iustrative exampe has shown the appicabiity of the proposed approach. References. Brin, S., Page, L.: The anatomy of a arge-scae hypertextua web search engine. Computer Networks and ISDN Systems 3(/7), 7 7 (998) 2. De Marsico, M., Leviadi, S.: Evauating web sites: expoiting user s expectations. Int. J. Human Computer Interactions 6(3), (24)

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