Discrete Structures & Algorithms Counting Counting I: One-To-One Correspondence and Choice Trees.
Fast counting of triangles in large networks without counting: Algorithms and laws
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Transcript of Fast counting of triangles in large networks without counting: Algorithms and laws
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CHARALAMPOS E. TSOURAKAKISSCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY
Fast counting of triangles in large networks without
counting:Algorithms and laws
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C. E. Tsourakakis
Triangle related problems
Given an undirected, simple graph G(V,E) a triangle is a set of three vertices such that any two of them are connected by an edge of the graph.
Related problems Decide if a graph is triangle-free. Count the total number of triangles Δ(G). Count the number of triangles Δ(v) that vertex
v participates in. List the triangles that each vertex v participates in.
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Generality
Our focus
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Why is Triangle Counting important?From the Graph Mining Perspective
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Clustering coefficient Transitivity ratio Social Network Analysis fact: “Friends of
friends are friends” [WF94]Other applications include:Hidden Thematic Structure of the Web [EM02]Motif Detection e.g. biological networks
[YPSB05]Web Spam Detection [BPCG08]
A
CB
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Outline
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Related WorkProposed Method
Theorems Algorithms Explaining efficiency
ExperimentsTriangle-related LawsTriangles in Kronecker GraphsConclusions
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Related Work
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Fast Low space
Time complexity
O(n2.37) O(n3)
Space complexity
O(n2) O(m)=O(n2)
Fast Low space
Time complexity
O(m0.7n1.2+n2+o(1)) e.g. O( n )
Space complexity
O(n2) (eventually) O(m)
2maxd
Dense graphs
S p a r s e g r a p h s
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Outline
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Related WorkProposed Method
Theorems Algorithms Explaining efficiency
ExperimentsTriangle-related LawsTriangles in Kronecker GraphsConclusions
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Theorem [EigenTriangle]
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Theorem 1
Δ(G) = # triangles in graph G(V,E) = eigenvalues of
adjacency matrix AG
||
1
3)(6V
iiG
||21 ... V
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Theorem [EigenTriangleLocal]
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Theorem 2
Δ(i) = #Δs vertex i participates at. = i-th eigenvector = j-th entry of
2||
1
3)(2 ij
V
jjui
ijuiu
iu
i
Δ(i) = 2
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Outline
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Related WorkProposed Method
Theorems Algorithms Explaining efficiency
ExperimentsTriangle-related LawsTriangles in Kronecker GraphsConclusions
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EigenTriangle Algorithm (interactively)
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I want to compute
the number of
triangles!
Use Lanczos to compute the first
two eigenvalues please!
Is the cube of the
second one significantly smaller than the cube of the first?
NOIterate
then!
After some iterations…(hopefully
few!)
Compute the k-th
eigenvalue.Is
much smaller than
?
3|| k
1
1
3k
i
YES!Algorithm
terminates! The estimated # of Δs
is the sum of cubes of λi’s divided by 6!
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EigenTriangle Algorithm
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EigenTriangleLocal Algorithm
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Why are these two
algorithms efficient on power law networks?
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Typical Spectra of Power Law Networks
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AirportsPolitical blogs
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1st Reason :Top Eigenvalues of Power-Law Graphs
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Very important for us because:Few eigenvalues contribute a lot!Cubes amplify this even more!Lanczos converges fast due to large spectral gaps [GL89]!
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1st Reason :Top Eigenvalues of Power-Law Graphs
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One of the first to observe that the top eigenvalues follow a power-law were Faloutsos, Faloutsos and Faloutsos [FFF99].
Some years later Mihail & Papadimitriou [MP02] and Chung, Lu and Vu [CLV03] gave an explanation of this fact.
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2nd Reason :Bulk of eigenvalues
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Almost symmetric around 0!
Sum of cubes almost cancels out!
Political Blogs
Omit!
Keep only 3!
3
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Outline
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Related WorkProposed Method
Theorems Algorithms Explaining efficiency
ExperimentsTriangle-related LawsTriangles in Kronecker GraphsConclusions
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Datasets
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Nodes Edges Description~75K ~405K Epinions network~404K ~2.1M Flickr~27K ~341K Arxiv Hep-Th~1K ~17K Political blogs~13K ~148K Reuters news~3M 35M Wikipedia 2006-Sep-05~3.15M
~37M Wikipedia 2006-Nov-04
~13.5K ~37.5K AS Oregon~23.5K ~47.5K CAIDA AS 2004 to 2008
(means over 151 timestamps)
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Datasets
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Nodes Edges Description~75K ~405K Epinions network~404K ~2.1M Flickr~27K ~341K Arxiv Hep-Th~1K ~17K Political blogs~13K ~148K Reuters news~3M 35M Wikipedia 2006-Sep-05~3.15M
~37M Wikipedia 2006-Nov-04
~13.5K ~37.5K AS Oregon~23.5K ~47.5K CAIDA AS 2004 to 2008
(means over 151 timestamps)
Social Networks
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Datasets
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Nodes Edges Description~75K ~405K Epinions network~404K ~2.1M Flickr~27K ~341K Arxiv Hep-Th~1K ~17K Political blogs~13K ~148K Reuters news~3M 35M Wikipedia 2006-Sep-05~3.15M
~37M Wikipedia 2006-Nov-04
~13.5K ~37.5K AS Oregon~23.5K ~47.5K CAIDA AS 2004 to 2008
(means over 151 timestamps)
Social Networks
Co-authorship network
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Datasets
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Nodes Edges Description~75K ~405K Epinions network~404K ~2.1M Flickr~27K ~341K Arxiv Hep-Th~1K ~17K Political blogs~13K ~148K Reuters news~3M 35M Wikipedia 2006-Sep-05~3.15M
~37M Wikipedia 2006-Nov-04
~13.5K ~37.5K AS Oregon~23.5K ~47.5K CAIDA AS 2004 to 2008
(means over 151 timestamps)
Social Networks
Co-authorship network
Information Networks
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Datasets
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Nodes Edges Description~75K ~405K Epinions network~404K ~2.1M Flickr~27K ~341K Arxiv Hep-Th~1K ~17K Political blogs~13K ~148K Reuters news~3M 35M Wikipedia 2006-Sep-05~3.15M
~37M Wikipedia 2006-Nov-04
~13.5K ~37.5K AS Oregon~23.5K ~47.5K CAIDA AS 2004 to 2008
(means over 151 timestamps)
Social Networks
Co-authorship network
Information Networks
Web Graphs
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Datasets
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Nodes Edges Description~75K ~405K Epinions network~404K ~2.1M Flickr~27K ~341K Arxiv Hep-Th~1K ~17K Political blogs~13K ~148K Reuters news~3M 35M Wikipedia 2006-Sep-05~3.15M
~37M Wikipedia 2006-Nov-04
~13.5K ~37.5K AS Oregon~23.5K ~47.5K CAIDA AS 2004 to 2008
(means over 151 timestamps)
Social Networks
Co-authorship network
Information Networks
Web Graphs
Internet Graphs
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Datasets
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~3.15M nodes~37M edges
Nodes Edges Description~75K ~405K Epinions network~404K ~2.1M Flickr~27K ~341K Arxiv Hep-Th~1K ~17K Political blogs~13K ~148K Reuters news~3M 35M Wikipedia 2006-Sep-05~3.15M
~37M Wikipedia 2006-Nov-04
~13.5K ~37.5K AS Oregon~23.5K ~47.5K CAIDA AS 2004 to 2008
(means over 151 timestamps)
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Competitor: Node Iterator 25
Node Iterator algorithm For each node, look at its neighbors, then
check how many edges among them.Complexity: O( )We report the results as the speedup vs.
Node Iterator.
2maxnd
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Results: #Eigenvalues vs. Speedup26
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Results: #Edges vs. Speedup 27
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Observe the trend
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Some interesting observations28
6.2 typical rank for at least 95%Speedups are between 33.7x and 1159x.
The mean speedup is 250.Notice the increasing speedup as the size of the network grows.
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Evaluating the Local Counting Method
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Triangles node i participatesTria
ngle
s no
de i
parti
cipa
tes
acco
rdin
g to
our
est
imat
ion
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#Eigenvalues vs. ρ for three networks
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2-3 eigenvaluesalmost ideal results!
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Outline
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Related WorkProposed Method
Theorems Algorithms Explaining efficiency
ExperimentsTriangle-related LawsTriangles in Kronecker GraphsConclusions
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Triangle Participation Power Law (TPPL)
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EPINIONS
δ = #TrianglesCou
nt o
f nod
es p
artic
ipat
ing
in δ
tria
ngle
s
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Triangle Participation Power Law (TPPL)
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HEP_TH (coauthorship)
Flickr
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Degree Triangle Power Law (DTPL)
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EPINIONS
d , all degrees appearing in the graph
Mea
n #Δ
s ov
er a
ll no
des
with
deg
ree
d
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Degree Triangle Power Law (DTPL)
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Flickr
Reuters
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Observations on TPPL & DTPL
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TTPL:Many nodes few triangles
Few nodes many triangles
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Observations on TPPL & DTPL
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DTPL: Power law fits nicely to the Degree-
Triangle plot. Slope is the opposite of the slope of the
degree distribution (slope complementarity).
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Outline
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Related WorkProposed Method
Theorems Algorithms Explaining efficiency
ExperimentsTriangle-related LawsTriangles in Kronecker GraphsConclusions
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Kronecker graphs
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Kronecker graphs is a model for generating graphs that mimic properties of real-world networks. The basic operation is the Kronecker product([LCKF05]).0 1 1
1 0 1
1 1 0
Initiator graph
Adjacency matrix A[0]
KroneckerProduct
Adjacency matrix A[1]Adjacency matrix A[2]
Repeat k times Adjacency matrix A[k]
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Triangles in Kronecker Graphs
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Theorem[KroneckerTRC ]Let B = A[k] k-th Kronecker product and Δ(GA),
Δ(GΒ) the total number of triangles in GA , GΒ . Then,
the following equality holds: 06 1 , k)Δ(G ) Δ(G k
Ak
B
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Outline
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Related WorkProposed Method
Theorems Algorithms Explaining efficiency
ExperimentsTriangle-related LawsTriangles in Kronecker GraphsConclusions
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Conclusions
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Triangles can be approximated with high accuracy in power law networks by taking a few, constant number of eigenvalues.
The method is easily parallelizable (matrix-vector multiplications only) and converges fast due to large spectral gaps.
New triangle-related power lawsClosed formula for triangles in Kronecker
graphs.
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Future Work
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Import in HADOOP
PEGASUS (Peta-Graph Mining)
On-going work with U Kang and Christos Faloutsos in collaboration with Yahoo! Research.
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Christos Faloutsos
Ioannis Koutis
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Acknowledgements
For the helpful discussions
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Maria Tsiarli
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Acknowledgements
For the PEGASUS logo
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References
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[WF94] Wasserman, Faust: “Social Network Analysis: Methods and Applications (Structural Analysis in the Social Sciences)”
[EM02] Eckmann, Moses: “Curvature of co-links uncovers hidden thematic layers in the World Wide Web”
[YPSB05] Ye, Peyser, Spencer, Bader: “Commensurate distances and similar motifs in genetic congruence and protein interaction networks in yeast”
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References
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[BPCG08] Becchetti, Boldi, Castillo, Gionis Efficient Semi-Streaming Algorithms for Local Triangle Counting in Massive Graphs
[LCKF05] Leskovec, Chakrabarti, Kleinberg, Faloutsos: “Realistic, Mathematically Tractable Graph Generation and Evolution using Kronecker Multiplication”
[FFF09] Faloutsos, Faloutsos, Faloutsos: “On power-law relationships of the Internet topology”
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References
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[MP02] Mihail, Papadimitriou: “On the Eigenvalue Power Law”
[CLV03] Chung, Lu, Vu: “Spectra of Random Graphs with given expected degrees”
[GL89] Golub, Van Loan: “Matrix Computations”
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References
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For more references, paper and slides:http://www.cs.cmu.edu/~ctsourak
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Questions?
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