Web Mining and Social Networking: Techniques and ... - tud.ttu.ee

5.3 PageRank and HITS Algorithms 91Fig. 5.2. Paper i and j cite paper k5.3 PageRank and HITS Algorithms5.3.1 PageRankIn 1998, Sergey Brin and Larry Page invented PageRank and presented it at the Seventh InternationalWorld Wide Web Conference (WWW7). The PageRank is a measure of pages qualityused in the Google search engine. PageRank utilizes hyperlinking as an indicator of the qualityof a Web page. In essence, PageRank interprets a hyperlink from page x to page y as aconveyance of prestige, by page x, for page y. However, PageRank does not just consider thesheer number of links that a page receives. It also takes into account the importance or qualityof the page that conveys the prestige. Hyperlinks from pages that are themselves importantweigh high and help to make other pages more important. This idea is based on the concept ofrank prestige in social network analysis [250].Algorithm: PageRankPageRank [196, 43] is a search query independent ranking algorithm for Web pages. It is basedon the measure of rank prestige in social network analysis. A PageRank value is calculatedfor each indexed page off-line. The main concepts underlying the PageRank algorithm can bedescribed as follows.1. A hyperlink from a page pointing to another page is an implicit conveyance of authorityto the target page. Thus, a Web page is important if it receives a lot of in-links.2. A hyperlink from a high prestige page is more important than a hyperlink from a lowprestige page. Thus, a Web page is important if it is pointed to by other important pages.To formulate the above concepts, we represent the Web as a directed graph G = (V, E),where V is the set of vertices or nodes, i.e. the set of all pages in the Web, and E is the set ofdirected edges between a pair of nodes, i.e. hyperlinks. Let the total number of pages on theWeb be n(n = |V|). The PageRank score of the page i, P(i), is defined by:P(i)=(1 − d)+d∑ ( j,i)∈EP( j)/outdegree( j) (5.3)where d is a damping factor usually set between 0.8 and 0.9; outdegree(q) is the numberof hyperlinks on page q.Alternately, the PageRank can be defined by the stationary distribution of the followinginfinite, random walk p 1 , p 2 , p 3 ,...,where each p i is a node in G: The walk starts at each nodewith equal probability. To determine node p i+1 a biased coin is flipped: With probability 1-dnode p i+1 is chosen uniformly at random from all nodes in G, with probability d it is chosenuniformly from all nodes q such that (p i ,q) exists in the graph G.Equation 5.1 can be rewritten in a matrix notation. Let P be a n-dimensional column vectorof PageRank values: P =(P(1),P(2),···,P(n)) T . Let A be the adjacency matrix of G with