104
CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

  • View
    223

  • Download
    0

Embed Size (px)

Citation preview

Page 1: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

CMU SCS

Graph Mining and Influence Propagation

Christos Faloutsos

CMU

Page 2: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 2

CMU SCS

Thank you!

• Adam Jatowt

Page 3: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 3

CMU SCS

Outline

• Problem definition / Motivation

• Static & dynamic laws; generators

• Tools: CenterPiece graphs; Tensors

• Other projects (Virus propagation, e-bay fraud detection)

• Conclusions

Page 4: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 4

CMU SCS

Motivation

Data mining: ~ find patterns (rules, outliers)

• Problem#1: How do real graphs look like?

• Problem#2: How do they evolve?

• Problem#3: How to generate realistic graphs

TOOLS

• Problem#4: Who is the ‘master-mind’?

• Problem#5: Track communities over time

Page 5: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 5

CMU SCS

Problem#1: Joint work with

Dr. Deepayan Chakrabarti (CMU/Yahoo R.L.)

Page 6: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 6

CMU SCS

Graphs - why should we care?

Internet Map [lumeta.com]

Food Web [Martinez ’91]

Protein Interactions [genomebiology.com]

Friendship Network [Moody ’01]

Page 7: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 7

CMU SCS

Graphs - why should we care?

• IR: bi-partite graphs (doc-terms)

• web: hyper-text graph

• ... and more:

D1

DN

T1

TM

... ...

Page 8: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 8

CMU SCS

Graphs - why should we care?

• network of companies & board-of-directors members

• ‘viral’ marketing

• web-log (‘blog’) news propagation

• computer network security: email/IP traffic and anomaly detection

• ....

Page 9: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 9

CMU SCS

Problem #1 - network and graph mining

• How does the Internet look like?• How does the web look like?• What is ‘normal’/‘abnormal’?• which patterns/laws hold?

Page 10: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 10

CMU SCS

Graph mining

• Are real graphs random?

Page 11: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 11

CMU SCS

Laws and patterns

• Are real graphs random?

• A: NO!!– Diameter– in- and out- degree distributions– other (surprising) patterns

Page 12: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 12

CMU SCS

Solution#1

• Power law in the degree distribution [SIGCOMM99]

log(rank)

log(degree)

-0.82

internet domains

att.com

ibm.com

Page 13: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 13

CMU SCS

Solution#1’: Eigen Exponent E

• A2: power law in the eigenvalues of the adjacency matrix

E = -0.48

Exponent = slope

Eigenvalue

Rank of decreasing eigenvalue

May 2001

Page 14: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 14

CMU SCS

Solution#1’: Eigen Exponent E

• [Mihail, Papadimitriou ’02]: slope is ½ of rank exponent

E = -0.48

Exponent = slope

Eigenvalue

Rank of decreasing eigenvalue

May 2001

Page 15: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 15

CMU SCS

But:

How about graphs from other domains?

Page 16: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 16

CMU SCS

The Peer-to-Peer Topology

• Count versus degree • Number of adjacent peers follows a power-law

[Jovanovic+]

Page 17: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 17

CMU SCS

More power laws:

citation counts: (citeseer.nj.nec.com 6/2001)

log(#citations)

log(count)

Ullman

Page 18: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 18

CMU SCS

More power laws:

• web hit counts [w/ A. Montgomery]

Web Site Traffic

log(in-degree)

log(count)

Zipf

userssites

``ebay’’

Page 19: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 19

CMU SCS

epinions.com• who-trusts-whom

[Richardson + Domingos, KDD 2001]

(out) degree

count

trusts-2000-people user

Page 20: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 20

CMU SCS

Motivation

Data mining: ~ find patterns (rules, outliers)

• Problem#1: How do real graphs look like?

• Problem#2: How do they evolve?

• Problem#3: How to generate realistic graphs

TOOLS

• Problem#4: Who is the ‘master-mind’?

• Problem#5: Track communities over time

Page 21: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 21

CMU SCS

Problem#2: Time evolution• with Jure Leskovec

(CMU/MLD)

• and Jon Kleinberg (Cornell – sabb. @ CMU)

Page 22: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 22

CMU SCS

Evolution of the Diameter

• Prior work on Power Law graphs hints at slowly growing diameter:– diameter ~ O(log N)– diameter ~ O(log log N)

• What is happening in real data?

Page 23: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 23

CMU SCS

Evolution of the Diameter

• Prior work on Power Law graphs hints at slowly growing diameter:– diameter ~ O(log N)– diameter ~ O(log log N)

• What is happening in real data?

• Diameter shrinks over time

Page 24: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 24

CMU SCS

Diameter – ArXiv citation graph

• Citations among physics papers

• 1992 –2003

• One graph per year

time [years]

diameter

Page 25: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 25

CMU SCS

Diameter – “Autonomous Systems”

• Graph of Internet

• One graph per day

• 1997 – 2000

number of nodes

diameter

Page 26: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 26

CMU SCS

Diameter – “Affiliation Network”

• Graph of collaborations in physics – authors linked to papers

• 10 years of data

time [years]

diameter

Page 27: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 27

CMU SCS

Diameter – “Patents”

• Patent citation network

• 25 years of data

time [years]

diameter

Page 28: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 28

CMU SCS

Temporal Evolution of the Graphs

• N(t) … nodes at time t

• E(t) … edges at time t

• Suppose thatN(t+1) = 2 * N(t)

• Q: what is your guess for E(t+1) =? 2 * E(t)

Page 29: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 29

CMU SCS

Temporal Evolution of the Graphs

• N(t) … nodes at time t• E(t) … edges at time t• Suppose that

N(t+1) = 2 * N(t)

• Q: what is your guess for E(t+1) =? 2 * E(t)

• A: over-doubled!– But obeying the ``Densification Power Law’’

Page 30: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 30

CMU SCS

Densification – Physics Citations• Citations among

physics papers • 2003:

– 29,555 papers, 352,807 citations

N(t)

E(t)

??

Page 31: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 31

CMU SCS

Densification – Physics Citations• Citations among

physics papers • 2003:

– 29,555 papers, 352,807 citations

N(t)

E(t)

1.69

Page 32: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 32

CMU SCS

Densification – Physics Citations• Citations among

physics papers • 2003:

– 29,555 papers, 352,807 citations

N(t)

E(t)

1.69

1: tree

Page 33: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 33

CMU SCS

Densification – Physics Citations• Citations among

physics papers • 2003:

– 29,555 papers, 352,807 citations

N(t)

E(t)

1.69clique: 2

Page 34: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 34

CMU SCS

Densification – Patent Citations

• Citations among patents granted

• 1999– 2.9 million nodes– 16.5 million

edges

• Each year is a datapoint N(t)

E(t)

1.66

Page 35: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 35

CMU SCS

Densification – Autonomous Systems

• Graph of Internet

• 2000– 6,000 nodes– 26,000 edges

• One graph per day

N(t)

E(t)

1.18

Page 36: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 36

CMU SCS

Densification – Affiliation Network

• Authors linked to their publications

• 2002– 60,000 nodes

• 20,000 authors

• 38,000 papers

– 133,000 edgesN(t)

E(t)

1.15

Page 37: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 37

CMU SCS

Motivation

Data mining: ~ find patterns (rules, outliers)

• Problem#1: How do real graphs look like?

• Problem#2: How do they evolve?

• Problem#3: How to generate realistic graphs

TOOLS

• Problem#4: Who is the ‘master-mind’?

• Problem#5: Track communities over time

Page 38: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 38

CMU SCS

Problem#3: Generation

• Given a growing graph with count of nodes N1, N2, …

• Generate a realistic sequence of graphs that will obey all the patterns

Page 39: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 39

CMU SCS

Problem Definition

• Given a growing graph with count of nodes N1, N2, …

• Generate a realistic sequence of graphs that will obey all the patterns – Static Patterns

Power Law Degree DistributionPower Law eigenvalue and eigenvector distributionSmall Diameter

– Dynamic PatternsGrowth Power LawShrinking/Stabilizing Diameters

Page 40: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 40

CMU SCS

Problem Definition

• Given a growing graph with count of nodes N1, N2, …

• Generate a realistic sequence of graphs that will obey all the patterns

• Idea: Self-similarity– Leads to power laws– Communities within communities– …

Page 41: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 41

CMU SCS

Adjacency matrix

Kronecker Product – a Graph

Intermediate stage

Adjacency matrix

Page 42: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 42

CMU SCS

Kronecker Product – a Graph• Continuing multiplying with G1 we obtain G4 and

so on …

G4 adjacency matrix

Page 43: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 43

CMU SCS

Kronecker Product – a Graph• Continuing multiplying with G1 we obtain G4 and

so on …

G4 adjacency matrix

Page 44: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 44

CMU SCS

Kronecker Product – a Graph• Continuing multiplying with G1 we obtain G4 and

so on …

G4 adjacency matrix

Page 45: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 45

CMU SCS

Properties:

• We can PROVE that– Degree distribution is multinomial ~ power law– Diameter: constant– Eigenvalue distribution: multinomial– First eigenvector: multinomial

• See [Leskovec+, PKDD’05] for proofs

Page 46: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 46

CMU SCS

Problem Definition

• Given a growing graph with nodes N1, N2, …

• Generate a realistic sequence of graphs that will obey all the patterns – Static Patterns

Power Law Degree Distribution

Power Law eigenvalue and eigenvector distribution

Small Diameter

– Dynamic PatternsGrowth Power Law

Shrinking/Stabilizing Diameters

• First and only generator for which we can prove all these properties

Page 47: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 47

CMU SCS

Stochastic Kronecker Graphs• Create N1N1 probability matrix P1

• Compute the kth Kronecker power Pk

• For each entry puv of Pk include an edge (u,v) with probability puv

0.4 0.2

0.1 0.3

P1

Instance

Matrix G2

0.16 0.08 0.08 0.04

0.04 0.12 0.02 0.06

0.04 0.02 0.12 0.06

0.01 0.03 0.03 0.09

Pk

flip biased

coins

Kronecker

multiplication

skip

Page 48: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 48

CMU SCS

Experiments

• How well can we match real graphs?– Arxiv: physics citations:

• 30,000 papers, 350,000 citations

• 10 years of data

– U.S. Patent citation network• 4 million patents, 16 million citations

• 37 years of data

– Autonomous systems – graph of internet• Single snapshot from January 2002

• 6,400 nodes, 26,000 edges

• We show both static and temporal patterns

Page 49: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 49

CMU SCS

(Q: how to fit the parm’s?)

A:

• Stochastic version of Kronecker graphs +

• Max likelihood +

• Metropolis sampling

• [Leskovec+, ICML’07]

Page 50: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 50

CMU SCS

Experiments on real AS graphDegree distribution Hop plot

Network valueAdjacency matrix eigen values

Page 51: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 51

CMU SCS

Conclusions

• Kronecker graphs have:– All the static properties

Heavy tailed degree distributions

Small diameter

Multinomial eigenvalues and eigenvectors

– All the temporal propertiesDensification Power Law

Shrinking/Stabilizing Diameters

– We can formally prove these results

Page 52: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 52

CMU SCS

Motivation

Data mining: ~ find patterns (rules, outliers)

• Problem#1: How do real graphs look like?

• Problem#2: How do they evolve?

• Problem#3: How to generate realistic graphs

TOOLS

• Problem#4: Who is the ‘master-mind’?

• Problem#5: Track communities over time

Page 53: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 53

CMU SCS

Problem#4: MasterMind – ‘CePS’

• w/ Hanghang Tong, KDD 2006

• htong <at> cs.cmu.edu

Page 54: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 54

CMU SCS

Center-Piece Subgraph(Ceps)

• Given Q query nodes• Find Center-piece ( )

• App.– Social Networks– Law Inforcement, …

• Idea:– Proximity -> random walk

with restarts

A C

B

A C

B

A C

B

b

Page 55: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 55

CMU SCS

Case Study: AND query

R. Agrawal Jiawei Han

V. Vapnik M. Jordan

Page 56: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 56

CMU SCS

Case Study: AND query

R. Agrawal Jiawei Han

V. Vapnik M. Jordan

H.V. Jagadish

Laks V.S. Lakshmanan

Heikki Mannila

Christos Faloutsos

Padhraic Smyth

Corinna Cortes

15 1013

1 1

6

1 1

4 Daryl Pregibon

10

2

11

3

16

Page 57: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 57

CMU SCS

Case Study: AND query

R. Agrawal Jiawei Han

V. Vapnik M. Jordan

H.V. Jagadish

Laks V.S. Lakshmanan

Heikki Mannila

Christos Faloutsos

Padhraic Smyth

Corinna Cortes

15 1013

1 1

6

1 1

4 Daryl Pregibon

10

2

11

3

16

Page 58: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 58

CMU SCS

R. Agrawal Jiawei Han

V. Vapnik M. Jordan

H.V. Jagadish

Laks V.S. Lakshmanan

Umeshwar Dayal

Bernhard Scholkopf

Peter L. Bartlett

Alex J. Smola

1510

13

3 3

5 2 2

327

42_SoftAnd query

ML/Statistics

databases

Page 59: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 59

CMU SCS

Conclusions

• Q1:How to measure the importance?

• A1: RWR+K_SoftAnd

• Q2:How to do it efficiently?

• A2:Graph Partition (Fast CePS)– ~90% quality

– 150x speedup (ICDM’06, b.p. award)

A C

B

Page 60: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 60

CMU SCS

Outline

• Problem definition / Motivation

• Static & dynamic laws; generators

• Tools: CenterPiece graphs; Tensors

• Other projects (Virus propagation, e-bay fraud detection)

• Conclusions

Page 61: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 61

CMU SCS

Motivation

Data mining: ~ find patterns (rules, outliers)

• Problem#1: How do real graphs look like?

• Problem#2: How do they evolve?

• Problem#3: How to generate realistic graphs

TOOLS

• Problem#4: Who is the ‘master-mind’?

• Problem#5: Track communities over time

Page 62: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 62

CMU SCS

Tensors for time evolving graphs

• [Jimeng Sun+ KDD’06]

• [ “ , SDM’07]• [ CF, Kolda, Sun,

SDM’07 tutorial]

Page 63: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 63

CMU SCS

Social network analysis

• Static: find community structures

DB

Aut

hors

Keywords1990

Page 64: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 64

CMU SCS

Social network analysis

• Static: find community structures

DB

Aut

hors

19901991

1992

Page 65: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 65

CMU SCS

Social network analysis

• Static: find community structures • Dynamic: monitor community structure evolution;

spot abnormal individuals; abnormal time-stamps

DB

Aut

hors

Keywords

DM

DB

1990

2004

Page 66: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 66

CMU SCS

DB

DM

Application 1: Multiway latent semantic indexing (LSI)

DB

2004

1990Michael

Stonebraker

QueryPattern

Ukeyword

authors

keyword

Uauthors

• Projection matrices specify the clusters

• Core tensors give cluster activation level

Philip Yu

Page 67: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 67

CMU SCS

Bibliographic data (DBLP)

• Papers from VLDB and KDD conferences• Construct 2nd order tensors with yearly

windows with <author, keywords> • Each tensor: 45843741 • 11 timestamps (years)

Page 68: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 68

CMU SCS

Multiway LSIAuthors Keywords Yearmichael carey, michaelstonebraker, h. jagadish,hector garcia-molina

queri,parallel,optimization,concurr,objectorient

1995

surajit chaudhuri,mitch cherniack,michaelstonebraker,ugur etintemel

distribut,systems,view,storage,servic,process,cache

2004

jiawei han,jian pei,philip s. yu,jianyong wang,charu c. aggarwal

streams,pattern,support, cluster, index,gener,queri

2004

• Two groups are correctly identified: Databases and Data mining

• People and concepts are drifting over time

DM

DB

Page 69: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 69

CMU SCS

Network forensics• Directional network flows

• A large ISP with 100 POPs, each POP 10Gbps link capacity [Hotnets2004]– 450 GB/hour with compression

• Task: Identify abnormal traffic pattern and find out the cause

normal trafficabnormal traffic

dest

inati

on

source

dest

inati

on

source(with Prof. Hui Zhang and Dr. Yinglian Xie)

Page 70: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 70

CMU SCS

MDL mining on time-evolving graph (Enron emails)

GraphScope [w. Jimeng Sun, Spiros Papadimitriou and Philip Yu, KDD’07]

Page 71: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 71

CMU SCS

Conclusions

Tensor-based methods (WTA/DTA/STA):

• spot patterns and anomalies on time evolving graphs, and

• on streams (monitoring)

Page 72: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 72

CMU SCS

Motivation

Data mining: ~ find patterns (rules, outliers)

• Problem#1: How do real graphs look like?

• Problem#2: How do they evolve?

• Problem#3: How to generate realistic graphs

TOOLS

• Problem#4: Who is the ‘master-mind’?

• Problem#5: Track communities over time

Page 73: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 73

CMU SCS

Outline

• Problem definition / Motivation

• Static & dynamic laws; generators

• Tools: CenterPiece graphs; Tensors

• Other projects (Virus propagation, e-bay fraud detection, blogs)

• Conclusions

Page 74: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 74

CMU SCS

Virus propagation

• How do viruses/rumors propagate?

• Blog influence?

• Will a flu-like virus linger, or will it become extinct soon?

Page 75: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 75

CMU SCS

The model: SIS

• ‘Flu’ like: Susceptible-Infected-Susceptible

• Virus ‘strength’ s= /

Infected

Healthy

NN1

N3

N2Prob.

Prob. β

Prob.

Page 76: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 76

CMU SCS

Epidemic threshold of a graph: the value of , such that

if strength s = / < an epidemic can not happen

Thus,

• given a graph

• compute its epidemic threshold

Page 77: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 77

CMU SCS

Epidemic threshold

What should depend on?

• avg. degree? and/or highest degree?

• and/or variance of degree?

• and/or third moment of degree?

• and/or diameter?

Page 78: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 78

CMU SCS

Epidemic threshold

• [Theorem] We have no epidemic, if

β/δ <τ = 1/ λ1,A

Page 79: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 79

CMU SCS

Epidemic threshold

• [Theorem] We have no epidemic, if

β/δ <τ = 1/ λ1,A

largest eigenvalueof adj. matrix A

attack prob.

recovery prob.epidemic threshold

Proof: [Wang+03]

Page 80: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 80

CMU SCS

Experiments (Oregon)

/ > τ (above threshold)

/ = τ (at the threshold)

/ < τ (below threshold)

Page 81: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 81

CMU SCS

Outline

• Problem definition / Motivation

• Static & dynamic laws; generators

• Tools: CenterPiece graphs; Tensors

• Other projects (Virus propagation, e-bay fraud detection, blogs)

• Conclusions

Page 82: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 82

CMU SCS

E-bay Fraud detection

w/ Polo Chau &Shashank Pandit, CMU

Page 83: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 83

CMU SCS

E-bay Fraud detection

• lines: positive feedbacks• would you buy from him/her?

Page 84: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 84

CMU SCS

E-bay Fraud detection

• lines: positive feedbacks• would you buy from him/her?

• or him/her?

Page 85: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 85

CMU SCS

E-bay Fraud detection - NetProbe

Page 86: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 86

CMU SCS

Outline

• Problem definition / Motivation

• Static & dynamic laws; generators

• Tools: CenterPiece graphs; Tensors

• Other projects (Virus propagation, e-bay fraud detection, blogs)

• Conclusions

Page 87: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 87

CMU SCS

Blog analysis

• with Mary McGlohon (CMU)

• Jure Leskovec (CMU)

• Natalie Glance (now at Google)

• Mat Hurst (now at MSR)

[SDM’07]

Page 88: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 88

CMU SCS

Cascades on the BlogosphereB1 B2

B4B3

a

b c

de

1

B1 B2

B4B3

11

2

3

1

Blogosphereblogs + posts

Blog networklinks among blogs

Post networklinks among posts

Q1: popularity-decay of a post?Q2: degree distributions?

Page 89: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 89

CMU SCS

Q1: popularity over time

Days after post

Post popularity drops-off – exponentially?

days after post

# in links

1 2 3

Page 90: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 90

CMU SCS

Q1: popularity over time

Days after post

Post popularity drops-off – exponentially?POWER LAW!Exponent?

# in links(log)

1 2 3 days after post(log)

Page 91: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 91

CMU SCS

Q1: popularity over time

Days after post

Post popularity drops-off – exponentially?POWER LAW!Exponent? -1.6 (close to -1.5: Barabasi’s stack model)

# in links(log)

1 2 3

-1.6

days after post(log)

Page 92: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 92

CMU SCS

Q2: degree distribution

44,356 nodes, 122,153 edges. Half of blogs belong to largest connected component.

blog in-degree

count

B

1

B

2

B

4

B

3

11

2

3

1

??

Page 93: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 93

CMU SCS

Q2: degree distribution

44,356 nodes, 122,153 edges. Half of blogs belong to largest connected component.

blog in-degree

count

B

1

B

2

B

4

B

3

11

2

3

1

Page 94: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 94

CMU SCS

Q2: degree distribution

44,356 nodes, 122,153 edges. Half of blogs belong to largest connected component.

blog in-degree

count

in-degree slope: -1.7out-degree: -3‘rich get richer’

Page 95: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 95

CMU SCS

Outline

• Problem definition / Motivation

• Static & dynamic laws; generators

• Tools: CenterPiece graphs; Tensors

• Other projects (Virus propagation, e-bay fraud detection)– And research directions

• Conclusions

Page 96: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 96

CMU SCS

Next steps:

• edges with

– categorical attributes and/or

– time-stamps and/or

– weights

• nodes with attributes [G-Ray, Tong et al]

• scalability (cloud computing)

Page 97: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 97

CMU SCS

E.g.: self-* system @ CMU

• >200 nodes• 40 racks of computing

equipment • 774kw of power. • target: 1 PetaByte• goal: self-correcting, self-

securing, self-monitoring, self-...

Page 98: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 98

CMU SCS

Cloud computing, D.I.S.C. and hadoop

• ‘Data Intensive Scientific Computing’ [R. Bryant, CMU]– ‘big data’ – http://www.cs.cmu.edu/~bryant/pubdir/cmu-cs-07-

128.pdf

• Yahoo: ~5Pb of data [Fayyad’07]• ‘M45’: 4K proc’s, 3Tb RAM, 1.5 Pb disk• Hadoop: open-source clone of map-reduce

http://hadoop.apache.org/

Page 99: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 99

CMU SCS

OVERALL CONCLUSIONS• Graphs pose a wealth of fascinating

problems

• self-similarity and power laws work, when textbook methods fail!

• New patterns (shrinking diameter!)

• New generator: Kronecker

• SVD / tensors / RWR: valuable tools

• Scalability / cloud computing -> PetaBytes

Page 100: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 100

CMU SCS

References• Hanghang Tong, Christos Faloutsos, and Jia-Yu

Pan Fast Random Walk with Restart and Its Applications ICDM 2006, Hong Kong.

• Hanghang Tong, Christos Faloutsos Center-Piece Subgraphs: Problem Definition and Fast Solutions, KDD 2006, Philadelphia, PA

• Hanghang Tong, Brian Gallagher, Christos Faloutsos, and Tina Eliassi-Rad Fast Best-Effort Pattern Matching in Large Attributed Graphs KDD 2007, San Jose, CA

Page 101: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 101

CMU SCS

References• Jure Leskovec, Jon Kleinberg and Christos

Faloutsos Graphs over Time: Densification Laws, Shrinking Diameters and Possible Explanations KDD 2005, Chicago, IL. ("Best Research Paper" award).

• Jure Leskovec, Deepayan Chakrabarti, Jon Kleinberg, Christos Faloutsos Realistic, Mathematically Tractable Graph Generation and Evolution, Using Kronecker Multiplication (ECML/PKDD 2005), Porto, Portugal, 2005.

Page 102: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 102

CMU SCS

References• Jure Leskovec and Christos Faloutsos, Scalable

Modeling of Real Graphs using Kronecker Multiplication, ICML 2007, Corvallis, OR, USA

• Shashank Pandit, Duen Horng (Polo) Chau, Samuel Wang and Christos Faloutsos NetProbe: A Fast and Scalable System for Fraud Detection in Online Auction Networks WWW 2007, Banff, Alberta, Canada, May 8-12, 2007.

• Jimeng Sun, Dacheng Tao, Christos Faloutsos Beyond Streams and Graphs: Dynamic Tensor Analysis, KDD 2006, Philadelphia, PA

Page 103: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 103

CMU SCS

References• Jimeng Sun, Yinglian Xie, Hui Zhang, Christos

Faloutsos. Less is More: Compact Matrix Decomposition for Large Sparse Graphs, SDM, Minneapolis, Minnesota, Apr 2007. [pdf]

• Jimeng Sun, Spiros Papadimitriou, Philip S. Yu, and Christos Faloutsos, GraphScope: Parameter-free Mining of Large Time-evolving Graphs ACM SIGKDD Conference, San Jose, CA, August 2007

Page 104: CMU SCS Graph Mining and Influence Propagation Christos Faloutsos CMU

WICOW 08 C. Faloutsos 104

CMU SCS

Contact info:

www. cs.cmu.edu /~christos

(w/ papers, datasets, code, etc)