Authors in a Markov matrix: Which author do people find most inspiring? (3)

Problem statement: Analyzing author relationships (2)

I would like to suggest we start with a simpler problem. Let's forget about the contents of a book and how much it influences others. Our model only concerns the existence of an influence.

Fortunately, there is a mathematical method based on the existence of a relationship, called graph theory. One of the tools used by graph theory is linear algebra. Graph theory has been widely used for a long time. When the World Wide Web era came, this method was also applied to the principles of web search to implement a basic idea. If you can find the most influential web page, you should list these results on top of the search results. There is also a lot of research to find a method to automatically compute the influence between web pages.

Probably the most frequently used method for such a web page ranking only considers the linking relationship between web pages. Since this method only uses link information, it doesn't need to know the contents of individual web pages. This method was a breakthrough in web search engine technology. The inventors of this method started a company called Google, using a computed influence rank for listing the search results[bib:pagerank]. Because the method doesn't need to understand the contents of a web page to evaluate that page's ranking, it can used for web pages in any language with any type of content. Furthermore, the ranking can be fully automated. The search engines of some other companies evaluated the rank by hand, but these companies were easily outperformed by Google. Ranking by hand may be more accurate than automated methods but the automated method is much better at updating speed and handling large numbers of web pages.

The method used for web page ranking by Google is called PageRank [bib:pagerank], though this method is based on the traditional technique of eigenanalysis in linear algebra.

In this article, I first provide an overview of graph theory. I then apply eigenanalysis to rank the influence of authors.

Therefore, my next blog entry will be ``introduction to graph theory.''

Comments

Why A^{T}A is invertible? (2) Linear Algebra

Why A^{T}A has the inverse Let me explain why A^{T}A has the inverse, if the columns of A are independent. First, if a matrix is n by n, and all the columns are independent, then this is a square full rank matrix. Therefore, there is the inverse. So, the problem is when A is a m by n, rectangle matrix. Strang's explanation is based on null space. Null space and column space are the fundamental of the linear algebra. This explanation is simple and clear. However, when I was a University student, I did not recall the explanation of the null space in my linear algebra class. Maybe I was careless. I regret that... Explanation based on null space This explanation is based on Strang's book. Column space and null space are the main characters. Let's start with this explanation. Assume x where x is in the null space of A . The matrices ( A^{T} A ) and A share the null space as the following: This means, if x is in the null space of A , x is also in the null spa

Gauss's quote for positive, negative, and imaginary number

Recently I watched the following great videos about imaginary numbers by Welch Labs. https://youtu.be/T647CGsuOVU?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF I like this article about naming of math by Kalid Azad. https://betterexplained.com/articles/learning-tip-idea-name/ Both articles mentioned about Gauss, who suggested to use other names of positive, negative, and imaginary numbers. Gauss wrote these names are wrong and that is one of the reason people didn't get why negative times negative is positive, or, pure positive imaginary times pure positive imaginary is negative real number. I made a few videos about explaining why -1 * -1 = +1, too. Explanation: why -1 * -1 = +1 by pattern https://youtu.be/uD7JRdAzKP8 Explanation: why -1 * -1 = +1 by climbing a mountain https://youtu.be/uD7JRdAzKP8 But actually Gauss's insight is much powerful. The original is in the Gauß, Werke, Bd. 2, S. 178 . Hätte man +1, -1, √-1) nicht positiv, negative, imaginäre (oder gar um

Why parallelogram area is |ad-bc|?

Here is my question. The area of parallelogram is the difference of these two rectangles (red rectangle - blue rectangle). This is not intuitive for me. If you also think it is not so intuitive, you might interested in my slides. I try to explain this for hight school students. Slides: A bit intuitive (for me) explanation of area of parallelogram (to my site, external link) .

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