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STING: Spatio-Temporal Interaction
Networks and Graphs for Intel Platforms
David Bader, Jason Riedy,
David Ediger, Rob McColl,
Timothy G. Mattson∗

                                24 July 2012
Outline



 Motivation

 STING for streaming, graph-structured data on Intel-based
 platforms

 World-leading community detection

 Community-building interactions

 Plans and direction


Note: Without hard limits, academics will use at least all the time...


                                                                         2 / 57
(Prefix)scale Data Analysis




    Health care Finding outbreaks, population epidemiology
Social networks Advertising, searching, grouping
    Intelligence Decisions at scale, regulating algorithms
Systems biology Understanding interactions, drug design
     Power grid Disruptions, conservation
     Simulation Discrete events, cracking meshes




                                                             3 / 57
Graphs are pervasive
   Graphs: things and relationships
      • Different kinds of things, different kinds of relationships, but
         graphs provide a framework for analyzing the relationships.
      • New challenges for analysis: data sizes, heterogeneity,
         uncertainty, data quality.


Astrophysics                Bioinformatics               Social Informatics
                            Problem Identifying target   Problem Emergent behavior,
Problem Outlier detection
                            proteins                     information spread
Challenges Massive data
                            Challenges Data              Challenges New analysis,
sets, temporal variation
                            heterogeneity, quality       data uncertainty
Graph problems Matching,
                            Graph problems Centrality,   Graph problems Clustering,
clustering
                            clustering                   flows, shortest paths




                                                                                 4 / 57
No shortage of data...

Existing (some out-of-date) data volumes
     NYSE 1.5 TB generated daily, maintain a 8 PB archive
    Google “Several dozen” 1PB data sets (CACM, Jan 2010)
  Wal-Mart 536 TB, 1B entries daily (2006)
      EBay 2 PB, traditional DB, and 6.5PB streaming, 17
           trillion records, 1.5B records/day, each web click is
           50-150 details.
           http://www.dbms2.com/2009/04/30/
           ebays-two-enormous-data-warehouses/
 Facebook 845 M users... and growing(?)

  • All data is rich and semantic (graphs!) and changing.
  • Base data rates include items and not relationships.


                                                                   5 / 57
General approaches

 • High-performance static graph analysis
     • Develop techniques that apply to unchanging massive
       graphs.
     • Provides useful after-the-fact information, starting points.
     • Serves many existing applications well: market research,
       much bioinformatics, ...
 • High-performance streaming graph analysis
     • Focus on the dynamic changes within massive graphs.
     • Find trends or new information as they appear.
     • Serves upcoming applications: fault or threat detection,
       trend analysis, ...


           Both very important to different areas.
            GT’s primary focus is on streaming.
Note: Not CS theory streaming, but analysis of streaming data.

                                                                      6 / 57
Why analyze data streams?

                         Data transfer
Data volumes               • 1 Gb Ethernet: 8.7TB daily at
     NYSE 1.5TB daily        100%, 5-6TB daily realistic
      LHC 41TB daily       • Multi-TB storage on 10GE: 300TB
Facebook, etc. Who           daily read, 90TB daily write
            knows?         • CPU ↔ Memory: QPI,HT:
                             2PB/day@100%

Data growth
                         Speed growth
  • Facebook: > 2×/yr
                           • Ethernet/IB/etc.: 4× in next 2
  • Twitter: > 10×/yr
                             years. Maybe.
  • Growing sources:
                           • Flash storage, direct: 10× write,
    Bioinformatics,          4× read. Relatively huge cost.
    µsensors, security

                                                                 7 / 57
Intel’s non-numeric computing
program (our perspective)


  Supporting analysis of massive graph under rapid change
       across the spectrum of Intel-based platforms.

STING on Intel-based platforms
  • Maintain a graph and metrics under large data changes
      • Performance, documentation, distribution
      • Greatly improved ingestion rate, improved example metrics
      • Available at http://www.cc.gatech.edu/stinger/.
  • Algorithm development
      • World-leading community detection (clustering) performance
        on Intel-based platforms
           • Good for focused or multi-level analysis, visualization, ...
      • Developments on community monitoring in dynamic graphs



                                                                            8 / 57
On to STING...


Motivation

STING for streaming, graph-structured data on Intel-based
platforms
   Assumptions about the data
    High-level architecture
   Algorithms and performance

World-leading community detection

Community-building interactions

Plans and direction


                                                            9 / 57
Overall streaming approach




   Yifan Hu’s (AT&T) visualization of the
           Livejournal data set
                                            Jason’s network via LinkedIn Labs



Assumptions
 • A graph represents some real-world phenomenon.
     • But not necessarily exactly!
     • Noise comes from lost updates, partial information, ...



                                                                                10 / 57
Overall streaming approach




   Yifan Hu’s (AT&T) visualization of the
           Livejournal data set
                                            Jason’s network via LinkedIn Labs



Assumptions
 • We target massive, “social network” graphs.
    • Small diameter, power-law degrees
    • Small changes in massive graphs often are unrelated.



                                                                                10 / 57
Overall streaming approach




   Yifan Hu’s (AT&T) visualization of the
           Livejournal data set
                                            Jason’s network via LinkedIn Labs



Assumptions
 • The graph changes but we don’t need a continuous view.
     • We can accumulate changes into batches...
     • But not so many that it impedes responsiveness.



                                                                                10 / 57
STING’s focus
           Control
  action                   prediction


                                             summary
 Source      data     Simulation / query                   Viz


 • STING manages queries against changing graph data.
     • Visualization and control often are application specific.
 • Ideal: Maintain many persistent graph analysis kernels.
     • One current snapshot of the graph, kernels keep smaller
       histories.
     • Also (a harder goal), coordinate the kernels’ cooperation.
 • STING components:
     • Framework for batching input changes.
     • Lockless data structure, STINGER, accumulating a typed
       graph with timestamps.

                                                                    11 / 57
STINGER

STING Extensible Representation:




  • Rule #1: No explicit locking.
      • Rely on atomic operations.
  • Massive graph: Scattered updates, scattered reads rarely
    conflict.
  • Use time stamps for some view of time.


                                                               12 / 57
STING’s year




Over the past year
  • Improved performance on Intel-based platforms by
    14×[HPEC12]
  • Evaluated multiple dynamic kernels across
    platforms[MTAAP12, ICASSP2012]
  • Documentation and tutorials
  • Assist projects with STING adoption on Intel-based
    platforms




                                                         13 / 57
STING in use

Where is it?
        http://www.cc.gatech.edu/stinger/
     Code, development, documentation, presentations...

Who is using it?
  • Center for Adaptive Supercomputing Software –
    Multithreaded Architectures (CASS-MT) directed by PNNL
  • PRODIGAL team for DARPA ADAMS (Anomaly Detection at
    Multiple Scales) including GTRI, SAIC, Oregon State U., U.
    Mass., and CMU
  • DARPA SMISC (Social Media in Strategic Communication)
    including GTRI
  • Well-attended tutorials given at PPoPP, in MD, and locally.


                                                                  14 / 57
Experimental setup
Unless otherwise noted
      Line       Model      Speed (GHz)       Sockets    Cores
   Nehalem       X5570           2.93             2        4
   Westmere      X5650           2.66             2        6
   Westmere     E7-8870          2.40             4        10
  • Large Westmere, mirasol, loaned by Intel (thank you!)
  • All memory: 1067MHz DDR3, installed appropriately
  • Implementations: OpenMP, gcc 4.6.1, Linux ≈ 3.2 kernel
  • Artificial graph and edge stream generated by R-MAT
    [Chakrabarti, Zhan, & Faloutsos].
      • Scale x, edge factor f ⇒ 2x vertices, ≈ f · 2x edges.
      • Edge actions: 7/8th insertions, 1/8th deletions
      • Results over five batches of edge actions.
  • Caveat: No vector instructions, low-level optimizations yet.
  • Portable: OpenMP, Cray XMT family
                                                                   15 / 57
STINGER insertion / removal rates

Edges per second

                                                                  E7−8870                                         X5570
   Aggregate STINGER updates per second


                                           107           q
                                                              q   q
                                                                  q   q
                                                                           q
                                                                           q   q
                                                                               q   q   q   q
                                                                                           q
                                                        q
                                                                                                                            q        q   Batch size
                                                      qqq                                                     q
                                                                                                              q        q
                                                                                                                       q             q
                                                      q       q   q        q       q
                                                                                   q                          q             q
                                            6.5                       q        q       q   q
                                          10          q
                                                      qq
                                                      q                    q                          q
                                                                                                      q
                                                                                                              q
                                                                                                              q
                                                                                                              q
                                                                                                              q
                                                                                                                       q    q
                                                                                                                            q
                                                                                                                                     q
                                                                                                                                     q
                                                                                                                                          q   3000000
                                                    qqqq  q                                           q
                                                                                                      q
                                                                                                                       q    q
                                                                                                                            q
                                                    q q       q                    q                  q       q             q        q
                                                    q     q   q   q   q
                                                                      q            q
                                                    q
                                                    qq
                                                    q
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                                                                  q   q
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                                                                           q
                                                                           q
                                                                               q   q
                                                                                   q   q
                                                                                       q   q      q
                                                                                                  q   q       q        q                  q   1000000
                                                  q q
                                                    q q       q            q   q   q              q           q
                                           106    q
                                                  q qqq
                                                  q qq
                                                  q q
                                                    q
                                                                  q
                                                                  q
                                                                  q   q
                                                                                       q
                                                                                       q   q    qq
                                                                                                q
                                                                                                  q                                  q
                                                                                                                                          q   300000
                                                  q q q
                                                  q
                                                          q           q    q
                                                                           q   q   q   q
                                                                                           q
                                                                                                q     q
                                                  q
                                                  q
                                                  q                            q   q            q             q
                                                  qq q        q                    q   q        q
                                                  qq
                                                  q
                                                  qq q        q                            q                                         q
                                                          q       q
                                                                  q                    q          q           q                           q   100000
                                            5.5         qqq           q                    q
                                          10                          q    q                          q       q
                                                                                                              q
                                                    qq
                                                    q         q
                                                                           q
                                                                           q   q
                                                                               q   q   q
                                                                                       q
                                                                                                                       q
                                                                                                                       q
                                                              q                                                                      q
                                                      qq
                                                        qq    q   q
                                                                                       q
                                                                                                                                     q
                                                                                                                                          q    30000
                                                      q q
                                                      qqq                  q               q                                q        q
                                           105                        q    q   q   q
                                                        qq    q
                                                                      q        q
                                                                               q   q
                                                                                   q
                                                                                       q
                                                                                       q
                                                                                           q                           q                  q    10000
                                                        qq    q                q   q   q   q
                                                        q         q            q   q   q
                                                          q           q    q   q                                                     q
                                                              q   q
                                                                  q
                                                                      q
                                                                      q    q
                                                                           q
                                                                           q       q
                                                                                   q   q   q
                                                                                           q
                                                                                                                                          q     3000
                                                                                       q   q                                q
                                          104.5                   q   q        q   q
                                                                      q    q   q                                            q
                                                                                           q                                              q     1000
                                                                                       q
                                                                                       q   q
                                                                                                                                          q       300
                                           104                                         q
                                                                                                                            q
                                                                                                                                          q       100
                                                                                                                            q



                                                          20          40       60          80             5       10       15   20
                                                                           Number of threads



                                                                                                                                                        16 / 57
STINGER insertion / removal rates

Seconds per batch, “latency”

                                                               E7−8870                                           X5570
                                           q
                                                                                             q
                                           q
                                                                                                 q
                                                                                                                                        Batch size
                                           q
    Seconds per batch processed




                                             q                                               q
                                       0                                                             q                                   q   3000000
                                  10       q
                                               q                                                 q           q
                                           qq q                                                                                     q
                                                                                             q                             q
                                           q
                                                       q   q   q
                                                               q   q
                                                                   q    q
                                                                        q   q
                                                                            q   q   q   q
                                                                                        q
                                                                                        q
                                                                                                     q                                   q   1000000
                                                qqq        q   q        q   q   q   q            q           q             q
                                                                                q                            q        q             q
                                           qq                      q    q   q   q
                                                                                q   q   q
                                                           q   q   q            q            q                                           q   300000
                                                                                                     q                     q
                                           q qqq                                                                           q
                                  10−1                                          q   q   q
                                                                                                 q           q        q             q
                                           qq
                                                           q   q   q
                                                                   q
                                                                   q    q   q   q   q
                                                                                                                           q
                                                                                                                                         q   100000
                                                q                                       q                                  q
                                           q qq                                     q
                                                                                    q
                                                                                        q
                                                                                        q                             q    q
                                                                                                                           q
                                                                        q   q   q   q   q
                                                                                        q        q                         q             q    30000
                                                               q   q                    q                                  q
                                           qq              q   q            q   q   q
                                                                        q   q   q   q
                                                                                    q        q       q
                                                   qq              q
                                                                   q    q
                                                                        q   q
                                                                            q   q   q
                                                                                    q   q
                                           q q                 q                q                            q
                                  10−2
                                                                        q   q       q                                               q
                                                           q
                                                               q
                                                               q
                                                               q
                                                               q
                                                                   q
                                                                   q    q
                                                                        q   q   q   q
                                                                                    q
                                                                                        q
                                                                                        q
                                                                                                             q        q                  q    10000
                                             q         q   q   q   q    q
                                                                        q           q
                                           q       q       q       q    q   q   q   q
                                                                                    q                q
                                               q       q   q
                                                           q   q            q
                                                                            q   q   q        q                                      q
                                           q           q
                                                       q   q
                                                           q                            q                             q                  q     3000
                                                   q
                                                   q               q
                                                                   q    q   q   q
                                             qq    q   q
                                                       q
                                                       q       q        q   q   q                q                                  q
                                                   q   q       q
                                                               q                q       q
                                                                                        q                                           q
                                           qq q
                                           q q
                                                   q
                                                   q   q
                                                       q
                                                           q                                                 q        q             q
                                           q q
                                             q
                                           q q
                                                   q
                                                   q
                                                   q   q
                                                       q
                                                                                                     q       q
                                                                                                                      q    q        q    q     1000
                                    −3       q
                                  10         qq
                                               q                                                 q
                                                                                                     q
                                                                                                             q
                                                                                                             q
                                                                                                             q
                                           qq
                                           q
                                           q                                                 q       q       q
                                                                                                                                         q       300
                                           q q
                                             q                                                                        q
                                                                                                 q   q       q
                                           q
                                           q                                                 q
                                                                                                 q           q                           q       100
                                                                                                 q



                                                       20          40       60          80               5       10       15   20
                                                                        Number of threads



                                                                                                                                                       16 / 57
Clustering coefficients

• Used to measure “small-world-ness”
  [Watts & Strogatz] and potential                     i             m
  community structure
• Larger clustering coefficient ⇒ more                            v
  inter-connected
                                                       j             n
• Roughly the ratio of the number of actual
  to potential triangles

  • Defined in terms of triplets.
  • i – v – j is a closed triplet (triangle).
  • m – v – n is an open triplet.
  • Clustering coefficient:
                    # of closed triplets / total # of triplets
  • Locally around v or globally for entire graph.

                                                                         17 / 57
STINGER clustering coefficient rates

Edges per second

                                                                      E7−8870                                           X5570
   Clustering coefficient updates per second


                                                                                                                                           q
                                                                                                                    q
                                                                                                                    q        q    q
                                                                                                                                  q
                                               105.5                                                        q       q        q             q   Batch size
                                                        q                                                                    q    q        q
                                                                                                        q   q       q
                                                       q                                                            q        q
                                                       q qqq              q                             q   q
                                                        q         q   q        q   q   q   q   q
                                                                                                    q
                                                                                                        q
                                                                                                            q                q
                                                                                                                             q    q        q
                                                                                                                                           q
                                                                                                                                                q   3000000
                                                       q                                                            q        q    q        q
                                                                                                                                  q
                                                   5   q q                                            q     q       q             q        q
                                                                                                                                           q
                                                10         qq     q   q   q    q   q   q       q    qq
                                                                                                      q
                                                                                                            q       q
                                                                                                                    q
                                                                                                                    q
                                                                                                                             q
                                                                                                                             q
                                                                                                                             q
                                                                                                                             q
                                                                                                                                  q
                                                                                                                                  q
                                                                                                                                           q    q   1000000
                                                                                                            q                q
                                                                                                                             q    q
                                                        q                                  q        qq      q
                                                                                                            q       q        q
                                                       qq                                           qqq             q        q    q
                                                                                                                                  q        q
                                                                                                                                           q
                                                       qq
                                                       q q                             q   q          q     q                q
                                                                                                                             q    q             q   300000
                                                       q
                                                       q qq
                                                       q              q   q    q
                                                                                   q           q    qqq     q
                                                                                                                    q             q
                                                       qq q q         q            q   q   q        qq                            q
                                                           qqq    q
                                                                  q   q   q
                                                                          q
                                                                               q
                                                                               q   q
                                                                                   q   q   q   q
                                                                                               q    qq                       q    q
                                               104.5   q qqq
                                                           q
                                                       q q q
                                                                  q
                                                                  q
                                                                      q
                                                                      q   q
                                                                          q    q   q   q   q
                                                                                           q   q    q
                                                                                                    q
                                                                                                    qq      q       q             q        q
                                                                                                                                                q   100000
                                                                                               q            q       q        q
                                                                                                                             q    q
                                                         q
                                                       q qqq      q       q                q        qq
                                                                                                    q
                                                                      q        q   q   q
                                                                                                    qq
                                                                                                                                                q    30000
                                                       q
                                                                                                    q
                                                104      q
                                                             qq
                                                                  q
                                                                      q        q               q
                                                                                               q
                                                                                                    q
                                                                                                        q
                                                                                                            q       q
                                                                                                                    q        q    q
                                                                                                                                           q
                                                                                                                                           q    q    10000
                                                       q                  q        q       q
                                                           q                           q                q
                                                       q                                            q                                           q     3000
                                                         q                                          q       q
                                                                                                            q       q
                                               103.5   q
                                                                                                        q
                                                                                                        q
                                                                                                        q
                                                                                                                    q        q    q        q    q     1000
                                                       q
                                                                                                    q
                                                                                                                                                q       300
                                                103    q                                                                                        q       100
                                                                                                                                  q



                                                              20          40       60          80               5       10       15   20
                                                                               Number of threads



                                                                                                                                                              18 / 57
STINGER clustering coefficient rates

Speed-up over static recalculation
    Clustering coefficient speedup over recalculation

                                                                                         E7−8870                                           X5570
                                                                                                                       q
                                                                                                                       q
                                                                                                                       qq
                                                                                                                       qq
                                                                   q                                                   qq      q
                                                                 q
                                                                 q
                                                                 qq q
                                                                 q
                                                                 q qq
                                                                 q                                                     qq
                                                                                                                         q     q
                                                                                                                               q
                                                                                                                                                                  Batch size
                                                                                                                               q
                                                        102      qq q q
                                                                 q      q
                                                                                     q                                   q     q
                                                                                                                               q
                                                                                                                                       q
                                                                                                                                       q
                                                                                                                                       q                           q   3000000
                                                                   qq                q   q                             qq              q        q
                                                                 q    q                           q                     q              q        q
                                                                        q                q   q
                                                                                             q        q       q   q                    q        q
                                                                                                                                                q
                                                                 q                                        q             q                                     q
                                                                   q
                                                                                     q
                                                                                         q
                                                                                                  q   q   q   q
                                                                                                                               q
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                                                                                                                                                q
                                                                                                                                                q
                                                                                                                                                q
                                                                                                                                                     q
                                                                                                                                                     q
                                                                                                                                                     q        q    q   1000000
                                                                 q q                              q                            q                q    q
                                                                                                                                                     q        q
                                                                                                                                                              q
                                                                                             q        q           q    q               q
                                                                                                                                       q
                                                                      q                                   q   q          q             q        q
                                                                 q      q                                         q      q     q                q    q        q
                                                                                                                                                                   q   300000
                                                                                     q                                                               q        q
                                                        101                              q   q    q       q   q                                 q
                                                                     q                                q           q            q
                                                                 q                                                     q
                                                                         q                                             q
                                                                                                                           q
                                                                                                                                       q
                                                                                                                                       q
                                                                                                                                       q
                                                                                                                                                     q
                                                                                                                                                                   q   100000
                                                                 q           q                                             q                         q        q
                                                                                                                                                              q
                                                                                                                                                q    q
                                                                                 q                                             q                q
                                                                                                                                                q
                                                                 qq q                q                                 q
                                                                                                                       qq
                                                                                                                               q
                                                                                                                                       q             q             q    30000
                                                                                         q        q   q   q   q                                      q
                                                                 q
                                                                 q                           q                         qq              q        q             q
                                                                                                                         q                           q        q
                                                                 q
                                                                 qq
                                                                             q
                                                                                 q                                q
                                                                                                                         q
                                                                                                                         q     q
                                                                                                                               q
                                                                                                                                                q
                                                                                                                                                q
                                                                                                                                                                   q    10000
                                                             0      q                                                  q               q
                                                        10        q                  q
                                                                                         q
                                                                                                                               q
                                                                                                                               q
                                                                                                                                                     q
                                                                 q
                                                                             q               q    q   q   q   q   q        q           q
                                                                                                                                       q        q
                                                                                                                                                     q
                                                                                                                                                     q
                                                                                                                                                     q        q    q     3000
                                                                     q           q                                             q
                                                                                                                               q                     q
                                                                         q                                                                      q    q        q
                                                                                                                                                              q
                                                                         q           q
                                                                                         q
                                                                                                                                       q
                                                                                                                                       q
                                                                                                                                       q        q
                                                                                                                                                q
                                                                                                                                                                   q     1000
                                                                          q                           q       q                                 q
                                                                                             q    q       q       q                                  q        q
                                                                         qqq                                                                         q
                                                          −1                q        q
                                                                                                                                                q    q
                                                                                                                                                     q
                                                                                                                                                     q
                                                                                                                                                                   q       300
                                                                          q
                                                        10                  q        q   q   q    q   q       q   q
                                                                                     q
                                                                                         q   q
                                                                                                  q   q
                                                                                                          q
                                                                                                          q       q
                                                                                                                                                                   q       100
                                                                                         q   q                q
                                                                                                  q   q   q   q   q



                                                                                 20          40       60          80               5       10       15   20
                                                                                                  Number of threads



                                                                                                                                                                                 18 / 57
Connected components


 • Maintain a mapping from vertex to
   component.
 • Global property, unlike triangle
   counts
 • In “scale free” social networks:
      • Often one big component, and
      • many tiny ones.
 • Edge changes often sit within
   components.
 • Remaining insertions merge
   components.
 • Deletions are more difficult...



                                       19 / 57
Connected components: Deleted
edges

The difficult case
  • Very few deletions matter.
  • Maintain a spanning tree, &
    ignore deletion if
      1 not in spanning tree,
      2 endpoints share a
        common neighbor∗ ,
      3 loose endpoint reaches
        root∗ .
  • In the last two (∗), also fix
    the spanning tree.
  • Rules out 99.7% of
    deletions.
  • (Long history, see related
    work in [MTAAP11].)
                                   20 / 57
STINGER component monitoring
rates

Edges per second

                                                                        E7−8870                                           X5570
   Connected component updates per second



                                            105.5       q
                                                        q   q   q
                                                                q
                                                                q   q
                                                                    q   q
                                                                        q   q    q   q
                                                                                     q
                                                                                     q
                                                                                         q
                                                                                         q
                                                                                         q
                                                                                             q
                                                                                             q
                                                                                             q
                                                                                                 q
                                                                                                 q
                                                                                                        q     q
                                                                                                              q
                                                                                                                      q
                                                                                                                      q
                                                                                                                      q
                                                                                                                               q
                                                                                                                               q
                                                                                                                               q
                                                                                                                                    q
                                                                                                                                    q
                                                                                                                                    q
                                                                                                                                    q
                                                                                                                                             q
                                                                                                                                             q
                                                                                                                                             q
                                                                                                                                             q
                                                                                                                                             q
                                                    q q
                                                      qq    q
                                                            q   q   q   q   q
                                                                            q    q   q           q      q     q       q        q
                                                                                                                               q             q
                                                    q q
                                                      qq
                                                    qq q
                                                    q       q
                                                            q
                                                                q
                                                                q
                                                                q
                                                                q
                                                                    q
                                                                    q
                                                                    q
                                                                    q
                                                                        q
                                                                        q
                                                                        q   q
                                                                            q
                                                                            q
                                                                                 q
                                                                                 q
                                                                                 q   q
                                                                                     q
                                                                                         q
                                                                                         q
                                                                                         q
                                                                                         q
                                                                                             q
                                                                                             q
                                                                                             q
                                                                                                 q
                                                                                                 q
                                                                                                 q
                                                                                                      qqq     q
                                                                                                              q
                                                                                                              q
                                                                                                                      q
                                                                                                                      q        q
                                                                                                                               q    q
                                                                                                                                    q
                                                                                                                                             q
                                                                                                                                                 Batch size
                                                    qq
                                                    q
                                                    q q     q   q       q
                                                                        q        q   q
                                                                                     q   q   q   q    qqq     q       q        q
                                                            q   q           q                                                  q
                                                    q q     q   q   q   q            q   q            qq
                                             105    q                            q       q   q        qq      q
                                                    q
                                                    q
                                                      q             q   q
                                                                                             q   q
                                                                                                 q
                                                                                                 q    q       q                q
                                                                                                                                    q
                                                                                                                                    q
                                                                                                                                                  q   3000000
                                                      q
                                                      q         q   q   q   q    q
                                                                                 q   q                q               q
                                                                                                                      q        q             q
                                                    q q     q               q
                                                                            q    q   q   q
                                                                                         q              q             q             q
                                                                                     q       q
                                                                                             q        q
                                                    q q
                                                                                                 q
                                                                                                              q
                                                                                                              q
                                                                                                                                                  q   1000000
                                                                                                      q                             q
                                            104.5    q
                                                    qq      qq
                                                            qq
                                                                    q
                                                                    q   q
                                                                        q
                                                                        q
                                                                            q
                                                                            q    q   q   q
                                                                                         q
                                                                                                 q
                                                                                                          q
                                                                                                                      q
                                                                                                                      q
                                                                                                                               q
                                                                                                                               q
                                                                                                                                    q
                                                                                                                                             q
                                                        q
                                                                    q
                                                                                     q   q   q
                                                                                             q   q
                                                                                                          q                                       q   300000
                                                    q q
                                                    q                                q       q                q
                                                                                                      q
                                             104    q
                                                    q
                                                      q
                                                              q     q
                                                                    q   q
                                                                        q   q
                                                                            q    q
                                                                                 q   q   q   q
                                                                                             q
                                                                                                 q
                                                                                                 q        q
                                                                                                                               q
                                                                                                                               q
                                                                                                                                    q             q   100000
                                                            qq      q       q        q   q                            q        q             q
                                                            q
                                                            q                                    q                                  q
                                                    q q
                                                    q q
                                                                                                      q       q
                                                                                                                                                  q    30000
                                                                        q        q   q
                                                                                     q   q       q                                  q
                                                                                                                                    q
                                                                    q                        q
                                            103.5   qq
                                                    q               q
                                                                    q   q   q            q   q
                                                                                             q                                 q
                                                            qq
                                                            qq                                                        q        q             q
                                                            q                                             q
                                                                                                          q
                                                                                                                                    q
                                                                                                                                             q    q    10000
                                                    q q
                                                    q q
                                                                                                 q            q
                                                                                                 q
                                                                                                 q            q
                                                                                                      q
                                                3   q
                                                     q                                                                                            q     3000
                                             10                                                           q
                                                    q
                                                    q
                                                                                                      q
                                                                                                                                                  q     1000
                                                    q
                                            102.5                                                                                                 q       300
                                                                                                                                    q
                                                                                                                                                  q       100
                                             102                                                                                    q



                                                                20          40       60          80               5       10       15   20
                                                                                 Number of threads


                                                                                                                                                                21 / 57
STINGER component monitoring
rates

Seconds per batch, “latency”

                                                            E7−8870                                           X5570
                                                                                          q
                                                                                              q
                                            q
                                            q q
                                                    q
                                                  qqq                                     qq
                                                                                                  q
                                                                                                  q                                  Batch size
                                                  qqq   q   q   q    q                     q              q        q
                                                        q       q        q           q                    q             q
    Seconds per batch processed




                                                            q        q   q   q   q
                                                                                 q   q                                  q
                                                                                                                        q
                                    101                                                                            q             q
                                                                                                                                 q    q   3000000
                                                                                          q
                                            qq                                                q
                                            q
                                            qq     q
                                                  qqq   q   q   q
                                                                q    q           q   q
                                                                                     q        q
                                                                                                  q
                                                                                                  q       q
                                                                                                                        q             q   1000000
                                                        q   q
                                                            q   q    q   q   q
                                                                             q   q                        q        q
                                                  q q                    q                    q           q        q    q        q
                                  100.5     q
                                            q                                             q
                                                                                                  q                     q        q
                                                                                                                                      q   300000
                                            q q
                                              q                          q           q        q
                                                                                              q
                                            q qqq       q
                                                        q   q                q   q
                                                                     q           q                q
                                              qq
                                                 q      q   q   q    q   q   q   q   q
                                                                                     q        q
                                                                                                          q
                                                                                                                   q    q
                                                                                                                        q
                                                                                                                        q        q
                                                                                                                                      q   100000
                                                 qq                                               q                              q
                                            q q                                                                         q        q
                                    100                                                   q                        q
                                                                                                                   q
                                                                                                          q
                                            q                                                 q
                                                                                                                                      q    30000
                                              qq                         q
                                                                         q       q   q
                                              q                      q
                                            qq q q q    q   q
                                                            q   q
                                                                q
                                                                q
                                                                     q       q
                                                                             q   q   q
                                                                                     q        q   q       q
                                                                                                                   q
                                                                                                                   q    q
                                            q qq
                                            q
                                                  qq    q   q                             q       q
                                                                                                          q        q
                                                                                                                                 q
                                                                                                                                 q    q    10000
                                  10−0.5    q
                                            q                                        q                    q             q
                                            q
                                            q                                             qq
                                                                                          q
                                            q q
                                              q         q       q        q
                                                                         q   q
                                                                             q
                                                                                 q   q
                                                                                     q
                                                                                          qq
                                                                                          q
                                                                                          q       q
                                                                                                                                      q     3000
                                            q
                                            q     qq        q   q        q       q          q                      q
                                            q
                                            q q                      q
                                                                     q                                    q
                                                                                                          q
                                              q     q   q   q                        q
                                       −1
                                              q
                                              q     q                                q
                                                                                     q
                                                                                            q
                                                                                            q
                                                                                            q     q
                                                                                                          q        q    q
                                                                                                                        q
                                                                                                                                 q
                                                                                                                                 q
                                                                                                                                      q     1000
                                              q                                  q                q
                                   10         q
                                              qqqq
                                              q qqq     q   q        q   q   q
                                                                             q
                                                                                 q   q
                                                                                     q
                                                                                     q
                                                                                                  q
                                                                                                                   q
                                                            q   q        q                        q       q
                                                q q
                                                q       q   q   q    q
                                                                         q       q   q
                                                                                                  q
                                                                                                  q
                                                                                                  q       q
                                                                                                                   q
                                                                                                                   q             q    q       300
                                                q
                                                qq              q    q   q   q
                                                                             q   q
                                                                                 q                q                     q
                                                                                                                        q
                                                qqq     q       q    q   q
                                                                         q   q   q   q                    q
                                                                                                          q                      q
                                                  q     q   q   q    q       q                            q        q
                                    −1.5          q
                                                  qq
                                                  qq    q
                                                        q   q
                                                            q
                                                            q
                                                                q
                                                                q
                                                                q
                                                                     q
                                                                     q
                                                                     q   q
                                                                         q   q   q
                                                                                                          q
                                                                                                          q
                                                                                                          q
                                                                                                                   q
                                                                                                                   q
                                                                                                                   q    q
                                                                                                                        q
                                                                                                                                 q
                                                                                                                                 q    q       100
                                  10                q   q
                                                        q
                                                        q   q
                                                            q        q
                                                                     q   q   q
                                                                             q
                                                                                 q
                                                                                 q   q
                                                                                     q
                                                                                                                   q
                                                                                                                   q    q
                                                                                                                        q
                                                                                                                        q
                                                                                                                        q
                                                                                                                        q
                                                                         q



                                                    20          40       60          80               5       10       15   20
                                                                    Number of threads


                                                                                                                                                    21 / 57
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms
STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms

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STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms

  • 1. STING: Spatio-Temporal Interaction Networks and Graphs for Intel Platforms David Bader, Jason Riedy, David Ediger, Rob McColl, Timothy G. Mattson∗ 24 July 2012
  • 2. Outline Motivation STING for streaming, graph-structured data on Intel-based platforms World-leading community detection Community-building interactions Plans and direction Note: Without hard limits, academics will use at least all the time... 2 / 57
  • 3. (Prefix)scale Data Analysis Health care Finding outbreaks, population epidemiology Social networks Advertising, searching, grouping Intelligence Decisions at scale, regulating algorithms Systems biology Understanding interactions, drug design Power grid Disruptions, conservation Simulation Discrete events, cracking meshes 3 / 57
  • 4. Graphs are pervasive Graphs: things and relationships • Different kinds of things, different kinds of relationships, but graphs provide a framework for analyzing the relationships. • New challenges for analysis: data sizes, heterogeneity, uncertainty, data quality. Astrophysics Bioinformatics Social Informatics Problem Identifying target Problem Emergent behavior, Problem Outlier detection proteins information spread Challenges Massive data Challenges Data Challenges New analysis, sets, temporal variation heterogeneity, quality data uncertainty Graph problems Matching, Graph problems Centrality, Graph problems Clustering, clustering clustering flows, shortest paths 4 / 57
  • 5. No shortage of data... Existing (some out-of-date) data volumes NYSE 1.5 TB generated daily, maintain a 8 PB archive Google “Several dozen” 1PB data sets (CACM, Jan 2010) Wal-Mart 536 TB, 1B entries daily (2006) EBay 2 PB, traditional DB, and 6.5PB streaming, 17 trillion records, 1.5B records/day, each web click is 50-150 details. http://www.dbms2.com/2009/04/30/ ebays-two-enormous-data-warehouses/ Facebook 845 M users... and growing(?) • All data is rich and semantic (graphs!) and changing. • Base data rates include items and not relationships. 5 / 57
  • 6. General approaches • High-performance static graph analysis • Develop techniques that apply to unchanging massive graphs. • Provides useful after-the-fact information, starting points. • Serves many existing applications well: market research, much bioinformatics, ... • High-performance streaming graph analysis • Focus on the dynamic changes within massive graphs. • Find trends or new information as they appear. • Serves upcoming applications: fault or threat detection, trend analysis, ... Both very important to different areas. GT’s primary focus is on streaming. Note: Not CS theory streaming, but analysis of streaming data. 6 / 57
  • 7. Why analyze data streams? Data transfer Data volumes • 1 Gb Ethernet: 8.7TB daily at NYSE 1.5TB daily 100%, 5-6TB daily realistic LHC 41TB daily • Multi-TB storage on 10GE: 300TB Facebook, etc. Who daily read, 90TB daily write knows? • CPU ↔ Memory: QPI,HT: 2PB/day@100% Data growth Speed growth • Facebook: > 2×/yr • Ethernet/IB/etc.: 4× in next 2 • Twitter: > 10×/yr years. Maybe. • Growing sources: • Flash storage, direct: 10× write, Bioinformatics, 4× read. Relatively huge cost. µsensors, security 7 / 57
  • 8. Intel’s non-numeric computing program (our perspective) Supporting analysis of massive graph under rapid change across the spectrum of Intel-based platforms. STING on Intel-based platforms • Maintain a graph and metrics under large data changes • Performance, documentation, distribution • Greatly improved ingestion rate, improved example metrics • Available at http://www.cc.gatech.edu/stinger/. • Algorithm development • World-leading community detection (clustering) performance on Intel-based platforms • Good for focused or multi-level analysis, visualization, ... • Developments on community monitoring in dynamic graphs 8 / 57
  • 9. On to STING... Motivation STING for streaming, graph-structured data on Intel-based platforms Assumptions about the data High-level architecture Algorithms and performance World-leading community detection Community-building interactions Plans and direction 9 / 57
  • 10. Overall streaming approach Yifan Hu’s (AT&T) visualization of the Livejournal data set Jason’s network via LinkedIn Labs Assumptions • A graph represents some real-world phenomenon. • But not necessarily exactly! • Noise comes from lost updates, partial information, ... 10 / 57
  • 11. Overall streaming approach Yifan Hu’s (AT&T) visualization of the Livejournal data set Jason’s network via LinkedIn Labs Assumptions • We target massive, “social network” graphs. • Small diameter, power-law degrees • Small changes in massive graphs often are unrelated. 10 / 57
  • 12. Overall streaming approach Yifan Hu’s (AT&T) visualization of the Livejournal data set Jason’s network via LinkedIn Labs Assumptions • The graph changes but we don’t need a continuous view. • We can accumulate changes into batches... • But not so many that it impedes responsiveness. 10 / 57
  • 13. STING’s focus Control action prediction summary Source data Simulation / query Viz • STING manages queries against changing graph data. • Visualization and control often are application specific. • Ideal: Maintain many persistent graph analysis kernels. • One current snapshot of the graph, kernels keep smaller histories. • Also (a harder goal), coordinate the kernels’ cooperation. • STING components: • Framework for batching input changes. • Lockless data structure, STINGER, accumulating a typed graph with timestamps. 11 / 57
  • 14. STINGER STING Extensible Representation: • Rule #1: No explicit locking. • Rely on atomic operations. • Massive graph: Scattered updates, scattered reads rarely conflict. • Use time stamps for some view of time. 12 / 57
  • 15. STING’s year Over the past year • Improved performance on Intel-based platforms by 14×[HPEC12] • Evaluated multiple dynamic kernels across platforms[MTAAP12, ICASSP2012] • Documentation and tutorials • Assist projects with STING adoption on Intel-based platforms 13 / 57
  • 16. STING in use Where is it? http://www.cc.gatech.edu/stinger/ Code, development, documentation, presentations... Who is using it? • Center for Adaptive Supercomputing Software – Multithreaded Architectures (CASS-MT) directed by PNNL • PRODIGAL team for DARPA ADAMS (Anomaly Detection at Multiple Scales) including GTRI, SAIC, Oregon State U., U. Mass., and CMU • DARPA SMISC (Social Media in Strategic Communication) including GTRI • Well-attended tutorials given at PPoPP, in MD, and locally. 14 / 57
  • 17. Experimental setup Unless otherwise noted Line Model Speed (GHz) Sockets Cores Nehalem X5570 2.93 2 4 Westmere X5650 2.66 2 6 Westmere E7-8870 2.40 4 10 • Large Westmere, mirasol, loaned by Intel (thank you!) • All memory: 1067MHz DDR3, installed appropriately • Implementations: OpenMP, gcc 4.6.1, Linux ≈ 3.2 kernel • Artificial graph and edge stream generated by R-MAT [Chakrabarti, Zhan, & Faloutsos]. • Scale x, edge factor f ⇒ 2x vertices, ≈ f · 2x edges. • Edge actions: 7/8th insertions, 1/8th deletions • Results over five batches of edge actions. • Caveat: No vector instructions, low-level optimizations yet. • Portable: OpenMP, Cray XMT family 15 / 57
  • 18. STINGER insertion / removal rates Edges per second E7−8870 X5570 Aggregate STINGER updates per second 107 q q q q q q q q q q q q q q q q Batch size qqq q q q q q q q q q q q q q 6.5 q q q q 10 q qq q q q q q q q q q q q q q q 3000000 qqqq q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q 1000000 q q q q q q q q q q 106 q q qqq q qq q q q q q q q q q q qq q q q q 300000 q q q q q q q q q q q q q q q q q q q q q qq q q q q q qq q qq q q q q q q q q q q q 100000 5.5 qqq q q 10 q q q q q qq q q q q q q q q q q q q q qq qq q q q q q 30000 q q qqq q q q q 105 q q q q qq q q q q q q q q q q q 10000 qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 3000 q q q 104.5 q q q q q q q q q q 1000 q q q q 300 104 q q q 100 q 20 40 60 80 5 10 15 20 Number of threads 16 / 57
  • 19. STINGER insertion / removal rates Seconds per batch, “latency” E7−8870 X5570 q q q q Batch size q Seconds per batch processed q q 0 q q 3000000 10 q q q q qq q q q q q q q q q q q q q q q q q q q q q q 1000000 qqq q q q q q q q q q q q q q qq q q q q q q q q q q q q q 300000 q q q qqq q 10−1 q q q q q q q qq q q q q q q q q q q q 100000 q q q q qq q q q q q q q q q q q q q q q q 30000 q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q q q q q q 10−2 q q q q q q q q q q q q q q q q q q q q q q 10000 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 3000 q q q q q q q qq q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q 1000 −3 q 10 qq q q q q q q qq q q q q q q 300 q q q q q q q q q q q q q 100 q 20 40 60 80 5 10 15 20 Number of threads 16 / 57
  • 20. Clustering coefficients • Used to measure “small-world-ness” [Watts & Strogatz] and potential i m community structure • Larger clustering coefficient ⇒ more v inter-connected j n • Roughly the ratio of the number of actual to potential triangles • Defined in terms of triplets. • i – v – j is a closed triplet (triangle). • m – v – n is an open triplet. • Clustering coefficient: # of closed triplets / total # of triplets • Locally around v or globally for entire graph. 17 / 57
  • 21. STINGER clustering coefficient rates Edges per second E7−8870 X5570 Clustering coefficient updates per second q q q q q q 105.5 q q q q Batch size q q q q q q q q q q q qqq q q q q q q q q q q q q q q q q q q q q 3000000 q q q q q q 5 q q q q q q q q 10 qq q q q q q q q qq q q q q q q q q q q q q q 1000000 q q q q q q qq q q q q qq qqq q q q q q q qq q q q q q q q q q q 300000 q q qq q q q q q q qqq q q q qq q q q q q q qq q qqq q q q q q q q q q q q q q qq q q 104.5 q qqq q q q q q q q q q q q q q q q q q q qq q q q q q 100000 q q q q q q q q qqq q q q qq q q q q q qq q 30000 q q 104 q qq q q q q q q q q q q q q q q q 10000 q q q q q q q q q q 3000 q q q q q 103.5 q q q q q q q q q 1000 q q q 300 103 q q 100 q 20 40 60 80 5 10 15 20 Number of threads 18 / 57
  • 22. STINGER clustering coefficient rates Speed-up over static recalculation Clustering coefficient speedup over recalculation E7−8870 X5570 q q qq qq q qq q q q qq q q q qq q qq q q q Batch size q 102 qq q q q q q q q q q q q q 3000000 qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 1000000 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 300000 q q q 101 q q q q q q q q q q q q q q q q q q q q 100000 q q q q q q q q q q q q qq q q q qq q q q q 30000 q q q q q q q q q qq q q q q q q q qq q q q q q q q q q q 10000 0 q q q 10 q q q q q q q q q q q q q q q q q q q q q q q 3000 q q q q q q q q q q q q q q q q q q q 1000 q q q q q q q q q q qqq q −1 q q q q q q q 300 q 10 q q q q q q q q q q q q q q q q q 100 q q q q q q q q 20 40 60 80 5 10 15 20 Number of threads 18 / 57
  • 23. Connected components • Maintain a mapping from vertex to component. • Global property, unlike triangle counts • In “scale free” social networks: • Often one big component, and • many tiny ones. • Edge changes often sit within components. • Remaining insertions merge components. • Deletions are more difficult... 19 / 57
  • 24. Connected components: Deleted edges The difficult case • Very few deletions matter. • Maintain a spanning tree, & ignore deletion if 1 not in spanning tree, 2 endpoints share a common neighbor∗ , 3 loose endpoint reaches root∗ . • In the last two (∗), also fix the spanning tree. • Rules out 99.7% of deletions. • (Long history, see related work in [MTAAP11].) 20 / 57
  • 25. STINGER component monitoring rates Edges per second E7−8870 X5570 Connected component updates per second 105.5 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q q q q q q q q q q Batch size qq q q q q q q q q q q q q q qqq q q q q q q q q q q q q q q q qq 105 q q q q qq q q q q q q q q q q q q q q q q 3000000 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 1000000 q q 104.5 q qq qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q 300000 q q q q q q q 104 q q q q q q q q q q q q q q q q q q q q q q q 100000 qq q q q q q q q q q q q q q q q q q q 30000 q q q q q q q q q q 103.5 qq q q q q q q q q q qq qq q q q q q q q q q 10000 q q q q q q q q q q 3 q q q 3000 10 q q q q q 1000 q 102.5 q 300 q q 100 102 q 20 40 60 80 5 10 15 20 Number of threads 21 / 57
  • 26. STINGER component monitoring rates Seconds per batch, “latency” E7−8870 X5570 q q q q q q qqq qq q q Batch size qqq q q q q q q q q q q q q q Seconds per batch processed q q q q q q q q q 101 q q q q 3000000 q qq q q qq q qqq q q q q q q q q q q q q q q 1000000 q q q q q q q q q q q q q q q q q q q 100.5 q q q q q q q 300000 q q q q q q q q qqq q q q q q q q q qq q q q q q q q q q q q q q q q q q q 100000 qq q q q q q q 100 q q q q q q q 30000 qq q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q qq q qq q q q q q q q q q 10000 10−0.5 q q q q q q q qq q q q q q q q q q q q q q qq q q q q 3000 q q qq q q q q q q q q q q q q q q q q q q −1 q q q q q q q q q q q q q q q q 1000 q q q 10 q qqqq q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 300 q qq q q q q q q q q q q qqq q q q q q q q q q q q q q q q q q q q −1.5 q qq qq q q q q q q q q q q q q q q q q q q q q q q q q q q 100 10 q q q q q q q q q q q q q q q q q q q q q q q 20 40 60 80 5 10 15 20 Number of threads 21 / 57