CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of...
-
Upload
alijah-ryals -
Category
Documents
-
view
221 -
download
0
Transcript of CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of...
![Page 1: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/1.jpg)
CHARALAMPOS E. TSOURAKAKISSCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY
Fast counting of triangles in large networks without
counting:Algorithms and laws
1
ICDM, Dec. '08
![Page 2: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/2.jpg)
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.
2
ICDM, Dec. '08
Generality
Our focus
![Page 3: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/3.jpg)
C. E. Tsourakakis
Why is Triangle Counting important?From the Graph Mining Perspective
ICDM, Dec. '08
3
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
![Page 4: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/4.jpg)
C. E. Tsourakakis
Outline
ICDM, Dec. '08
4
Related WorkProposed Method
Theorems Algorithms Explaining efficiency
ExperimentsTriangle-related LawsTriangles in Kronecker GraphsConclusions
![Page 5: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/5.jpg)
C. E. Tsourakakis
Related Work
ICDM, Dec. '08
5
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
![Page 6: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/6.jpg)
C. E. Tsourakakis
Outline
ICDM, Dec. '08
6
Related WorkProposed Method
Theorems Algorithms Explaining efficiency
ExperimentsTriangle-related LawsTriangles in Kronecker GraphsConclusions
![Page 7: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/7.jpg)
C. E. Tsourakakis
Theorem [EigenTriangle]
ICDM, Dec. '08
7
Theorem 1
Δ(G) = # triangles in graph G(V,E) = eigenvalues of
adjacency matrix AG
||
1
3)(6V
iiG
||21 ... V
![Page 8: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/8.jpg)
C. E. Tsourakakis
Theorem [EigenTriangleLocal]
ICDM, Dec. '08
8
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
![Page 9: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/9.jpg)
C. E. Tsourakakis
Outline
ICDM, Dec. '08
9
Related WorkProposed Method
Theorems Algorithms Explaining efficiency
ExperimentsTriangle-related LawsTriangles in Kronecker GraphsConclusions
![Page 10: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/10.jpg)
C. E. Tsourakakis
EigenTriangle Algorithm (interactively)
ICDM, Dec. '08
10
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!
![Page 11: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/11.jpg)
C. E. Tsourakakis
EigenTriangle Algorithm
ICDM, Dec. '08
11
![Page 12: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/12.jpg)
C. E. Tsourakakis
EigenTriangleLocal Algorithm
ICDM, Dec. '08
12
Why are these two
algorithms efficient on power law networks?
![Page 13: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/13.jpg)
C. E. Tsourakakis
Typical Spectra of Power Law Networks
ICDM, Dec. '08
13
AirportsPolitical blogs
![Page 14: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/14.jpg)
C. E. Tsourakakis
1st Reason :Top Eigenvalues of Power-Law Graphs
ICDM, Dec. '08
14
Very important for us because:Few eigenvalues contribute a lot!Cubes amplify this even more!Lanczos converges fast due to large spectral gaps [GL89]!
![Page 15: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/15.jpg)
C. E. Tsourakakis
1st Reason :Top Eigenvalues of Power-Law Graphs
ICDM, Dec. '08
15
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.
![Page 16: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/16.jpg)
C. E. Tsourakakis
2nd Reason :Bulk of eigenvalues
ICDM, Dec. '08
16
Almost symmetric around 0!
Sum of cubes almost cancels out!
Political Blogs
Omit!
Keep only 3!
3
![Page 17: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/17.jpg)
C. E. Tsourakakis
Outline
ICDM, Dec. '08
17
Related WorkProposed Method
Theorems Algorithms Explaining efficiency
ExperimentsTriangle-related LawsTriangles in Kronecker GraphsConclusions
![Page 18: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/18.jpg)
C. E. Tsourakakis
Datasets
ICDM, Dec. '08
18
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)
![Page 19: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/19.jpg)
C. E. Tsourakakis
Datasets
ICDM, Dec. '08
19
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
![Page 20: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/20.jpg)
C. E. Tsourakakis
Datasets
ICDM, Dec. '08
20
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
![Page 21: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/21.jpg)
C. E. Tsourakakis
Datasets
ICDM, Dec. '08
21
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
![Page 22: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/22.jpg)
C. E. Tsourakakis
Datasets
ICDM, Dec. '08
22
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
![Page 23: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/23.jpg)
C. E. Tsourakakis
Datasets
ICDM, Dec. '08
23
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
![Page 24: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/24.jpg)
C. E. Tsourakakis
Datasets
ICDM, Dec. '08
24
~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)
![Page 25: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/25.jpg)
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
ICDM, Dec. '08
![Page 26: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/26.jpg)
C. E. Tsourakakis
Results: #Eigenvalues vs. Speedup26
ICDM, Dec. '08
![Page 27: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/27.jpg)
C. E. Tsourakakis
Results: #Edges vs. Speedup 27
ICDM, Dec. '08
Observe the tre
nd
![Page 28: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/28.jpg)
C. E. Tsourakakis
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.
ICDM, Dec. '08
![Page 29: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/29.jpg)
C. E. Tsourakakis
Evaluating the Local Counting Method
ICDM, Dec. '08
29
Triangles node i participatesTria
ngle
s no
de i
part
icip
ates
acco
rdin
g to
our
est
imat
ion
![Page 30: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/30.jpg)
C. E. Tsourakakis
#Eigenvalues vs. ρ for three networks
30
ICDM, Dec. '08
2-3 eigenvaluesalmost ideal results!
![Page 31: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/31.jpg)
C. E. Tsourakakis
Outline
ICDM, Dec. '08
31
Related WorkProposed Method
Theorems Algorithms Explaining efficiency
ExperimentsTriangle-related LawsTriangles in Kronecker GraphsConclusions
![Page 32: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/32.jpg)
C. E. Tsourakakis
Triangle Participation Power Law (TPPL)
ICDM, Dec. '08
32
EPINIONS
δ = #TrianglesCou
nt o
f nod
es p
artic
ipat
ing
in δ
tria
ngle
s
![Page 33: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/33.jpg)
C. E. Tsourakakis
Triangle Participation Power Law (TPPL)
ICDM, Dec. '08
33
HEP_TH (coauthorship)
Flickr
![Page 34: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/34.jpg)
C. E. Tsourakakis
Degree Triangle Power Law (DTPL)
ICDM, Dec. '08
34
EPINIONS
d , all degrees appearing in the graph
Mea
n #Δ
s ov
er a
ll no
des
with
deg
ree
d
![Page 35: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/35.jpg)
C. E. Tsourakakis
Degree Triangle Power Law (DTPL)
ICDM, Dec. '08
35
Flickr
Reuters
![Page 36: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/36.jpg)
C. E. Tsourakakis
Observations on TPPL & DTPL
ICDM, Dec. '08
36
TTPL:Many nodes few triangles
Few nodes many triangles
![Page 37: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/37.jpg)
C. E. Tsourakakis
Observations on TPPL & DTPL
ICDM, Dec. '08
37
DTPL: Power law fits nicely to the Degree-
Triangle plot. Slope is the opposite of the slope of the
degree distribution (slope complementarity).
![Page 38: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/38.jpg)
C. E. Tsourakakis
Outline
ICDM, Dec. '08
38
Related WorkProposed Method
Theorems Algorithms Explaining efficiency
ExperimentsTriangle-related LawsTriangles in Kronecker GraphsConclusions
![Page 39: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/39.jpg)
C. E. Tsourakakis
Kronecker graphs
ICDM, Dec. '08
39
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]
![Page 40: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/40.jpg)
C. E. Tsourakakis
Triangles in Kronecker Graphs
ICDM, Dec. '08
40
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
![Page 41: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/41.jpg)
C. E. Tsourakakis
Outline
ICDM, Dec. '08
41
Related WorkProposed Method
Theorems Algorithms Explaining efficiency
ExperimentsTriangle-related LawsTriangles in Kronecker GraphsConclusions
![Page 42: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/42.jpg)
C. E. Tsourakakis
Conclusions
ICDM, Dec. '08
42
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.
![Page 43: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/43.jpg)
C. E. Tsourakakis
Future Work
ICDM, Dec. '08
43
Import in HADOOP
PEGASUS (Peta-Graph Mining)
On-going work with U Kang and Christos Faloutsos in collaboration with Yahoo! Research.
![Page 44: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/44.jpg)
C. E. Tsourakakis
Christos Faloutsos
Ioannis Koutis
ICDM, Dec. '08
44
Acknowledgements
For the helpful discussions
![Page 45: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/45.jpg)
C. E. Tsourakakis
Maria Tsiarli
ICDM, Dec. '08
45
Acknowledgements
For the PEGASUS logo
![Page 46: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/46.jpg)
C. E. Tsourakakis
46
ICDM, Dec. '08
![Page 47: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/47.jpg)
C. E. Tsourakakis
References
ICDM, Dec. '08
47
[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”
![Page 48: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/48.jpg)
C. E. Tsourakakis
References
ICDM, Dec. '08
48
[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”
![Page 49: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/49.jpg)
C. E. Tsourakakis
References
ICDM, Dec. '08
49
[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”
![Page 50: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/50.jpg)
C. E. Tsourakakis
References
ICDM, Dec. '08
50
For more references, paper and slides:http://www.cs.cmu.edu/~ctsourak
![Page 51: CHARALAMPOS E. TSOURAKAKIS SCHOOL OF COMPUTER SCIENCE CARNEGIE MELLON UNIVERSITY Fast counting of triangles in large networks without counting: Algorithms.](https://reader036.fdocuments.net/reader036/viewer/2022062423/56649ca65503460f9496813c/html5/thumbnails/51.jpg)
C. E. Tsourakakis
Questions?
ICDM, Dec. '08
51