PageRank vs. Katz Status Index, a Theoretical Approach

Abstract

In World Wide Web search engines, it is important to have a good ranking system. One of the most famous ranking components is the PageRank system by Google. However, PageRank is protected by patents and it is impossible for other companies to use it in their search engines. There is an old model, called Katz status index, that is reported to work very similar to PageRank. If the quality of Katz status index turns out to be similar to or better than that of PageRank, it could become a patent-free alternative to PageRank. We consider the problem of comparing Katz status index to PageRank in this paper with some preliminary results on the theoretical comparison and give a proposal for practical comparison of the two models.

Page Rank Versus Katz: Is the Centrality Algorithm Choice Relevant to Measure User Influence in Twitter?

“…In this paper, we analyze how two well known network analysis algorithms, PageRank and Katz, affect the computation of mention-based user influence in Twitter. Although these two methods have previously been compared [11] and found to have been equivalent, we show that the same conclusion does not apply in the context of social networks, and that PageRank is indeed more adequate. We base our conclusions on a real world case study of the 2011 London Riots, since it was an important social event where Twitter users were said to have played a role in its origin and dissemination.…”

“…It was shown in [11] that "Katz status index may be considered a more general form of PageRank because in can be modified, within a reasonable range, to be equivalent to PageRank" and that under a "relaxed definition of equivalence (...) PageRank and Katz status index is practically equivalent to each other" as long as the number of outgoing links from any vertex is the same throughout the graph, which is very unlikely for graph modeled from a social network. On the other hand, "it is also possible to modify PageRank to become completely equivalent to Katz status index", however, in that case, "the modified PageRank is no long a random work model because it can no longer be modeled from a probabilistic standpoint" [11].…”

“…It is required to reduce the effects on the PageRank computation of loops and dangling links in the Web." [11]. Dangling links are "simply links that point to any page with no outgoing links (...) they affect the model because it is not clear where their weight should be distributed" [9].…”

“…Another well known method is the Katz algorithm [7]. It is a generalization of a back-link counting method where the weight of each node is "determined by the number of directed paths that ends in the page, where the influence of longer paths is attenuated by a decay factor" and "the length of a path is defined to be the number of edges it contains" [11]. It is defined by Equation 2 "where N (v i , k) is the number of paths of length k that starts at any page and ends at v i and α is the decay factor.…”

“…It is defined by Equation 2 "where N (v i , k) is the number of paths of length k that starts at any page and ends at v i and α is the decay factor. Solutions for all the pages are guaranteed to exist as long as α is smaller than λ > 1, where 1/λ is the maximum in-degree of any page" [11].…”