Category: Mathematics

Google Pagerank – An emprical analysis

Pagerank is an algorithm used by google to assign importance of each page in world wide web. The order of search results  depends on the page rank assigned to each page.  A web site has multiple pages and page rank assigned to each page in the web site. The page rank of the page depends on number of external links coming to the page and number of links going out of the page.  The page rank of an incoming link also plays a role on determining the page rank of the page.

Page rank is used to decide the sequence of search results for a google search. Companies want their page to be at the top of the search result. The easiest way to be at top of the search result is to pay google for the adwords. The sponsored web sites appear above and to the right hand side of its regular search results. 

Not all companies can afford to pay google for their Adwords. There is cost effective alternative which requires an understanding of how google works.  Google’s internal working knowledge will aid to define  analytical (or mathematical) strategies and implemented in the web site to improve the page rank and eventually with more visitors to the web site.

To validate the theory with emprical analysis, a plan with following steps drafted.

  1. Find a key word that is not in google’s index server
  2. Create a graph (random graph) and find the initial transition probability
  3. Evaluate the steady state of transition probability matrix using power methods – Which is the page rank of the graph (I will post the technical/mathematical details in a pdf. It is time-consuming to write matrix and other math notation in word press editor)
  4. Develop a set of web sites (pages) adhering to the random graph and each web site to contain the new key word
  5. Allow google crawler to include the new sites in their index server for the new key words
  6. Search for the new key word and absorb the order of search results
  7. Report the results

1. A new key word was selected and  is given below.   There is no google search result for the key word.

2. The random graph  (a representation of how web sites are linked to each other) and the graph will be implemented  with various blog post and  each blog post will have the key word  adhering to the graph. Each node of the graph denote a blog posting. The links connecting the nodes are the hyperlinks connecting the nodes.

Note: This page is used for google’s page rank emprical analysis. The links will be created based on the random graph created.  This is node #1 which has the key word:  xysivabodzinyx , xysivabodzinxy . As per the graph, it links out to page 2, page 4

Theory (conjecture) of Object Symmetry

Like NDPL, it is another incomplete work of Praba Siva. Being curious is a strength and also it is a weakness. Strength is, you will learn something new all the time and the weakness is, once you know the basics, you are not motivated to go to end of it. I belong to that category. Once up on a time, I used to spend 15 hours a day thinking about the Praba’s theory (conjecture) of object symmetry. I still remember, at Toronto airport, after having one or two molson, I had thought about how to mathematically prove it by writing numerous equation in a napkin. I believe that napkin is still hanging in my home office.

Let me state the theory (conjecture) . That is, what is Praba’s theory (conjecture) of object symmetry. Theory (conjecture) of object symmetry states that every objects (signals) has a perfect symmetry with respect to a finite set of line of axes.

It may not be clear & obvious what the theory states. To simplify the statement, look into the natural objects like stone, trees, flowers, insects, humans, you split them with respect to different set of line of axes, you would see a perfect symmetry. Most of the man made object obey this theory obviously and all of them should obey mathematically. In another simpler words, take a tree and break it into smaller pieces such way that each smaller piece has a perfect symmetry with respect to different line of axes. Image is a signal. It applies to the other types of signals like voice. What ever language you speak, English or Tamil, that is a signal which can be found a perfect symmetry by finding the smaller line of axes. The bigger (or major) line of axes more the application you could find. The smaller the line of axes, lesser practical applications since it is not worth it.

I spent almost 8-12 months proving this theory using Harmonic analysis. I could not do it. It must be because I do not fully understand the concept of harmonic analysis.

I assumed that the conjecture (theory with out proof) is true and took an empirical approach. I applied it for pose estimation problem. That is, for improving the car safety, identification of the driver position is very important. If the driver is sleeping nor paying full attention to driving, the safety system should warn them that they are not fully paying attention to avoid any potential accidents.

The national high way traffic safety administration (http://www.nhtsa.dot.gov) concluded that the car/truck driver position can be classified into 6 different set of position. If the driver is not focusing on one position (ie. position 3, looking forward ) for more than significant time (15 seconds), then the system should alarm the driver. I applied the theory (conjecture) of object symmetry principle to this problem and compared it with other solutions. Other solutions available (like Eigen faces, Neural network, Gabor Jets) are not practical since it takes at least 30 seconds to classify each frame of the video. Where as, by applying the theory of object symmetry, it took only 1-2 seconds per image.

Empirically, the theory is proven and it works. I studied cognitive psychology for a brief time to see how the human mind does the discrimination naturally and simply. There are some theory in cognitive psychology believes that human brain does discrimination fast by identifying the symmetry and corresponding line of axes. I also understood, when a human being is considered as beautiful, they have a symmetry with major line of axes. ie. take a women who is considered as most beautiful. You could see her face (other parts) have a symmetry with major line of axes.

I believe this theory could be applied to credit risk management, signal processing, scene analysis, image analysis, astronomy, etc.

I lost interest since I know it works and now I’m interested in something I do not know. This could easily be an another Ph.d thesis in the field of computer engineering or applied mathematics. As I understand Princeton Mathematics department works on the symmetry problems.

If any one working on this subject and want to work together to mathematically prove this.. I’m all for it. I can provide all the work I have done so far.