scalable community detection benchmark generation
TRANSCRIPT
Scalable Community Detection
Benchmark Generation
Jonathan Berry1 Cynthia Phillips1
Siva Rajamanickam1 George M. Slota2
1Sandia National Labs, 2Rensselaer Polytechnic [email protected], [email protected], [email protected], [email protected]
SIAM CSC 6 June 2018
Sandia National Laboratories is a multimission laboratory managed and operated by National Technology &Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S.
Department of Energys National Nuclear Security Administration under contract DE-NA0003525
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Motivation
Better evaluate community detection algorithmsprocessing O(Billion)-sized graphs on HPC resources
Small-scale state-of-the-art: “LFR”Lancichinetti, Fortunato, Radicchi, 2008With >1600 citations, this is a de facto standardGenerates ground truth to test againstHas a tunable parameter for community coherence: µLimited scalability: best implementation takes ∼17hrs togenerate O(1B) edges (Hamann et al., 2017)
Large-scale state-of-the-artWithout a reliable ground truth, parallel algorithms testwith modularity or similar measuresThis approach is flawed in several ways
Goal: evaluate at HPC scale against ground truth2 / 13
Overview
Primary results of this work:
We develop a novel method for generating large-scalegraphs with a tunable ground truth community structure
We utilize the scalable BTER generator (Kolda et al.,2014) as a core step
Our approach generates large-scale communitybenchmarking graphs at a rate of 1B edge/minute onKNL– Orders-of-magnitude faster than state-of-the-art
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BTERBlock Two-level Erdos-Reyni Graph Generator
Step 0: Input degree (nd) and clustering coefficient (cd) distributions
Step 1: With ordered degree sequence, group d + 1 vertices v of degree d(v) >= d into affinity blocksStep 2: Use Erdos-Renyi probability pd = 3√cd to create intra-block edges via G(n, p) processStep 3: Create inter-block edges via Chung-Lu process
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BTERBlock Two-level Erdos-Reyni Graph Generator
Step 0: Input degree (nd) and clustering coefficient (cd) distributionsStep 1: With ordered degree sequence, group d + 1 vertices v of degree d(v) >= d into affinity blocks
Step 2: Use Erdos-Renyi probability pd = 3√cd to create intra-block edges via G(n, p) processStep 3: Create inter-block edges via Chung-Lu process
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BTERBlock Two-level Erdos-Reyni Graph Generator
Step 0: Input degree (nd) and clustering coefficient (cd) distributionsStep 1: With ordered degree sequence, group d + 1 vertices v of degree d(v) >= d into affinity blocks
Step 2: Use Erdos-Renyi probability pd = 3√cd to create intra-block edges via G(n, p) processStep 3: Create inter-block edges via Chung-Lu process
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BTERBlock Two-level Erdos-Reyni Graph Generator
Step 0: Input degree (nd) and clustering coefficient (cd) distributionsStep 1: With ordered degree sequence, group d + 1 vertices v of degree d(v) >= d into affinity blocksStep 2: Use Erdos-Renyi probability pd = 3√cd to create intra-block edges via G(n, p) process
Step 3: Create inter-block edges via Chung-Lu process
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BTERBlock Two-level Erdos-Reyni Graph Generator
Step 0: Input degree (nd) and clustering coefficient (cd) distributionsStep 1: With ordered degree sequence, group d + 1 vertices v of degree d(v) >= d into affinity blocksStep 2: Use Erdos-Renyi probability pd = 3√cd to create intra-block edges via G(n, p) processStep 3: Create inter-block edges via Chung-Lu process
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Our Implementation - For Community DetectionwBTER – wrapped BTER
How we wrap the baseline BTER process for generatinggraphs for community detection benchmarking:
Treat affinity blocks as ground truth communities
We have a native µn, based on ratio of inter- tointra-block edges generated from the original distributions
Can shift µn to some target goal µg via a Linear Programsolve (to be described) – we use Pyomo and CBC
Our BTER implementation: fully-parallelized inshared-memory with OpenMP/C++
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Linear ProgramShifting the native µ of a graph’s CC distribution
Minimally shift the input clustering coefficient (CC) distribution suchthat the output graph has a desired goal µg considering both definitions:
µg =1
N
∑d
dinterd
µg =1
2M
∑d
nddinter
minimize∑d
|pd − pd|
subject to∑d
ndpd = N(1− µg)∑d
dndpd = 2M(1− µg)
0 ≤ pd ≤ 1output cd = p3d
pd is G(n, p) probabilities per degree from CC distribution cd, pd = 3√cd
pd is output probabilities to get new CC distribution cd, cd = p3dnd is degree distribution, n vertices of d degreedinter is expected number of inter-community edges for vertex of degree dN is number of vertices in graph, M is number of edges
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Experimental SetupTest system and test graphs
Test System: Bowman at Sandia Labs – each node has aKNL with 68 cores, 96 GB DDR, and 16 GB MCDRAM
Test Graphs:
Network n m davg dmax D
LJ-fp 4.2 M 27 M 18 20 K 18uk-2002 18 M 261 M 28 195 K 28Wikilinks 26 M 332 M 23 39 K 170RMAT 26 67 M 1.1 B 16 6.7 K 8Friendster 66 M 1.8 B 27 5.2 K 34
Graphs are from the SNAP, Koblenz, and LAW databases.LiveJournal-fp is a parsed version of LiveJournal from SNAP.
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Shifting DistributionHow the CC distribution shifts for varying µ
Only every 5th value plotted for better visualizationGenerally, distribution is most “accurate” near native µBetter smoothing of distribution via LP constraints is future work
0
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2 100 500 1000 2000
Vertex Degree
Clu
ste
rin
g C
oe
ffic
ien
t
Native 0.05 0.3 0.5 0.7
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Hitting Target MuAccuracy of LP for generating desired µ
Generation accuracy is comparable to LFRLess than 5% error in most instancesError is greatest at lower µ targets
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Target−mu
Ou
tpu
t−m
u
LJ−fp uk−2002 WikiLinks RMAT_26 Friendster
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Generation time vs. target µ(Left) Time vs. µ – (Right) Time vs. graph scale
Strong scaling generally good up to 2 threads/coreTime decreases with increasing µ, due to coupon collectors edgegeneration scaling - higher CC requires more attempts for each edgeGeneration time a function of scale and complexity (max degree)
Average ∼2 minutes for 1.8B unique edgesOriginal BTER code: ∼4 min. for 1.2B edges on 32 node Hadoop cluster
Fastest LFR implementation: 17 hours for 1B edges in shared-memory
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500
17 32 68 136 204 272
Threads
Ge
ne
ratio
n T
ime
(s)
LJ−fp uk−2002 WikiLinks RMAT_26 Friendster
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Number of Edges (Millions)
Ge
ne
ratio
n T
ime
(s)
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A Note on BTER Assortativity
An issue with our approach so far is the degreehomogeneity of communities
We propose the following addition:
Consider intra-comm edge count of each vertexPermute community assignments of all vertices withsame countObservation: won’t affect µ, de-homogenizescommunities in terms of degree
This approach might also be applied to baseline BTERgeneration
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Timing BreakdownFull wBTER approach with community permutation, µ = 0.5
Time costs of majorwBTER steps withcommunity assignmentpermutation
Work Complexity:d = Dmax, n = |V |,m = |E|
LP: expected to scaleas O(d log d)EdgeGen: O(m log d)Finalize: O(n+m)CSR: O(n+m)Swap: O(n log n+m)
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30
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LJ−fp
uk−2002
WikiLinks
RM
AT_26
Friendster
Graph
Tim
e (
s)
Parts
LP
EdgeGen
Finalize
CSR
Swap
LP: linear programEdgeGen: primary BTER phaseFinalize: remove 0-degree vertices & cleanupCSR: create graph representationSwap: community degree permutation
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Conclusionsand future work
We shift a graph’s CCD to fit a µ generated by BTER
Our approach can output graphs for community detectionorder-of-magnitudes faster than commonly-usedgenerators, e.g., LFR
Our approach can output graphs with more realisticdegree and CC distributions than commonly-usedgenerators
Future Work:Better develop LP to reduce noise in output CC distributionShift graph scale – i.e., output equivalent distributions for agraph with 2×, 12× original scaleDevelop generation methods for hierarchical communities
www.gmslota.com, [email protected] / 13