Google: An Application of Linear Algebra (The mathematics of a great success) Peter Butkovic Peter Butkovic Google: An Application of Linear Algebra Peter Butkovic Google: An Application of Linear Algebra Peter Butkovic Google: An Application of Linear Algebra Hyun’sMap of the Web Peter Butkovic Google: An Application of Linear Algebra Peter Butkovic Google: An Application of Linear Algebra Peter Butkovic Google: An Application of Linear Algebra Founders of Google: Larry Page and Sergey Brin 1995: Research students at Stanford 1996: Started a student project on search engines 1998: Google incorporates as a company (initial investment: $1.1million) and …les patent for PageRank Google (~googol = 10100) Peter Butkovic Google: An Application of Linear Algebra 1995: Research students at Stanford 1996: Started a student project on search engines 1998: Google incorporates as a company (initial investment: $1.1million) and …les patent for PageRank Google (~googol = 10100) Founders of Google: Larry Page and Sergey Brin Peter Butkovic Google: An Application of Linear Algebra 1996: Started a student project on search engines 1998: Google incorporates as a company (initial investment: $1.1million) and …les patent for PageRank Google (~googol = 10100) Founders of Google: Larry Page and Sergey Brin 1995: Research students at Stanford Peter Butkovic Google: An Application of Linear Algebra 1998: Google incorporates as a company (initial investment: $1.1million) and …les patent for PageRank Google (~googol = 10100) Founders of Google: Larry Page and Sergey Brin 1995: Research students at Stanford 1996: Started a student project on search engines Peter Butkovic Google: An Application of Linear Algebra Google (~googol = 10100) Founders of Google: Larry Page and Sergey Brin 1995: Research students at Stanford 1996: Started a student project on search engines 1998: Google incorporates as a company (initial investment: $1.1million) and …les patent for PageRank Peter Butkovic Google: An Application of Linear Algebra 2001: Patent granted 2004: Total capital reached $23billion 2007: Google web search engine covers 53% of the market (Yahoo 20%) 2008: Google uses 450,000 servers in 25 locations around the world to index > 8 billion websites 2000: Selling advertisements began Peter Butkovic Google: An Application of Linear Algebra 2004: Total capital reached $23billion 2007: Google web search engine covers 53% of the market (Yahoo 20%) 2008: Google uses 450,000 servers in 25 locations around the world to index > 8 billion websites 2000: Selling advertisements began 2001: Patent granted Peter Butkovic Google: An Application of Linear Algebra 2007: Google web search engine covers 53% of the market (Yahoo 20%) 2008: Google uses 450,000 servers in 25 locations around the world to index > 8 billion websites 2000: Selling advertisements began 2001: Patent granted 2004: Total capital reached $23billion Peter Butkovic Google: An Application of Linear Algebra 2008: Google uses 450,000 servers in 25 locations around the world to index > 8 billion websites 2000: Selling advertisements began 2001: Patent granted 2004: Total capital reached $23billion 2007: Google web search engine covers 53% of the market (Yahoo 20%) Peter Butkovic Google: An Application of Linear Algebra 2000: Selling advertisements began 2001: Patent granted 2004: Total capital reached $23billion 2007: Google web search engine covers 53% of the market (Yahoo 20%) 2008: Google uses 450,000 servers in 25 locations around the world to index > 8 billion websites Peter Butkovic Google: An Application of Linear Algebra A hyperlink is a "recommendation": A webpage is important if other webpages point to it (via hyperlinks) The PageRank of the recommender matters (the higher the better) The generosity of the recommender matters (the higher the worse) KEY IDEAS Each webpage is assigned a measure of importance ... PageRank Peter Butkovic Google: An Application of Linear Algebra The PageRank of the recommender matters (the higher the better) The generosity of the recommender matters (the higher the worse) KEY IDEAS Each webpage is assigned a measure of importance ... PageRank A hyperlink is a "recommendation": A webpage is important if other webpages point to it (via hyperlinks) Peter Butkovic Google: An Application of Linear Algebra The generosity of the recommender matters (the higher the worse) KEY IDEAS Each webpage is assigned a measure of importance ... PageRank A hyperlink is a "recommendation": A webpage is important if other webpages point to it (via hyperlinks) The PageRank of the recommender matters (the higher the better) Peter Butkovic Google: An Application of Linear Algebra KEY IDEAS Each webpage is assigned a measure of importance ... PageRank A hyperlink is a "recommendation": A webpage is important if other webpages point to it (via hyperlinks) The PageRank of the recommender matters (the higher the better) The generosity of the recommender matters (the higher the worse) Peter Butkovic Google: An Application of Linear Algebra Peter Butkovic Google: An Application of Linear Algebra r (P2) = 1 1 2 r (P1) + 3 r (P3) 1 r (P3) = 2 r (P1) 1 r (P4) = 2 r (P5) + r (P6) r (P5) = 1 1 3 r (P3) + 2 r (P4) r (P6) = 1 1 2 r (P4) + 2 r (P5) A.N.Langville and C.D.Meyer: Google’sPageRank and Beyond (PUP 2006) 1 2 1 r (P1) = 3 r (P3) 3 6 5 4 Node 2 - dangling Peter Butkovic Google: An Application of Linear Algebra 1 r (P3) = 2 r (P1) 1 r (P4) = 2 r (P5) + r (P6) r (P5) = 1 1 3 r (P3) + 2 r (P4) r (P6) = 1 1 2 r (P4) + 2 r (P5) A.N.Langville and C.D.Meyer: Google’sPageRank and Beyond (PUP 2006) 1 2 1 r (P1) = 3 r (P3) 3 r (P2) = 1 1 2 r (P1) + 3 r (P3) 6 5 4 Node 2 - dangling Peter Butkovic Google: An Application of Linear Algebra 1 r (P4) = 2 r (P5) + r (P6) r (P5) = 1 1 3 r (P3) + 2 r (P4) r (P6) = 1 1 2 r (P4) + 2 r (P5) A.N.Langville and C.D.Meyer: Google’sPageRank and Beyond (PUP 2006) 1 2 1 r (P1) = 3 r (P3) 3 r (P2) = 1 1 2 r (P1) + 3 r (P3) 1 r (P3) = 2 r (P1) 6 5 4 Node 2 - dangling Peter Butkovic Google: An Application of Linear Algebra r (P5) = 1 1 3 r (P3) + 2 r (P4) r (P6) = 1 1 2 r (P4) + 2 r (P5) A.N.Langville and C.D.Meyer: Google’sPageRank and Beyond (PUP 2006) 1 2 1 r (P1) = 3 r (P3) 3 r (P2) = 1 1 2 r (P1) + 3 r (P3) 1 r (P3) = 2 r (P1) 1 r (P4) = r (P5) + r (P6) 6 5 2 4 Node 2 - dangling Peter Butkovic Google: An Application of Linear Algebra r (P6) = 1 1 2 r (P4) + 2 r (P5) A.N.Langville and C.D.Meyer: Google’sPageRank and Beyond (PUP 2006) 1 2 1 r (P1) = 3 r (P3) 3 r (P2) = 1 1 2 r (P1) + 3 r (P3) 1 r (P3) = 2 r (P1) 1 r (P4) = r (P5) + r (P6) 6 5 2 r (P5) = 1 1 3 r (P3) + 2 r (P4) 4 Node 2 - dangling Peter Butkovic Google: An Application of Linear Algebra A.N.Langville and C.D.Meyer: Google’sPageRank and Beyond (PUP 2006) 1 2 1 r (P1) = 3 r (P3) 3 r (P2) = 1 1 2 r (P1) + 3 r (P3) 1 r (P3) = 2 r (P1) 1 r (P4) = r (P5) + r (P6) 6 5 2 r (P5) = 1 1 3 r (P3) + 2 r (P4) r (P6) = 1 1 4 2 r (P4) + 2 r (P5) Node 2 - dangling Peter Butkovic Google: An Application of Linear Algebra r0 r1 ! r1 r2 ! ... rk rk+1 ! ... 1 1 r0 = n , n , ... Peter Butkovic Google: An Application of Linear Algebra r1 r2 ! ... rk rk+1 ! ... 1 1 r0 = n , n , ... r0 r1 ! Peter Butkovic Google: An Application of Linear Algebra ... rk rk+1 ! ... 1 1 r0 = n , n , ... r0 r1 ! r1 r2 ! Peter Butkovic Google: An Application of Linear Algebra rk rk+1 ! ... 1 1 r0 = n , n , ... r0 r1 ! r1 r2 ! ... Peter Butkovic Google: An Application of Linear Algebra ... 1 1 r0 = n , n , ... r0 r1 ! r1 r2 ! ... rk rk+1 ! Peter Butkovic Google: An Application of Linear Algebra 1 1 r0 = n , n , ... r0 r1 ! r1 r2 ! ... rk rk+1 ! ... Peter Butkovic Google: An Application of Linear Algebra 1 1 r1 (P1) = 3 r0 (P3) = 18 r1 (P2) = 1 1 5 2 r0 (P1) + 3 r0 (P3) = 36 1 1 r1 (P3) = 2 r0 (P1) = 12 r1 (P4) = 1 1 2 r0 (P5) + r0 (P6) = 4 r1 (P5) = 1 1 5 3 r0 (P3) + 2 r0 (P4) = 36 r1 (P6) = 1 1 1 2 r0 (P4) + 2 r0 (P5) = 6 1 1 1 1 1 1 1 2 r0 = 6 , 6 , 6 , 6 , 6 , 6 3 6 5 4 Peter Butkovic Google: An Application of Linear Algebra r1 (P2) = 1 1 5 2 r0 (P1) + 3 r0 (P3) = 36 1 1 r1 (P3) = 2 r0 (P1) = 12 r1 (P4) = 1 1 2 r0 (P5) + r0 (P6) = 4 r1 (P5) = 1 1 5 3 r0 (P3) + 2 r0 (P4) = 36 r1 (P6) = 1 1 1 2 r0 (P4) + 2 r0 (P5) = 6 1 1 1 1 1 1 1 2 r0 = 6 , 6 , 6 , 6 , 6 , 6 1 1 r1 (P1) = 3 r0 (P3) = 18 3 6 5 4 Peter Butkovic Google: An Application of Linear Algebra 1 1 r1 (P3) = 2 r0 (P1) = 12 r1 (P4) = 1 1 2 r0 (P5) + r0 (P6) = 4 r1 (P5) = 1 1 5 3 r0 (P3) + 2 r0 (P4) = 36 r1 (P6) = 1 1 1 2 r0 (P4) + 2 r0 (P5) = 6 1 1 1 1 1 1 1 2 r0 = 6 , 6 , 6 , 6 , 6 , 6 1 1 r1 (P1) = 3 r0 (P3) = 18 r1 (P2) = 3 1 1 5 2 r0 (P1) + 3 r0 (P3) = 36 6 5 4 Peter Butkovic Google: An Application of Linear Algebra r1 (P4) = 1 1 2 r0 (P5) + r0 (P6) = 4 r1 (P5) = 1 1 5 3 r0 (P3) + 2 r0 (P4) = 36 r1 (P6) = 1 1 1 2 r0 (P4) + 2 r0 (P5) = 6 1 1 1 1 1 1 1 2 r0 = 6 , 6 , 6 , 6 , 6 , 6 1 1 r1 (P1) = 3 r0 (P3) = 18 r1 (P2) = 3 1 1 5 2 r0 (P1) + 3 r0 (P3) = 36 1 1 r1 (P3) = 2 r0 (P1) = 12 6 5 4 Peter Butkovic Google: An Application of Linear Algebra r1 (P5) = 1 1 5 3 r0 (P3) + 2 r0 (P4) = 36 r1 (P6) = 1 1 1 2 r0 (P4) + 2 r0 (P5) = 6 1 1 1 1 1 1 1 2 r0 = 6 , 6 , 6 , 6 , 6 , 6 1 1 r1 (P1) = 3 r0 (P3) = 18 r1 (P2) = 3 1 1 5 2 r0 (P1) + 3 r0 (P3) = 36 1 1 r1 (P3) = 2 r0 (P1) = 12 r1 (P4) = 1 r (P ) + r (P ) = 1 6 5 2 0 5 0 6 4 4 Peter Butkovic Google: An Application of Linear Algebra r1 (P6) = 1 1 1 2 r0 (P4) + 2 r0 (P5) = 6 1 1 1 1 1 1 1 2 r0 = 6 , 6 , 6 , 6 , 6 , 6 1 1 r1 (P1) = 3 r0 (P3) = 18 r1 (P2) = 3 1 1 5 2 r0 (P1) + 3 r0 (P3) = 36 1 1 r1 (P3) = 2 r0 (P1) = 12 r1 (P4) = 1 r (P ) + r (P ) = 1