MAS 801 It's a Discreetly Discrete World Ã¢Â€Â“ Mathematics in Real-life ...

Lecture 9: 22 October 2008 

MAS 801 

It’s a Discreetly Discrete World – 

Mathematics in Real-life Applications


Proving Optimality: Method of Tight Restrictions 

• For simple 2-variable LPs, we have seen how the graphical method 

can be used to find the optimal solution rather easily 

• To prove that the answer obtained by the graphical method is actually 

optimal, we need more than just drawing diagrams 

• For example, we have obtained (x,y) = (100,100) as the optimal solution 

to the LP 

maximize z = 1000x + 3000y 

subject to x + 2y ≤ 300 

x + 5y ≤ 600 

x ≥ 0 

y ≥ 0 

• How do we know (100,100) is really the optimal solution



• What does it take to show that (x,y) is really an optimal solution 

• We need to show two things: 

• That (x,y) is a feasible solution 

• No other feasible solution gives a better objective function value 

• To show that (x,y) is a feasible solution is easy 

• Just plug the values of x and y into the constraints and show that 

they are all satisfied 

• To show the second is harder



• The method of tight restrictions tries to combine the constraints to 

show that the objective function value must be greater than some value 

p or less than some value q 

• Let’s illustrate this with an example 

maximize 

z = 1000x + 3000y 


x + 5y ≤ 600 

x ≥ 0 

y ≥ 0 

• First note that the point (100,100) gives an objective function value of 

400000 

• If we can combine the constraints to show that 1000x + 3000y ≤ 400000, 

then (100,100) must be optimal



• Let’s multiply the 1st constraint throughout by a nonnegative number a 

ax + 2ay ≤ 300a 

• Let’s multiply the 2nd constraint throughout by a nonnegative number b 

• Adding these two constraints give 

maximize z = 1000x + 3000y 


x + 5y ≤ 600 

x ≥ 0 

y ≥ 0 

bx + 5by ≤ 600b 

(a+b)x + (2a+5b)y ≤ 300a + 600b 

• If we want the left hand side to be 1000x + 3000y we need 

a + b = 1000 and 2a + 5b = 3000



• Solving 

a + b = 1000 and 2a + 5b = 3000 

gives a = 2000/3 and b = 1000/3 

• Hence, plugging these values into 

gives 

(a+b)x + (2a+5b)y ≤ 300a + 600b 

1000x + 3000y ≤ 400000 

• This shows that the objective function value can never exceed 400000 

• Since the point (100,100) gives an objective function value of 400000 

and it is a feasible solution, it must be optimal


Exercise 7 

• Using the method of tight restrictions, show that the solution you 

obtained from Exercise 5 is optimal 

minimize 

z = -2x + y 

subject to -x + y ≥ -1 

-x + 2y ≤ 2 

x ≥ 0 

y ≥ 0


Tight Inequalities 

• An inequality is tight at a point P if equality holds at P 

• Example 

2x + 3y ≤ 6 is tight at the point (0,2) 

Theorem 

If the optimal solution of an LP occurs at an extreme point P, then 

optimality can be proved by combining constraints that are tight at P

The Mathematics of 

Lecture 9: 22 October 2008


The Web Graph 

• Primary focus is mathematics surrounding a real-world network called 

the web graph - denoted W 

• vertices of W represent web pages 

• edges of W represent the links between pages 

• The terms “Internet” and “World Wide Web” are often treated as 

synonymous, but they are actually different 

• Internet is a vast network of smaller interconnected networks, 

exchanging data by certain protocols 

• The World Wide Web consists of information stored and available 

on the Internet 

• Most of these information are web pages in the form of HTML 

documents identified with strings called universal resource 

locators (URL) 

• HTML documents are joined by links, represented as arcs in W


How Big is the Web (Graph) 

• Even a casual surf will convince you that the Web is huge! 

• Rapid growth 

• 1997: 320 million web pages 

• 1999: 800 million web pages 

• 2005: 53.7 billion web pages (34.7 billion indexed by Google)






Searching the Web 

• We have seen the web contains a huge amount of information 

• Traversing the web unaided would be difficult if not impossible 

• Search engines are an invaluable tool 

• Modern search engines exploit the graph structure of the web 

• Link analysis


Overview of Search Engines 

• How does a search engine work 

• A search engine contains the following components 

• Crawler 

• Indexer 

• Query engine


Crawler 

• Program that browses WWW in a methodical, automated fashion 

• The process is called crawling or spidering 

• downloads a copy of all the visited pages for later use by an indexer 

• Characteristics of WWW that makes crawling difficult 

• WWW has large volume 

• can only download a fraction of the web pages in a given time; 

need to prioritize which page to download 

• WWW has fast rate of change 

• by the time the pages of a site are downloaded, new pages may 

have been added, or content of pages may have changed 

• Behaviour of a crawler is governed by its policies 

• Selection policy: which page to download 

• Revisit policy: when to check for changes


Indexer 

• The indexer builds an index from the downloaded web pages 

• The index contains keywords and key phrases that points back to the 

web pages containing them


Query Engine 

• When given input keywords or key phrases, responds in realtime with 

relevant web pages containing keywords or key phrases 

• As part of the query engine, a ranking algorithm attempts to rank web 

pages in order of their relevance to the query


Ranking Algorithm 

• Early search engines rank web pages according to the number of query 

keywords and/or key phrases they contain 

• This is susceptible to keyword spamming 

• A bookstore who wants to attract people to its website will just 

create one that contains lots of the word “book” 

• Modern search engines analyzes the links in W to rank a page 

• Most famous of this is the PageRank algorithm of Google, invented 

by Larry Page and Sergey Brin in 1998


PageRank 

• Assigns a rank to every web page 

• Influences the order in which Google displays search results 

• Based on link structure of W 

• Does not depend on content of web pages 

• Does not depend on query 

• Higher PageRank = more visitors


PageRank 

• Links are the currency of the web 

• Exchanging and buying of links 

• Backlink obsession


PageRank: First Try 

• We want to assign an “importance” to every web page 

• Let xi be the importance of web page i 

• We can just measure the importance of a web page by the number of 

web pages that links to it 

• Example 

2 3 4 

• x1 = 3 

1 

• Given the web graph W, how do we compute the importance of each 

node in W


Adjacency Matrix of a Digraph 

Adjacency Matrix 

• Given a digraph G with n vertices, its adjacency matrix is the n x n 

matrix A whose entry in row i and column j is given by 

aij = 

{ 1, 

if (i, j) is an arc of G 

0, otherwise


Adjacency Matrix of a Digraph 

• Example 

2 3 4 

• Adjacency matrix of the graph above is 

1 

( ) 

0 0 0 0 

1 0 0 0 

1 0 0 0 

1 0 0 0


Exercise 1 

• What is the adjacency matrix of the digraph below 

2 3 4 

1


Matrix Transpose 

Matrix Transpose 

• Given a matrix A, its transpose A T is the matrix created by writing the 

rows of A as the columns of A T . 

• Example


PageRank: First Try 

• Let A be the adjacency matrix of the web graph W 

• What is the sum of entries in the i-th row of A T 

• The number of nodes pointing to node i 

• This is precisely the importance of node (web page) i 

• Computing the importance of every node is just computing 

x = A T J 

where J is the vector of all 1’s


Exercise 2 

• Compute the importance of all the nodes for the “web graph” below 

2 3 4 

1


Exercise 2: Answer 

2 3 4 

) 

1 0 0 0 

• The adjacency matrix for the graph is A = 

0 0 0 0 (1 

( 

0 1 0 1 

) 

0 1 0 

0 0 0 0 

( ( ) ) 

1 

• The transpose of A is A T = 

• A T J = 

0 1 0 1 

0 0 0 0 

1 0 0 1 

0 0 0 0 

) 

1 

1 

1 

1 

0 0 0 0 

1 0 0 1 

= 

(2 

0 

2 

0 

0 0 1 0 

• Importance of 

• node 1 = 2 

• node 2 = 0 

• node 3 = 2 

• node 4 = 0


PageRank: An Improvement 

• The previous definition of “importance” of a web page suffers from a 

problem 

• Can you see what it is 

• To be included in a short list should mean more than being included 

in a long list 

• Suggests that a link from i to j should be weighted by the number of 

pages linked to by i 

• Importance of a node i is the sum of the weights of the links to i


Markov Matrix of a Digraph 

Markov Matrix 

• Given a digraph G with n vertices, its Markov matrix is the n x n matrix 

M whose entry in row i and column j is given by 

mij = 

{ 1/d i, if (i, j) is an arc of G, and di is the outdegree of i 

0, otherwise


Markov Matrix of a Digraph 

• Example 

2 3 4 

• Markov matrix of the graph above is 

1 

( ) 

0 0 1 0 

1 0 0 0 

0 0 0 0 

1/2 0 1/2 0


Exercise 3 

• What is the Markov matrix of the digraph below 

1 2 

3 

4 5 

6



1 2 

3 

4 5 

6 

• The Markov matrix is P = 

0 1/3 1/3 1/3 0 0 

1/2 0 1/2 0 0 0 

1/4 1/4 0 1/4 1/4 0 

1/3 0 0 0 1/3 1/3 

0 1/3 0 1/3 0 1/3 

0 0 0 0 0 0



• Let P be the Markov matrix of the web graph W 

• What is the sum of entries in the i-th row of P T 

• The sum of weights of links pointing to node i 

• This is precisely the importance of node (web page) i 

• Computing the importance of every node is just computing 

x = P T J 

where J is the vector of all 1’s


Exercise 4 

• Compute the importance of all the nodes in the “web graph” 

1 2 

3 

4 5 

6



1 2 

3 

4 5 

0 1/3 1/3 1/3 0 0 

T 

1 

6 

0 1/2 1/4 1/3 0 0 

1 

13/12 

1/2 0 1/2 0 0 0 

1 

1/3 0 1/4 0 1/3 0 

1 

11/12 

1/4 1/4 0 1/4 1/4 0 

1/3 0 0 0 1/3 1/3 

1 

1 

= 

1/3 1/2 0 0 0 0 

1/3 0 1/4 0 1/3 0 

1 

1 

= 

10/12 

11/12 

0 1/3 0 1/3 0 1/3 

1 

0 0 1/4 1/3 0 0 

1 

7/12 

0 0 0 0 0 0 

1 

0 0 0 1/3 1/3 0 

1 

8/12



• The previous definition of “importance” of a web page still suffers from 

a problem 

• Can you see what it is 

• Many fictitious web pages can be created to link to a web page i if 

we want to make web page i important (link spamming) 

• Suggests that a link from i to j should also be weighted by the 

importance of the web page i 

• Previously the weight of a link from i to j is given by 

1/di (di is the outdegree of node i) 

• We now modify this so that the weight of a link from i to j is give by 

xi/di (di is the outdegree of node i) 

• where xi is the importance of node i



• We seem to be running into a circular problem here 

• To determine the importance of a web page, we need to know the 

importance of other web pages 

• Where do we start 

• Linear algebra to the rescue!


Some Linear Algebra 

• Given an n x n matrix A, we can find nonzero vectors x such that 

• Ax = λx 

• x is called an eigenvector of A 

• λ is called an eigenvalue of A 

• To tie x and λ together, we say that x is the eigenvector corresponding 

to λ 

• Example 

A x λ x 

3 1 

1 3 

1 

1 

4 

= = 4 

4 

1 

1 

• 4 is an eigenvalue of A and is an eigenvector corresponding to 4 

1 

1


Eigenvalues & Eigenvectors 

• How can we find the eigenvalues and eigenvectors of a given n x n 

matrix A 

• We want to find λ and x such that Ax = λx 

• This can be rewritten as Ax - λx = 0 or (A - λI)x = 0 

• Turns out that λ is an eigenvalue of A if and only if 

det(A - λI) = 0 

The above equation is called the characteristic equation of A 

Example 

• If A = 3 1 , then det(A - λI) = 3-λ 1 = (3 - λ) 2 - 1 = λ 2 - 6λ + 8 

1 3 

1 3-λ 

• Solving the quadratic equation λ 2 - 6λ + 8 = 0 gives λ = 2 and λ = 4 

• The eigenvalues of A are therefore 2 and 4


Exercise 5 

• Find the eigenvalues of the 3 x 3 matrix B = 

0 -1 -1 

1 2 1 

1 1 2


Eigenvalues & Eigenvectors 

• How do we find eigenvectors corresponding to eigenvalues 

• This is more difficult than finding eigenvalues 

• Eigenvectors are not unique 

• For A = 3 1 , any vector of the form k (k ≠ 0) is an 

1 3 

k 

eigenvector corresponding to the eigenvalue 4 

• Check! 

• If x is an eigenvector of A, so is kx for every k ≠ 0 

• To find eigenvectors, we must solve systems of linear equations


Example 

• Find all the eigenvectors of A = 

3 1 

1 3 

• We know the eigenvalues are 2 and 4 

• Let’s first try to find the eigenvectors corresponding to eigenvalue 2 

• We have to find x such that Ax = 2x 

3 1 

1 3 

x1 

x2 

= 

2x1 

2x2 

 

3x1 + x2 

x1 + 3x2 

= 

2x1 

2x2 

• We get x1 + x2 = 0; if we let x1 = k then x2 = -k 

• The eigenvectors are given by k , where k ≠ 0 

-k


Exercise 6 

• Find the eigenvectors of the 3 x 3 matrix B = 

0 -1 -1 

1 2 1 

1 1 2


Further Properties of Matrices 

• Let A be an m x n matrix and let B be an n x k matrix. Then 

(AB) T = B T A T 

• If A is an n x n matrix, then 

det(A T ) = det(A) 

• If A is an n x n matrix, then A and A T have the same eigenvalues

MAS 801 It's a Discreetly Discrete World Ã¢Â€Â“ Mathematics in Real-life ...

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