Large Graph Mining

CMU SCS

Large Graph Mining

Christos Faloutsos

CMU

CMU SCS

ICDM-LDMTA 2009 C. Faloutsos 2

Thank you!

• Dr. Yan Liu

• Dr. Jimeng Sun

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Outline

• Introduction – Motivation

• Problem#1: Patterns in graphs

• Problem#2: Generators

• Problem#3: Scalability

• Conclusions

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Graphs - why should we care?

Internet Map [lumeta.com]

Food Web [Martinez ’91]

Protein Interactions [genomebiology.com]

Friendship Network [Moody ’01]

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• IR: bi-partite graphs (doc-terms)

• web: hyper-text graph

• ... and more:

D1

DN

T1

TM

... ...

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• network of companies & board-of-directors members

• ‘viral’ marketing

• web-log (‘blog’) news propagation

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

• ....

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Outline


• Problem#1: Patterns in graphs– Static graphs– Weighted graphs– Time evolving graphs



• Conclusions

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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?

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Graph mining

• Are real graphs random?

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Laws and patterns• Are real graphs random?• A: NO!!

– Diameter– in- and out- degree distributions– other (surprising) patterns

• So, let’s look at the data • (‘it is amazing what you hear, when you

listen’)

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Solution# S.1

• Power law in the degree distribution [SIGCOMM99]

log(rank)

log(degree)

-0.82

internet domains

att.com

ibm.com

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Solution# S.2: 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

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Solution# S.2: Eigen Exponent E

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

E = -0.48

Exponent = slope

Eigenvalue

Rank of decreasing eigenvalue

May 2001

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But:

How about graphs from other domains?

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More power laws:

• web hit counts [w/ A. Montgomery]

Web Site Traffic

log(in-degree)

log(count)

Zipf

userssites

``ebay’’

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epinions.com• who-trusts-whom

[Richardson + Domingos, KDD 2001]

(out) degree

count

trusts-2000-people user

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Outline


• Problem#1: Patterns in graphs– Static graphs

• degree, diameter, eigen*,

• triangles

• cliques

– Weighted graphs– Time evolving graphs


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Solution# S.3: Triangle ‘Laws’

• Real social networks have a lot of triangles

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Triangle ‘Laws’

• Real social networks have a lot of triangles– Friends of friends are friends

• Any patterns?

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Triangle Law: #S.3 [Tsourakakis ICDM 2008]

ASNHEP-TH

Epinions X-axis: # of Triangles a node participates inY-axis: count of such nodes

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Triangle Law: #S.4 [Tsourakakis ICDM 2008]

SNReuters

EpinionsX-axis: degreeY-axis: mean # trianglesNotice: slope ~ degree exponent (insets)

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Triangle Law: Computations [Tsourakakis ICDM 2008]

But: triangles are expensive to compute(3-way join; several approx. algos)

Q: Can we do that quickly?

details

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But: triangles are expensive to compute(3-way join; several approx. algos)

Q: Can we do that quickly?A: Yes!

#triangles = 1/6 Sum ( i3 )

(and, because of skewness, we only need the top few eigenvalues!

details

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1000x+ speed-up, high accuracy

details

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Outline




• triangles

• cliques



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Large Human Communication NetworksPatterns and a Utility-Driven Generator

Nan Du, Christos Faloutsos, Bai Wang, Leman Akoglu

KDD 2009

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Cliques• Clique is a complete subgraph.• If a clique can not be

contained by any largerclique, it is called the maximal clique.

20

1 3

4

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Clique• Clique is a complete subgraph.• If a clique can not be


20

1 3

4

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20

1 3

4

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• {0,1,2}, {0,1,3}, {1,2,3}{2,3,4}, {0,1,2,3} are cliques;

• {0,1,2,3} and {2,3,4} are the maximal cliques.

20

1 3

4

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S.5: Clique-Degree Power-Law

• Power law:

More friends, even more social circles !

# maximalcliques of node i

degreeof node i

Dataset: who-calls-whomanonymized,Over several time units€

C∝da

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S.5 Clique-Degree Power-Law

• Outlier Detection

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1.5 Clique-Degree Power-Law

• Outlier Detection

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Outline




• triangles

• cliques



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Observation W.1

Question : Nodes in a triangle are topologically equivalent. Will they also give equal number of phone calls to each other ?

Max

Wei

ght M

in Weight

Mid Weight

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Observation W.1:Triangle Weight Law

€

Max ∝mediuma

Periods S1 – S3

€

Max ∝minb

€

medium∝minc

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Other observations on weighted graphs?

• A: yes - even more ‘laws’!

M. McGlohon, L. Akoglu, and C. Faloutsos Weighted Graphs and Disconnected Components: Patterns and a Generator. SIG-KDD 2008

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Observation W.2: fortification

Q: How do the weights

of nodes relate to degree?

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Edges (# donors)

In-weights($)


Observation W.2: fortification:Snapshot Power Law

• weight of a node is super-linear on the in-degree with PL exponent ‘iw’:

– i.e. 1.01 < iw < 1.26, super-linear

Orgs-Candidates

e.g. John Kerry, $10M received,from 1K donors

More donors, even more $

$10

$5

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Are there deviations from P.L.?

• A: yes – but they also correspond to very skewed distributions (‘black/gray swans’)

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“Mobile Call Graphs: Beyond Power Law and Lognormal Distributions”

K D D ’ 0 8

Mukund Seshadri , Sridhar Machiraju,

Ashwin Sridharan, Jean Bolot

Christos Faloutsos, Jure Leskovec

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Dataset• Who-calls-whom (anonymized)

• Degree distribution, in green

Area S1Time period T11 monthCount

Degree of Mobile-phone call graph

.... Observed Data

--- Power Law Fit+ Observed Data

.... LogNormal Fit

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A poor fit?• Power Laws: p(x) ~ x^(-a)

• Lognormal: log(x) is Normal

Count

Degree of Mobile-phone call graph

.... Observed Data

--- Power Law Fit+ Observed Data

.... LogNormal Fit

Area S1Time period T11 month

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Solution: DPLN • Double Pareto Log Normal (Reed, 2003)

• 4 parameters: [α,β,ν,τ]• Linear head and tail.

Degree

Area S1, Time Period T1

β

α

“Rich get Richer”BUT

Population lifetimesNOT identical

count

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Datasets• Anonymized monthly aggregates of call metrics

per user – Collected at 4 coverage areas S1,S2,S3,S4– Collected for two month-long periods T1 and

T2, separated by 6 months – Total coverage:

• >1 million users

• >10 million calls

• ~7000 square miles.

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• Degree (No. of Call Partners)• Calls• Talk Time

Total over a month,per user

Area S1Time-period T1Degree Distribution

Metrics

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Persistence –

Across Metric, Time, Space

Partners

S2, T2

Partners

S1, T2

CallsArea S1, Time T1

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• Outlier detection– Many un-answered calls => multiples

of 27 sec!

Applications

Per-user distribution of total talk time (@ T1, S1)

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Outline


• Problem#1: Patterns in graphs– Static graphs – Weighted graphs– Time evolving graphs


• …

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Problem: Time evolution• with Jure Leskovec (CMU ->

Stanford)

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

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T.1 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?

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T.1 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

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T.1 Diameter – “Patents”

• Patent citation network

• 25 years of data

time [years]

diameter

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T.2 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)

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T.2 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’’

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T.2 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

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Outline


• Problem#1: Patterns in graphs– Static graphs – Weighted graphs– Time evolving graphs


• …

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More on Time-evolving graphs

M. McGlohon, L. Akoglu, and C. Faloutsos Weighted Graphs and Disconnected Components: Patterns and a Generator. SIG-KDD 2008

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Observation T.3: NLCC behaviorQ: How do NLCC’s emerge and join with

the GCC?

(``NLCC’’ = non-largest conn. components)

– Do they continue to grow in size?

– or do they shrink?

– or stabilize?

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Observation T.4: NLCC behavior

• After the gelling point, the GCC takes off, but NLCC’s remain ~constant (actually, oscillate).

IMDB

CC size

Time-stamp

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T.5 How do new edges appear?

[LBKT’08] Microscopic Evolution of Social Networks Jure Leskovec, Lars Backstrom, Ravi Kumar, Andrew Tomkins. (ACM KDD), 2008.

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Edge gap δ(d): inter-arrival time between dth and d+1st edge

T.5 How do edges appear in time?[LBKT’08]

What is the PDF of Poisson?

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)()(),);(( dg eddp

Edge gap δ(d): inter-arrival time between dth and d+1st edge

LinkedIn

T.5 How do edges appear in time?[LBKT’08]

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Similarly for Blogs

• with Mary McGlohon (CMU)

• Jure Leskovec (CMU->Stanford)

• Natalie Glance (now at Google)

• Mat Hurst (now at MSR)

[SDM’07]

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T.6 : popularity over time

Post popularity drops-off – exponentially?

lag: days after post

# in links

1 2 3

@t

@t + lag

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

# in links(log)

1 2 3 days after post(log)

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Post popularity drops-off – exponentially?POWER LAW!Exponent? -1.6 • close to -1.5: Barabasi’s stack model• and like the zero-crossings of a random walk

# in links(log)

1 2 3

-1.6

days after post(log)

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Outline





• Conclusions

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Problem#2: Generation

Goal: Generate a realistic sequence of graphs that

1. obey all the patterns• including weights on the edges and• timestamps on the edges

2. needs no randomness

3. needs no parameters (or as few as possible)

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PaC generator

• Why do we need generators?– What-if scenarios

– How to attract/retain customers/members

– Stress-testing algorithms, when real graphs are unavailable (eg., due to privacy, national/corporate security, etc)

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(NUMEROUS earlier generators)

• preferential attachment [Barabasi+]

• copying model [Kleinberg+]

• winner does not take all [Giles+]

• Kronecker graphs

• ...

• (survey: [Chakrabarti+, ’06])

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Problem#2: Generation

Goal: Generate a realistic sequence of graphs that

1. obey all the patterns• including weights on the edges and

• timestamps on the edges

2. needs no randomness

3. needs no parameters (or as few as possible)

Proposed ‘PaC’ model – idea:• design ‘agents’, who try to max utility of

contacts

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PaC Model• People benefit from calling each other.

• A Pay and Call game = PaC Model

• The payoffs are measured as “emotional dollars”.

agent

1iF 0iF

Prob pl to stay in the game

Friendliness 0<=Fi<=1

capital

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Outline of Agent Behavior• Step 1: decide to stay (PL)

• Step 2: if stay, call the most profitable person(s)– Existing friend (‘exploit’)

– Stranger (‘explore’) or ask for recommendation (if available) to maximize benefits

Exponential lifetime

Rich get richer

Closing Triangle

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Validation of PaC

• Ran 35 simulations

• 100,000 agents per simulation

• Variation of the parameters does not change the shape of the distribution

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Goals of Validation

• G1: Skewed degree/node weight distribution

• G2: Snapshot Power-Law

• G3: Skewed connected components distribution

• G4: Clique-Degree Power-Law

• G5: Clique-Participation Law

• G6: Triangle Weight Law

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Validation of PaC• G1: Skewed Degree / Node Weight Distribution

Real Network

Synthetic Network

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Validation of PaC• G2: Snapshot Power Law [McGlohon, Akoglu,

Faloutsos 08] “more partners, even more calls”

Real Network Synthetic Network

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Validation of PaC• G3: Skewed distribution of the connected

components


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Validation of PaC

• G4: Clique Degree Power Law


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Validation of PaC

• G5: Clique Participation Law


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Validation of PaC

• G6: Triangle Weight Law

Real Network

Synthetic Network

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Validation of PaCG1: Skewed degree/node weight

distributionG2: Snapshot Power-LawG3: Skewed connected components

distributionG4: Clique-Degree Power-LawG5: Clique-Participation LawG6: Triangle Weight Law

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Outline





• Conclusions

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Scalability

• How about if graph/tensor does not fit in core?

• How about handling huge graphs?

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Scalability

• How about if graph/tensor does not fit in core?

• [‘MET’: Kolda, Sun, ICMD’08, best paper award]

• How about handling huge graphs?

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Scalability• Google: > 450,000 processors in clusters of ~2000

processors each [Barroso, Dean, Hölzle, “Web Search for a Planet: The Google Cluster Architecture” IEEE Micro 2003]

• Yahoo: 5Pb of data [Fayyad, KDD’07]• Problem: machine failures, on a daily basis• How to parallelize data mining tasks, then?• A: map/reduce – hadoop (open-source clone)

http://hadoop.apache.org/

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2’ intro to hadoop

• master-slave architecture; n-way replication (default n=3)

• ‘group by’ of SQL (in parallel, fault-tolerant way)• e.g, find histogram of word frequency

– compute local histograms– then merge into global histogram

select course-id, count(*)from ENROLLMENTgroup by course-id

details

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2’ intro to hadoop

• master-slave architecture; n-way replication (default n=3)

• ‘group by’ of SQL (in parallel, fault-tolerant way)• e.g, find histogram of word frequency

– compute local histograms– then merge into global histogram

select course-id, count(*)from ENROLLMENTgroup by course-id map

reduce

details

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UserProgram

Reducer

Reducer

Master

Mapper

Mapper

Mapper

fork fork fork

assignmap

assignreduce

readlocalwrite

remote read,sort

Output

File 0

Output

File 1

write

Split 0Split 1Split 2

Input Data(on HDFS)

By default: 3-way replication;Late/dead machines: ignored, transparently (!)

details

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PEGASUS: A Peta-Scale Graph Mining System, ICDM’09

U Kang, Charalampos Tsourakakis, C. Faloutsos

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the PageRank distribution: power-law

Analysis of Real Networks

YahooWeb: 1.4B nodes and 6.6B edges, 120Gb(largest public graph analyzed)

M45 (Y!):• 1.5Pb disk• 3Tb RAM• 4000 cores• among top 500

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Spikes?


YahooWeb: 1.4 B nodes and 6.6 B edges -Connected components:

size

count

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Spikes: due to replications of pages.


YahooWeb: 1.4 B nodes and 6.6 B edges -Connected components:

size

count

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Evolution of connected components of LinkedIn: stable, *after* the gelling point (@2004)

LinkedIn: 7.5M nodes and 58M edges

details

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OVERALL CONCLUSIONS – low level:

• Several new patterns (fortification, triangle-laws, etc)

• New generator: ‘PaC’ (agent based)

• Scalability / cloud computing -> PetaBytes

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OVERALL CONCLUSIONS – high level

• good news: unprecedented opportunities (huge datasets, available on-line)

• better news: cross-disciplinary work:

– DB/sys -> scalability

– ML/stat -> tools

– Econ, Soc -> experience; what questions to ask

– phys, math -> phase transitions, tensors, eigenvalues

– …

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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

• T. G. Kolda and J. Sun. Scalable Tensor Decompositions for Multi-aspect Data Mining. In: ICDM 2008, pp. 363-372, December 2008.

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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.

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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

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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

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Contact info:

www. cs.cmu.edu /~christos

Thanks again to the organizers:

Jimeng Sun, Yan LiuFunding sources:• NSF IIS-0705359, IIS-0534205, DBI-0640543• LLNL, PITA, IBM, INTEL, HP

Documents

Large Graph Mining