1 Tools for Large Graph Mining Thesis Committee: Christos Faloutsos Chris Olston Guy Blelloch ...

Tools for Large Graph Mining

Thesis Committee:

Christos Faloutsos

Chris Olston

Guy Blelloch

Jon Kleinberg (Cornell)

- Deepayan Chakrabarti

Introduction

Internet Map [lumeta.com]

Food Web [Martinez ’91]

Protein Interactions [genomebiology.com]

Friendship Network [Moody ’01]

► Graphs are ubiquitous

Introduction

What can we do with graphs? How quickly will a disease

spread on this graph?

“Needle exchange” networks of drug users

[Weeks et al. 2002]

Introduction

What can we do with graphs? How quickly will a disease

spread on this graph? Who are the “strange

bedfellows”? Who are the key people? …

► Graph analysis can have great impact

Hijacker network [Krebs ‘01]

“Key” terrorist

Graph Mining: Two Paths

Specific applications

• Node grouping

• Viral propagation

• Frequent pattern mining

• Fast message routing

General issues

• Realistic graph generation

• Graph patterns and “laws”

• Graph evolution over time?

Our Work

General issues

• Node grouping

Our Work

General issues

• Node grouping

Node Grouping Find “natural” partitions and outliers

automatically. Viral Propagation

Will a virus spread and become an epidemic?

Graph Generation How can we mimic a given real-world

graph?

Roadmap

• Node grouping

General issues

• Graph patterns and “laws”31

4 Conclusions

Find “natural” partitions and outliers automatically

Focus of this talk

Node Grouping [KDD 04]

Products

Product Groups

Simultaneously group customers and products, or, documents and words, or, users and preferences …

Customers

Products

Node Grouping [KDD 04]

Product Groups

Row and column groups

• need not be along a diagonal, and

• need not be equal in number

Product Groups

Both are fine

Motivation

Visualization

Summarization

Detection of outlier nodes and edges

Compression, and others…

Node Grouping

Desiderata:

1. Simultaneously discover row and column groups

2. Fully Automatic: No “magic numbers”

3. Scalable to large matrices

4. Online: New data should not require full recomputations

Closely Related Work

Information Theoretic Co-clustering [Dhillon+/2003] Number of row and column groups must be

specified

Desiderata:

Simultaneously discover row and column groups

Fully Automatic: No “magic numbers”

Scalable to large graphs

Online

Other Related Work

K-means and variants: [Pelleg+/2000, Hamerly+/2003]

“Frequent itemsets”: [Agrawal+/1994]

Information Retrieval:[Deerwester+1990, Hoffman/1999]

Graph Partitioning:[Karypis+/1998]

Do not cluster rows and cols simultaneously

User must specify “support”

Choosing the number of “concepts”

Number of partitions

Measure of imbalance between clusters

What makes a cross-association “good”?

versus

Column groups Column groups

Good Clustering

1. Similar nodes are grouped together

2. As few groups as necessary

A few, homogeneous

blocks

Good Compression

Why is this better?

implies

Main Idea

Good Compression

Good Clusteringimplies

Column groups

density pi1 = % of dots

size * H(pi1) Cost of describing

ni1, ni

0 and groups

Code Cost Description Cost

Binary Matrix

Examples

One row group, one column group

high low

m row group, n column group

highlow

Total Encoding Cost = size * H(pi1) Cost of describing

ni1, ni

0 and groups

Σi + Σi

What makes a cross-association “good”?

Why is this better?

low low

Total Encoding Cost = size * H(pi1) Cost of describing

ni1, ni

0 and groups

Σi + Σi

versus

Column groups Column groups

Formal problem statement

Given a binary matrix,

Re-organize the rows and columns into groups, and

Choose the number of row and column groups, to

Minimize the total encoding cost.

Formal problem statement

Given a binary matrix,

Re-organize the rows and columns into groups, and

Choose the number of row and column groups, to

Minimize the total encoding cost.

Note: No Parameters

Algorithmsk =

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

Algorithmsl = 5

Start with initial matrix

Find good groups for fixed k and l

Choose better values for k and l

Final cross-association

Lower the encoding cost

Fixed k and ll = 5

Fixed k and lRe-assign: for each row x

re-assign it to the row group which minimizes the code cost

Column groups

ups 1.Row re-assigns

2.Column re-assigns

3. and repeat …

Column groups

Choosing k and ll = 5

Choosing k and l

Split:1. Find the most “inhomogeneous” group.

2. Remove the rows/columns which make it inhomogeneous.

3. Create a new group for these rows/columns.

Column groups

Algorithmsl = 5

Re-assigns

Splits

Experiments

l = 5 col groups

k = 5 row

groups

“Customer-Product” graph with Zipfian sizes, no noise

Experiments

“Quasi block-diagonal” graph with Zipfian sizes, noise=10%

l = 8 col groups

k = 6 row

groups

Experiments

“White Noise” graph: we find the existing spurious patterns

l = 3 col groups

k = 2 row

groups

Experiments“CLASSIC”

• 3,893 documents

• 4,303 words

• 176,347 “dots”

Combination of 3 sources:

• MEDLINE (medical)

• CISI (info. retrieval)

• CRANFIELD (aerodynamics)

Experiments

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

Experiments

MEDLINE(medical)

insipidus, alveolar, aortic, death, …

blood, disease, clinical, cell, …

Experiments

MEDLINE(medical)

CISI(Information Retrieval)

providing, studying, records, development, …

abstract, notation, works, construct, …

Experiments

MEDLINE(medical)

CRANFIELD (aerodynamics)

shape, nasa, leading, assumed, …

Experiments

MEDLINE(medical)

CRANFIELD (aerodynamics)

paint, examination, fall, raise, leave, based, …

ExperimentsN

Words in abstract

“GRANTS”

• 13,297 documents

• 5,298 words

• 805,063 “dots”

Experiments

“GRANTS” graph of documents & words: k=41, l=28

Words in abstract

Experiments

The Cross-Associations refer to topics:

• Genetics

encoding, characters, bind, nucleus

Experiments

• Genetics

• Physics

coupling, deposition, plasma, beam

Experiments

• Genetics

• Physics

• Mathematics

• …

manifolds, operators, harmonic

Experiments

Number of “dots”

Splits

Re-assigns

Linear on the number of “dots”: Scalable

Summary of Node Grouping

Desiderata:

Simultaneously discover row and column groups

Fully Automatic: No “magic numbers”

Scalable to large matrices

Online: New data does not need full recomputation

Extensions

We can use the same MDL-based framework for other problems:

1. Self-graphs

2. Detection of outlier edges

Extension #1 [PKDD 04]

Self-graphs, such as Co-authorship graphs Social networks The Internet, and the World-wide Web

Customers

Products

Authors

Bipartite graph Self-graph

Self-graphs Rows and columns represent the same nodes so row re-assigns affect column re-assigns…

Bipartite graph Self-graph

Authors

Customers

Products

Experiments

Authors

DBLP dataset

• 6,090 authors in:• SIGMOD

• ICDE

• VLDB

• PODS

• ICDT

• 175,494 co-citation or co-authorship links

Experiments

Authors

Author groups

k=8 author groups found

Stonebraker, DeWitt, Carey

Outlier edges Which links should not exist?

(illegal contact/access?) Which links are missing?

(missing data?)

Outliers Deviations from “normality”

Lower quality compression

Find edges whose removal maximally reduces cost

Node Groups

Outlier edges

Roadmap

• Node grouping

General issues

4 Conclusions

Will a virus spread and become an epidemic?

The SIS (or “flu”) model

(Virus) birth rate β : probability than an infected neighbor attacks

(Virus) death rate δ : probability that an infected node heals

Cured = Susceptible

Infected

Healthy

N2Prob. β

Prob. β

Prob. δ

Undirected network

The SIS (or “flu”) model

Competition between virus birth and death Epidemic or extinction?

depends on the ratio β/δ but also on the network topology

Epidemicor

Extinction

Example of the effect of network topology

Epidemic threshold

The epidemic threshold τ is the value such that If β/δ < τ there is no epidemic where β = birth rate, and δ = death

Previous models

Question: What is the epidemic threshold?

Answer #1: 1/<k>[Kephart and White ’91, ’93]

Answer #2: <k>/<k2>[Pastor-Satorras and Vespignani ’01]

Homogeneity assumption: All nodes have the same degree(but most graphs have power laws)

Mean-field assumption: All nodes of the same degree are equally affected(but susceptibility should depend on position in network too)

The full solution is intractable! The full Markov Chain

has 2N states intractable so, a simplification is needed.

Independence assumption: Probability that two neighbors are infected =

Product of individual probabilities of infection This is a point estimate of the full Markov Chain.

Our model

A non-linear dynamical system (NLDS) which makes no assumptions about the topology

1-pi,t = [1-pi,t-1 + δpi,t-1] . ∏ (1-β.Aji.pj,t-1)j=1

Probability of being infected

Adjacency matrix

Healthy at time t

Healthy at time t-1

Infected but cured

No infection received from another node

Epidemic threshold [Theorem 1] We have no epidemic if:

β/δ < τ = 1/ λ1,A

(Virus) Birth rate

(Virus) Death rate

Epidemic threshold

largest eigenvalueof adj. matrix A

► λ1,A alone decides viral epidemics!

Recall the definition of eigenvalues

A X X= λA

eigenvalue

λ1,A = largest eigenvalue

≈ size of the largest “blob”

Experiments (100-node Star)

β/δ = τ (close to the threshold)

β/δ < τ (below threshold)

β/δ > τ (above threshold)

……

Experiments (Oregon)

β/δ > τ (above threshold)

β/δ = τ (at the threshold)

β/δ < τ (below threshold)

10,900 nodes and 31,180 edges

Extensions

This dynamical-systems framework can exploited further

1. The rate of decay of the infection

2. Information survival thresholds in sensor/P2P networks

Extension #1

Below the threshold:How quickly does an infection die out?

[Theorem 2] Exponentially quickly

Experiment (10K Star Graph)

“Score” s = β/δ * λ1,A = “fraction” of threshold

Time-steps (linear-scale)

Linear on log-lin scale exponential decay

Experiment (Oregon Graph)

“Score” s = β/δ * λ1,A = “fraction” of threshold

Time-steps (linear-scale)

Linear on log-lin scale exponential decay

Extension #2

• Sensors gain new information

Information survival insensor networks[+ Leskovec, Faloutsos, Guestrin, Madden]

Extension #2

• but they may die due to harsh environment or battery failure

• so they occasionally try to transmit data to nearby sensors

• and failed sensors are occasionally replaced.

Extension #2

• but they may die due to harsh environment or battery failure

• so they occasionally try to transmit data to nearby sensors

• and failed sensors are occasionally replaced.

• Under what conditions does the information survive?

Extension #2

[Theorem 1] The information dies out exponentially quickly if

Retransmission rate

Resurrection rate

Failure rate of sensors

Largest eigenvalue of the “link quality” matrix

Roadmap

• Node grouping

General issues

4 Conclusions

How can we generate a “realistic” graph, that mimics

a given real-world?

Experiments (Clickstream bipartite graph)

In-degree

Websites

Some personal webpage

Yahoo, Google and others

ClickstreamR-MAT

Websites

Email-checking surfers

“All-night” surfers

Out-degree

ClickstreamR-MAT

Count vs Out-degree Count vs In-degree Hop-plot

Left “Network value” Right “Network value”

►R-MAT can match real-world graphs

Singular value vs Rank

Roadmap

• Node grouping

General issues

4 Conclusions

Conclusions

Two paths in graph mining: Specific applications:

Viral Propagation non-linear dynamical system, epidemic depends on largest eigenvalue

Node Grouping MDL-based approach for automatic grouping

General issues: Graph Patterns Marks of “realism” in a graph Graph Generators R-MAT, a scalable generator

matching many of the patterns

Software

http://www-2.cs.cmu.edu/~deepay/#Sw CrossAssociations

To find natural node groups. Used by “anonymous” large accounting firm. Used by Intel Research, Cambridge, UK. Used at UC, Riverside (net intrusion detection). Used at the University of Porto, Portugal

NetMine To extract graph patterns quickly + build realistic graphs. Used by Northrop Grumman corp.

F4 A non-linear time series forecasting package.

===CROSS-ASSOCIATIONS=== Why simultaneous groupin

g? Differences from co-cluster

ing and others? Other parameter-fitting crit

eria? Cost surface Exact cost function Exact complexity, wall-

clock times Soft clustering Different weights for code

and description costs?

Precision-recall for CLASSIC

Inter-group “affinities” Collaborative filtering and r

ecommendation systems? CA versus bipartite cores Extras General comments on CA

communities

===Viral Propagation===

Comparison with previous methods Accuracy of dynamical system Relationship with full Markov chain Experiments on information survival threshold Comparison with Infinite Particle Systems Intuition behind the largest eigenvalue Correlated failures

===R-MAT===

Graph patterns Generator desiderata Description of R-MAT Experiments on a directed graph R-MAT communities via Cross-Associations? R-MAT versus tree-based generators

===Graphs in general===

Relational learning Graph Kernels

Simultaneous grouping is useful Sparse blocks, with little

in common between rows

Grouping rows first would

collapse these two into one!

82Index

Cross-Associations ≠ Co-clustering !Information-theoretic

co-clustering Cross-Associations

1. Lossy Compression.

2. Approximates the original matrix, while trying to minimize KL-divergence.

3. The number of row and column groups must be given by the user.

1. Lossless Compression.

2. Always provides complete information about the matrix, for any number of row and column groups.

3. Chosen automatically using the MDL principle.

83Index

Other parameter-fitting methods The Gap statistic [Tibshirani+ ’01]

Minimize the “gap” of log-likelihood of intra-cluster distances from the expected log-likelihood.

But Needs a distance function between graph nodes Needs a “reference” distribution Needs multiple MCMC runs to remove “variance

due to sampling” more time.

Other parameter-fitting methods Stability-based method [Ben-Hur+ ’02, ‘03]

Run clustering multiple times on samples of data, for several values of “k”

For low k, clustering is stable; for high k, unstable Choose this transition point.

But Needs many runs of the clustering algorithm Arguments possible over definition of transition

Precision-Recall for CLASSIC

Cost surface (total cost)

Surface plot

Contour plot

With increasing k and l: Total cost decays very rapidly initially, but then starts increasing slowly

Cost surface (code cost only)

With increasing k and l: Code cost decays very rapidly

Surface plot Contour plot

Encoding Cost Function

Total encoding cost =log*(k) + log*(l) + (cluster number)

N.log(N) + M.log(M) + (row/col order)

Σ log(ai) + Σ log(bj) + (cluster sizes) ΣΣ log(aibj+1) + (block densities)

ΣΣ aibj . H(pi,j)

Code cost

Complexity of CA

O(E. (k2+l2)) ignoring the number of re-assign iterations, which is typically low.

Complexity of CA

Number of edges

Inter-group distances

Node Groups

Two groups are “close”

Merging them does not increase cost by much

distance(i,j) = relative increase in cost on merging i and j

Inter-group distances

Node Groups

Two groups are “close”

Merging them does not increase cost by much

distance(i,j) = relative increase in cost on merging i and j

Grp1 Grp2

4.55.1

Experiments

Author groups

Grp8Grp1

Inter-group distances can aid in visualization

Stonebraker, DeWitt, Carey

Collaborative filtering and recommendation systems Q: If someone likes a product X, will (s)he like

product Y? A: Check if others who liked X also liked Y.

Focus on distances between people, typically cosine similarity

and not on clustering

CA and bipartite cores: related but different

A 3x2 bipartite core

Hubs Authorities

Kumar et al. [1999] say that bipartite cores correspond to communities.

CA and bipartite cores: related but different

CA finds two communities there: one for hubs, and one for authorities.

We gracefully handle cases where a few links are missing.

CA considers connections between all sets of clusters, and not just two sets.

Not each node need belong to a non-trivial bipartite core.

CA is (informally) a generalization

Comparison with soft clustering Soft clustering each node belongs to each

cluster with some probability Hard clustering one cluster per node

Comparison with soft clustering1. Far more degrees of freedom

1. Parameter fitting is harder

2. Algorithms can be costlier

2. Hard clustering is better for exploratory data analysis

3. Some real-world problems require hard clustering e.g., fraud detection for accountants

Weights for code cost vs description cost Total = 1. (code cost) + 1. (description cost) Physical meaning: Total number of bits

Total = α. (code cost) + β. (description cost) Physical meaning: Number of encoding bits

under some prior

Re-assign: for each row x

Formula for re-assigns

Column groups

Choosing k and ll = 5

Split:1. Find the row group R with the maximum entropy per row

2. Choose the rows in R whose removal reduces the entropy per row in R

3. Send these rows to the new row group, and set k=k+1

Experiments

User groups

Epinions dataset

• 75,888 users

• 508,960 “dots”, one “dot” per “trust” relationship

k=19 groups foundSmall dense “core”

Comparison with previous methods Our threshold subsumes the homogeneous

model Proof We are more accurate than the Mean-Field

Assumption model.

Comparison with previous methods 10K Star Graph

Comparison with previous methods Oregon Graph

Accuracy of dynamical system 10K Star Graph

Accuracy of dynamical system Oregon Graph

Accuracy of dynamical system 10K Star Graph

Accuracy of dynamical system Oregon Graph

Relationship with full Markov Chain The full Markov Chain is of the form:

Prob(infection at time t) = Xt-1 + Yt-1 – Zt-1

Independence assumption leads to a point estimate for Zt-1 non-linear dynamical system.

Still non-linear, but now tractable

Non-linear component

Experiments: Information survival INTEL sensor map (54 nodes) MIT sensor map (40 nodes) and others…

Experiments: Information survival

INTEL sensor map

Survival threshold on INTEL

Experiments: Information survival

MIT sensor map

Survival threshold on MIT

Infinite Particle Systems

“Contact Process” ≈ SIS model Differences:

Infinite graphs only the questions asked are different

Very specific topologies lattices, trees Exact thresholds have not been found for these;

proving existence of thresholds is important Our results match those on the finite line

graph [Durrett+ ’88]

Intuition behind the largest eigenvalue Approximately size of the largest “blob” Consider the special case of a “caveman”

graph Largest eigenvalue = 4

Intuition behind the largest eigenvalue Approximately size of the largest “blob”

Largest eigenvalue = 4.016

Graph Patterns

Power Laws

Count vs Outdegree

Count vs Indegree

The “epinions” graph with 75,888 nodes and508,960 edges

Graph Patterns

Power Laws

Count vs Outdegree

Count vs Indegree

The “epinions” graph with 75,888 nodes and508,960 edges

Graph Patterns

Power Laws and deviations (DGX/Lognormals [Bi+ ’01])

Degree

Count vs Indegree

Graph Patterns

Power Lawsand deviations

Small-world “Community” effect …

Effective Diameter

Graph Generator Desiderata

Power Lawsand deviations

Small-world “Community” effect …

Most current graph generators fail to match some of these.

Other desiderata Few parameters Fast parameter-fitting Scalable graph

generation Simple extension to

undirected, bipartite and weighted graphs

The R-MAT generator

[SIAM DM’04]

Subdivide the adjacency matrix

and choose one quadrant with probability (a,b,c,d)

a (0.5)

d (0.25)

c (0.15)

b (0.1)

From To

Intuition: The “80-20 law”

The R-MAT generator

[SIAM DM’04]

Subdivide the adjacency matrix

and choose one quadrant with probability (a,b,c,d)

Recurse till we reach a 1*1 cell

where we place an edge and repeat for all edges.

The R-MAT generator

[SIAM DM’04]

Only 3 parameters a, b and c (d = 1-a-b-c).

We have a fast parameter fitting algorithm.

Experiments (Epinions directed graph)

Count vs Indegree Count vs Outdegree Hop-plot

Eigenvalue vs Rank “Network value” Count vs Stress

Effective Diameter

►R-MAT matches directed graphs

R-MAT communities and Cross-Associations R-MAT builds communities in graphs, and

Cross-Associations finds them. Relationship?

R-MAT builds a hierarchy of communities, while CA finds a flat set of communities

Linkage in the sizes of communities found by CA: When the R-MAT parameters are very skewed, the

community sizes for CA are skewed and vice versa

R-MAT and tree-based generators Recursive splitting in R-MAT ≈ following a

tree from root to leaf.

Relationship with other tree-based generators [Kleinberg ’01, Watts+ ’02]? The R-MAT tree has edges as leaves, the others

have nodes Tree-distance between nodes is used to connect

nodes in other generators, but what does tree-distance between edges mean?

Comparison with relational learningRelational Learning

(typical) Graph Mining

(typical)

1. Aims to find small structure/patterns at the local level

2. Labeled nodes and edges

3. Semantics of labels are important

4. Algorithms are typically costlier

1. Emphasis on global aspects of large graphs

2. Unlabeled graphs

3. More focused on topological structure and properties

4. Scalability is more important

===OTHER WORK===

OTHER WORK

Other Work

Time Series Prediction[CIKM 2002] We use the fractal dimension of the data This is related to chaos theory and Lyapunov exponents…

Other Work

Time Series Prediction[CIKM 2002]

Logistic Parabola

Other Work

Lorenz attractor

Other Work

Laser fluctuations

Other Work Adaptive histograms with error guarantees

[+ Ashraf Aboulnaga, Yufei Tao, Christos Faloutsos]

Salary

• Maintain count probabilities for buckets

• to give statistically correct query result-size estimation

• and query feedback

• + …

Insertions, deletions

Other Work

User-personalization Patent number 6,611,834 (IBM)

Relevance feedback in multimedia image search Filed for patent (IBM)

Building 3D models using robot camera and rangefinder data [ICML 2001]

===EXTRAS===

Conclusions Two paths in graph mining:

Specific applications: Viral Propagation Resilience testing, information

dissemination, rumor spreading Node Grouping automatically grouping nodes, AND

finding the correct number of groups

References:1. Fully automatic Cross-Associations,

by Chakrabarti, Papadimitriou, Modha and Faloutsos, in KDD 20042. AutoPart: Parameter-free graph partitioning and Outlier detection,

by Chakrabarti, in PKDD 20043. Epidemic spreading in real networks: An eigenvalue viewpoint,

by Wang, Chakrabarti, Wang and Faloutsos, in SRDS 2003

Conclusions Two paths in graph mining:

Specific applications General issues:

Graph Patterns Marks of “realism” in a graph Graph Generators R-MAT, a fast, scalable generator

matching many of the patterns

References:1. R-MAT: A recursive model for graph mining,

by Chakrabarti, Zhan and Faloutsos in SIAM Data Mining 2004.2. NetMine: New mining tools for large graphs,

by Chakrabarti, Zhan, Blandford, Faloutsos and Blelloch, in the SIAM 2004 Workshop on Link analysis, counter-terrorism and privacy

Other References

F4: Large Scale Automated Forecasting using Fractals,by D. Chakrabarti and C. Faloutsos, in CIKM 2002.

Using EM to Learn 3D Models of Indoor Environments with Mobile Robots,by Y. Liu, R. Emery, D. Chakrabarti, W. Burgard and S. Thrun, in ICML 2001

Graph Mining: Laws, Generators and Algorithms,by D. Chakrabarti and C. Faloutsos, under submission to ACM Computing Surveys

References --- graphs

1. R-MAT: A recursive model for graph mining, by D. Chakrabarti, Y. Zhan, C. Faloutsos in SIAM Data Mining 2004.

2. Epidemic spreading in real networks: An eigenvalue viewpoint, by Y. Wang, D. Chakrabarti, C. Wang and C. Faloutsos, in SRDS 2003

3. Fully automatic Cross-Associations, by D. Chakrabarti, S. Papadimitriou, D. Modha and C. Faloutsos, in KDD 2004

4. AutoPart: Parameter-free graph partitioning and Outlier detection, by D. Chakrabarti, in PKDD 2004

5. NetMine: New mining tools for large graphs, by D. Chakrabarti, Y. Zhan, D. Blandford, C. Faloutsos and G. Blelloch, in the SIAM 2004 Workshop on Link analysis, counter-terrorism and privacy

Roadmap

• Node grouping

General issues

4 Other Work

5 Conclusions

In-degree

Websites

Some personal webpage

Yahoo, Google and others

Clickstream +

Websites

Email-checking surfers

“All-night” surfers

Out-degree

Clickstream +

Websites

ClickstreamR-MAT

Graph Generation

Important for: Simulations of new algorithms Compression using a good graph generation

model Insight into the graph formation process

Our R-MAT (Recursive MATrix) generator can match many common graph patterns.

Recall the definition of eigenvalues

β/δ < τ = 1/ λ1,A

A X X= λA

λA = eigenvalue of A

λ1,A = largest eigenvalue

Tools for Large Graph Mining

Deepayan Chakrabarti Carnegie Mellon University

1 Tools for Large Graph Mining Thesis Committee: Christos Faloutsos Chris Olston Guy Blelloch ...

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