ASAP: Fast, Approximate Graph Pattern Mining at Scale

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ASAP: Fast, Approximate Graph Pattern Mining at Scale Anand Iyer
⋆, Zaoxing Liu ⬩, Xin Jin⬩, Shivaram Venkataraman✢, Vladimir Braverman⬩, Ion Stoica ⋆ ⋆UC Berkeley ⬩Johns Hopkins University ✢University of Wisconsin & Microsoft OSDI, October 10, 2018

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Graphs popular in big data analytics Social networks

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Graphs popular in big data analytics Metabolic network of a
single cell organism Social networks

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Graphs popular in big data analytics Metabolic network of a
single cell organism Social networks Tuberculosis

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Graphs popular in big data analytics *“The Ubiquity of Large
Graphs and Surprising Challenges of Graph Processing” ,Sahu et. al, VLDB 2018 (best paper) Also popular in traditional enterprises*

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Graphs popular in big data analytics Products and customers *“The
Ubiquity of Large Graphs and Surprising Challenges of Graph Processing” ,Sahu et. al, VLDB 2018 (best paper) Also popular in traditional enterprises* P P P P P

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Graphs popular in big data analytics Products and customers *“The
Ubiquity of Large Graphs and Surprising Challenges of Graph Processing” ,Sahu et. al, VLDB 2018 (best paper) Transactions and involved entities Also popular in traditional enterprises* P P P P P D D D D D W W D

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Graphs popular in big data analytics Products and customers Which
(classes of) products are frequently bought together? *“The Ubiquity of Large Graphs and Surprising Challenges of Graph Processing” ,Sahu et. al, VLDB 2018 (best paper) Transactions and involved entities Also popular in traditional enterprises* P P P P P D D D D D W W D

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(classes of) products are frequently bought together? *“The Ubiquity of Large Graphs and Surprising Challenges of Graph Processing” ,Sahu et. al, VLDB 2018 (best paper) Transactions and involved entities Also popular in traditional enterprises* Small deposits followed by large withdrawal P P P P P D D D D D W W D

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(classes of) products are frequently bought together? *“The Ubiquity of Large Graphs and Surprising Challenges of Graph Processing” ,Sahu et. al, VLDB 2018 (best paper) Transactions and involved entities Also popular in traditional enterprises* Small deposits followed by large withdrawal P P P P P P P D D D D D W W D

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(classes of) products are frequently bought together? *“The Ubiquity of Large Graphs and Surprising Challenges of Graph Processing” ,Sahu et. al, VLDB 2018 (best paper) Transactions and involved entities Also popular in traditional enterprises* Small deposits followed by large withdrawal P P P P P P P D D D D D W W D

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Graph Pattern Mining Discover structural patterns in the underlying graph

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Graph Pattern Mining Motifs Cliques Discover structural patterns in the
underlying graph Frequent Subgraphs

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underlying graph Frequent Subgraphs Standard approach: Iterative expansion

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underlying graph Frequent Subgraphs 0 1 2 3 Standard approach: Iterative expansion

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underlying graph Frequent Subgraphs 0 1 2 3 0 1 2 3 Standard approach: Iterative expansion

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underlying graph Frequent Subgraphs 0 1 2 3 0 1 2 3 0 1 0 2 0 3 1 2 1 0 2 3 2 0 2 1 3 1 3 1 3 2 1 3 Standard approach: Iterative expansion

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underlying graph Frequent Subgraphs Huge intermediate data Quickly intractable in large graphs 0 1 2 3 0 1 2 3 0 1 0 2 0 3 1 2 1 0 2 3 2 0 2 1 3 1 3 1 3 2 1 3 Standard approach: Iterative expansion

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underlying graph Frequent Subgraphs Huge intermediate data Quickly intractable in large graphs 0 1 2 3 0 1 2 3 0 1 0 2 0 3 1 2 1 0 2 3 2 0 2 1 3 1 3 1 3 2 1 3 Standard approach: Iterative expansion Challenging to mine patterns in large graphs

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Graph Pattern Mining Log scale # Edges Computation Time

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Graph Pattern Mining Log scale # Edges Computation Time Arabesque
(SOSP ‘15)

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Graph Pattern Mining ~1 billion 11 hours Motifs with size
= 3 Log scale # Edges Computation Time Arabesque (SOSP ‘15)

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Graph Pattern Mining 150 s 1.5 billion ~1 billion 11
hours Motifs with size = 3 This work: Log scale # Edges Computation Time Arabesque (SOSP ‘15)

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hours Motifs with size = 3 This work: Log scale # Edges Computation Time Arabesque (SOSP ‘15) 258x faster

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hours Motifs with size = 3 This work: Log scale # Edges Computation Time Arabesque (SOSP ‘15) 258x faster 5x less CPU & Memory

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hours Motifs with size = 3 This work: Log scale # Edges Computation Time Arabesque (SOSP ‘15) 258x faster 5x less CPU & Memory <5% error

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Many mining tasks do not need exact answers

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Leverage approximation for pattern mining Many mining tasks do not
need exact answers

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General approach: Apply algorithm on subset(s) (sample) of the input
data Approximate Analytics

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data Approximate Analytics 0 1 4 2 3 graph

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data Approximate Analytics 0 1 4 2 3 edge sampling (p=0.5) graph 0 1 4 2 3

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data Approximate Analytics 0 1 4 2 3 edge sampling (p=0.5) graph e = 1 0 1 4 2 3 triangle counting

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data Approximate Analytics 0 1 4 2 3 edge sampling (p=0.5) graph e = 1 0 1 4 2 3 triangle counting result $ % 2 = 2

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Answer: 10 General approach: Apply algorithm on subset(s) (sample) of
the input data Approximate Analytics 0 1 4 2 3 edge sampling (p=0.5) graph e = 1 0 1 4 2 3 triangle counting result $ % 2 = 2

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the input data Approximate Analytics 0 1 4 2 3 edge sampling (p=0.5) graph e = 1 0 1 4 2 3 triangle counting result $ % 2 = 2 0 20 40 60 80 100 0 10 20 30 40 50 60 70 80 90 0 2 4 6 8 10 12 Error (%) Speedup Edges Dropped (%) Error Speedup

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the input data Approximate Analytics 0 1 4 2 3 edge sampling (p=0.5) graph e = 1 0 1 4 2 3 triangle counting result $ % 2 = 2 Applying exact algorithm on sampled graph(s) not the right approach for pattern mining

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ASAP leverages existing work in graph approximation theory and makes
it practical

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Graph Pattern Mining Theory Sample instances of the pattern from
the graph stream

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Graph Pattern Mining Theory 0 1 4 2 3 graph
edge stream: (0,1), (0,2), (0,3), (0,4), (1,2), (1,3), (1,4), (2,3), (2,4), (3,4) Sample instances of the pattern from the graph stream

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E0 Graph Pattern Mining Theory 0 1 4 2 3
graph edge stream: (0,1), (0,2), (0,3), (0,4), (1,2), (1,3), (1,4), (2,3), (2,4), (3,4) Sample instances of the pattern from the graph stream

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! = 1 10 ∗ 1 4 E0 Graph Pattern
Mining Theory 0 1 4 2 3 graph edge stream: (0,1), (0,2), (0,3), (0,4), (1,2), (1,3), (1,4), (2,3), (2,4), (3,4) Sample instances of the pattern from the graph stream

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Mining Theory 0 1 4 2 3 graph edge stream: (0,1), (0,2), (0,3), (0,4), (1,2), (1,3), (1,4), (2,3), (2,4), (3,4) '( = 40 Sample instances of the pattern from the graph stream

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Mining Theory 0 1 4 2 3 graph E1 E2 E3 edge stream: (0,1), (0,2), (0,3), (0,4), (1,2), (1,3), (1,4), (2,3), (2,4), (3,4) '( = 40 Sample instances of the pattern from the graph stream

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Mining Theory 0 1 4 2 3 graph E1 E2 E3 edge stream: (0,1), (0,2), (0,3), (0,4), (1,2), (1,3), (1,4), (2,3), (2,4), (3,4) '( = 40 result 1 ) * +,( -./ '+ = 10 '/ = 0 '0 = 0 '1 = 0 Sample instances of the pattern from the graph stream

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Mining Theory 0 1 4 2 3 estimator (r=4) neighborhood sampling graph E1 E2 E3 edge stream: (0,1), (0,2), (0,3), (0,4), (1,2), (1,3), (1,4), (2,3), (2,4), (3,4) '( = 40 result 1 ) * +,( -./ '+ = 10 '/ = 0 '0 = 0 '1 = 0 Pavan et al. Counting and sampling triangles from a graph stream, VLDB 2013 Sample instances of the pattern from the graph stream

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A Swift Approximate Pattern miner

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graphA.patterns(“a->b->c”, “100s”) graphB.fourClique(“5.0%”,“95.0%”) 1 A Swift Approximate Pattern miner

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… Graphs stored on disk or main memory graphA.patterns(“a->b->c”, “100s”)
graphB.fourClique(“5.0%”,“95.0%”) 1 Apache Spark Generalized Approximate Pattern Mining 2 A Swift Approximate Pattern miner

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graphB.fourClique(“5.0%”,“95.0%”) 1 Estimator Count Selection 3 Apache Spark Generalized Approximate Pattern Mining 2 A Swift Approximate Pattern miner

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graphB.fourClique(“5.0%”,“95.0%”) 1 Estimator Count Selection 3 1 2 3 0 0.5M 1M 1.5M 2.1M Runtime (min) No. of Estimators Twitter Graph Profiling 0 5 10 15 20 25 30 35 40 0 0.5m 1m 1.5m 2.1m Error Rate (%) No. of Estimators Twitter Graph Profiling 1 2 3 0 0.5M 1M 1.5M 2.1M Runtime (min) No. of Estimators Twitter Graph Profiling 0 5 10 15 20 25 30 35 40 0 0.5m 1m 1.5m 2.1m Error Rate (%) No. of Estimators Twitter Graph Profiling 4 Apache Spark Generalized Approximate Pattern Mining 2 A Swift Approximate Pattern miner

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graphB.fourClique(“5.0%”,“95.0%”) 1 Estimator Count Selection 3 1 2 3 0 0.5M 1M 1.5M 2.1M Runtime (min) No. of Estimators Twitter Graph Profiling 0 5 10 15 20 25 30 35 40 0 0.5m 1m 1.5m 2.1m Error Rate (%) No. of Estimators Twitter Graph Profiling 1 2 3 0 0.5M 1M 1.5M 2.1M Runtime (min) No. of Estimators Twitter Graph Profiling 0 5 10 15 20 25 30 35 40 0 0.5m 1m 1.5m 2.1m Error Rate (%) No. of Estimators Twitter Graph Profiling 4 Error-Latency Profile (ELP) 5 Apache Spark Generalized Approximate Pattern Mining 2 A Swift Approximate Pattern miner

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graphB.fourClique(“5.0%”,“95.0%”) 1 Estimator Count Selection 3 1 2 3 0 0.5M 1M 1.5M 2.1M Runtime (min) No. of Estimators Twitter Graph Profiling 0 5 10 15 20 25 30 35 40 0 0.5m 1m 1.5m 2.1m Error Rate (%) No. of Estimators Twitter Graph Profiling 1 2 3 0 0.5M 1M 1.5M 2.1M Runtime (min) No. of Estimators Twitter Graph Profiling 0 5 10 15 20 25 30 35 40 0 0.5m 1m 1.5m 2.1m Error Rate (%) No. of Estimators Twitter Graph Profiling 4 Estimates:{error: <5%, time: 95s} Estimates:{error: <5%, time: 60s} 6 Error-Latency Profile (ELP) 5 Apache Spark Generalized Approximate Pattern Mining 2 A Swift Approximate Pattern miner

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graphB.fourClique(“5.0%”,“95.0%”) 1 Estimator Count Selection 3 1 2 3 0 0.5M 1M 1.5M 2.1M Runtime (min) No. of Estimators Twitter Graph Profiling 0 5 10 15 20 25 30 35 40 0 0.5m 1m 1.5m 2.1m Error Rate (%) No. of Estimators Twitter Graph Profiling 1 2 3 0 0.5M 1M 1.5M 2.1M Runtime (min) No. of Estimators Twitter Graph Profiling 0 5 10 15 20 25 30 35 40 0 0.5m 1m 1.5m 2.1m Error Rate (%) No. of Estimators Twitter Graph Profiling 4 Estimates:{error: <5%, time: 95s} Estimates:{error: <5%, time: 60s} 6 count: 21453 +/- 14 confidence: 95%, time: 92s Embeddings (optional) 7 Error-Latency Profile (ELP) 5 Apache Spark Generalized Approximate Pattern Mining 2 A Swift Approximate Pattern miner

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Graph updates … … Graphs stored on disk or main
memory graphA.patterns(“a->b->c”, “100s”) graphB.fourClique(“5.0%”,“95.0%”) 1 Estimator Count Selection 3 1 2 3 0 0.5M 1M 1.5M 2.1M Runtime (min) No. of Estimators Twitter Graph Profiling 0 5 10 15 20 25 30 35 40 0 0.5m 1m 1.5m 2.1m Error Rate (%) No. of Estimators Twitter Graph Profiling 1 2 3 0 0.5M 1M 1.5M 2.1M Runtime (min) No. of Estimators Twitter Graph Profiling 0 5 10 15 20 25 30 35 40 0 0.5m 1m 1.5m 2.1m Error Rate (%) No. of Estimators Twitter Graph Profiling 4 Estimates:{error: <5%, time: 95s} Estimates:{error: <5%, time: 60s} 6 count: 21453 +/- 14 confidence: 95%, time: 92s Embeddings (optional) 7 Error-Latency Profile (ELP) 5 Apache Spark Generalized Approximate Pattern Mining 2 A Swift Approximate Pattern miner

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Error-Latency Profile (ELP) Apache Spark Generalized Approximate Pattern Mining graphA.patterns(“a->b->c”,
“100s”) graphB.fourClique(“5.0%”,“95.0%”) Estimator Count Selection … Graphs stored on disk or main memory Estimates:{error: <5%, time: 95s} Estimates:{error: <5%, time: 60s} … Graph updates 1 2 3 0 0.5M 1M 1.5M 2.1M Runtime (min) No. of Estimators Twitter Graph Profiling 0 5 10 15 20 25 30 35 40 0 0.5m 1m 1.5m 2.1m Error Rate (%) No. of Estimators Twitter Graph Profiling 1 2 3 0 0.5M 1M 1.5M 2.1M Runtime (min) No. of Estimators Twitter Graph Profiling 0 5 10 15 20 25 30 35 40 0 0.5m 1m 1.5m 2.1m Error Rate (%) No. of Estimators Twitter Graph Profiling count: 21453 +/- 14 confidence: 95%, time: 92s Embeddings (optional) 1 3 4 6 7 5 2

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“100s”) graphB.fourClique(“5.0%”,“95.0%”) Estimator Count Selection … Graphs stored on disk or main memory Estimates:{error: <5%, time: 95s} Estimates:{error: <5%, time: 60s} … Graph updates 1 2 3 0 0.5M 1M 1.5M 2.1M Runtime (min) No. of Estimators Twitter Graph Profiling 0 5 10 15 20 25 30 35 40 0 0.5m 1m 1.5m 2.1m Error Rate (%) No. of Estimators Twitter Graph Profiling 1 2 3 0 0.5M 1M 1.5M 2.1M Runtime (min) No. of Estimators Twitter Graph Profiling 0 5 10 15 20 25 30 35 40 0 0.5m 1m 1.5m 2.1m Error Rate (%) No. of Estimators Twitter Graph Profiling count: 21453 +/- 14 confidence: 95%, time: 92s Embeddings (optional) 1 3 4 6 7 5 2 Contributions: • Extends neighborhood sampling to general patterns • Provides a unified API • Applies approximate pattern mining in distributed settings

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Generalized Approximate Pattern Mining Under submission. Please do not distribute.
API Description sampleVertex: ()!(v, p) Uniformly sample one vertex from the graph. SampleEdge: ()!(e, p) Uniformly sample one edge from the graph. ConditionalSampleVertex: (subgraph)!(v, p) Uniformly sample a vertex that appears after a sampled subgraph. ConditionalSampleEdge: (subgraph)!(e, p) Uniformly sample an edge that is adjacent to the given subgraph and comes after the subgraph in the order. ConditionalClose: (subgraph, subgraph)!boolean Given a sampled subgraph, check if another subgraph that appears later in the order can be formed. Table 1: ASAP’s Approximate Pattern Mining API. SampleThreeNodeChain SampleTriangle Developers write a single estimator using ASAP’s API

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Using ASAP’s API 0 1 4 2 3 edge stream:
(0,1), (0,2), (0,3), (0,4), (1,2), (1,3), (1,4), (2,3), (2,4), (3,4) ditionalSampleEdge: (subgraph)!(e, p) Uniformly sample an edge that is adjacent to the give subgraph and comes after the subgraph in the order. ditionalClose: (subgraph, subgraph)!boolean Given a sampled subgraph, check if another subgrap that appears later in the order can be formed. Table 1: ASAP’s Approximate Pattern Mining API. leThreeNodeChain p1) = SampleEdge() p2) = ConditionalSampleEdge(Subgraph(e1)) e2) rn 0 rn 1/(p1.p2) SampleTriangle 1 (e1, p1) = sampleEdge() 2 (e2, p2) = conditionalSampleEdge(Subgraph(e1)) 3 if (!e2) return 0 4 subgraph1 = Subgraph(e1, e2) 5 subgraph2 = Triangle(e1, e2)-subgraph1 6 if conditionalClose(subgraph1, subgraph2) 7 return 1/(p1.p2) 8 else return 0 leFourCliqueType1 p1) = SampleEdge() p2) = ConditionalSampleEdge(Subgraph(e1)) e2) return 0 p3) = ConditionalSampleEdge(Subgraph(e1, e2)) e3) return 0 SampleFourCliqueType2 1 (e1, p1) = SampleEdge() 2 (e2, p2) = SampleEdge() 3 if (isAdjacent(e1, e2) == true) 4 return 0 5 subgraph1 = Subgraph(e1, e2)

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(0,1), (0,2), (0,3), (0,4), (1,2), (1,3), (1,4), (2,3), (2,4), (3,4) ditionalSampleEdge: (subgraph)!(e, p) Uniformly sample an edge that is adjacent to the give subgraph and comes after the subgraph in the order. ditionalClose: (subgraph, subgraph)!boolean Given a sampled subgraph, check if another subgrap that appears later in the order can be formed. Table 1: ASAP’s Approximate Pattern Mining API. leThreeNodeChain p1) = SampleEdge() p2) = ConditionalSampleEdge(Subgraph(e1)) e2) rn 0 rn 1/(p1.p2) SampleTriangle 1 (e1, p1) = sampleEdge() 2 (e2, p2) = conditionalSampleEdge(Subgraph(e1)) 3 if (!e2) return 0 4 subgraph1 = Subgraph(e1, e2) 5 subgraph2 = Triangle(e1, e2)-subgraph1 6 if conditionalClose(subgraph1, subgraph2) 7 return 1/(p1.p2) 8 else return 0 leFourCliqueType1 p1) = SampleEdge() p2) = ConditionalSampleEdge(Subgraph(e1)) e2) return 0 p3) = ConditionalSampleEdge(Subgraph(e1, e2)) e3) return 0 SampleFourCliqueType2 1 (e1, p1) = SampleEdge() 2 (e2, p2) = SampleEdge() 3 if (isAdjacent(e1, e2) == true) 4 return 0 5 subgraph1 = Subgraph(e1, e2) 0 3

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(0,1), (0,2), (0,3), (0,4), (1,2), (1,3), (1,4), (2,3), (2,4), (3,4) ditionalSampleEdge: (subgraph)!(e, p) Uniformly sample an edge that is adjacent to the give subgraph and comes after the subgraph in the order. ditionalClose: (subgraph, subgraph)!boolean Given a sampled subgraph, check if another subgrap that appears later in the order can be formed. Table 1: ASAP’s Approximate Pattern Mining API. leThreeNodeChain p1) = SampleEdge() p2) = ConditionalSampleEdge(Subgraph(e1)) e2) rn 0 rn 1/(p1.p2) SampleTriangle 1 (e1, p1) = sampleEdge() 2 (e2, p2) = conditionalSampleEdge(Subgraph(e1)) 3 if (!e2) return 0 4 subgraph1 = Subgraph(e1, e2) 5 subgraph2 = Triangle(e1, e2)-subgraph1 6 if conditionalClose(subgraph1, subgraph2) 7 return 1/(p1.p2) 8 else return 0 leFourCliqueType1 p1) = SampleEdge() p2) = ConditionalSampleEdge(Subgraph(e1)) e2) return 0 p3) = ConditionalSampleEdge(Subgraph(e1, e2)) e3) return 0 SampleFourCliqueType2 1 (e1, p1) = SampleEdge() 2 (e2, p2) = SampleEdge() 3 if (isAdjacent(e1, e2) == true) 4 return 0 5 subgraph1 = Subgraph(e1, e2) 0 3 4

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(0,1), (0,2), (0,3), (0,4), (1,2), (1,3), (1,4), (2,3), (2,4), (3,4) ditionalSampleEdge: (subgraph)!(e, p) Uniformly sample an edge that is adjacent to the give subgraph and comes after the subgraph in the order. ditionalClose: (subgraph, subgraph)!boolean Given a sampled subgraph, check if another subgrap that appears later in the order can be formed. Table 1: ASAP’s Approximate Pattern Mining API. leThreeNodeChain p1) = SampleEdge() p2) = ConditionalSampleEdge(Subgraph(e1)) e2) rn 0 rn 1/(p1.p2) SampleTriangle 1 (e1, p1) = sampleEdge() 2 (e2, p2) = conditionalSampleEdge(Subgraph(e1)) 3 if (!e2) return 0 4 subgraph1 = Subgraph(e1, e2) 5 subgraph2 = Triangle(e1, e2)-subgraph1 6 if conditionalClose(subgraph1, subgraph2) 7 return 1/(p1.p2) 8 else return 0 leFourCliqueType1 p1) = SampleEdge() p2) = ConditionalSampleEdge(Subgraph(e1)) e2) return 0 p3) = ConditionalSampleEdge(Subgraph(e1, e2)) e3) return 0 SampleFourCliqueType2 1 (e1, p1) = SampleEdge() 2 (e2, p2) = SampleEdge() 3 if (isAdjacent(e1, e2) == true) 4 return 0 5 subgraph1 = Subgraph(e1, e2) 0 3 4 !" = 40

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(0,1), (0,2), (0,3), (0,4), (1,2), (1,3), (1,4), (2,3), (2,4), (3,4) ditionalSampleEdge: (subgraph)!(e, p) Uniformly sample an edge that is adjacent to the give subgraph and comes after the subgraph in the order. ditionalClose: (subgraph, subgraph)!boolean Given a sampled subgraph, check if another subgrap that appears later in the order can be formed. Table 1: ASAP’s Approximate Pattern Mining API. leThreeNodeChain p1) = SampleEdge() p2) = ConditionalSampleEdge(Subgraph(e1)) e2) rn 0 rn 1/(p1.p2) SampleTriangle 1 (e1, p1) = sampleEdge() 2 (e2, p2) = conditionalSampleEdge(Subgraph(e1)) 3 if (!e2) return 0 4 subgraph1 = Subgraph(e1, e2) 5 subgraph2 = Triangle(e1, e2)-subgraph1 6 if conditionalClose(subgraph1, subgraph2) 7 return 1/(p1.p2) 8 else return 0 leFourCliqueType1 p1) = SampleEdge() p2) = ConditionalSampleEdge(Subgraph(e1)) e2) return 0 p3) = ConditionalSampleEdge(Subgraph(e1, e2)) e3) return 0 SampleFourCliqueType2 1 (e1, p1) = SampleEdge() 2 (e2, p2) = SampleEdge() 3 if (isAdjacent(e1, e2) == true) 4 return 0 5 subgraph1 = Subgraph(e1, e2) 0 3 4 Sampling phase fixes the vertices for a particular instance of a pattern and closing phase waits for remaining edges !" = 40

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(0,1), (0,2), (0,3), (0,4), (1,2), (1,3), (1,4), (2,3), (2,4), (3,4) ditionalSampleEdge: (subgraph)!(e, p) Uniformly sample an edge that is adjacent to the give subgraph and comes after the subgraph in the order. ditionalClose: (subgraph, subgraph)!boolean Given a sampled subgraph, check if another subgrap that appears later in the order can be formed. Table 1: ASAP’s Approximate Pattern Mining API. leThreeNodeChain p1) = SampleEdge() p2) = ConditionalSampleEdge(Subgraph(e1)) e2) rn 0 rn 1/(p1.p2) SampleTriangle 1 (e1, p1) = sampleEdge() 2 (e2, p2) = conditionalSampleEdge(Subgraph(e1)) 3 if (!e2) return 0 4 subgraph1 = Subgraph(e1, e2) 5 subgraph2 = Triangle(e1, e2)-subgraph1 6 if conditionalClose(subgraph1, subgraph2) 7 return 1/(p1.p2) 8 else return 0 leFourCliqueType1 p1) = SampleEdge() p2) = ConditionalSampleEdge(Subgraph(e1)) e2) return 0 p3) = ConditionalSampleEdge(Subgraph(e1, e2)) e3) return 0 SampleFourCliqueType2 1 (e1, p1) = SampleEdge() 2 (e2, p2) = SampleEdge() 3 if (isAdjacent(e1, e2) == true) 4 return 0 5 subgraph1 = Subgraph(e1, e2) 0 3 4 Sampling phase fixes the vertices for a particular instance of a pattern and closing phase waits for remaining edges ASAP computes the right expectations, runs many instances of the estimator and aggregates results !" = 40

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(0,1), (0,2), (0,3), (0,4), (1,2), (1,3), (1,4), (2,3), (2,4), (3,4) ditionalSampleEdge: (subgraph)!(e, p) Uniformly sample an edge that is adjacent to the give subgraph and comes after the subgraph in the order. ditionalClose: (subgraph, subgraph)!boolean Given a sampled subgraph, check if another subgrap that appears later in the order can be formed. Table 1: ASAP’s Approximate Pattern Mining API. leThreeNodeChain p1) = SampleEdge() p2) = ConditionalSampleEdge(Subgraph(e1)) e2) rn 0 rn 1/(p1.p2) SampleTriangle 1 (e1, p1) = sampleEdge() 2 (e2, p2) = conditionalSampleEdge(Subgraph(e1)) 3 if (!e2) return 0 4 subgraph1 = Subgraph(e1, e2) 5 subgraph2 = Triangle(e1, e2)-subgraph1 6 if conditionalClose(subgraph1, subgraph2) 7 return 1/(p1.p2) 8 else return 0 leFourCliqueType1 p1) = SampleEdge() p2) = ConditionalSampleEdge(Subgraph(e1)) e2) return 0 p3) = ConditionalSampleEdge(Subgraph(e1, e2)) e3) return 0 SampleFourCliqueType2 1 (e1, p1) = SampleEdge() 2 (e2, p2) = SampleEdge() 3 if (isAdjacent(e1, e2) == true) 4 return 0 5 subgraph1 = Subgraph(e1, e2) 0 3 4 See paper for more examples & proof Sampling phase fixes the vertices for a particular instance of a pattern and closing phase waits for remaining edges ASAP computes the right expectations, runs many instances of the estimator and aggregates results !" = 40

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Applying to Distributed Settings graph

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Applying to Distributed Settings graph subgraph 0 partial count c0
(using r estimators) subgraph 1 partial count c1 (using r estimators) subgraph 2 partial count c2 (using r estimators)

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Applying to Distributed Settings graph map: w(=3) workers subgraph 0
partial count c0 (using r estimators) subgraph 1 partial count c1 (using r estimators) subgraph 2 partial count c2 (using r estimators)

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Applying to Distributed Settings graph map: w(=3) workers subgraph 0
partial count c0 (using r estimators) subgraph 1 partial count c1 (using r estimators) subgraph 2 partial count c2 (using r estimators)

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Applying to Distributed Settings graph ! "#$ %&' (" map:
w(=3) workers subgraph 0 partial count c0 (using r estimators) subgraph 1 partial count c1 (using r estimators) subgraph 2 partial count c2 (using r estimators)

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w(=3) workers reduce subgraph 0 partial count c0 (using r estimators) subgraph 1 partial count c1 (using r estimators) subgraph 2 partial count c2 (using r estimators)

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w(=3) workers reduce subgraph 0 partial count c0 (using r estimators) subgraph 1 partial count c1 (using r estimators) subgraph 2 partial count c2 (using r estimators) Random Vertex-cut Partitioning

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w(=3) workers reduce subgraph 0 partial count c0 (using r estimators) subgraph 1 partial count c1 (using r estimators) subgraph 2 partial count c2 (using r estimators) )(+) Random Vertex-cut Partitioning

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w(=3) workers reduce subgraph 0 partial count c0 (using r estimators) subgraph 1 partial count c1 (using r estimators) subgraph 2 partial count c2 (using r estimators) Upper bounds on f(w) can be proved using Hajnal-Szemerédi theorem )(+) Random Vertex-cut Partitioning

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“100s”) graphB.fourClique(“5.0%”,“95.0%”) Estimator Count Selection … Graphs stored on disk or main memory Estimates:{error: <5%, time: 95s} Estimates:{error: <5%, time: 60s} … Graph updates 1 2 3 0 0.5M 1M 1.5M 2.1M Runtime (min) No. of Estimators Twitter Graph Profiling 0 5 10 15 20 25 30 35 40 0 0.5m 1m 1.5m 2.1m Error Rate (%) No. of Estimators Twitter Graph Profiling 1 2 3 0 0.5M 1M 1.5M 2.1M Runtime (min) No. of Estimators Twitter Graph Profiling 0 5 10 15 20 25 30 35 40 0 0.5m 1m 1.5m 2.1m Error Rate (%) No. of Estimators Twitter Graph Profiling count: 21453 +/- 14 confidence: 95%, time: 92s Embeddings (optional) 1 3 4 6 7 5 2

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“100s”) graphB.fourClique(“5.0%”,“95.0%”) Estimator Count Selection … Graphs stored on disk or main memory Estimates:{error: <5%, time: 95s} Estimates:{error: <5%, time: 60s} … Graph updates 1 2 3 0 0.5M 1M 1.5M 2.1M Runtime (min) No. of Estimators Twitter Graph Profiling 0 5 10 15 20 25 30 35 40 0 0.5m 1m 1.5m 2.1m Error Rate (%) No. of Estimators Twitter Graph Profiling 1 2 3 0 0.5M 1M 1.5M 2.1M Runtime (min) No. of Estimators Twitter Graph Profiling 0 5 10 15 20 25 30 35 40 0 0.5m 1m 1.5m 2.1m Error Rate (%) No. of Estimators Twitter Graph Profiling count: 21453 +/- 14 confidence: 95%, time: 92s Embeddings (optional) 1 3 4 6 7 5 2 Contribution: • Novel way to build ELP very fast without the need to know the ground truth or running mining on the full graph.

105.

Building Error-Latency Profile Given a time / error bound, how
many estimators should ASAP use?

106.

many estimators should ASAP use? Number of estimators Time Time vs Estimators

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many estimators should ASAP use? Number of estimators Time Time vs Estimators Error Number of estimators Error vs Estimators

108.

Building Estimators vs Time Profile Time complexity linear in number
of estimators

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Building Estimators vs Time Profile 1 2 3 0.5M 1M
1.5M 2M Runtime (min) No. of Estimators Twitter Graph Time complexity linear in number of estimators

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1.5M 2M Runtime (min) No. of Estimators Twitter Graph Time complexity linear in number of estimators ASAP sets a profiling cost and picks maximum points within the budget

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1.5M 2M Runtime (min) No. of Estimators Twitter Graph Time complexity linear in number of estimators 1 2 3 0 0.5M 1M 1.5M 2.1M Runtime (min) Twitter Graph Profiling

112.

Building Estimators vs Error Profile 0 5 10 15 20
25 30 35 40 50k 1m 1.5m 2.1m Error Rate (%) No. of Estimators Twitter Graph Error complexity non-linear in number of estimators

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25 30 35 40 50k 1m 1.5m 2.1m Error Rate (%) No. of Estimators Twitter Graph Error complexity non-linear in number of estimators Key idea: Use a very small sample of the graph to build the ELP § Chernoff analysis provides a loose upper bound on the number of estimators. § In small graphs, a large number of estimators can get us very close to ground truth.

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25 30 35 40 50k 1m 1.5m 2.1m Error Rate (%) No. of Estimators Twitter Graph Error complexity non-linear in number of estimators 0 5 10 15 20 25 30 35 40 0 0.5m 1m 1.5m 2.1m Error Rate (%) No. of Estimators Twitter Graph Profiling

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Advanced Mining Predicate Matching • Find patterns where vertices are
of type “electronics” • ASAP allows simple edge and vertex predicates Motif Mining • Some patterns are building blocks for other patterns • ASAP caches state of the estimators and reuses them Accuracy Refinement • Users may require more accurate answer later • ASAP can checkpoint and reuse estimators More details in the paper

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Implementation & Evaluation § Implemented on Apache Spark § Not
limited to it, only relies on simple dataflow operators § Evaluated in a 16 node cluster § Twitter: 1.47B edges § Friendster: 1.8B edges § UK: 3.73B edges § Comparison using representative patterns: § 3 (2 patterns), 4 (6 patterns) and 5 motifs (21 patterns)

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Performance on Small Graphs 12.1 162 291.4 3161 7.3 14.9
18.1 41.6 1 10 100 1000 10000 CiteSeer Mico Youtube LiveJournal Time (s) Arabesque ASAP 4-Motifs (6 patterns)

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Performance on Small Graphs 12.1 162 291.4 3161 7.3 14.9
18.1 41.6 1 10 100 1000 10000 CiteSeer Mico Youtube LiveJournal Time (s) Arabesque ASAP 4-Motifs (6 patterns)

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Performance on Small Graphs 77 x <5% error 12.1 162
291.4 3161 7.3 14.9 18.1 41.6 1 10 100 1000 10000 CiteSeer Mico Youtube LiveJournal Time (s) Arabesque ASAP 4-Motifs (6 patterns)

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Large Graphs & Simple Patterns 645 2.5 5 5.9 1
10 100 1000 0.9 1.5 1.8 3.7 Time (min) # Edges (Billions) 3-Motifs (2 patterns) Arabesque ASAP

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10 100 1000 0.9 1.5 1.8 3.7 Time (min) # Edges (Billions) Proprietary graph, 20 machines (256GB each) 3-Motifs (2 patterns) Arabesque ASAP

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10 100 1000 0.9 1.5 1.8 3.7 Time (min) # Edges (Billions) Proprietary graph, 20 machines (256GB each) 258 x <5% error 3-Motifs (2 patterns) Twitter Friendster UK Arabesque ASAP

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10 100 1000 0.9 1.5 1.8 3.7 Time (min) # Edges (Billions) Proprietary graph, 20 machines (256GB each) 258 x <5% error 3-Motifs (2 patterns) Twitter Friendster UK Arabesque ASAP

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Large Graphs & Complex Patterns 4-Motifs 22 47 0 10
20 30 40 50 Twitter UK Time (min)

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Large Graphs & Complex Patterns 12.3 22.1 5.6 14.2 0
5 10 15 20 25 Twitter UK Time (min) 5% 10% 5-House 4-Motifs 22 47 0 10 20 30 40 50 Twitter UK Time (min)

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Summary § Pattern mining important & challenging problem § Applications
in many domains § ASAP uses approximation for fast pattern mining § Leverages graph mining theory & makes it practical § Simple API for developers § ASAP outperforms existing solutions § Can handle much larger graphs with fewer resources http://www.cs.berkeley.edu/~api api@cs.berkeley.edu

ASAP: Fast, Approximate Graph Pattern Mining at Scale

ASAP: Fast, Approximate Graph Pattern Mining at Scale

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