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Eigenvalues and Vectors: Exploring the POWER Method in HITS Implementation

This presentation by John Yankowski and Larry Phillips delves into computing dominant eigenvectors for Authority and Hub Matrices using the POWER Method. Discover the benefits of this approach and how it refines scores efficiently. Dive into Mexican Hats, Sombreros, and Bibliometricity to understand the real-world application of matrix calculations in web searches. Explore the concept of Bibliometricity and see Mexican Hats in action through examples like Teoma.com. An insightful exploration of HITS technology.

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Eigenvalues and Vectors: Exploring the POWER Method in HITS Implementation

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  1. HITs Implementation Presented by the Amazingly Brilliant John Yankowski and the slightly less brilliant Larry Phillips

  2. Eigen Values and Vectors • Av = λv (λ is the Eigenvalue) • Each λ corresponds to one Eigenvector v I don’t know what this means, but Google seems to think its related to Eigen somehow.

  3. The POWER Method!!!! • x(k+1) = Ax(k) • xk -> Dominant Eigenvector • Hey John, What about other methods??

  4. Computing the ultimate authority and hub scores x and y

  5. Steps • Step 1 Initialize y(0) = e; e is a column vector of all ones • Step 2 take x(k) = Lt y(k-1) , y(k) = Lx(k) and simplify to get…

  6. x(k) = Lt L x(k-1)y(k) = L Lt y(k-1) • Computes the dominant eigenvector for the matrices LT L (Authority matrix) and L LT (Hub Matrix)

  7. Benefits of using the dominant eigenvectors of LTL and LLT • Incurs a small cost in comparison with using scores from all documents on Web • Only one document eigenvector needs to be computed: (LTL or LLT)

  8. Authoritative and Hub Matrices • Authoritative means the links are to the website • Hub means the the links shoot out from the website

  9. Mexican Hats? • Yes, Mexican hats. • We submit a query that results in pages 1 and 6, where 1 happens to point to 6

  10. But Hey, What about Sombreros?? • Related nodes can be added to a limited extent to make the search more comprehensive

  11. I need Mexican Hats! • The query results in Matrix L

  12. MSPaint Matrices are Awesome! • From L, we can find the Authoritative and Hub Matrices.

  13. HITs successfully refines the score by computing • Xi(k) = Σ yj(k-1) • Can be written as X(k) = LTy(k-1) which is the power method that will give you the dominate eigenvector

  14. We have vectors, weee!!! • xT = (0 0 .3660 .1340 .5 0) • yT = (.3660 0 .2113 0 .2113 .2113) • Why John, Don’t those add up to 1? • Why yes they do, and thank you for asking. • These numbers give you the ranking for all your Mexican hat web pages. • Auth. Ranking = (6 3 5 1 2 10) • Hub Ranking = (1 3 6 10 2 5) Dangerously close to a Mexican hat, so we’ll count it

  15. Bibliometricity • Yeah, it’s a big word, and we know it • Refers to two documents that are in-laws (related through association).

  16. How does Bibliometricity apply to mexican hats? • LTL = Din + Ccit • LLT = Dout + Cref Mexican Hat in action

  17. How does this apply to the real world? • http://www.teoma.com is a search engine that uses hits technology.

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