Graph Mining: patterns and tools for static and time-evolving graphs

1
Graph Mining patterns and tools for static and
time-evolving graphs

• Christos Faloutsos
• CMU

2
Outline

• Problem definition / Motivation
• Static dynamic laws generators
• Tools CenterPiece graphs Tensors
• Other projects (Virus propagation, e-bay fraud
  detection)
• Conclusions

3
Motivation

• Data mining find patterns (rules, outliers)
• Problem1 How do real graphs look like?
• Problem2 How do they evolve?
• Problem3 How to generate realistic graphs
• TOOLS
• Problem4 Who is the master-mind?
• Problem5 Track communities over time

4
Problem1 Joint work with

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

5
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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Graphs why should we care?

• IR bi-partite graphs (doc-terms)
• web hyper-text graph
• ... and more

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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
• ....

8
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

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Solution1

• Power law in the degree distribution SIGCOMM99

internet domains
att.com
ibm.com
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Solution1 Eigen Exponent E
Eigenvalue
Exponent slope
E -0.48
May 2001
Rank of decreasing eigenvalue

• A2 power law in the eigenvalues of the adjacency
  matrix

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But

• How about graphs from other domains?

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The Peer-to-Peer Topology
Jovanovic

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

15
More power laws

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

log(count)
Ullman
log(citations)
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Swedish sex-web
Nodes people (Females Males) Links sexual
relationships
Albert Laszlo Barabasi http//www.nd.edu/networks
/ Publication20Categories/ 0420Talks/2005-norway
-3hours.ppt
4781 Swedes 18-74 59 response rate.
Liljeros et al. Nature 2001
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More power laws

• web hit counts w/ A. Montgomery

Web Site Traffic
log(count)
Zipf
ebay
log(in-degree)
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epinions.com

• who-trusts-whom Richardson Domingos, KDD 2001

count
trusts-2000-people user
(out) degree
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Outline

• Problem definition / Motivation
• Static dynamic laws generators
• Tools CenterPiece graphs Tensors
• Other projects (Virus propagation, e-bay fraud
  detection)
• Conclusions

20
Motivation

• Data mining find patterns (rules, outliers)
• Problem1 How do real graphs look like?
• Problem2 How do they evolve?
• Problem3 How to generate realistic graphs
• TOOLS
• Problem4 Who is the master-mind?
• Problem5 Track communities over time

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Problem2 Time evolution

• with Jure Leskovec (CMU/MLD)
• and Jon Kleinberg (Cornell sabb. _at_ CMU)

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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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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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Diameter ArXiv citation graph
diameter

• Citations among physics papers
• 1992 2003
• One graph per year

time years
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Diameter Autonomous Systems
diameter

• Graph of Internet
• One graph per day
• 1997 2000

number of nodes
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Diameter Affiliation Network
diameter

• Graph of collaborations in physics authors
  linked to papers
• 10 years of data

time years
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Diameter Patents
diameter

• Patent citation network
• 25 years of data

time years
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Temporal Evolution of the Graphs

• N(t) nodes at time t
• E(t) edges at time t
• Suppose that
• N(t1) 2 N(t)
• Q what is your guess for
• E(t1) ? 2 E(t)

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Temporal Evolution of the Graphs

• N(t) nodes at time t
• E(t) edges at time t
• Suppose that
• N(t1) 2 N(t)
• Q what is your guess for
• E(t1) ? 2 E(t)
• A over-doubled!
• But obeying the Densification Power Law

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Densification Physics Citations

• Citations among physics papers
• 2003
• 29,555 papers, 352,807 citations

E(t)
??
N(t)
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Densification Physics Citations

• Citations among physics papers
• 2003
• 29,555 papers, 352,807 citations

E(t)
1.69
N(t)
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Densification Physics Citations

• Citations among physics papers
• 2003
• 29,555 papers, 352,807 citations

E(t)
1.69
1 tree
N(t)
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Densification Physics Citations

• Citations among physics papers
• 2003
• 29,555 papers, 352,807 citations

E(t)
1.69
clique 2
N(t)
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Densification Patent Citations

• Citations among patents granted
• 1999
• 2.9 million nodes
• 16.5 million edges
• Each year is a datapoint

E(t)
1.66
N(t)
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Densification Autonomous Systems

• Graph of Internet
• 2000
• 6,000 nodes
• 26,000 edges
• One graph per day

E(t)
1.18
N(t)
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Densification Affiliation Network

• Authors linked to their publications
• 2002
• 60,000 nodes
• 20,000 authors
• 38,000 papers
• 133,000 edges

E(t)
1.15
N(t)
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Outline

• Problem definition / Motivation
• Static dynamic laws generators
• Tools CenterPiece graphs Tensors
• Other projects (Virus propagation, e-bay fraud
  detection)
• Conclusions

38
Motivation

• Data mining find patterns (rules, outliers)
• Problem1 How do real graphs look like?
• Problem2 How do they evolve?
• Problem3 How to generate realistic graphs
• TOOLS
• Problem4 Who is the master-mind?
• Problem5 Track communities over time

39
Problem3 Generation

• Given a growing graph with count of nodes N1, N2,
  
• Generate a realistic sequence of graphs that will
  obey all the patterns

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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 Distribution
• Power Law eigenvalue and eigenvector
  distribution
• Small Diameter
• Dynamic Patterns
• Growth Power Law
• Shrinking/Stabilizing Diameters

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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

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Recursive Graph Generation

• There are many obvious (but wrong) ways
• Does not obey Densification Power Law
• Has increasing diameter
• Kronecker Product is exactly what we need

Recursive expansion
Initial graph
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Kronecker Product a Graph
Intermediate stage
Adjacency matrix
Adjacency matrix
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Kronecker Product a Graph

• Continuing multiplying with G1 we obtain G4 and
  so on

G4 adjacency matrix
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Kronecker Graphs Formally

• We create the self-similar graphs recursively
• Start with a initiator graph G1 on N1 nodes and
  E1 edges
• The recursion will then product larger graphs G2,
  G3, Gk on N1k nodes
• Since we want to obey Densification Power Law
  graph Gk has to have E1k edges

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Kronecker Product Definition

• The Kronecker product of matrices A and B is
  given by
• We define a Kronecker product of two graphs as a
  Kronecker product of their adjacency matrices

N x M
K x L
NK x ML
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Kronecker Graphs

• We propose a growing sequence of graphs by
  iterating the Kronecker product
• Each Kronecker multiplication exponentially
  increases the size of the graph

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Kronecker Graphs Intuition

• Intuition
• Recursive growth of graph communities
• Nodes get expanded to micro communities
• Nodes in sub-community link among themselves and
  to nodes from different communities

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Properties

• We can PROVE that
• Degree distribution is multinomial power law
• Diameter constant
• Eigenvalue distribution multinomial
• First eigenvector multinomial
• See Leskovec, PKDD05 for proofs

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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 Patterns
• Growth Power Law
• Shrinking/Stabilizing Diameters
• First and only generator for which we can prove
  all these properties

?
?
?
?
?
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Stochastic Kronecker Graphs
skip

• Create N1?N1 probability matrix P1
• Compute the kth Kronecker power Pk
• For each entry puv of Pk include an edge (u,v)
  with probability puv

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
Kronecker multiplication
0.4 0.2
0.1 0.3
Instance Matrix G2
P1
flip biased coins
Pk
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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

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Arxiv Degree Distribution
Real graph
Deterministic Kronecker
Stochastic Kronecker
count
degree
degree
degree
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Arxiv Scree Plot
Real graph
Deterministic Kronecker
Stochastic Kronecker
Eigenvalue
Rank
Rank
Rank
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Arxiv Densification
Real graph
Deterministic Kronecker
Stochastic Kronecker
Edges
Nodes(t)
Nodes(t)
Nodes(t)
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Arxiv Effective Diameter
Real graph
Deterministic Kronecker
Stochastic Kronecker
Diameter
Nodes(t)
Nodes(t)
Nodes(t)
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Arxiv citation network
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U.S. Patent citations
Static patterns
Temporal patterns
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Autonomous Systems
Static patterns
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(Q how to fit the parms?)

• A
• Stochastic version of Kronecker graphs
• Max likelihood
• Metropolis sampling
• Leskovec, 07, under review

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Experiments on real AS graph
Degree distribution
Hop plot
Network value
Adjacency matrix eigen values
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Conclusios

• Kronecker graphs have
• All the static properties
• Heavy tailed degree distributions
• Small diameter
• Multinomial eigenvalues and eigenvectors
• All the temporal properties
• Densification Power Law
• Shrinking/Stabilizing Diameters
• We can formally prove these results

?
?
?
?
?
63
Outline

• Problem definition / Motivation
• Static dynamic laws generators
• Tools CenterPiece graphs Tensors
• Other projects (Virus propagation, e-bay fraud
  detection)
• Conclusions

64
Motivation

• Data mining find patterns (rules, outliers)
• Problem1 How do real graphs look like?
• Problem2 How do they evolve?
• Problem3 How to generate realistic graphs
• TOOLS
• Problem4 Who is the master-mind?
• Problem5 Track communities over time

65
Problem4 MasterMind CePS

• w/ Hanghang Tong, KDD 2006
• htong ltatgt cs.cmu.edu

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Center-Piece Subgraph(Ceps)

• Given Q query nodes
• Find Center-piece ( )
• App.
• Social Networks
• Law Inforcement,
• Idea
• Proximity -gt random walk with restarts

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Case Study AND query
R
.
Agrawal
Jiawei Han
V
.
Vapnik
M
.
Jordan
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Case Study AND query
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(No Transcript)
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Conclusions

• Q1How to measure the importance?
• A1 RWRK_SoftAnd
• Q2 How to find connection subgraph?
• A2Extract Alg.
• Q3How to do it efficiently?
• A3Graph Partition (Fast CePS)
• 90 quality
• 61 speedup 150x speedup (ICDM06, b.p. award)

71
Outline

• Problem definition / Motivation
• Static dynamic laws generators
• Tools CenterPiece graphs Tensors
• Other projects (Virus propagation, e-bay fraud
  detection)
• Conclusions

72
Motivation

• Data mining find patterns (rules, outliers)
• Problem1 How do real graphs look like?
• Problem2 How do they evolve?
• Problem3 How to generate realistic graphs
• TOOLS
• Problem4 Who is the master-mind?
• Problem5 Track communities over time

73
Tensors for time evolving graphs

• Jimeng Sun KDD06
• , SMD07
• CF, Kolda, Sun, SDM07 tutorial

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Social network analysis

• Static find community structures

1990
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Social network analysis

• Static find community structures
• Dynamic monitor community structure evolution
  spot abnormal individuals abnormal time-stamps

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Application 1 Multiway latent semantic indexing
(LSI)
Philip Yu
2004
Michael Stonebreaker
Uauthors
1990
authors
Ukeyword
keyword
Pattern
Query

• Projection matrices specify the clusters
• Core tensors give cluster activation level

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Bibliographic data (DBLP)

• Papers from VLDB and KDD conferences
• Construct 2nd order tensors with yearly windows
  with ltauthor, keywordsgt
• Each tensor 4584?3741
• 11 timestamps (years)

78
Multiway LSI
Authors Keywords Year
michael carey, michael stonebreaker, h. jagadish, hector garcia-molina queri,parallel,optimization,concurr, objectorient 1995
surajit chaudhuri,mitch cherniack,michael stonebreaker,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
DB
DM

• Two groups are correctly identified Databases
  and Data mining
• People and concepts are drifting over time

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Application 2Network Anomaly Detection

• Anomaly detection
• Reconstruction error driven
• Multiple resolution
• Data
• TCP flows collected at CMU backbone
• Raw data 500GB with compression
• Construct 3rd order tensors with hourly windows
  with ltsource, destination, port gt
• 1200 timestamps (hours)

80
with

• Hui Zhang
• Yinglian Xie
• (Vyas Sekar)

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Network anomaly detection
scanners
error

• Identify when and where anomalies occurred.
• Prominent difference between normal and abnormal
  ones is mainly due to unusual scanning activity
  (confirmed by the campus admin).

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Conclusions

• Tensor-based methods (WTA/DTA/STA)
• spot patterns and anomalies on time evolving
  graphs, and
• on streams (monitoring)

83
Outline

• Problem definition / Motivation
• Static dynamic laws generators
• Tools CenterPiece graphs Tensors
• Other projects (Virus propagation, e-bay fraud
  detection)
• Conclusions

84
Virus propagation

• How do viruses/rumors propagate?
• Will a flu-like virus linger, or will it become
  extinct soon?

85
The model SIS

• Flu like Susceptible-Infected-Susceptible
• Virus strength s b/d

Healthy
N2
N
N1
Infected
N3
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Epidemic threshold t

• of a graph the value of t, such that
• if strength s b / d lt t
• an epidemic can not happen
• Thus,
• given a graph
• compute its epidemic threshold

87
Epidemic threshold t

• What should t depend on?
• avg. degree? and/or highest degree?
• and/or variance of degree?
• and/or third moment of degree?
• and/or diameter?

88
Epidemic threshold

• Theorem We have no epidemic, if

ß/d ltt 1/ ?1,A
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Epidemic threshold

• Theorem We have no epidemic, if

epidemic threshold
recovery prob.
ß/d ltt 1/ ?1,A
largest eigenvalue of adj. matrix A
attack prob.
Proof Wang03
90
Experiments (Oregon)
b/d gt t (above threshold)
b/d t (at the threshold)
b/d lt t (below threshold)
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Outline

• Problem definition / Motivation
• Static dynamic laws generators
• Tools CenterPiece graphs Tensors
• Other projects (Virus propagation, e-bay fraud
  detection)
• Conclusions

92
E-bay Fraud detection
w/ Polo Chau Shashank Pandit, CMU
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E-bay Fraud detection - NetProbe
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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

95
Philosophical observation

• Graph mining brings together
• ML/AI / IR
• Stat, Num. analysis,
• DB (Gb/Tb),
• Systems (Networks),
• sociology,

96
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

97
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.

98
References

• Jimeng Sun, Dacheng Tao, Christos Faloutsos
  Beyond Streams and Graphs Dynamic Tensor
  Analysis, KDD 2006, Philadelphia, PA
• Jimeng Sun, Yinglian Xie, Hui Zhang, Christos
  Faloutsos. Less is More Compact Matrix
  Decomposition for Large Sparse Graphs, SDM,
  Minneapolis, Minnesota, Apr 2007. pdf

99
THANK YOU!

• Contact info WeH 7107
• christos, htong, jimeng, jure ltatgt cs.cmu.edu
• www. cs.cmu.edu /christos
• (w/ papers, datasets, code, etc)