fast counting of triangles in large networks without counting: algorithms and laws

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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ICDM, Dec. '08

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

C. E. Tsourakakis

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

C. E. Tsourakakis

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

C. E. Tsourakakis

Outline

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Related WorkProposed Method

Theorems Algorithms Explaining efficiency

ExperimentsTriangle-related LawsTriangles in Kronecker GraphsConclusions

C. E. Tsourakakis

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

C. E. Tsourakakis

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!

C. E. Tsourakakis

EigenTriangle Algorithm

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C. E. Tsourakakis

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.

C. E. Tsourakakis

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!

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Outline

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Related WorkProposed Method

Theorems Algorithms Explaining efficiency

ExperimentsTriangle-related LawsTriangles in Kronecker GraphsConclusions

C. E. Tsourakakis

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

C. E. Tsourakakis

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)

C. E. Tsourakakis

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

C. E. Tsourakakis

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

C. E. Tsourakakis

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

C. E. Tsourakakis

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.

C. E. Tsourakakis

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

C. E. Tsourakakis

Questions?

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