Piyush Kumar (Lecture 2: PageRank) Welcome to COT5405

Quick Recap: Linear Algebra Matrices Source:

3 Square matrices  When m = n, i.e., 1.1 Matrices

4 Sums of matrices 1.2 Operations of matrices Example: if and Evaluate A + B and A – B.

5 Scalar multiplication 1.2 Operations of matrices Example:. Evaluate 3A.

6 Properties 1.2 Operations of matrices Matrices A, B and C are conformable,  A + B = B + A  A + (B +C) = (A + B) +C  (A + B) = A + B, where is a scalar (commutative law) (associative law) Can you prove them? (distributive law)

7 Matrix multiplication 1.2 Operations of matrices  If A = [a ij ] is a m  p matrix and B = [b ij ] is a p  n matrix, then AB is defined as a m  n matrix C = AB, where C= [c ij ] with Example:, and C = AB. Evaluate c 21. for 1  i  m, 1  j  n.

8 Matrix multiplication 1.2 Operations of matrices Example:,, Evaluate C = AB.

9 Properties 1.2 Operations of matrices Matrices A, B and C are conformable,  A(B + C) = AB + AC  (A + B)C = AC + BC  A(BC) = (AB) C  AB  BA in general  AB = 0 NOT necessarily imply A = 0 or B = 0  AB = AC NOT necessarily imply B = C However

Identity Matrix Examples of identity matrices: and

11 The transpose of a matrix  The matrix obtained by interchanging the rows and columns of a matrix A is called the transpose of A (write A T ). Example: The transpose of A is  For a matrix A = [a ij ], its transpose A T = [b ij ], where b ij = a ji. 1.3 Types of matrices

12  If matrices A and B such that AB = BA = I, then B is called the inverse of A (symbol: A -1 ); and A is called the inverse of B (symbol: B -1 ). The inverse of a matrix Show B is the the inverse of matrix A. Example: Ans: Note that Can you show the details? 1.3 Types of matrices

13 Symmetric matrix  A matrix A such that A T = A is called symmetric, i.e., a ji = a ij for all i and j.  A + A T must be symmetric. Why? Example: is symmetric.  A matrix A such that A T = -A is called skew- symmetric, i.e., a ji = -a ij for all i and j.  A - A T must be skew-symmetric. Why? 1.3 Types of matrices

14  (AB) -1 = B -1 A -1  (A T ) T = A and ( A) T =  A T  (A + B) T = A T + B T  (AB) T = B T A T 1.4 Properties of matrix

3.15 The determinant of a 2 × 2 matrix: Note: 1. For every square matrix, there is a real number associated with this matrix and called its determinant 2. It is common practice to delete the matrix brackets Source:

3.16 Historically, the use of determinants arose from the recognition of special patterns that occur in the solutions of linear systems: Note: 1. a 11 a 22 - a 21 a 12 ≠0 2. x 1 and x 2 have the same denominator, and this quantity is called the determinant of the coefficient matrix A

3.17 Ex. 1: (The determinant of a matrix of order 2) Note: The determinant of a matrix can be positive, zero, or negative

Determinants 1.If every element of a row (column) is zero, e.g.,, then |A| = |A T | = |A| 3. |AB| = |A||B| determinant of a matrix = that of its transpose The following properties are true for determinants of any order.

Eigenvalues and Eigenvectors Ax = λx Should not exist? det(A − λI) = 0. Fact: A and transpose(A) have the same eigenvalues. Why?

Task of search engines Crawl Build indices so that one can search keywords efficiently. Rate the importance of pages. One example is the simple algorithm named pagerank.

The basic idea Mimic democracy! Use the brains of all people collectively.

The basic idea Mimic democracy! Use the brains of all people collectively for the ranking. What’s wrong with counting backlinks? Should page 1 be ranked above page 4?

Voting using backlinks? But then we don’t want an individual to cast more than one vote? Normalize?

Normalized Voting?

Link Matrix (for the given web):

Most important node = 1?

Definition A square matrix is called column stochastic if all of its entries are non-negative and the entries in each column sum to 1. Lemma: Every column stochastic matrix has 1 as an eigenvalue. Proof: A and A’ = transpose of A, have the same eigenvalues: Why?

Two shortcomings Nonunique Rankings. Dangling nodes : Nodes with no outgoing edges. The matrix is no longer column stochastic. Can we transform it into one easily?

Nonunique Rankings Not clear: Which linear combination should we pick for the ranking?

Nonunique Rankings

Modification of the Link Matrix The value of m used by google (1998) was.15 For any m between 0,1; M is column stochastic. M can be used to compute unambiguous importance scores (in the absence of dangling nodes) For m = 1, the only normalized eigenvector with eigenvalue 1 is ?

Modification of the Link Matrix

Example 1 For our first example graph, m = 0.15.

Example 2 Still, m = 0.15.

Towards the proof For real numbers Proof by Contradiction? -> Let x be an eigenvector with mixed signs for the eigenvalue 1.

Towards the proof

A punchline

The Algorithm (aka Power Method)

c ?

One last lemma…

Why does it converge?

The main theorem For figure 2:

First Example Do we need any modifications to A?

Calculations

Another Example

Random Surfer Model The Rule: Assume that 85 per cent of the time the random surfer clicks a random link on the current page (each link chosen with equal probability) 15 percent of the time the random surfer goes directly to a random page (all pages on the web chosen with equal probability).

Random Surfer Model Cons No one chooses links or pages with equal probability. There is no real potential to surf directly to each page on the web. The (or any fixed) breakdown is just a guess. Back Button? Bookmarks? Despite these flaws, the model is good enough that we have learnt a great deal about the web using it.

Related stuff to explore Random walks and Markov Chains. Random Graph construction using Random walks. Absorbing Markov Chains. Ranking with not too many similar items at the top. Dynamical Systems point of view. Equilibrium or Stationary Distributions. Rate of convergence. Perron-Frobenius Theorem Intentional Surfer model. Markov Chain Slides:

Homework 1 Implementation: Parse wikipedia pages and find pageranks of top 1000 pages of the given input. (TBA) Theory: Solve Exercises in the given paper. (Online) There are 24 questions in total in the paper (including subproblems marked with a filled disc, Example, problem 6 has 3 subproblems). Pick the first two characters of your fsu.edu address. Example “pk” for (all Represent in hex : “706B” = x = Your hex number goes here. Calculate f1 = ((x mod 3D) mod 18)+1 Calculate f2 = (f1 + 12) mod 18 Solve those two exercises in the paper. Write the problems you solve (including problem numbers) and the solution in Latex. Submit.