![MAPREDUCE
• 1st mapper:
• 𝑀 𝑖, 𝑗, 𝑀𝑖𝑗 → { 𝛼, 𝛽 ; "M", 𝑖, 𝑗, 𝑀𝑖𝑗 }
where 𝛼 = 𝑖/∆, 𝛽 = 𝑗/∆, where the
∆ represents interval.
𝑉 𝑗, 𝑉𝑗 → { 𝛼, 𝛽 ; ("𝑉", 𝑗, 𝑉𝑗)}
where ∀𝛼 ∈ [0, 𝐺 − 1], 𝛽 = 𝑗/∆,
𝐺 = 𝑐𝑒𝑖𝑙(
𝑁
∆
) represents the group number.
• 1st reducer gets input as:
{ (𝛼, 𝛽); [ "M", 𝑖, 𝑗, 𝑀𝑖𝑗 , ("𝑉", 𝑗, 𝑉𝑗) ] }
∀ 𝑖 ∈ partion 𝛼
∀ 𝑗 ∈ partion 𝛽
• 1st reducer outputs:
{ 𝑖 ; 𝑆 𝛽 = ∀ 𝑗 ∈ partion 𝛽 𝑀𝑖𝑗 ∗ 𝑉𝑗 }
• 2nd mapper: Pass
• 2nd reducer gets input as:
{ 𝑖; [𝑆0, 𝑆1, 𝑆2,… , 𝑆 𝐺−1] }
• 2nd reducer outputs { 𝑖 ; 𝛽=0
𝐺−1
𝑆 𝛽 }](https://image.slidesharecdn.com/ahadoopimplementationofpagerank-160404000339/85/A-hadoop-implementation-of-pagerank-9-320.jpg)
A hadoop implementation of pagerank
- 1. A HADOOP IMPLEMENTATION OF PAGERANK C H E N G E N G M A 2 0 1 6 / 0 2 / 0 2
- 2. HOW DOES GOOGLE FIGHT WITH SPAMMERS ? • The old version search engine usually relies on the information (e.g.,word frequency) shown on each page itself. • A spammer who want to sell hisT-shirt may create his own web page which has words like“movie” 1000 times.But he can make these words invisible by setting the same color as the background. • When you search“movie”,the old search engine will find this page unbelievably important,so you click it and only find his ad forT-shirt. • “While Google was not the first search engine,it was the first able to defeat the spammers who had made search almost useless. ” • The key innovation that Google has introduced is a measurement of web page importance,called PageRank.
- 3. PAGERANK IS ABOUT WEB LINKS. WHY WEB LINKS? • People usually like to add a tag or a link to a page he/she thinks is correct,useful or reliable. • For spammers,they can create their own page as whatever they like,but it’s usually hard for them to ask other pages to link to them. • Even though he can create a link farm where thousands of pages link to one particular page which he want to emphasis, that thousands of pages he has control are still not linked by billions of web pages in the out side of world. For example,a Chinese web user who see the left picture on site will probably add a tag as “MilkTea Beauty” (A Chinese young celebrity whose reputation is disputed).
- 4. WHAT IS PAGERANK?• PageRank is a vector whose j th element is the probability that a random surfer is travelling at the j th web page at the final static state. • At the beginning,you can set each page onto the same value ( Vj=1/N ). Then you multiply the PageRank vectorV with transition matrix M to get the next moment’s probability distribution X. • In the final state,PageRank will converge and vector X will be the same as vectorV. For web that does not contains dead end or spider trap,vectorV now represents the PageRank. A B C D A B C D J: from I: to
- 5. SPIDER TRAP • Once you come to page C, you have no way to leave C. The random surfer get trapped at page C, so that everything becomes not random. • Finally all the PageRank will be taken by page C.
- 6. DEAD END • In the real situation,a page can be a dead end (does not link to any other pages).Once the random surfer comes to a dead end,it stops travelling and has no more chance to go out to other pages,so the random assumption is violated. • The column correspond to it in transition matrix will be an empty column,for the previous definition. • Keeping on multiplying this matrix will leave nothing left.
- 7. TAXATION For loop iterations: 𝑉1 = 𝜌 ∗ 𝑀 ∗ 𝑉0 𝑉1 = 𝑉1 + (1 − 𝑠𝑢𝑚(𝑉1))/𝑁 𝑉0 = 𝑉1 The modified version algorithm: • The modification to solve the above 2 problems is adding a possibility 𝜌 that the surfer will keep on going through the links, so there is (1 − 𝜌) possibility the surfer will teleport to random pages. • This method is called taxation.
- 8. HOWEVER, THE REAL WEB HAS BILLIONS OF PAGES, MULTIPLICATION BETWEEN MATRIX AND VECTOR IS OVERHEAD. • By using partitioned matrix and vector,the calculation can be paralleled onto a computing cluster that has more than thousands of nodes. • And such large a magnitude of computing is usually managed by a mapreduce system, like Hadoop. beta=0 1 2 3 4 alpha=0 1 2 3 4 beta= 0 1 2 3 4
- 9. MAPREDUCE • 1st mapper: • 𝑀 𝑖, 𝑗, 𝑀𝑖𝑗 → { 𝛼, 𝛽 ; "M", 𝑖, 𝑗, 𝑀𝑖𝑗 } where 𝛼 = 𝑖/∆, 𝛽 = 𝑗/∆, where the ∆ represents interval. 𝑉 𝑗, 𝑉𝑗 → { 𝛼, 𝛽 ; ("𝑉", 𝑗, 𝑉𝑗)} where ∀𝛼 ∈ [0, 𝐺 − 1], 𝛽 = 𝑗/∆, 𝐺 = 𝑐𝑒𝑖𝑙( 𝑁 ∆ ) represents the group number. • 1st reducer gets input as: { (𝛼, 𝛽); [ "M", 𝑖, 𝑗, 𝑀𝑖𝑗 , ("𝑉", 𝑗, 𝑉𝑗) ] } ∀ 𝑖 ∈ partion 𝛼 ∀ 𝑗 ∈ partion 𝛽 • 1st reducer outputs: { 𝑖 ; 𝑆 𝛽 = ∀ 𝑗 ∈ partion 𝛽 𝑀𝑖𝑗 ∗ 𝑉𝑗 } • 2nd mapper: Pass • 2nd reducer gets input as: { 𝑖; [𝑆0, 𝑆1, 𝑆2,… , 𝑆 𝐺−1] } • 2nd reducer outputs { 𝑖 ; 𝛽=0 𝐺−1 𝑆 𝛽 }
- 10. BEFORE THE PAGERANK CALCULATING TRANSLATING THE WEB TO NUMBERS • 𝐴 → 𝐵 • 𝐴 → 𝐶 • 𝐴 → 𝐷 • 𝐵 → 𝐴 • 𝐵 → 𝐷 • 𝐶 → 𝐴 • 𝐷 → 𝐵 • 𝐷 → 𝐶 • A 0 • B 1 • C 2 • D 3 LINKS ID • Performing Inner Join twice, where the 1st time’s key is FromNodeID,the 2nd time’s key isToNodeID. • 𝐴, 𝐵, 0 • 𝐴, 𝐶, 0 • 𝐴, 𝐷, 0 • 𝐵, 𝐴, 1 • 𝐵, 𝐷, 1 • 𝐶, 𝐴, 2 • 𝐷, 𝐵, 3 • 𝐷, 𝐶, 3 • 𝐴, 𝐵, 0, 1 • 𝐴, 𝐶, 0, 2 • 𝐴, 𝐷, 0, 3 • 𝐵, 𝐴, 1, 0 • 𝐵, 𝐷, 1, 3 • 𝐶, 𝐴, 2, 0 • 𝐷, 𝐵, 3, 1 • 𝐷, 𝐶, 3, 2 After 1st inner join After 2nd inner join After the PageRank is calculated,the same thing can be done to translate index back to node names. From Node ID To Node ID Web Node ID in data Index used in program
- 11. 2002 GOOGLE PROGRAMMING CONTEST WEB GRAPH DATA • 875713 pages, 5105039 edges • 72 MB txt file • Hadoop program iterates 75 times (“For the web itself, 50-75 iterations are sufficient to converge to within the error limits of double precision”). • 𝜌 = 0.85 as the possibility to follow the web links and 0.15 possibility to teleport. • The program has a structure of for loop, each of which has 4 map-reduce job inside. • The first 2 MR job are for matrix multiplying vector. • The 3rd MR job is to calculate the sum of the product vector beta*M*V. • And the final MR job does the shifting.
- 12. PAGERANK RESULT • A Python program is written to compare the result from Hadoop:
- 13. RESULT ANALYSIS • The value not sorted is noisy and hard to see. • But sorting by PageRank value and plotting in log-log provides a linear line.
- 14. RESULT ANALYSIS • The histogram has exponentially decaying counts for large PageRankvalue. • The largest 1/9 web pages contains 60% of PageRank importance over the whole dataset.
- 15. REFERENCE • Mining of Massive Datasets, Chapter 5 Jure Leskovec, Anand Rajaraman and Jeffrey D. Ullman The code will be attached as the following files. FINALLY, A TOP K PROGRAM IN HADOOP • 1st column is the index used in this program; • 2nd column is the web node ID within the original data; • 3rd column is the PageRank value. The right table shows the top 15 PageRank value.