XXL Graph Algorithms__HadoopSummit2010

XXL Graph Algorithms Sergei Vassilvitskii Yahoo! Research With help from Jake Hofman, Siddharth Suri, Cong Yu and many others

Introduction XXL Graphs are everywhere: – Web graph – Friend graphs – Advertising graphs... 2

Introduction XXL Graphs are everywhere: – Web graph – Friend graphs – Advertising graphs... But we have Hadoop! – Few algorithms have been ported (no Hadoop Algorithms book) – Few general algorithmic approaches – Active area of research 3

Outline Today: – Act 1: Crawl before you walk • Counting connected components – Act 2: The curse of the last reducer • Finding tight knit friend groups 4

Act 1: Connected Components Given a graph, how many components does it have? f b a g c e h d 5

Act 1: Connected Components Given a graph, how many components does it have? f b (b,c) 1 a (f,h) 1 g (b,d) 1 (a,c) 1 (a,b) 1 (c,d) 1 c (c,e) 1 (f,g) 1 e h (d,e) 1 (d,e) 1 d (b,e) 1 (g,h) 1 Data too big to fit on one reducer! 6

CC Overview Outline for Connected Components – Partition the input into several chunks (map 1) – Summarize the connectivity on each chunk (reduce 1) – Combine all of the (small) summaries (map 2) – Find the number of connected components 7

Connected Components 1. Partition (randomly): f b a g c e h d 8

Connected Components 1. Partition (randomly): f b b a g c c e h d Reduce 1 Reduce 2 9

Connected Components 1. Partition: 2. Summarize (retain < n edges): f b b a g c c e h d Reduce 1 Reduce 2 10

Connected Components 1. Partition: 2. Summarize (retain < n edges): f b b a g c c e h d Reduce 1 Reduce 2 11

Connected Components 1. Partition: 2. Summarize: 3. Recombine: f b b a g c c e h d Reduce 1 Reduce 2 12

Connected Components 1. Partition: 2. Summarize: 3. Recombine: b f a g c e h d Round 2 13

Connected Components 1. Partition: 2. Summarize: 3. Recombine: b f (b,c) 1 a (f,h) 1 (b,d) 1 g (a,c) 1 (a,b) 1 (c,d) 1 c (c,e) 1 (f,g) 1 (d,e) 1 e h (d,e) 1 (b,e) 1 d (g,h) 1 Round 2 14

Connected Components 1. Partition: 2. Summarize: 3. Recombine: b f a g (a,c) 1 (a,b) 1 (c,d) 1 c (f,g) 1 e h (d,e) 1 d (g,h) 1 Round 2 Small enough to fit! 15

Connected Components The summarization does not affect connectivity – Drops redundant edges – Dramatically reduces data size – Takes two MapReduce rounds 16

Connected Components The summarization does not affect connectivity – Drops redundant edges – Dramatically reduces data size – Takes two MapReduce rounds Similar approach works in other situations: – Consider vertices connected only if k edges between vertices – Consider vertices connected if similarity score above a threshold • E.g. approximate Jaccard similarity when computing for recommendation systems – Find minimum spanning trees • Summarize by computing an MST on the subset graph – Clustering • Cluster each partition, then aggregate the clusters 17

Outline Today: – Act 1: Crawl before you walk • Counting connected components – Act 2: The curse of the last reducer • Finding tight knit friend groups 18

Act 2: Clustering Coefficient Finding tight knit groups of friends 19

Act 2: Clustering Coefficient Finding tight knit groups of friends vs. 19

Act 2: Clustering Coefficient Finding tight knit groups of friends vs. 2/15 ≈ 0.13 8/15 ≈ 0.53 CC(v) = Fraction of v’s friends who know each other – Count: number of triangles incident on v 20

Finding CC For Each Node Attempt 1: – Look at each node – Enumerate all possible triangles (Pivot) 21

Finding CC For Each Node Attempt 1: – Look at each node – Enumerate all possible triangles (Pivot) 22

Finding CC For Each Node Attempt 1: – Look at each node – Enumerate all possible triangles (Pivot) – Check which of those edges exist: ∩ = 15 edges possible 2 edges present 23

Finding CC For Each Node Attempt 1: – Look at each node – Enumerate all possible triangles (Pivot) – Check which of those edges exist 24

Finding CC For Each Node Attempt 1: – Look at each node – Enumerate all possible triangles – Check which of those edges exist Amount of intermediate data – Quadratic in the degree of the nodes – 6 friends: 15 possible triangles – n friends, n(n-1)/2 possible triangles 25

Finding CC For Each Node Attempt 1: – Look at each node – Enumerate all possible triangles – Check which of those edges exist Amount of intermediate data – Quadratic in the degree of the nodes – 6 friends: 15 possible triangles – n friends, n(n-1)/2 possible triangles There’s always “that guy”: – tens of thousands of friends – tens of thousands of movie ratings (really!) – millions of followers 26

Finding CC For Each Node Attempt 1: – Look at each node a le Sc triangles ot – Enumerate all possible sn oe – Check which of those edges exist D 27

Finding CC For Each Node Attempt 1: – Look at each node a le Sc triangles ot – Enumerate all possible sn oe – Check which of those edges exist D Attempt 2: – There is a limited number of High degree nodes – Count LLL, LLH, LHH, and HHH triangles differently – If a triangle has at least one Low node – Pivot on Low node to count the triangles – If a triangle has all High nodes – Pivot but only on other neighboring High nodes (not all nodes) 28

Algorithm in Pictures When looking at Low degree nodes – Check for all triangles 29

Algorithm in Pictures When looking at Low degree nodes – Check for all triangles When looking at High degree nodes – Check for triangles with other High degree nodes 30

Clustering Coefficient Discussion Attempt 2: – Main idea: treat High and Low degree nodes differently • Limit the amount of data generated (No more than O(n) per node) – All triangles accounted for – Can set High-Low threshold to balance the two cases • Rule of thumb: threshold around square root of number of vertices – A bit more complex, but still easy to code • Doesn’t suffer from the one high degree node problem 31

XXL Graphs: Conclusions Algorithm Design – Prove performance guarantees independent of input data • Input skew (e.g. high degree nodes) should not severely affect algorithm performance • Number of rounds fixed (and hopefully small) 32

XXL Graphs: Conclusions Algorithm Design – Prove performance guarantees independent of input data • Input skew (e.g. high degree nodes) should not severely affect algorithm performance • Number of rounds fixed (and hopefully small) Rethink graph algorithms: – Connected Components: Two round approach – Clustering Coefficient: High-Low node decomposition – (Breaking News) Matchings: Two round sampling technique 33

Thank You sergei@yahoo-inc.com

XXL Graph Algorithms__HadoopSummit2010

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