Asking Hard Graph Questions. Paul Burkhardt. February 3, 2014


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1 Beyond Watson: Predictive Analytics and Big Data U.S. National Security Agency Research Directorate  R6 Technical Report February 3, 2014
2 300 years before Watson there was Euler! The first (Jeopardy!) graph question? A path crossing each of the Seven Bridges of Königsberg exactly once is not possible because of this type of graph.
3 300 years before Watson there was Euler! The first (Jeopardy!) graph question? A path crossing each of the Seven Bridges of Königsberg exactly once is not possible because of this type of graph. Answer: What is a graph with more than two, odd degree vertices?
4 Asking a question is a search for an answer... Can we find X in Y? This is a graph search problem! What is a graph? A network of pairwise relationships... vertices connected by edges Brain network of C. elegans Watts, Strogatz, Nature 393(6684), 1998
5 Search the Web Graph! Google it! In 1998 Google published the PageRank algorithm... Treat the Web as a Big Graph Web Pages are vertices and hyperlinks are edges... Initialize all pages with a starting rank Imitate web surfer randomly clicking on hyperlinks Random Walk (Markov Chain) on a graph! Rank pages by quantity and quality of links Important/Relevant websites rank higher and websites referenced by important websites rank higher
6 Search the Social Graph! Facebook Graph Search index a social graph of 1 trillion edges complex queries based on the social connections between users in Facebook What questions can it answer? The Facebook Graph Search can answer questions like: which restaurants did my friends like? did people like my comments about the latest movie?
7 Search the Knowledge Graph! Semantic Graphs The meaning of a graph is encoded in the graph nodes are linked by semantics, e.g. dog is a mammal semantics are machinereadable... RDF, OWL, Open Graph Google Knowledge Graph Added to Google search engine in 2012 Microsoft Satori Announced in 2013 for the Microsoft Bing! search engine
8 Graphs are great, so what s the problem? Locality of Reference Conventional implementations expect O(1) random access, but the realworld is different... Random Access Memory (RAM) Typically takes 100 nanoseconds (ns) to load a memory reference.
9 Back to Basics... BreadthFirst Search Search can be a walk on a graph... Traversal by breadthfirst expansion can answer the question: Starting from A can we find K? A B C D E F G H I J K L M N O
10 Back to Basics... BreadthFirst Search Search can be a walk on a graph... Traversal by breadthfirst expansion can answer the question: Starting from A can we find K? A B C D E F G H I J K L M N O
11 Back to Basics... BreadthFirst Search Search can be a walk on a graph... Traversal by breadthfirst expansion can answer the question: Starting from A can we find K? A B C D E F G H I J K L M N O
12 Back to Basics... BreadthFirst Search Search can be a walk on a graph... Traversal by breadthfirst expansion can answer the question: Starting from A can we find K? A B C D E F G H I J K L M N O
13 Back to Basics... BreadthFirst Search Search can be a walk on a graph... Traversal by breadthfirst expansion can answer the question: Starting from A can we find K? A B C D E F G H I J K L M N O
14 Realworld costs for simple BFS Preliminaries (U) For a simple, undirected graph G = (V, E) with n = V vertices and m = E edges where... N(v) = {u V (v, u) E} is the neighborhood of v, d v = N(v) is the degree of v A is the adjacency matrix of G where A T = A Cost of BreadthFirst Search { Θ(2n) Θ() Θ(n + 2m) algorithm storage memory references
15 Why locality matters! Example The cost of BFS on the 2002 Yahoo! Web Graph (n = , m = ): Θ() { Θ(2n) 8 = Θ(n + 2m) = bytes of storage memory references If each memory reference took 100 ns the minimum time to complete BFS would be more than 24 minutes!
16 What s next... Bigger? Big Data begets Big Graphs Big Data challenges conventional algorithms... can t store it all in memory but... disks are 1000x slower need a scalable BreadthFirst Search... An NSA Big Graph experiment Tech Report NSARD v1 1 Petabyte RMAT graph 19.5x more than cluster memory linear performance from 1 trillion to 70 trillion edges... Problem Class Problem Class Memory = 57.6 TB Traversed Edges (trillion) Time (h) Terabytes
17 ... Harder graph questions? Beyond BreadthFirst Search Hard questions aren t always linear time... Example How similar is each vertex to all other vertices in the graph? Realworld use case Find all duplicate or nearduplicate web pages...
18 Similarity on Graphs Vertex Similarity Similarity between a pair of vertices can be defined by the overlap of common neighbors; i.e. structural similarity depends only on adjacency information does not require transitivity, i.e. triangles
19 Jaccard Similarity Jaccard Coefficient Ratio of intersection to union cardinalities of two sets. Properties J ij = N(i) N(j) N(i) N(j) range in [0, 1]; 0 disjoint sets and 1 identical sets nonzero if (i, j) are endpoints of paths of length two, i.e. 2paths Jaccard Distance, 1 J ij, satisfies the triangle inequality
20 Computing exact, allpairs Jaccard Similarity is hard! Cubic upper bound! O(n 3 ) worst case complexity! Count (i, j) pairs where i, j N(v) Let γ ij denote count of (i, j) 2paths and δ ij = d i + d j then, γ ij J ij = δ ij γ ij Runtime complexity Neighbor pairing 1: for all v V do 2: for all {i, j} N(v) do 3: γ ij γ ij + 1 4: end for 5: end for ( ( ) dv ) ( O O 2 v v d 2 v ) ) O (d max d v O(md max ) O(mn) O(n 3 ) v
21 MapReduce Jaccard Similarity (memorybound) Round 1 Map: Identity u, (v, dv ) u, (v, d v ) Reduce: Create ( d v 2 ) ordered pairs of neighbors as compound keys u, {(v, dv ) v N(u)} (v w, w), d v + d w v, w N(u) Round 2 Map: Identity Reduce function: Calculate the Jaccard Coefficient (i, j), {δij, δ ij,...} (i, j), J ij = γ ij /(δ ij γ ij ) δ ij = d i + d j γ ij = {δ ij, δ ij,...}
22 Must loadbalance v ( d(v) ) 2 pair construction Parallel, pairwise combinations construct neighbor pairs for each adjacency set; for each v i N(u) { (u, } (u 1, v 1 ) (u 2, v 2 ) (u 3, v 3 ) (u 4, v 4 ) j), vi (u 1, v 2 ) (u 2, v 3 ) (u 3, v 4 ) j=1..i (u 1, v 3 ) (u 2, v 4 ) (u 1, v 4 ) pair first element in each column with the remaining elements to generate all ( ) d(v) 2 pairs Benefit Enables distributed pair construction in O(1) memory
23 Jaccard Similarity in O(1) rounds and memory Round 1 Map: Identity Reduce: Label edges with ordinal Du, (v i, d vi ) v i N(u) E n (u, i), (vi, d vi ) o i=1..d(u) Round 2 Map: Prepare edges for pairing (u, i), (v, dv ) n (u, j), (v, dv ) o j=1..i Reduce: Create ordered neighbor pairs as compound keys Du, (v, d v ) v N(u) E (v w, w), d v + d w v, w N(u) Round 3 Map: Identity Reduce: Calculate and output J ij (i, j), {δij, δ ij,...} (i, j), J ij = γ ij /(δ ij γ ij ) δ ij = d i + d j γ ij = {δ ij, δ ij,...}
24 Exact, AllPairs Jaccard Similarity benchmarks Experiments Verify scalability on synthetic datasets Graph500 RMAT graphs (A=.57,B=C=.19,D=0.05) n = 2 SCALE, m = 16n Cluster 1000 nodes 12 cores per node 64 GB RAM per node Constantmemory MapReduce Job parameters prestep round to annotate undirected edges with degree, e.g. u, (v, d v ) edge preparation is blockdistributed total of 5 rounds; one additional round inserted to randomize output from round 1
25 AllPairs Jaccard Similarity on Graph500 datasets Graph500 RMAT Graphs (undirected) n (10 6 ) m (10 6 ) 2paths (10 9 ) J ij (10 9 ) RMAT RMAT RMAT RMAT RMAT RMAT Time (minutes) Fast Total Edges Time (minutes) RMAT RMAT RMAT RMAT RMAT RMAT
26 Results Exact, allpairs Jaccard Similarity in O(1) memory and rounds neighbor pairing similar to Node Iterator for triangle listing loadbalance O( v d(v)2 ) pair generation scales well with increasing 2path count MapReduce AllPairs Jaccard Similarity performance 9 billion Jaccard coefficients per minute! (RMAT scale trillion J ij )
27 What s the next Big question? Beyond...?
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