The goal of this part is to implement the PageRank algorithm of the Google searc

Question

The goal of this part is to implement the PageRank algorithm of the Google search engine with the Power method and its variants, and apply them to a simplified graph simulating the World Wide Web. The ranking is based on finding the dominant eigenvector of a matrix which describes the connection of all webpages in one network. We start with some basic concepts from the graph theory. A directed graph is a set of vertices connected by edges where each edge has a direction. For example, the internet with various webpages can be considered as a directed graph where each vertex and directed edge stand for a webpage and a link, respectively. The adjacency matrix of a directed graph is defined as an n × n matrix B whose entries are bij = ( 1, if there is a link from the page i to the page j; 0, otherwise. From B, we can construct a modified adjacency matrix M = (mij ) which tells us the probability of visiting the webpage j from the webpage i. Suppose that when vising a webpage, one surfer clicks one link in this webpage with probability p and jumps to a completely random webpage in the network with probability 1 p. The probability p is also known as the damping factor in the PageRank theory and is usually set around 0.85. Furthermore, if there are n pages in the graph, we define the modified adjacency matrix M whose entries are mij = p P bij n k=1 bik + 1 p n . In fact, M is a stochastic matrix satisfying that Pn j=1 mij = 1 and thereby 1 is the dominant eigenvalue of M with algebraic multiplicity 1 by the Perron’s Theorem. The Markov theory states that if ~v with Pn i=1 |vi | = 1 is a left dominant eigenvector of M associated with the eigenvalue 1, i.e., ~vTM = ~vT , then vi is the probability that one visits the page i at the stationary state independent of the starting page, which determines the rank of the ith webpage. Now we use the power method, scaled power method and inverse power method to find the left dominant eigenvector of M, respectively. The initial vector can be randomly generated by randn(n,1)1 . Requirements Submit to CCLE a file lastname_firstname_hw8.zip containing the following files: • A MATLAB function power1.m that implements the Power Method with l-norm scaling, power2.m implements the Power Method with l2-norm scaling, invpower.m that implements the Inverse Power Method, and a MATLAB script main.m that finds the rank of each webpage in the network shown in Figure 1 with 15 webpages. (Hint: Use the graph to construct the corresponding adjacency matrix B first and then build the modified adjacency matrix M. The exact eigenvalues can be obtained via eig.) 1http://www.mathworks.com/help/matlab/ref/randn.html. 1 Math 151B Winter 2017 • A PDF report that describes what you have explored, lists the ranks of all webpages by each method, compares the performance of the aforementioned algorithms (e.g., convergence speed, robustness with respect to the initial guess), and concludes what you have discovered. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Figure 1: Simulated network with 15 nodes.

Explanation / Answer

The Random natator Model
In their publications, Lawrence Page and Sergey Brin provides a terribly easy intuitive justification for the PageRank algorithmic program. They contemplate PageRank as a model of user behaviour, wherever a natator clicks on links haphazardly with no regard towards content.
The random natator visits an internet page with a particular likelihood that derives from the page's PageRank. The likelihood that the random natator clicks on one link is entirely given by the amount of links on it page. this can be why one page's PageRank isn't utterly passed on to a page it links to, however is devided by the amount of links on the page.
So, the likelihood for the random natator reaching one page is that the total of chances for the random natator following links to the present page. Now, this likelihood is reduced by the damping issue d. The justification at intervals the Random natator Model, therefore, is that the natator doesn't click on associate infinite range of links, however gets bored typically and jumps to a different page haphazardly.
The likelihood for the random natator not stopping to click on links is given by the damping issue d, which is, counting on the degree of likelihood so, set between zero and one. the upper d is, the a lot of doubtless can the random natator keep clicking links. Since the natator jumps to a different page haphazardly once he stopped clicking links, the likelihood so is enforced as a continuing (1-d) into the algorithmic program. despite inward links, the likelihood for the random natator jumping to a page is usually (1-d), thus a page has invariably a minimum PageRank.
A Different Notation of the PageRank algorithmic program
Lawrence Page and Sergey Brin have printed 2 totally different versions of their PageRank algorithmic program in numerous papers. within the second version of the algorithmic program, the PageRank of page A is given as
PR(A) = (1-d) / N + d (PR(T1)/C(T1) + ... + PR(Tn)/C(Tn))
where N is that the total range of all pages on the online. The second version of the algorithmic program, indeed, doesn't disagree basically from the primary one. relating to the Random natator Model, the second version's PageRank of a page is that the actual likelihood for a natator reaching that page once clicking on several links. The PageRanks then type a likelihood distribution over websites, therefore the total of all pages' PageRanks are one.
Contrary, within the 1st version of the algorithmic program the likelihood for the random natator reaching a page is weighted by the whole range of websites. So, during this version PageRank is associate first moment for the random natator visiting a page, once he restarts this procedure as usually because the net has pages. If the online had a hundred pages and a page had a PageRank price of two, the random natator would reach that page in a median double if he restarts a hundred times.
As mentioned higher than, the 2 versions of the algorithmic program don't disagree basically from one another. A PageRank that has been calculated by mistreatment the second version of the algorithmic program needs to be increased by the whole range of websites to induce the according PageRank that might are caculated by mistreatment the primary version. Even Page and Brin caught up the 2 algorithmic program versions in their most well-liked paper "The Anatomy of a Large-Scale Hypertextual net Search Engine", wherever they claim the primary version of the algorithmic program to create a likelihood distribution over websites with the total of all pages' PageRanks being one.
In the following, we'll use the primary version of the algorithmic program. the rationale is that PageRank calculations by means that of this algorithmic program square measure easier to cipher, as a result of we are able to disregard the whole range of websites.

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