Indian Journal of Science and Technology, Vol 8(34), DOI: 10.17485/ijst/2015/v8i34/86120, December 2015 ISSN (Print) : 0974-6846 ISSN (Online) : 0974-5645 eighted Page Rank Algorithm based on In-Out eight of ebages B. Jaganathan* and Kalyani Desikan Division of Mathematics, School of Advanced Sciences (SAS), VIT-Chennai, Chennai 600048, Tamil Nadu, India; jaganathan.b@vit.ac.in, kalyanidesikan@vit.ac.in Abstract In its classical formulation, the well known age rank algorithm ranks web ages only based on in-links between web ages. e roose a new in- weight based age rank algorithm. In this aer, we have introduced a new weight matrix based on both the in-links and -links between web ages to comute the age ranks. e have illustrated the working of our algorithm using a web grah. e notice that the age rank values of the web ages comuted using the original age rank algorithm and our roosed algorithm are comarable. Moreover, our algorithm is found to be efficient with resect to the time taken to comute the age rank values. Keywords: Algorithm, In-eight, Out-eight, Page Rank, eb Pages 1. Introduction orld ide eb () comrises of billions of web ages that hold a huge amount of information. Search engines hel users to retrieve relevant information from this large collection of information. However, user s interest for high quality information search services is not fully satisfied by the current search engines. This oses challenges for information retrieval and to navigate within the search results various ranking methods are alied. Page rank algorithms are well known for ordering web ages. Ranking methods have become an imortant tool to sort and fetch the relevant web ages based on the user s interest. The structure of this aer is as follows: Section 1 resents the need for ranking algorithms, Section 2 deals with the different tyes of ranking algorithms. In Section 3, the relationshi between eb ages and eb grah is resented. In Section 4, the adjacency matrix based age rank algorithm is resented. In Section 5, weighted age rank algorithm is resented. In Section 6 we resent our roosed in- weight based age ranking method and its illustration. In Section 7, the comarison between the age rank algorithms is given. In Section 8, the conclusion and the ossible future work are resented. 2. Different Tyes of Ranking Algorithms Sergey Brin and Larry Page roosed the age rank algorithm 1,2 at Stanford University. A new aroach known as weighted age rank algorithm 3 was ut forth by enu Xing et al. This algorithm is an extension of the original age rank algorithm. Recently, we develoed two age rank algorithms: Category-based age rank method 4 and enalty-based age rank algorithm 5. In this aer, we roose an efficient age ranking algorithm based on in- weights of web ages for finding more relevant web ages to users query. *Author for corresondence
eighted Page Rank Algorithm based on In-Out eight of ebages 3. eb Page and Grah Relationshi Before roceeding further, we must understand the relation between web ages and a web grah. The orld ide eb () is generally reresented as a directed web grah. The vertices are the web ages and directed edges reresent the hyerlinks between web ages (link/in-link) 5, 6. A degenerate edge of a grah which joins a vertex to itself is called a loo. A web grah with no loos is called a simle directed grah. An examle of a simle directed grah reresenting 5 web ages connected by hyerlinks is given in Figure 1. In grah theory, an adjacency matrix 7 for a directed grah is a matrix for reresenting adjacent vertices (or nodes) of the grah. For any directed grah G with n vertices, the adjacency matrix is a n n matrix with matrix elements being 1 for vertices (or nodes) which oint to other vertices and 0 otherwise. This can be mathematically reresented as: a i,j 1 if i j and v i is ointing/links to v j 0 otherwise (1) And it is denoted by A (G). In simle grahs with no loos the adjacency matrix consists of only zeros and ones with diagonal entries being zero. Adjacency Matrix for Figure 1 is as follows: 0 0 1 1 0 1 0 0 0 0 AG ( ) 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 (2) 4. Page Rank Algorithm The age rank algorithm 1,2 was originally roosed by Larry Page and Sergey Brin. A brief exlanation of adjacency matrix based age rank algorithm used in the Google search engine is given below. Consider the web as a directed grah G = {V,E}, where V is the vertex (node) set, that is, the set of all web ages and E is the set of all directed edges in G, that is, hyerlinks of the web grah. Let n be the total number of ages in the web grah. Adjacency matrix A (G) for a web grah G can be calculated using equation 1. The initial age rank for each of the n web ages is PR 0 = (PR 0 (1), PR 0 (2), PR 0 (n)) and their value is set as 1. The formula for comuting the m th iteration of the age rank is given by: PR m = (1- d)+ d * A(G)* PR m-1 (3) here d is the daming factor 8,9. The value of the daming factor ranges between 0 and 1 and it is usually taken as 0.85. At each iteration, the age rank value of each web age is calculated using equation 3. To determine the final age rank of a web age, iterations are carried until they converge. 5. eighted Page Rank Algorithm of eb Pages The weighted age rank algorithm 3 was roosed by enu Xing and Ghorbani Ali. A brief exlanation of weighted age rank algorithm is given below. In the weighted age rank algorithm, more imortant (oular) web ages are assigned larger age rank values. The oularity of a web age deends on the number of its in links and links and each web age gets a roortional age rank value. The oularity of each age can be in obtained using the in and weights, and, given by equations 4 and 5, resectively. in Iu (4) I r(v) = Ou O r(v) (5) Figure 1. eb grah. Here r (v) is the set of all eb ages that have in links from node v (reference age list of age v). These weights 2 Indian Journal of Science and Technology
B. Jaganathan and Kalyani Desikan deend on the number of in links and links of age u and the sum of the number of in links and links of all reference ages of age v, resectively. The initial age rank for each of the n eb ages is given by PR 0 = (PR 0 (1), PR 0 (2), PR 0 (n)) and their value is set as 1. The formula for comuting the weighted age rank of eb age v is given by: PR(v) = (1 d)+ d PR(v) (6) in vb(u) ' here B (u) is the set of all web ages that oint to u and d denotes the daming factor 8,9. 6. Proosed eighted Page Rank Algorithm based on In-Out eight of eb Pages e roose a new weighted age rank algorithm. In this algorithm we first calculate the in-weight and -weight of the web ages using equations 7 and 8. in Iu = (7) I = R(v) R(v) Ou O (8) here I u and I in equation 7 denote the number of in links of age u and age resectively. Also, O u and O in equation 8 are the number of links of age u and age resectively. R (v) is the set of all web ages that oint to v (reference age list of age v). e calculate the weight matrix (G) using equation 9. (G) (9) in And its denoted as (G), where u, v reresent the row and column resectively of the weight matrix (G). e make use of the weight matrix given in 9 to comute the m th iteration of the age rank as given below: PR m(1d) d * (G)* A(G)* PRm 1. (10) here d is the daming factor 8,9 that always lies between 0 and 1 and it is usually set as 0.85. The initial age rank for each of the n web ages is PR 0 = (PR 0 (1), PR 0 (2), PR 0 (n)) and their value is originally set as 1. The age rank value of each age, at each iteration, is calculated using equation 10. To determine the final age rank of a web age iterations are carried until they converge. e now exlain the working of the original and our roosed in- weighted age rank algorithms by considering the hyerlinked web grah shown in Figure 1, consisting of five ages A, B, C, D and E. The in and weights for ages A, B, C, D and E are calculated using equations 7 and 8 resectively. e now illustrate, for a few web ages, how the in and weights are comuted. For examle: I 1 1 I +I 1+1 2 I 2 1 in C AC = = = B E in D AD = = = 1 I B+IE 1+1 1 O 1 1 O +O 1+1 2 O 1 1 O +O 1+1 2 C AC = = = B E D AC = = = B E In the same way we can comute the in and weights for the remaining ages and we get the following matrices: 0 0 1/2 1 0 2 0 0 0 0 0 0 0 0 1/3 1 0 0 0 0 in 0 1/2 0 1/2 0 0 0 1/2 1/2 0 1 0 0 0 0 0 0 0 0 1/4 2 0 0 0 0 0 1/2 0 1/2 0 From the above two matrices we can form the weight matrix (G) using equation 9 as follows: 0 0 1 3/2 0 3 0 0 0 0 G ( ) 0 1 0 1 0 0 0 0 0 7/12 3 0 0 0 0 Indian Journal of Science and Technology 3
eighted Page Rank Algorithm based on In-Out eight of ebages After comuting the weight matrix (G), we make use of equation 10 to comute the age ranks. e have comuted the age ranks for three different values of the daming factor viz., d = 0.85, d = 0.7 and d = 0.5. The values to which the age ranks converge in these three cases are given in Table 1. Table 1. Page rank comutations using our roosed age rank algorithm for various daming factor (d) values for web grah in Figure 1 eb age Proosed Page Rank Algorithm Page rank (d = 0.85) Page rank (d = 0.7) Page rank (d = 0.5) A 0.5343 0.8149 1.0043 B 0.2783 0.4580 0.6371 C 0.5314 0.7885 0.9446 D 0.4368 0.6603 0.8172 E 0.2783 0.4580 0.6371 7. Comarison between Page Rank Algorithms e have calculated the age rank for the web ages for the web grah (Figure 1.) using the adjacency matrix based original age rank algorithm 1,2, weighted age rank algorithm method and our roosed in- weighted matrix based age rank algorithm. The Tables 2, 3 and 4 shows the age ranks comuted for the web ages using the adjacency matrix, weighted age rank algorithm method and in- weighted matrix based age rank algorithm for the three different values of the daming factor. The web ages (vertices) are arranged in the tables in increasing order of age rank value. The age rank algorithm and our roosed in- weighted based age rank algorithm method give the same rank to the web ages. Calculating age rank using our roosed age rank method takes lesser time comared to the original age rank algorithm. Time taken to Table 2. Page rank comutations using original age rank algorithm, weighted age rank algorithm and our roosed age rank algorithm for d=0.85 in the web grah (shown in Figure 1) Original Page Rank Algorithm eighted Page Rank Algorithm Proosed Page Rank Algorithm eb age Page rank eb age Page rank eb age Page rank A 1.4773 A 0.7008 A 0.5343 C 1.1559 E 0.4580 C 0.5314 D 0.8112 D 0.3624 D 0.4368 B 0.7778 C 0.2824 B 0.2783 E 0.7778 B 0.1900 E 0.2783 Table 3. Page rank comutations using original age rank algorithm, weighted age rank algorithm and our roosed age rank algorithm taking d = 0.7 for the web grah (Figure 1.) Original Page Rank Algorithm eighted Page Rank Algorithm Proosed Page Rank Algorithm eb age Page rank eb age Page rank eb age Page rank A 1.4068 A 1.0364 A 0.8149 C 1.1538 E 0.6982 C 0.7885 D 0.8547 D 0.5688 D 0.6603 B 0.7924 C 0.4612 B 0.4580 E 0.7924 B 0.3638 E 0.4580 4 Indian Journal of Science and Technology
B. Jaganathan and Kalyani Desikan Table 4. Page rank comutations using original age rank algorithm, weighted age rank algorithm and our roosed age rank algorithm taking d=0.5 for the web grah (Figure 1.) Original Page Rank Algorithm eighted Page Rank Algorithm Proosed Page Rank Algorithm eb age Page rank eb age Page rank eb age Page rank A 1.2982 A 1.2115 A 1.0043 C 1.1404 E 0.8702 C 0.9446 D 0.9123 D 0.7404 D 0.8172 B 0.8246 C 0.6346 B 0.6371 E 0.8246 B 0.5529 E 0.6371 comute age rank values, with d = 0.85, using age rank algorithm is 15.00292 seconds and for our roosed age rank method it is 10.79334 seconds. The same behavior is observed for d = 0.70 and d = 0.5. 8. Conclusions and Future ork From the revious section on in- weight based age rank algorithm we notice that our algorithm is more efficient when comared to the original age rank algorithm with resect to time. It can also be seen that the ranking of the web ages using our algorithm agrees with that obtained by using the original age rank algorithm. hile the ranking obtained using the weighted age rank algorithm does not agree with the original age rank algorithm. In our future work, based on this algorithm, we envisage to work with bigger web grahs. e also roose to introduce other weight based techniques for calculating the ranks of web ages. Figure 3. Shows the age rank values obtained using the three algorithms for d = 0.7 for the web grah in Figure 1. Figure 4. Shows the age rank values obtained using the three algorithms for d=0.5 for the web grah in Figure 1. 9. References Figure 2. Shows the age rank values obtained using the three algorithms for d=0.85 for the web grah in Figure 1. 1. Page L, Brin S, Motwani R, inograd T. The age rank citation ranking: Bringing order to the eb. Technical reort. Stanford Digital Library Technologies Project; 1998 Jan.. 1 17. Indian Journal of Science and Technology 5
eighted Page Rank Algorithm based on In-Out eight of ebages 2. Page L, Brin S. The anatomy of a large scale hyer-textual web search engine. Comuter Networks and ISDN Systems.1998 Ar; 30(1-7):107 17. 3. Xing, Ghorbani A. eighted age rank alogithm. Proceedings of the Second Annual Conference on Communication Networks and Services Research (CNSR 04); IEEE. 2004 May 19-21.. 305 14. 4. Jaganathan B, Kalyani D. Category-based age rank algorithm. International Journal of Pure and Alied Mathematics. 2015 Aug; 101(5):811-820. 5. Jaganathan B, Kalyani D. Penalty-based age rank algorithm. ARPN Journal of Engineering and Alied Sciences. 2015 Mar; 10(5):2000 3. 6. Kleinberg JM. Authoritative sources in a hyerlinked environment. Journal of the ACM. 1999 Se; 46(5):604 32. 7. Langville AN, Meyer CD. Google s age rank and beyond: The science of search engine rankings. Princeton, NJ: Princeton University Press; 2006. 8. Bressen M, Peserico E. Choose the daming, choose the ranking? Journal of Discrete Algorithms. 2010 Jun; 8(2):199 213. 9. Ali D, Faqir M, Hassan D, Hussain D. Ranking cricket teams. International Journal Information Processing & Management. 2015; 5:62 73. 6 Indian Journal of Science and Technology