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tors that correspond to λ2 = c are associated with subgraphs of the web link graph which may have incoming edges but no outgoing edges, a structure often generated by link spammers to increase their rank. In their 2015 paper, Sangers and van Gijzen built on that to show how one could deal with link spamming using the eigenvectors related to the second eigenvalue of the Google Matrix. They found that a set of states S is a closed subset of the Google matrix G only if i ∈ S and j ∈/ S, then that implies that pij = 0. In other words, that there are no outgoing connections from the subset to the rest of the web. S is an irreducible closed subset if there is no proper sub- set of S that is a closed subset. Irreducible closed subsets correspond to the structures built by link spammers to hoard PageRank. The eigenvectors corre- sponding to the second eigenvalues of the Google matrix have a non-zero value corresponding to the irreducible closed subsets and a zero in other nodes. One solution proposed by them was to utilize the personalization vector v to lower the PageRank of suspected pages, by giving a small value in the correspond- ing node. This personalization vector allows for adjustments to the PageRank values without changing the basic formula. 4.3 Other Search Engines Google was built using the PageRank to weigh web pages in relevance to searches. Google uses PageRank in combination with weighing the text and contents of the page, link text, and even capitalization of words to determine relevance of the page to the search terms. Other search services also started using similar methods to rank web pages for searches. This paper explores more of the variations that other services may use, and how Google may have changed what they have done in the past 20 years (Facts about Google and Competition, n.d.). 21PDF Image | MATHEMATICS BEHIND GOOGLE PAGERANK ALGORITHM
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