Subgraph Isomorphism Graph Challenge
- draft Siddharth Samsi, Vijay Gadepally, Jeremy Kepner, Albert Reuther
http://GraphChallenge.org
Outline
•  Introduction
•  Data Sets
•  Static Graph Isomorphism Challenge
•  Metrics
•  Summary
Graph Challenge - 2
Introduction
•  Previous challenges in machine learning, High Performance Computing and visual
analytics include
–  YOHO, MNIST, HPC Challenge, ImageNet, VAST
•  GraphChallenge encourages community approaches, such as DARPA HIVE, to
develop new solutions for analyzing graphs derived from social media, sensor feeds,
and scientific data to enable relationships between events to be discovered as they
unfold in the field
•  GraphChallenge organizers will provide specifications, data sets, data generators, and
serial implementations in various languages
•  GraphChallenge participants are encouraged to apply innovative hardware, software,
and algorithm techniques to push the envelop of power efficiency, computational
efficiency, and scale
•  Submissions will be in the form of full conference write ups to IEEE HPEC which will
allow participants to be evaluated on their complete solution
Graph Challenge - 3
Graph Challenge
•  GraphChallenge seeks input from diverse communities to develop graph challenges
that take the best of what has been learned from groundbreaking efforts such as
GraphAnalysis, Graph500, FireHose, MiniTri, and GraphBLAS to create a new set of
challenges to move the community forward
•  Initial Graph Challenges
–  Static Graph Challenge: Sub-Graph Isomorphism
•  This challenge seeks to identify a given sub-graph in a larger graph
–  Streaming Graph Challenge: Stochastic Block Optimization
•  This challenge seeks to identify optimal blocks (or clusters) in a larger graph
Graph Challenge - 4
Static versus Streaming Mode
•  Static graph processing
–  Given a large graph G
–  Evaluate ƒ(G)
•  Two classes of streaming: stateless and stateful
–  Stateless: process data g as it goes by ƒ(g)
–  Stateful: add data g to a larger corpus G and then process corpus ƒ(G + g)
•  Graph processing is often in the stateful category
–  Easy to simulate by partitioning G into streaming pieces g
Graph Challenge - 5
Labels and Filtering
•  Filtering on edge/vertex labels is often used when available to reduce the search
space
–  Filtering can be applied at initialization, during intermediate steps, or at the vertex level
–  Very problem dependent
•  Some Graph Challenge data sets have labels and some data sets are unlabeled
–  Some participants will want to filter on labels
•  Initial example implementations will work without labels
–  Labels can be used if they choose
•  GraphChallenge is judged by a panel
–  Panel will value the variations that are submitted
Graph Challenge - 6
Outline
•  Introduction
•  Data Sets
•  Static Graph Isomorphism Challenge
•  Metrics
•  Summary
Graph Challenge - 7
Publicly Available Datasets
Initial Datasets in Blue
Stanford Large Network Dataset Collection snap.stanford.edu/data
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Social networks (10 data sets up to 4.8M vertices & 69M edges)
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Nodes represent scientists, edges represent collaborations
Nodes represent webpages and edges are hyperlinks
Amazon networks (5 data sets up to 548K vertices & 3.4M edges)
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Nodes represent products and edges link commonly co-purchased
products
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Nodes represent computers and edges communication
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Data from online communities such as Reddit and Flickr
Online reviews (6 data sets up to 34M product reviews)
– 
Graph Challenge - 8
Memetracker phrases, links and Tweets
Online communities (3 data sets up to 2.3M images)
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Internet networks (9 data sets up to 62K vertices & 147K edges)
Talk, editing, voting, and article data from Wikipedia
Twitter and Memetracker (4 data sets up to 96M vertices & 476M edges)
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Social networks with geographic check-ins
Wikipedia networks, articles, and metadata (7 data sets up to 3.5M
vertices & 250M edges)
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Networks with positive and negative edges (friend/foe, trust/distrust)
Location-based online social networks (2 data sets up to 198K vertices
& 950K edges)
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Graphs of the internet
Signed networks (10 data sets up to 4.8M vertices & 69M edges)
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Web graphs (4 data sets up to 875K vertices & 5.1M edges)
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Nodes represent intersections and edges roads connecting the
intersections
Autonomous systems (5 data sets up to 26K vertices & 106K edges)
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Nodes represent papers, edges represent citations
Collaboration networks (5 data sets up to 23K vertices & 198K edges)
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Email communication networks with edges representing communication
Citation networks (3 data sets up to 3.7M vertices & 16M edges)
Road networks (3 data sets up to 1.9M vertices & 2.8M edges)
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Ground-truth network communities in social and information networks
Communication networks (3 data sets up to 2.3M vertices & 5M edges)
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Online social networks, edges represent interactions between people
Networks with ground-truth (6 data sets up to 66M vertices & 1.8B edges,
e.g. Friendster)
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Data from online review systems such as BeerAdvocate and Amazon
Publicly Available Datasets
Initial Datasets in Blue
Yahoo! Webscope Datasets webscope.sandbox.yahoo.com
AWS Public Data Sets aws.amazon.com/public-data-sets
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Astronomy (1 data set 180 GB)
–  Sloan Digital Sky Survey SQL MDF files
Biology (4 data sets up to 200 TB)
–  Genome sequence data
Climate (3 data sets size growing daily)
–  Satellite imagery data
Economics (10 data sets up to 220 GB)
–  Census, transaction, and transportation data
• 
Encyclopedic (10 data sets up to 541 TB - Common Crawl Corpus)
–  Various online encyclopedia data
Geographic (3 data sets up to 125 GB)
• 
–  Street maps
Mathematics (1 data sets 160 GB)
• 
–  Yahoo!'s auction-based platform for selling advertising space
• 
Competition Data (3 data sets up to 1.5 GB)
–  Data challenges run by Yahoo
• 
Computing Systems Data (5 data sets up to 8.8 GB)
–  Computer systems log data
• 
Graph and Social Data (3 data sets up to 5 GB)
–  Graph data from search, groups, and webpages
• 
Image Data (3 data sets up to 14 GB)
–  Flickr imagery and metadata
• 
Language Data (29 data sets up to 166 GB - Answers browsing behavior)
–  Wide range of question/answer data sets
–  University of Florida Sparse Matrix Collection
• 
Graph500.org Data Generator
Advertising and Market Data (4 data sets up to 3.7 GB)
Ratings and Classification Data (10 data sets up to 83 GB — 1.5 TB
dataset withdrawn)
–  Community preferences and data
•  Used to generate world’s largest power law graphs
•  Can be modeled with Kronecker Product of a Recursive
MATrix (R-MAT): G⊗k = G⊗k–1 ⊗ G
– 
Where “⊗”denotes the Kronecker product of two matrices
Graph Challenge - 9
Tools to generate (at various scales and parameters)
data sets will be provided as part of the challenge
•  R-MAT: A Recursive Model for Graph Mining, Chakrabarti, Zhan & Faloutsos (2004)
•  Mathematically Tractable Graph Generation and Evolution, Leskovec, Chakrabarti, Kleinberg & Faloutsos, (2005)
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Twitter Data
•  Public data is available in a variety of formats
–  Linked list, tab separated, labeled/unlabeled
•  Requires parsing and standardization
•  Proposed formats
–  Tab separated triples in ASCII file with labels removed
–  MMIO ASCII format: math.nist.gov/MatrixMarket
Amazon cloud services will be used to host particularly large data sets
Graph Challenge - 10
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GraphChallenge Data Formats
Outline
•  Introduction
•  Data Sets
•  Static Graph Isomorphism Challenge
•  Metrics
•  Summary
Graph Challenge - 11
Static Subgraph Isomorphism Problem
Vertex-Based
Array-Based
•  Given a sub-graph H and
a larger graph G
•  Given sub-graph adjacency array B and
a larger graph adjacency array A
•  Is there a 1-to-1 mapping of the vertices
in H to vertices in G such that every
edge in H is also in G ?
•  A subgraph isomorphism exists iff there
is a permutation array P where
B = PT A P
Bold uppercase indicates a matrix, Bold lowercase indicates a vector, lowercase indicates scalars, sets, ...
Graph Challenge - 12
Scaling Observations
•  Two key parameters govern the complexity of subgraph isomorphism
–  Size of G and H
–  Nominal complexity is |G||H|
•  Large G and H is may be too challenging
•  Large G or H is more practical
•  Small G and G and H are of same order is less interesting
–  Focus of program is large graphs
•  Large G and small H (G is much larger than H) is more interesting
–  Example: H is a triangle (i.e., triangle finding algorithm)
–  Example: H is a k-truss (i.e., k-truss algorithm)
Graph Challenge - 13
Triangle Counting
•  Triangles are the most basic non-trivial subgraph
–  Set of three mutually adjacent vertices in a graph
•  Number of triangles in a graph is an important metric used in applications such as
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Social network mining (e.g.: clustering coefficient calculation)
Link classification and recommendation
Cyber security
Intelligence
3
Functional biology
1
Spam detection
•  Example of triangles in a Graph
–  Triangle 1 : Nodes 3,4,5
–  Triangle 2 : Nodes 2,4,5
Graph Challenge - 14
Counting and Sampling Triangles from a Graph Stream
A. Pavan, K. Tangwongsan, S. Tirthapura & Kun-Lung Wu, VLDB 2013
4
2
5
2
1
Triangle Counting and Enumeration
- Example Algorithm -
•  Lower complexity array approach
gle listing
results for
SNAP datasets.
•  Triangle
enumeration
has big I/O
•  Triangle counting has minimal I/O
Article
Graphing trillions of triangles
Information Visualization
1–10
! The Author(s) 2016
Reprints and permissions:
sagepub.co.uk/journalsPermissions.nav
DOI: 10.1177/1473871616666393
ivi.sagepub.com
Paul Burkhardt
rkhardt
hich every vertex is connected to every other vertex,
d a cycle, i.e. a path with identical endpoints.
iangles in graphs are useful for complex network
alysis based solely on structural, i.e. adjacency,
ormation. Let D(v) be the local count of triangles
volving vertex v and D(G) be the global count of trigles in G. There can be many more triangles than
rtices and edges, and if the graph is completely concted, where G itself is a clique, the total number of
Graph Challenge - 15
5
Abstract
The increasing size of Big Data is often heralded but how data are transformed and represented is also profoundly important to knowledge discovery, and this is exemplified in Big Graph analytics. Much attention has
been placed on the scale of the input graph but the product of a graph algorithm can be many times larger
than the input. This is true for many graph problems, such as listing all triangles in a graph. Enabling scalable graph
to algorithms, architectures, and visual analy2 s exploration for Big Graphs requires new
1 approaches 2
s
tics. A brief tutorial is given to aid the argument for
thoughtful
representation
v of data in the context of graph
v
2
analysis.
Then a new
1
2 s algebraic method to reduce the arithmetic operations in counting and listing triangles in
graphs
a scalable triangle listing algorithm in the MapReduce model will be preij
6 isijintroduced. Additionally,
sented followed by a description of the experiments with that algorithm that led to the current largest and
fastest triangle listing benchmarks to date. Finally, a method for identifying triangles in new visual graph
exploration technologies is proposed.
Theorem 1. Given G P
and the Hadamard product
(A
(A A) and D(G) =
P A), then D(v) =
(A A) .
Proof. The paths of length two, i.e. 2-paths, are identified by A2 ; then element-wise multiplication A A2
Keywords
retains
only those
fv, ugvisualendpoints
2-paths
that are
Graph,
scalable algorithms,
triangle counting,
analytics, parallelof
programming,
MapReduce
also the fv, ug 2 E endpoints of edges. Then the sum
connected triplets in a graph2or network of entities,
Introduction
th
column
vector
of AwhereAeachisvertex
the
of
elements
in
the
v
often depicted
as a triangle
of the
Data, data, data, and more data! Obtaining data is just
s
s
triangle represents an entity that is connected
1 3to the
Triangle Counting using miniTri
- Example Algorithm •  Array based algorithm
–  Given a graph adjacency array A and incidence array E
3
C = AE
nT = nnz(C)/3
4
5
•  Multiplication is overloaded such that
2
C(i,j) = {i,x,y} iff
A(i,x) = A(i,y) = 1 and ET(x,j) = ET(y,j) = 1
0
B
B
A
=
A B
B
@
0
0
0
0
1
Graph Challenge - 16
0
0
0
1
1
0
0
0
1
1
0
1
1
0
1
1
1
1
1
0
1
C
C
C
C
A
0
B
B
T
B
=
E B
B
@
1
0
0
0
1
0
1
0
0
1
0
0
1
0
1
0
0
1
1
0
0
1
0
1
0
0
0
0
1
1
1
C
C
C
C
A
0
B
B
C=AET ==BB@
;
;
;
;
;
;
;
;
;
;
;
{2, 4, 5}
;
;
;
;
;
{3, 4, 5}
; {4, 2, 5} {4, 3, 5}
;
;
;
;
;
;
{5, 3, 4} {5, 2, 4}
;
miniTri represented in compact linear algebra-based operations
Kokkos/Qthreads Task-Parallel Approach to Linear Algebra Based Graph Analytics
Wolf, Edwards & Olivier, IEEE HPEC 2016
1
1
C
C
C
C
A
K-Truss
•  A graph is a k-truss if each edge is part of at least k-2 triangles
•  A generalization of a clique (a k-clique is a k-truss), ensuring a minimum level of
connectivity within the graph
•  Traditional technique to find a k-truss subgraph:
–  Compute the support for every edge
–  Remove any edges with support less than k-2 and update the list of edges
–  When all edges have support of at least k-2, we have a k-truss
Example 3-truss
Graph Challenge - 17
Graphulo: Linear algebra graph kernels for NoSQL databases
Gadepally, Bolewski, Hook, Hutchison, Miller & Kepner, IPDPS GABB 2015
K Truss: Array Formulation
- Example Algorithm -
•  If E is the unoriented incidence array
(rows are edges and columns are
vertices) of graph G, and A is the
adjacency array
Algorithm
R=EA
x = find( (R == 2 ) 1 < k − 2 )
while x
Ex = E(x,∶)
•  If G is a k-truss, the following must
be satisfied
–  (E A == 2) 1 > (k – 2)
E = E(xc,∶)
R = E(xc,∶) A
R = R − E ( ExExT − diag(ExExT) )
x = find( (R == 2 ) 1 < k − 2 )
Graph Challenge - 18
Graphulo: Linear algebra graph kernels for NoSQL databases,
Gadepally, Bolewski, Hook, Hutchison, Miller & Kepner, IPDPS GABB 2015
Outline
•  Introduction
•  Data Sets
•  Static Graph Isomorphism Challenge
•  Metrics
•  Summary
Graph Challenge - 19
Metrics
•  Correctness
–  Number of triangles is exact for a given graph
–  Enumerating all k-trusses is exact for a given graph
•  Performance
– 
– 
– 
– 
– 
Total number of edges in graph (edges)
Execution time (seconds)
Rate (edges/second)
Energy (Watts)
Rate per energy (edges/second/Watt)
Graph Challenge - 20
Summary
•  Static sub-graph isomorphism graph challenge consists of
–  Triangle finding
–  K-Truss
•  An implementation will perform these algorithms on provided data sets and report the
given metrics
•  A submission to the IEEE HPEC Graph Challenge consists of a conference style paper
describing the approach, implementation, innovations, and results
•  Both hardware and software should be described in solution
–  Innovative hardware solutions are of interest (in addition to best algorithm for hardware)
–  Special interest in performance for very large scale data sets (1-100 trillion edges)
Graph Challenge - 21