CMU SCS Graph Analytics wkshpC. Faloutsos (CMU) 1 Graph Analytics Workshop: Tools Christos Faloutsos...

CMU SCS

Graph Analytics wkshp C. Faloutsos (CMU) 1

Graph Analytics Workshop:Tools

Christos FaloutsosCMU

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Welcome!

Tue We Thu

9:00-10:30 Tools Laplacians Parallelism

11:00-12:30 NELL Rich graphs Communities

1:30-3:00 Exercises Panel Scalability

3:30-5:00 Graph. models Posters Graph ‘Laws’

Reception

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Roadmap• Introduction – Motivation• Task 1: Node importance • Task 2: Community detection• Task 3: Mining graphs over time – Tensors• Task 4: Theory – intro to Laplacians• Conclusions

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C. Faloutsos (CMU) 4

Graphs - why should we care?

Internet Map [lumeta.com]

Food Web [Martinez ’91]

>$10B revenue

>0.5B users

Graph Analytics wkshp

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Graphs - why should we care?• IR: bi-partite graphs (doc-terms)

• ‘NELL’: ‘merkel’ ‘chancellor’ ‘germany’ - <S><V><O> facts -> tensors

• web: hyper-text graph• ... and more:

... ...

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Graphs - why should we care?• ‘viral’ marketing• web-log (‘blog’) news propagation• computer network security: email/IP traffic and anomaly

detection• ....• Any M:N relationship -> Graph• Any subject-verb-object construct: -> Graph/Tensor

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Graphs and matrices• Closely related• Powerful tools from matrix algebra, for

graph mining

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Examples of Matrices: Graph - social network

John Peter Mary Nick...

JohnPeterMaryNick

0 11 22 55 ...5 0 6 7 ...

... ... ... ... ...

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Examples of Matrices: Market basket

• market basket as in Association Rules

milk bread choc. wine ...JohnPeterMaryNick

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Examples of Matrices: Documents and terms

13 11 22 55 ...5 4 6 7 ...

... ... ... ... ...

Paper#1

Paper#2

Paper#3Paper#4

data mining classif. tree ...

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Examples of Matrices:Authors and terms

13 11 22 55 ...5 4 6 7 ...

... ... ... ... ...

data mining classif. tree ...JohnPeterMaryNick

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Node importance - Motivation:

• Given a graph (eg., web pages containing the desirable query word)

• Q: Which node is the most important?

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Node importance - Motivation:

• Q: Which node is the most important?• A1: HITS (SVD = Singular Value

Decomposition)• A2: eigenvector (PageRank) ‘I am important,

if my friends are important’ ->Fixed point / eigenvector

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Node importance - motivation

• SVD and eigenvector analysis: very closely related

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Task 1 - SVD - Detailed outline

• Motivation• Definition - properties• Interpretation• Complexity• Case Studies

– HITS– PageRank

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SVD - Motivation

• problem #1: text - LSI: find ‘concepts’• problem #2: compression / dim. reduction

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SVD - Motivation

• problem #1: text - LSI: find ‘concepts’

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SVD - Motivation

• Customer-product, for recommendation system:

1 1 1 0 0

2 2 2 0 0

1 1 1 0 0

5 5 5 0 0

0 0 0 2 2

0 0 0 3 30 0 0 1 1

vegetarians

meat eaters

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SVD - Motivation

• problem #2: compress / reduce dimensionality

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Problem - specs

• Visualize customers

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SVD - Motivation

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SVD - Motivation

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SVD - Definition

• A = U L VT - example:

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SVD - Definition

A[n x m] = U[n x r] L [ r x r] (V[m x r])T

• A: n x m matrix (eg., n documents, m terms)

• U: n x r matrix (n documents, r concepts)• L: r x r diagonal matrix (strength of each

‘concept’) (r : rank of the matrix)• V: m x r matrix (m terms, r concepts)

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SVD - Properties

THEOREM [Press+92]: always possible to decompose matrix A into A = U L VT , where

• U, ,L V: unique (*)• U, V: column orthonormal (ie., columns are unit

vectors, orthogonal to each other)– UT U = I; VT V = I (I: identity matrix)

• L: singular are positive, and sorted in decreasing order

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SVD - Example

1 1 1 0 0

2 2 2 0 0

1 1 1 0 0

5 5 5 0 0

0 0 0 2 2

0 0 0 3 30 0 0 1 1

datainf.

retrieval

brain lung

0.18 0

0.36 0

0.18 0

0.90 0

0 0.53

0 0.800 0.27

9.64 0

0 5.29x

0.58 0.58 0.58 0 0

0 0 0 0.71 0.71

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SVD - Example

1 1 1 0 0

2 2 2 0 0

1 1 1 0 0

5 5 5 0 0

0 0 0 2 2

0 0 0 3 30 0 0 1 1

datainf.

retrieval

brain lung

0.18 0

0.36 0

0.18 0

0.90 0

0 0.53

0 0.800 0.27

9.64 0

0 5.29x

0.58 0.58 0.58 0 0

0 0 0 0.71 0.71

CS-conceptMD-concept

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SVD - Example

1 1 1 0 0

2 2 2 0 0

1 1 1 0 0

5 5 5 0 0

0 0 0 2 2

0 0 0 3 30 0 0 1 1

datainf.

retrieval

brain lung

0.18 0

0.36 0

0.18 0

0.90 0

0 0.53

0 0.800 0.27

9.64 0

0 5.29x

0.58 0.58 0.58 0 0

0 0 0 0.71 0.71

CS-conceptMD-concept

doc-to-concept similarity matrix

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SVD - Example

1 1 1 0 0

2 2 2 0 0

1 1 1 0 0

5 5 5 0 0

0 0 0 2 2

0 0 0 3 30 0 0 1 1

datainf.

retrieval

brain lung

0.18 0

0.36 0

0.18 0

0.90 0

0 0.53

0 0.800 0.27

9.64 0

0 5.29x

0.58 0.58 0.58 0 0

0 0 0 0.71 0.71

‘strength’ of CS-concept

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SVD - Example

1 1 1 0 0

2 2 2 0 0

1 1 1 0 0

5 5 5 0 0

0 0 0 2 2

0 0 0 3 30 0 0 1 1

datainf.

retrieval

brain lung

0.18 0

0.36 0

0.18 0

0.90 0

0 0.53

0 0.800 0.27

9.64 0

0 5.29x

0.58 0.58 0.58 0 0

0 0 0 0.71 0.71

term-to-conceptsimilarity matrix

CS-concept

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SVD - Example

1 1 1 0 0

2 2 2 0 0

1 1 1 0 0

5 5 5 0 0

0 0 0 2 2

0 0 0 3 30 0 0 1 1

datainf.

retrieval

brain lung

0.18 0

0.36 0

0.18 0

0.90 0

0 0.53

0 0.800 0.27

9.64 0

0 5.29x

0.58 0.58 0.58 0 0

0 0 0 0.71 0.71

term-to-conceptsimilarity matrix

CS-concept

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• Motivation• Definition - properties• Interpretation• Complexity• Case studies• Additional properties

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SVD - Interpretation #1

‘documents’, ‘terms’ and ‘concepts’:• U: document-to-concept similarity matrix• V: term-to-concept sim. matrix• L: its diagonal elements: ‘strength’ of each

concept

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SVD – Interpretation #1

‘documents’, ‘terms’ and ‘concepts’:Q: if A is the document-to-term matrix, what

is AT A?A:Q: A AT ?A:

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Graph Analytics wkshp C. Faloutsos (CMU) 38Copyright: Faloutsos, Tong (2009) 2-38

SVD – Interpretation #1

‘documents’, ‘terms’ and ‘concepts’:Q: if A is the document-to-term matrix, what

is AT A?A: term-to-term ([m x m]) similarity matrixQ: A AT ?A: document-to-document ([n x n]) similarity

matrix

ICDE’09

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Graph Analytics wkshp C. Faloutsos (CMU) 39Copyright: Faloutsos, Tong (2009) 2-39

SVD properties

• V are the eigenvectors of the covariance matrix ATA

• U are the eigenvectors of the Gram (inner-product) matrix AAT

Further reading:1. Ian T. Jolliffe, Principal Component Analysis (2nd ed), Springer, 2002.2. Gilbert Strang, Linear Algebra and Its Applications (4th ed), Brooks Cole, 2005.

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• best axis to project on: (‘best’ = min sum of squares of projection errors)

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SVD - Motivation

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SVD - interpretation #2

• minimum RMS error

SVD: givesbest axis to project

first singular

vector

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1 1 1 0 0

2 2 2 0 0

1 1 1 0 0

5 5 5 0 0

0 0 0 2 2

0 0 0 3 30 0 0 1 1

0.18 0

0.36 0

0.18 0

0.90 0

0 0.53

0 0.800 0.27

=9.64 0

0 5.29x

0.58 0.58 0.58 0 0

0 0 0 0.71 0.71

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1 1 1 0 0

2 2 2 0 0

1 1 1 0 0

5 5 5 0 0

0 0 0 2 2

0 0 0 3 30 0 0 1 1

0.18 0

0.36 0

0.18 0

0.90 0

0 0.53

0 0.800 0.27

=9.64 0

0 5.29x

0.58 0.58 0.58 0 0

0 0 0 0.71 0.71

variance (‘spread’) on the v1 axis

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• A = U L VT - example:– U L gives the coordinates of the points in the

projection axis

1 1 1 0 0

2 2 2 0 0

1 1 1 0 0

5 5 5 0 0

0 0 0 2 2

0 0 0 3 30 0 0 1 1

0.18 0

0.36 0

0.18 0

0.90 0

0 0.53

0 0.800 0.27

=9.64 0

0 5.29x

0.58 0.58 0.58 0 0

0 0 0 0.71 0.71

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• More details• Q: how exactly is dim. reduction done?

1 1 1 0 0

2 2 2 0 0

1 1 1 0 0

5 5 5 0 0

0 0 0 2 2

0 0 0 3 30 0 0 1 1

0.18 0

0.36 0

0.18 0

0.90 0

0 0.53

0 0.800 0.27

=9.64 0

0 5.29x

0.58 0.58 0.58 0 0

0 0 0 0.71 0.71

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• More details• Q: how exactly is dim. reduction done?• A: set the smallest singular values to zero:

1 1 1 0 0

2 2 2 0 0

1 1 1 0 0

5 5 5 0 0

0 0 0 2 2

0 0 0 3 30 0 0 1 1

0.18 0

0.36 0

0.18 0

0.90 0

0 0.53

0 0.800 0.27

=9.64 0

0 5.29x

0.58 0.58 0.58 0 0

0 0 0 0.71 0.71

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1 1 1 0 0

2 2 2 0 0

1 1 1 0 0

5 5 5 0 0

0 0 0 2 2

0 0 0 3 30 0 0 1 1

0.18 0

0.36 0

0.18 0

0.90 0

0 0.53

0 0.800 0.27

~9.64 0

0.58 0.58 0.58 0 0

0 0 0 0.71 0.71

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1 1 1 0 0

2 2 2 0 0

1 1 1 0 0

5 5 5 0 0

0 0 0 2 2

0 0 0 3 30 0 0 1 1

0.18 0

0.36 0

0.18 0

0.90 0

0 0.53

0 0.800 0.27

~9.64 0

0.58 0.58 0.58 0 0

0 0 0 0.71 0.71

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1 1 1 0 0

2 2 2 0 0

1 1 1 0 0

5 5 5 0 0

0 0 0 2 2

0 0 0 3 30 0 0 1 1

0.58 0.58 0.58 0 0

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1 1 1 0 0

2 2 2 0 0

1 1 1 0 0

5 5 5 0 0

0 0 0 2 2

0 0 0 3 30 0 0 1 1

1 1 1 0 0

2 2 2 0 0

1 1 1 0 0

5 5 5 0 0

0 0 0 0 0

0 0 0 0 00 0 0 0 0

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Exactly equivalent:‘spectral decomposition’ of the matrix:

1 1 1 0 0

2 2 2 0 0

1 1 1 0 0

5 5 5 0 0

0 0 0 2 2

0 0 0 3 30 0 0 1 1

0.18 0

0.36 0

0.18 0

0.90 0

0 0.53

0 0.800 0.27

=9.64 0

0 5.29x

0.58 0.58 0.58 0 0

0 0 0 0.71 0.71

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1 1 1 0 0

2 2 2 0 0

1 1 1 0 0

5 5 5 0 0

0 0 0 2 2

0 0 0 3 30 0 0 1 1

= x xu1 u2

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1 1 1 0 0

2 2 2 0 0

1 1 1 0 0

5 5 5 0 0

0 0 0 2 2

0 0 0 3 30 0 0 1 1

= u1l1 vT1 u2l2 vT

2+ +...n

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1 1 1 0 0

2 2 2 0 0

1 1 1 0 0

5 5 5 0 0

0 0 0 2 2

0 0 0 3 30 0 0 1 1

= u1l1 vT1 u2l2 vT

2+ +...n

n x 1 1 x m

r terms

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approximation / dim. reduction:by keeping the first few terms (Q: how many?)

1 1 1 0 0

2 2 2 0 0

1 1 1 0 0

5 5 5 0 0

0 0 0 2 2

0 0 0 3 30 0 0 1 1

= u1l1 vT1 u2l2 vT

2+ +...n

assume: l1 >= l2 >= ...

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A (heuristic - [Fukunaga]): keep 80-90% of ‘energy’ (= sum of squares of li ’s)

1 1 1 0 0

2 2 2 0 0

1 1 1 0 0

5 5 5 0 0

0 0 0 2 2

0 0 0 3 30 0 0 1 1

= u1l1 vT1 u2l2 vT

2+ +...n

assume: l1 >= l2 >= ...

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Pictorially: matrix form of SVD

– Best rank-k approximation in L2

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Pictorially: Spectral form of SVD

1u1v1 2u2v2

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• Motivation• Definition - properties• Interpretation

– #1: documents/terms/concepts– #2: dim. reduction– #3: picking non-zero, rectangular ‘blobs’

• Complexity• Case studies• Additional properties

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• finds non-zero ‘blobs’ in a data matrix

1 1 1 0 0

2 2 2 0 0

1 1 1 0 0

5 5 5 0 0

0 0 0 2 2

0 0 0 3 30 0 0 1 1

0.18 0

0.36 0

0.18 0

0.90 0

0 0.53

0 0.800 0.27

=9.64 0

0 5.29x

0.58 0.58 0.58 0 0

0 0 0 0.71 0.71

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• finds non-zero ‘blobs’ in a data matrix

1 1 1 0 0

2 2 2 0 0

1 1 1 0 0

5 5 5 0 0

0 0 0 2 2

0 0 0 3 30 0 0 1 1

0.18 0

0.36 0

0.18 0

0.90 0

0 0.53

0 0.800 0.27

=9.64 0

0 5.29x

0.58 0.58 0.58 0 0

0 0 0 0.71 0.71

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• finds non-zero ‘blobs’ in a data matrix =• ‘communities’ (bi-partite cores, here)

1 1 1 0 0

2 2 2 0 0

1 1 1 0 0

5 5 5 0 0

0 0 0 2 2

0 0 0 3 30 0 0 1 1

Col 4Row 5

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SVD - Complexity

• O( n * m * m) or O( n * n * m) (whichever is less)

• less work, if we just want singular values• or if we want first k singular vectors• or if the matrix is sparse [Berry]• Implemented: in any linear algebra package

(LINPACK, matlab, Splus, mathematica ...)

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SVD - conclusions so far

• SVD: A= U L VT : unique (*)• U: document-to-concept similarities• V: term-to-concept similarities• L: strength of each concept• dim. reduction: keep the first few strongest

singular values (80-90% of ‘energy’)– SVD: picks up linear correlations

• SVD: picks up non-zero ‘blobs’

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Kleinberg’s algo (HITS)

Kleinberg, Jon (1998). Authoritative sources in a hyperlinked environment. Proc. 9th ACM-SIAM Symposium on Discrete Algorithms.

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Recall: problem dfn

• Q: Which node is the most important?

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Kleinberg’s algorithm• Problem dfn: given the web and a query• find the most ‘authoritative’ web pages for

this query

Step 0: find all pages containing the query terms

Step 1: expand by one move forward and backward

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Kleinberg’s algorithm• Step 1: expand by one move forward and

backward

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Kleinberg’s algorithm• on the resulting graph, give high score (=

‘authorities’) to nodes that many important nodes point to

• give high importance score (‘hubs’) to nodes that point to good ‘authorities’)

hubs authorities

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Kleinberg’s algorithm

observations• recursive definition!• each node (say, ‘i’-th node) has both an

authoritativeness score ai and a hubness score hi

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Let E be the set of edges and A be the adjacency matrix: the (i,j) is 1 if the edge from i to j exists

Let h and a be [n x 1] vectors with the ‘hubness’ and ‘authoritativiness’ scores.

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ai = hk + hl + hm

that is

ai = Sum (hj) over all j that (j,i) edge exists

a = AT h

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symmetrically, for the ‘hubness’:

hi = an + ap + aq

that is

hi = Sum (qj) over all j that (i,j) edge exists

h = A a

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In conclusion, we want vectors h and a such that:

h = A a

a = AT hSVD properties:

A [n x m] v1 [m x 1] = l1 u1 [n x 1]

u1T A = l1 v1

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Kleinberg’s algorithmIn short, the solutions to

h = A a

a = AT h

are the left- and right- singular-vectors of the adjacency matrix A.

Starting from random a’ and iterating, we’ll eventually converge

(Q: to which of all the singular-vectors? why?)

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(Q: to which of all the singular-vectors? why?)

A: to the ones of the strongest singular-value:(AT

A ) k v’ ~ (constant) v1

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Kleinberg’s algorithm - results

Eg., for the query ‘java’:

0.328 www.gamelan.com

0.251 java.sun.com

0.190 www.digitalfocus.com (“the java developer”)

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Kleinberg’s algorithm - discussion• ‘authority’ score can be used to find ‘similar

pages’ (how?)

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PageRank (google)

•Brin, Sergey and Lawrence Page (1998). Anatomy of a Large-Scale Hypertextual Web Search Engine. 7th Intl World Wide Web Conf.

LarryPage

SergeyBrin

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Problem: PageRank

Given a directed graph, find its most interesting/central node

A node is important,if it is connected with important nodes(recursive, but OK!)

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Problem: PageRank - solution

Given a directed graph, find its most interesting/central node

Proposed solution: Random walk; spot most ‘popular’ node (-> steady state prob. (ssp))

A node has high ssp,if it is connected with high ssp nodes(recursive, but OK!)

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(Simplified) PageRank algorithm

• Let A be the adjacency matrix;• let B be the transition matrix: transpose, column-normalized - then

To From

1/2 1/2

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(Simplified) PageRank algorithm• B p = p

B p = p

1/2 1/2

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Definitions

A Adjacency matrix (from-to)

D Degree matrix = (diag ( d1, d2, …, dn) )

B Transition matrix: to-from, column normalized

B = AT D-1

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(Simplified) PageRank algorithm• B p = 1 * p• thus, p is the eigenvector that corresponds

to the highest eigenvalue (=1, since the matrix is

column-normalized)• Why does such a p exist?

– p exists if B is nxn, nonnegative, irreducible [Perron–Frobenius theorem]

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(Simplified) PageRank algorithm• In short: imagine a particle randomly

moving along the edges• compute its steady-state probabilities (ssp)

Full version of algo: with occasional random jumps

Why? To make the matrix irreducible

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Full Algorithm• With probability 1-c, fly-out to a random

node• Then, we have

p = c B p + (1-c)/n 1 =>

p = (1-c)/n [I - c B] -1 1

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Alternative notation

M Modified transition matrix

M = c B + (1-c)/n 1 1T

p = M p

That is: the steady state probabilities =

PageRank scores form the first eigenvector of the ‘modified transition matrix’

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Parenthesis: intuition behind eigenvectors

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Formal definition

If A is a (n x n) square matrix(l , x) is an eigenvalue/eigenvector pair of A if A x = l x

CLOSELY related to singular values:

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Property #1: Eigen- vs singular-values

B[n x m] = U[n x r] L [ r x r] (V[m x r])T

then A = (BTB) is symmetric and

C(4): BT B vi = li2 vi

ie, v1 , v2 , ...: eigenvectors of A = (BTB)

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Property #2

• If A[nxn] is a real, symmetric matrix

• Then it has n real eigenvalues

(if A is not symmetric, some eigenvalues may be complex)

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Property #3

• If A[nxn] is a real, symmetric matrix

• Then it has n real eigenvalues• And they agree with its n singular values,

except possibly for the sign

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Intuition

• A as vector transformation

2 11 3

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Intuition

• By defn., eigenvectors remain parallel to themselves (‘fixed points’)

2 11 3

0.853.62 *

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Convergence

• Usually, fast:

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Convergence

• Usually, fast:

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Convergence

• Usually, fast:• depends on ratio

l1 : l2l1

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Kleinberg/google - conclusions

SVD helps in graph analysis:

hub/authority scores: strongest left- and right- singular-vectors of the adjacency matrix

random walk on a graph: steady state probabilities are given by the strongest eigenvector of the (modified) transition matrix

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Conclusions• SVD: a valuable tool• given a document-term matrix, it finds

‘concepts’ (LSI)• ... and can find fixed-points or steady-state

probabilities (google/ Kleinberg/ Markov Chains)

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Conclusions cont’d

(We didn’t discuss/elaborate, but, SVD• ... can reduce dimensionality (KL)• ... and can find rules (PCA; RatioRules)• ... and can solve optimally over- and under-

constraint linear systems (least squares / query feedbacks)

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References

• Berry, Michael: http://www.cs.utk.edu/~lsi/• Brin, S. and L. Page (1998). Anatomy of a

Large-Scale Hypertextual Web Search Engine. 7th Intl World Wide Web Conf.

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References

• Christos Faloutsos, Searching Multimedia Databases by Content, Springer, 1996. (App. D)

• Fukunaga, K. (1990). Introduction to Statistical Pattern Recognition, Academic Press.

• I.T. Jolliffe Principal Component Analysis Springer, 2002 (2nd ed.)

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References cont’d• Kleinberg, J. (1998). Authoritative sources

in a hyperlinked environment. Proc. 9th ACM-SIAM Symposium on Discrete Algorithms.

• Press, W. H., S. A. Teukolsky, et al. (1992). Numerical Recipes in C, Cambridge University Press. www.nr.com

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PART 2: Communities

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Task 2 – Communities - Detailed outline

• Motivation• Hard clustering – k pieces• Hard clustering – optimal # pieces• Observations

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Problem

• Given a graph, and k• Break it into k (disjoint) communities

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Problem

• Given a graph, and k• Break it into k (disjoint) communities

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Solution #1: METIS

• Arguably, the best algorithm• Open source, at

– http://www.cs.umn.edu/~metis

• and *many* related papers, at same url• Main idea:

– coarsen the graph; – partition; – un-coarsen

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Solution #1: METIS

• G. Karypis and V. Kumar. METIS 4.0: Unstructured graph partitioning and sparse matrix ordering system. TR, Dept. of CS, Univ. of Minnesota, 1998.

• <and many extensions>

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Solution #2

(problem: hard clustering, k pieces)

Spectral partitioning:• Consider the 2nd smallest eigenvector of the

(normalized) Laplacian

See details in ‘Task 7’, later

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Solutions #3, …

Many more ideas:• Clustering on the A2 (square of adjacency

matrix) [Zhou, Woodruff, PODS’04]• Minimum cut / maximum flow [Flake+,

KDD’00]• …

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Cross-association

Desiderata:

Simultaneously discover row and column groups

Fully Automatic: No “magic numbers”

Scalable to large matrices

Reference:1. Chakrabarti et al. Fully Automatic Cross-Associations, KDD’04

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What makes a cross-association “good”?

versus

Column groups

Why is this better?

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versus

Column groups

Why is this better?

simpler; easier to describeeasier to compress!

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Problem definition: given an encoding scheme• decide on the # of col. and row groups k and l• and reorder rows and columns,• to achieve best compression

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Main Idea

sizei * H(xi) +Cost of describing cross-associations

Code Cost Description Cost

Σi Total Encoding Cost =

Good Compression

Better Clustering

Minimize the total cost (# bits)

for lossless compression

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Algorithmk =

5 row groups

k=1, l=2

k=2, l=2

k=2, l=3

k=3, l=3

k=3, l=4

k=4, l=4

k=4, l=5

l = 5 col groups

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Experiments

“CLASSIC”

• 3,893 documents

• 4,303 words

• 176,347 “dots”

Combination of 3 sources:

• MEDLINE (medical)

• CISI (info. retrieval)

• CRANFIELD (aerodynamics)

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Experiments

“CLASSIC” graph of documents & words: k=15, l=19

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Experiments

MEDLINE(medical)

insipidus, alveolar, aortic, death, prognosis, intravenous

blood, disease, clinical, cell, tissue, patient

CMU SCS

Experiments

CISI(Information Retrieval)

providing, studying, records, development, students, rules

abstract, notation, works, construct, bibliographies

MEDLINE(medical)

CMU SCS

Experiments

CRANFIELD (aerodynamics)

shape, nasa, leading, assumed, thin

MEDLINE(medical)

CMU SCS

Experiments

paint, examination, fall, raise, leave, based

CRANFIELD (aerodynamics)

MEDLINE(medical)

CMU SCS

AlgorithmCode for cross-associations (matlab):

www.cs.cmu.edu/~deepay/mywww/software/CrossAssociations-01-27-2005.tgz

Variations and extensions:• ‘Autopart’ [Chakrabarti, PKDD’04]• www.cs.cmu.edu/~deepay

CMU SCS

Algorithm• Hadoop implementation [ICDM’08]

Spiros Papadimitriou, Jimeng Sun: DisCo: Distributed Co-clustering with Map-Reduce: A Case Study towards Petabyte-Scale End-to-End Mining. ICDM

2008: 512-521

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Observation #1

• Skewed degree distributions – there are nodes with huge degree (>O(10^4), in facebook/linkedIn popularity contests!)

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Observation #2

• Maybe there are no good cuts: ``jellyfish’’ shape [Tauro+’01], [Siganos+,’06], strange behavior of cuts [Chakrabarti+’04], [Leskovec+,’08]

CMU SCS

Observation #2

• Maybe there are no good cuts: ``jellyfish’’ shape [Tauro+’01], [Siganos+,’06], strange behavior of cuts [Chakrabarti+,’04], [Leskovec+,’08]

CMU SCS

Jellyfish model [Tauro+]

A Simple Conceptual Model for the Internet Topology, L. Tauro, C. Palmer, G. Siganos, M. Faloutsos, Global Internet, November 25-29, 2001

Jellyfish: A Conceptual Model for the AS Internet Topology G. Siganos, Sudhir L Tauro, M. Faloutsos, J. of Communications and Networks, Vol. 8, No. 3, pp 339-350, Sept. 2006.

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Strange behavior of min cuts

• ‘negative dimensionality’ (!)

NetMine: New Mining Tools for Large Graphs, by D. Chakrabarti, Y. Zhan, D. Blandford, C. Faloutsos and G. Blelloch, in the SDM 2004 Workshop on Link Analysis, Counter-terrorism and Privacy

Statistical Properties of Community Structure in Large Social and Information Networks, J. Leskovec, K. Lang, A. Dasgupta, M. Mahoney. WWW 2008.

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“Min-cut” plot• Do min-cuts recursively.

log (# edges)

log (mincut-size / #edges)

N nodes

Mincut size = sqrt(N)

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log (# edges)

N nodes

New min-cut

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log (# edges)

N nodes

New min-cut

Slope = -0.5

For a d-dimensional grid, the slope is -1/d

CMU SCS

“Min-cut” plot

log (# edges)

Slope = -1/d

For a d-dimensional grid, the slope is -1/d

log (# edges)

For a random graph, the slope is 0

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“Min-cut” plot• What does it look like for a real-world

graph?

log (# edges)

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Experiments• Datasets:– Google Web Graph: 916,428 nodes and

5,105,039 edges– Lucent Router Graph: Undirected graph of

network routers from www.isi.edu/scan/mercator/maps.html; 112,969 nodes and 181,639 edges

– User Website Clickstream Graph: 222,704 nodes and 952,580 edges

NetMine: New Mining Tools for Large Graphs, by D. Chakrabarti, Y. Zhan, D. Blandford, C. Faloutsos and G. Blelloch, in the SDM 2004 Workshop on Link Analysis, Counter-terrorism and Privacy

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Experiments• Used the METIS algorithm [Karypis, Kumar,

log (# edges)

• Google Web graph

• Values along the y-axis are averaged

• “lip” for large edges

• Slope of -0.4, corresponds to a 2.5-dimensional grid!

Slope~ -0.4

CMU SCS

Experiments• Same results for other graphs too…

log (# edges) log (# edges)

Lucent Router graph Clickstream graph

Slope~ -0.57 Slope~ -0.45

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Task 2 – Communities Conclusions – Practitioner’s guide

• Hard clustering – k pieces• Hard clustering – optimal # pieces• Observations

Cross-associations

‘jellyfish’: no good cuts

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PART 3: Tensors

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Task 3 – Tensors - Detailed roadmap

• Motivation• Definitions: PARAFAC• Case study: web mining

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Examples of Matrices:Authors and terms

13 11 22 55 ...5 4 6 7 ...

... ... ... ... ...

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But: if it changes over time??

• A: treat it as ‘tensor’

13 11 22 55 ...5 4 6 7 ...

... ... ... ... ...

KDD’08

KDD’07

KDD’09

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Motivation: Why tensors?

• Q: what is a tensor?

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• A: N-D generalization of matrix:

13 11 22 55 ...5 4 6 7 ...

... ... ... ... ...

KDD’09

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• A: N-D generalization of matrix:

13 11 22 55 ...5 4 6 7 ...

... ... ... ... ...

KDD’08

KDD’07

KDD’09

CMU SCS

Tensors are useful for 3 or more modes

Terminology: ‘mode’ (or ‘aspect’):

13 11 22 55 ...5 4 6 7 ...

... ... ... ... ...

data mining classif. tree ...

Mode (== aspect) #1

Mode#2

Mode#3

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Notice

• 3rd mode does not need to be time• we can have more than 3 modes

13 11 22 55 ...5 4 6 7 ...

... ... ... ... ...

IP destination

Dest. port

IP source

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Background: Tensors

• Tensors (=multi-dimensional arrays) are everywhere– Sensor stream (time, location, type)– Predicates (subject, verb, object) in knowledge base

“Barrack Obama is the president of U.S.”

“Eric Clapton playsguitar”

NELL (Never Ending Language Learner) data

Nonzeros =144M

Graph Analytics wkshp 159C. Faloutsos (CMU)

CMU SCS

Task 3 – Tensors - Detailed roadmap

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Tensor basics

• Multi-mode extensions of SVD – recall that:

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Reminder: SVD

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Reminder: SVD

1u1v1 2u2v2

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Extension to (>=)3 modes

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Main points:

• 2 major types of tensor decompositions: PARAFAC and Tucker (not examined here)

• both can be solved with ``alternating least squares’’ (ALS)

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Task 3 – Tensors - Detailed outline

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Discoveries: Problem Definition

Most important concepts and synonyms?

NELL (Never Ending Language Learner) data

Nonzeros =144M

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A1: Concept Discovery

• Concept Discovery in Knowledge Base

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A2.1: Concept Discovery

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A2: Synonym Discovery

• Synonym Discovery in Knowledge Base

a1a2 aR…

(Given) noun phrase

(Discovered) synonym 1

(Discovered) synonym 2

CMU SCS

171C. Faloutsos (CMU)

A2: Synonym Discovery

Graph Analytics wkshp

CMU SCS

GigaTensor: Scaling Tensor Analysis Up By 100 Times –

Algorithms and Discoveries

U Kang

ChristosFaloutsos

KDD 2012

EvangelosPapalexakis

AbhayHarpale

CMU SCS

Experiments

• GigaTensor solves 100x larger problem

Number of nonzero= I / 50

GigaTensor

Tensor

Toolbox Out ofMemory

CMU SCS

Conclusions

• Real data may have multiple aspects (modes)

• Tensors provide elegant theory and algorithms– PARAFAC (and Tucker): discover groups

• GigaTensor: scales up (hadoop/PEGASUS)– www.cs.cmu.edu/~pegasus

CMU SCS

References

• T. G. Kolda, B. W. Bader and J. P. Kenny. Higher-Order Web Link Analysis Using Multilinear Algebra. In: ICDM 2005, Pages 242-249, November 2005.

• Jimeng Sun, Spiros Papadimitriou, Philip Yu. Window-based Tensor Analysis on High-dimensional and Multi-aspect Streams, Proc. of the Int. Conf. on Data Mining (ICDM), Hong Kong, China, Dec 2006

CMU SCS

Resources

• See tutorial on tensors, KDD’07 (w/ Tamara Kolda and Jimeng Sun):

www.cs.cmu.edu/~christos/TALKS/KDD-07-tutorial

CMU SCS

Tensor tools - resources

• Toolbox: from Tamara Kolda:csmr.ca.sandia.gov/~tgkolda/TensorToolbox

2-177Copyright: Faloutsos, Tong (2009) 2-177ICDE’09

• T. G. Kolda and B. W. Bader. Tensor Decompositions and Applications. SIAM Review, Volume 51, Number 3, September 2009

csmr.ca.sandia.gov/~tgkolda/pubs/bibtgkfiles/TensorReview-preprint.pdf

• T. Kolda and J. Sun: Scalable Tensor Decomposition for Multi-Aspect Data Mining (ICDM 2008)

CMU SCS

PART 4: Theory

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Task 4 – Theory - Detailed roadmap• Adjacency matrix• Laplacian

– Connected Components– Intuition: 2nd smallest eigenvalue -> ‘good cut’

180Graph Analytics wkshp 180C. Faloutsos (CMU)

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Adjacency matrix

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Adjacency matrix

1-step-awaypaths

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Adjacency matrix

4 Obvious extensions,for directed and/or weighted cases

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Main upcoming result

the second smallest eigenvector of the Laplacian (u2)

gives a good cut:Nodes with positive scores should go to one

And the rest to the other

CMU SCS

Laplacian

L= D-A=1

Diagonal matrix, dii=di

CMU SCS

Connected Components• Lemma: Let G be a graph with n vertices

and c connected components. If L is the Laplacian of G, then rank(L)= n-c.

• Proof: see p.279, Godsil-Royle

CMU SCS

Connected Components

G(V,E)

eig(L)=

CMU SCS

G(V,E)

eig(L)=

#zeros = #components

CMU SCS

G(V,E)

eig(L)=

CMU SCS

G(V,E)

eig(L)=

#zeros = #components

CMU SCS

G(V,E)

eig(L)=

Indicates a “good cut”

CMU SCS

Task 4 – Theory - Detailed roadmap• Reminders

• Adjacency matrix• Laplacian

CMU SCS

Example: Spectral Partitioning

• K500• K500

dumbbell graph

? Montagues

Capulets

Juliet

http://en.wikipedia.org/wiki/File:Romeo_and_juliet_brown.jpg

CMU SCS

Example: Spectral Partitioning• This is how adjacency matrix of B looks

spy(B)

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• 2nd eigenvector u2 of B: B u2 = l u2

L = diag(sum(B))-B;[u v] = eigs(L,2,'SM');

plot(u(:,1),’x’)

Not so much information yet…

Node-id ‘i’

u2,i score

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• 2nd eigenvector after sorting on x2,i score

[ign ind] = sort(u(:,1));plot(u(ind),'x')

x2,i score

Node-id ‘i’

CMU SCS

• 2nd eigenvector after sorting on x2,i score

[ign ind] = sort(u(:,1));plot(u(ind),'x')

But now we seethe two communities!

x2,i score

Node-id ‘i’

CMU SCS

Example: Spectral Partitioning• This is how adjacency matrix of B looks

spy(B(ind,ind))

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Why λ2?

Each ball 1 unit of mass xLx x1 xnOSCILLATE

Dfn of eigenvector

Matrix viewpoint:

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Why λ2?

Each ball 1 unit of mass xLx x1 xnOSCILLATE

Force due to neighbors displacement

Hooke’s constantPhysics viewpoint:

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Why λ2?

Each ball 1 unit of mass

Eigenvector value

Node idxLx x1 xnOSCILLATE

For the first eigenvector:All nodes: same displacement (= value)

CMU SCS

Why λ2?

Each ball 1 unit of mass

Eigenvector value

Node idxLx x1 xnOSCILLATE

CMU SCS

Conclusions

Spectrum tells us a lot about the graph:• Adjacency: #Paths• Laplacian: Sparse Cut

CMU SCS

References• Fan R. K. Chung: Spectral Graph Theory (AMS) • Chris Godsil and Gordon Royle: Algebraic Graph

Theory (Springer) • Bojan Mohar and Svatopluk Poljak: Eigenvalues

in Combinatorial Optimization, IMA Preprint Series #939

• Gilbert Strang: Introduction to Applied

Mathematics (Wellesley-Cambridge Press)

CMU SCS

PART 5: Conclusions

CMU SCS

Graph Analytics wkshp C. Faloutsos (CMU) P9-208

Summary• Task 1: Node importance• Task 2: Community detection• Task 3: Mining graphs over

time• Task 4: Spectral graph theory

CMU SCS

Summary• Task 1: Node importance• Task 2: Community detection• Task 3: Mining graphs over

time• Task 4: Spectral graph theory

->SVD, PageRank, HITS -> METIS; ‘no good cuts’ -> Tensors

-> Laplacians

CMU SCS

AcknowledgementsFunding:

IIS-0705359, IIS-0534205, DBI-0640543, CNS-0721736

THANK YOU!Christos Faloutsoswww.cs.cmu.edu/~christoswww.cs.cmu.edu/~pegasus

http://www.cs.cmu.edu/~epapalex/gmc/

CMU SCS Graph Analytics wkshpC. Faloutsos (CMU) 1 Graph Analytics Workshop: Tools Christos Faloutsos...

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