Bridging the GAP: Towards Approximate Graph Analytics

Transcript

1. Bridging the GAP: Towards
   Approximate Graph Analytics
   Anand Iyer ⋆, Aurojit Panda ▪, Shivaram Venkataraman⬩,
   Mosharaf Chowdhury▴, Aditya Akella⬩, Scott Shenker ⋆, Ion Stoica ⋆
   ⋆ UC Berkeley ▪ NYU ⬩ University of Wisconsin ▴ University of Michigan
   June 10, 2018
   

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2. Graphs popular in big data analytics
   

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3. Graphs popular in big data analytics
   

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4. Graphs popular in big data analytics
   

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5. Graphs popular in big data analytics
   

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7. Can benefit from timely analysis
   

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8. Can benefit from timely analysis
   Often do not require exact answers
   

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9. Cellular Network Analytics
   

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10. Financial Network Analytics
    Image courtesy: Neo4J
    

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11. Graph Analytics
    

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12. Graph Analytics
    Takes several minutes to produce exact answers
    

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13. Can we leverage approximation for
    graph analytics?
    

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14. Apply query on samples of the input data
    Approximate Analytics
    

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15. Apply query on samples of the input data
    Approximate Analytics
    ID City Buff Ratio
    1 NYC 0.78
    2 NYC 0.13
    3 Berkeley 0.25
    4 NYC 0.19
    5 NYC 0.11
    6 Berkeley 0.09
    7 NYC 0.18
    8 NYC 0.15
    9 Berkeley 0.13
    10 Berkeley 0.49
    11 NYC 0.19
    12 Berkeley 0.10
    

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16. Apply query on samples of the input data
    Approximate Analytics
    ID City Buff Ratio
    1 NYC 0.78
    2 NYC 0.13
    3 Berkeley 0.25
    4 NYC 0.19
    5 NYC 0.11
    6 Berkeley 0.09
    7 NYC 0.18
    8 NYC 0.15
    9 Berkeley 0.13
    10 Berkeley 0.49
    11 NYC 0.19
    12 Berkeley 0.10
    What is the average buffering
    ratio in the table?
    

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17. Apply query on samples of the input data
    Approximate Analytics
    ID City Buff Ratio
    1 NYC 0.78
    2 NYC 0.13
    3 Berkeley 0.25
    4 NYC 0.19
    5 NYC 0.11
    6 Berkeley 0.09
    7 NYC 0.18
    8 NYC 0.15
    9 Berkeley 0.13
    10 Berkeley 0.49
    11 NYC 0.19
    12 Berkeley 0.10
    What is the average buffering
    ratio in the table?
    0.2325
    

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18. Apply query on samples of the input data
    Approximate Analytics
    ID City Buff Ratio
    1 NYC 0.78
    2 NYC 0.13
    3 Berkeley 0.25
    4 NYC 0.19
    5 NYC 0.11
    6 Berkeley 0.09
    7 NYC 0.18
    8 NYC 0.15
    9 Berkeley 0.13
    10 Berkeley 0.49
    11 NYC 0.19
    12 Berkeley 0.10
    What is the average buffering
    ratio in the table?
    ID City Buff Ratio Sampling Rate
    2 NYC 0.13 1/4
    6 Berkeley 0.25 1/4
    8 NYC 0.19 1/4
    Uniform
    Sample
    

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19. Apply query on samples of the input data
    Approximate Analytics
    ID City Buff Ratio
    1 NYC 0.78
    2 NYC 0.13
    3 Berkeley 0.25
    4 NYC 0.19
    5 NYC 0.11
    6 Berkeley 0.09
    7 NYC 0.18
    8 NYC 0.15
    9 Berkeley 0.13
    10 Berkeley 0.49
    11 NYC 0.19
    12 Berkeley 0.10
    What is the average buffering
    ratio in the table?
    ID City Buff Ratio Sampling Rate
    2 NYC 0.13 1/4
    6 Berkeley 0.25 1/4
    8 NYC 0.19 1/4
    Uniform
    Sample
    0.19
    0.2325
    

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20. Apply query on samples of the input data
    Approximate Analytics
    ID City Buff Ratio
    1 NYC 0.78
    2 NYC 0.13
    3 Berkeley 0.25
    4 NYC 0.19
    5 NYC 0.11
    6 Berkeley 0.09
    7 NYC 0.18
    8 NYC 0.15
    9 Berkeley 0.13
    10 Berkeley 0.49
    11 NYC 0.19
    12 Berkeley 0.10
    What is the average buffering
    ratio in the table?
    ID City Buff Ratio Sampling Rate
    2 NYC 0.13 1/4
    6 Berkeley 0.25 1/4
    8 NYC 0.19 1/4
    Uniform
    Sample
    0.19
    0.2325
    +/- 0.05
    

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21. Apply query on samples of the input data
    Approximate Analytics
    ID City Buff Ratio
    1 NYC 0.78
    2 NYC 0.13
    3 Berkeley 0.25
    4 NYC 0.19
    5 NYC 0.11
    6 Berkeley 0.09
    7 NYC 0.18
    8 NYC 0.15
    9 Berkeley 0.13
    10 Berkeley 0.49
    11 NYC 0.19
    12 Berkeley 0.10
    What is the average buffering
    ratio in the table?
    ID City Buff Ratio Sampling Rate
    2 NYC 0.13 1/2
    3 Berkeley 0.25 1/2
    5 NYC 0.11 1/2
    6 Berkeley 0.09 1/2
    8 NYC 0.15 1/2
    12 Berkeley 0.10 1/2
    Uniform
    Sample
    

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22. Apply query on samples of the input data
    Approximate Analytics
    ID City Buff Ratio
    1 NYC 0.78
    2 NYC 0.13
    3 Berkeley 0.25
    4 NYC 0.19
    5 NYC 0.11
    6 Berkeley 0.09
    7 NYC 0.18
    8 NYC 0.15
    9 Berkeley 0.13
    10 Berkeley 0.49
    11 NYC 0.19
    12 Berkeley 0.10
    What is the average buffering
    ratio in the table?
    ID City Buff Ratio Sampling Rate
    2 NYC 0.13 1/2
    3 Berkeley 0.25 1/2
    5 NYC 0.11 1/2
    6 Berkeley 0.09 1/2
    8 NYC 0.15 1/2
    12 Berkeley 0.10 1/2
    Uniform
    Sample
    0.19 +/- 0.05
    0.2325
    0.22 +/- 0.02
    

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23. Can we use the same idea on graphs?
    Approximate Analytics on Graphs
    

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24. Can we use the same idea on graphs?
    Approximate Analytics on Graphs
    0
    1 4
    2 3
    graph
    

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25. Can we use the same idea on graphs?
    Approximate Analytics on Graphs
    0
    1 4
    2 3
    edge sampling
    (p=0.5)
    graph
    0
    1 4
    2 3
    

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26. Can we use the same idea on graphs?
    Approximate Analytics on Graphs
    0
    1 4
    2 3
    edge sampling
    (p=0.5)
    graph
    e = 1
    0
    1 4
    2 3
    triangle
    counting
    

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27. Can we use the same idea on graphs?
    Approximate Analytics on Graphs
    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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28. Answer: 10
    Can we use the same idea on graphs?
    Approximate Analytics on Graphs
    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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29. Challenge: Non-linear relation between sample
    size and runtime / error
    Approximate Analytics on Graphs
    

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30. Challenge: Non-linear relation between sample
    size and runtime / error
    Approximate Analytics on Graphs
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31. Challenge: Non-linear relation between sample
    size and runtime / error
    Approximate Analytics on Graphs
    How to sample graphs?
    What is the right sample size?
    How to compute the error for a given (iterative) graph query?
    

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32. Our Proposal: GAP
    Run A within T sec Result, Error
    Graph Algorithms
    Sparsification Selector
    Models
    Sparsifier
    

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33. Our Proposal: GAP
    Run A within T sec Result, Error
    Graph Algorithms
    Sparsification Selector
    Models
    Sparsifier
    

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34. Our Proposal: GAP
    Run A within T sec Result, Error
    Graph Algorithms
    Sparsification Selector
    Models
    Sparsifier
    

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35. Our Proposal: GAP
    Run A within T sec Result, Error
    Graph Algorithms
    Sparsification Selector
    Models
    Sparsifier
    

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36. * Daniel A. Spielman and Shang-Hua Teng. “Spectral Sparsification of Graphs”
    Sampling for Graph Approximation
    § Sparsification extensively studied in graph theory
    § Idea: approximate the graph using a sparse, much smaller graph
    § Many computationally intensive
    § Not amenable to distributed implementation
    § Build on Spielman & Teng’s work*
    § Keep edges with probability
    cial properties of the input graph.
    While several proposals on the type of sparsier exists, many o
    m are either computationally intensive, or are not amenable to
    stributed implementation (which is the focus of our work)3. A
    nitial solution, we developed a simple sparsier adapted from
    work of Spielman and Teng [31] that is based on vertex degree
    sparsier uses the following probability to decide to keep an
    e between vertex a and b:
    dAV G ⇥ s
    min(d
    o
    a,d
    i
    b
    )
    (1
    o
    

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37. * Daniel A. Spielman and Shang-Hua Teng. “Spectral Sparsification of Graphs”
    Sampling for Graph Approximation
    § Sparsification extensively studied in graph theory
    § Idea: approximate the graph using a sparse, much smaller graph
    § Many computationally intensive
    § Not amenable to distributed implementation
    § Build on Spielman & Teng’s work*
    § Keep edges with probability
    cial properties of the input graph.
    While several proposals on the type of sparsier exists, many o
    m are either computationally intensive, or are not amenable to
    stributed implementation (which is the focus of our work)3. A
    nitial solution, we developed a simple sparsier adapted from
    work of Spielman and Teng [31] that is based on vertex degree
    sparsier uses the following probability to decide to keep an
    e between vertex a and b:
    dAV G ⇥ s
    min(d
    o
    a,d
    i
    b
    )
    (1
    o
    Exploring other sparsification techniques
    

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38. Estimating the Error / Latency
    What is the error / speedup due to sparsification?
    Many approaches in approximate processing literature:
    • Exhaustively run every possible point
    • Theoretical closed-bound solutions
    • Experiment design / Bayesian techniques
    

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39. Estimating the Error / Latency
    What is the error / speedup due to sparsification?
    Many approaches in approximate processing literature:
    • Exhaustively run every possible point
    • Theoretical closed-bound solutions
    • Experiment design / Bayesian techniques
    None applicable for graph approximation
    

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40. Building a model for s
    Use machine learning to build a model.
    

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41. Building a model for s
    Model
    Builder
    Use machine learning to build a model.
    

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42. Building a model for s
    Model
    Builder
    Use machine learning to build a model.
    Learn the relation between s and error / latency
    

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43. Building a model for s
    Model
    Builder
    Input
    features
    Use machine learning to build a model.
    Learn the relation between s and error / latency
    

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44. Building a model for s
    Model
    Builder
    Input
    features
    Model
    Use machine learning to build a model.
    Learn the relation between s and error / latency
    

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45. Building a model for s
    Model
    Builder
    Input
    features
    Model
    Use machine learning to build a model.
    Learn the relation between s and error / latency
    “The most important determinant of graph workload
    characteristics is typically the input graph and surprisingly not
    the implementation or even the graph kernel.” Beamer et. al.
    Indistributed graph processing, communication (shuffles)
    dominate execution time.
    

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46. Building a model for s
    Model
    Builder
    Input
    features
    Model
    Use machine learning to build a model.
    Learn the relation between s and error / latency
    

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47. Building a model for s
    Model
    Builder
    Learn H: (s, a, g) => e/l
    Input
    features
    Model
    Use machine learning to build a model.
    Learn the relation between s and error / latency
    

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48. Building a model for s
    Model
    Builder
    Learn H: (s, a, g) => e/l
    Input
    features
    Model
    Random Forests
    Use machine learning to build a model.
    Learn the relation between s and error / latency
    

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49. Building a model for s
    Model Builder
    

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50. Building a model for s
    Model Builder
    Input Graph
    

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51. Building a model for s
    Model Builder Models
    Input Graph
    

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52. Building a model for s
    Model Builder Models
    Input Graph
    Benchmark
    Graphs/Queries
    (e.g., Graph500)
    

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53. Building a model for s
    Model Builder
    0
    1 4
    2 3
    Models
    Input Graph
    Benchmark
    Graphs/Queries
    (e.g., Graph500)
    

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54. Building a model for s
    Model Builder
    Model Mapper
    0
    1 4
    2 3
    Models
    Input Graph
    Benchmark
    Graphs/Queries
    (e.g., Graph500)
    

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55. Building a model for s
    Model Builder
    Model Mapper
    0
    1 4
    2 3
    Models
    Input Graph
    Benchmark
    Graphs/Queries
    (e.g., Graph500)
    

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56. Building a model for s
    Model Builder
    Model Mapper
    0
    1 4
    2 3
    Models
    Input Graph
    Benchmark
    Graphs/Queries
    (e.g., Graph500)
    

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57. Building a model for s
    Model Builder
    Model Mapper
    0
    1 4
    2 3
    Model
    Models
    Input Graph
    Benchmark
    Graphs/Queries
    (e.g., Graph500)
    

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58. Preliminary Feasibility Evaluation
    § Implemented sparsifier on Apache Spark
    § Not limited to it
    § Evaluated on a few real-world graphs
    § Largest: UK 3.73B edges
    § Goal: check if our assumptions hold
    

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59. Preliminary Feasibility Evaluation
    0
    0.5
    1
    1.5
    2
    2.5
    0.9
    0.85
    0.8
    0.75
    0.7
    0.65
    0.6
    0.55
    0.5
    0.45
    0.4
    0.35
    0.3
    0.25
    0.2
    0.15
    0.1
    Speedup
    Sparsifcation Parameter
    AstroPh
    Facebook
    0
    0.5
    1
    1.5
    2
    2.5
    0.9
    0.85
    0.8
    0.75
    0.7
    0.65
    0.6
    0.55
    0.5
    0.45
    0.4
    0.35
    0.3
    0.25
    0.2
    0.15
    0.1
    Speedup
    Sparsifcation Parameter
    Epinions
    Wikivote
    

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60. Preliminary Feasibility Evaluation
    0
    0.5
    1
    1.5
    2
    2.5
    0.9
    0.85
    0.8
    0.75
    0.7
    0.65
    0.6
    0.55
    0.5
    0.45
    0.4
    0.35
    0.3
    0.25
    0.2
    0.15
    0.1
    Speedup
    Sparsifcation Parameter
    AstroPh
    Facebook
    0
    0.5
    1
    1.5
    2
    2.5
    0.9
    0.85
    0.8
    0.75
    0.7
    0.65
    0.6
    0.55
    0.5
    0.45
    0.4
    0.35
    0.3
    0.25
    0.2
    0.15
    0.1
    Speedup
    Sparsifcation Parameter
    Epinions
    Wikivote
    Performance trends similar for graphs that are similar
    

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61. Preliminary Feasibility Evaluation
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62. Preliminary Feasibility Evaluation
    1
    2
    3
    4
    5
    6
    7
    8
    9
    0.9
    0.8
    0.7
    0.6
    0.5
    0.4
    0.3
    0.2
    0.1
    0
    10
    20
    30
    40
    50
    60
    70
    Speedup
    Error (%)
    Sparsifcation Parameter
    Speedup
    Error
    

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63. Preliminary Feasibility Evaluation
    1
    2
    3
    4
    5
    6
    7
    8
    9
    0.9
    0.8
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    Speedup
    Error (%)
    Sparsifcation Parameter
    Speedup
    Error
    Bigger benefits achievable in large graphs
    

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64. Ongoing/Future Work
    § Deep Learning
    § Build better models.
    § Better sparsifiers
    § Can we cherry pick sparsifiers?
    § Programming Language techniques
    § Can we synthesize approximate versions of an exact
    graph-parallel program?
    

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65. Conclusion
    § Approximate graph analytics challenging
    § Unlike approximate query processing, no direct relation
    between graph size and latency/error.
    § Our proposal GAP:
    § Uses sparsification theory to reduce input to graph algorithms,
    and ML to learn the relation between input latency/error.
    § Initial results are encouraging.
    http://www.cs.berkeley.edu/~api
    api@cs.berkeley.edu
    

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