[PPT] - DS504/CS586: Big Data Analytics Graph Mining Prof. Yanhua Li Time: PowerPoint Presentation

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DS504/CS586: Big Data Analytics Graph Mining

Prof. Yanhua Li

Welcome to

Time: 6:00pm –8:50pm R Location: AK 233 Spring 2018

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Urban Sensing & Data Acquisition

Participatory Sensing, Crowd Sensing, Mobile Sensing Traffic Road Networks POIs Air Quality Human mobility Meteorolo gy Social Media Energy

Urban Data Management

Spatio-temporal index, streaming, trajectory, and graph data management,...

Urban Data Analytics

Data Mining, Machine Learning, Visualization

Service Providing

Improve urban planning, Ease Traffic Congestion, Save Energy, Reduce Air Pollution, ...

Urban Computing: concepts, methodologies, and applications. Zheng, Y., et al. ACM transactions on Intelligent Systems and Technology.

Acquisition Cleaning Management

Big Graph Data Mining Big Data Clustering Recommender systems Applications

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Graph Data: Social Networks

J. Leskovec, A. Rajaraman, J. Ullman:

Mining of Massive Datasets, http:// www.mmds.org 3

Facebook social graph

4-degrees of separation [Backstrom-Boldi-Rosa-Ugander-Vigna, 2011]

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Graph Data: Media Networks

J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org

4

Connections between political blogs

Polarization of the network [Adamic-Glance, 2005]

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Graph Data: Information Nets

J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org

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Citation networks and Maps of science

[Börner et al., 2012]

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domain2 domain1 domain3 router

Internet

Graph Data: Communication Nets

J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org

6

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Questions?

Partial map of the Internet based on the January 15, 2005 data found on

pte.org. (from http://atheistuniverse.net/group/internet)

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Graph Data: Topological Networks

J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org

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Seven Bridges of Königsberg

[Euler, 1735]

Return to the starting point by traveling each link of the graph once and only once.

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Graph representation of networks

+

+ +

Trust & distrust Repulsion & cohesion Friend & foe Following One-way road Resistance Wireless channel Friendship Co-authorship Undirected links Directed links Signed links Multi-relational links Hyperlinks … …

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Mining in Big Graphs

v Network Statistic Analysis (this lecture)

§ Network Size § Degree distribution.

v Node Ranking (Next lecture)

§ Identifying most influential nodes § Viral Marketing, resource allocation

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Graph Data: Social Networks

J. Leskovec, A. Rajaraman, J. Ullman:

Mining of Massive Datasets, http:// www.mmds.org 11

Facebook social graph

4-degrees of separation [Backstrom-Boldi-Rosa-Ugander-Vigna, 2011]

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12

Sampling graphs

Random sampling (uniform & independent)

crawling

} vertex sampling } BFS sampling

12

} random walk sampling } edge sampling

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Random Walks on Graphs

Random walk sampling Random Walk Routing Influence diffusion Molecule in liquid

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Undirected Graphs

1 2 6 4 5 3

Undirected !!

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Random Walk

v Adjacency matrix v Transition Probability Matrix v |E|: number of links v Stationary Distribution

1 4 3 2

D = 3 2 3 2 ! " # # # # $ % & & & &

Undirected

A = 1 1 1 1 1 1 1 1 1 1 ! " # # # # $ % & & & &

Symmetric

P = A•D−1 = 1/ 3 1/ 3 1/ 3 1/ 2 1/ 2 1/ 3 1/ 3 1/ 3 1/ 2 1/ 2 " # $ $ $ $ % & ' ' ' '

πi = di 2 E

P

ij =

1 ki if i is not equal to j 0 if i=j ⎧ ⎨ ⎪ ⎩ ⎪

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Metropolis-Hastings Random Walk

v Adjacency matrix v Transition Probability Matrix v |E|: number of links v Stationary Distribution

1 4 3 2

D = 3 2 3 2 ! " # # # # $ % & & & &

Undirected

A = 1 1 1 1 1 1 1 1 1 1 ! " # # # # $ % & & & &

Symmetric

P = A•D−1 = 1/ 3 1/ 3 1/ 3 1/ 3 1/ 3 1/ 3 1/ 3 1/ 3 1/ 3 1/ 3 1/ 3 1/ 3 " # $ $ $ $ % & ' ' ' '

P

υ,w MH =

1 kυ min(1, kυ kw ) if w neighbor of υ 1− P

υ,y MH if w=υ y≠υ

∑

⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪

1 V

υ

π =

1 4 3 2

Undirected

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Minas Gjoka, UC Irvine Walking in Facebook 17

Walking in Facebook: A Case Study of Unbiased Sampling of OSNs

Minas Gjoka, Maciej Kurant ‡, Carter Butts, Athina Markopoulou UC Irvine, EPFL ‡

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Minas Gjoka, UC Irvine Walking in Facebook 18

Outline

v Motivation and Problem Statement v Sampling Methodology v Data Analysis v Conclusion

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Minas Gjoka, UC Irvine Walking in Facebook 19

Online Social Networks (OSNs)

v A network of declared friendships

between users

v Allows users to maintain relationships v Many popular OSNs with different focus

§ Facebook, LinkedIn, Flickr, …

C A E G F B D H

Social Graph

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Minas Gjoka, UC Irvine Walking in Facebook 20

Why Sample OSNs?

v Representative samples desirable

§ study properties § test algorithms § We use node distribution in this study

v Obtaining complete dataset difficult

§ companies usually unwilling to share data § tremendous overhead to measure all (~100TB for Facebook)

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Minas Gjoka, UC Irvine Walking in Facebook 21

Problem statement

v Obtain a representative sample of

users in a given OSN by exploration

f the social graph.

§ in this work we sample Facebook (FB) § explore graph using various crawling techniques

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Minas Gjoka, UC Irvine Walking in Facebook 22

Related Work

v Graph traversal (BFS)

§ A. Mislove et al, IMC 2007 § Y. Ahn et al, WWW 2007 § C. Wilson, Eurosys 2009

v Random walks (MHRW, RDS)

§ M. Henzinger et al, WWW 2000 § D. Stutbach et al, IMC 2006 § A. Rasti et al, Mini Infocom 2009

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Minas Gjoka, UC Irvine Walking in Facebook 23

Outline

v Motivation and Problem Statement v Sampling Methodology

§ crawling methods § data collection § convergence evaluation § method comparisons

v Data Analysis v Conclusion

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Walking in Facebook 24

(1) Breadth-First-Search (BFS)

C A E G F B D H

Unexplored Explored Visited

v Starting from a seed, explores all

neighbor nodes. Process continues iteratively without replacement.

v BFS leads to bias towards high

degree nodes

Lee et al, “Statistical properties of Sampled Networks”, Phys Review E, 2006

v Early measurement studies of

OSNs use BFS as primary sampling technique

i.e [Mislove et al], [Ahn et al], [Wilson et al.]

1 | | ( ) 1 | |

i

u A i i u V

A p A V

∈ ∈

= =

∑ ∑

Subset of sampled nodes with value i All sampled nodes

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Minas Gjoka, UC Irvine Walking in Facebook 25

(2) Random Walk (RW)

C A E G F B D H

1/3 1 / 3 1/3

Next candidate Current node

Explores graph one node at

a time with replacement

In the stationary distribution

,

1

RW w

P k

υ υ

=

2 k E

υ υ

π = ⋅

Degree of node υ Number of edges

1 | | ( ) 1 | |

i

u A i i u V

A p A V

∈ ∈

= =

∑ ∑

Subset of sampled nodes with value i All sampled nodes

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Minas Gjoka, UC Irvine Walking in Facebook 26

(3) Re-Weighted Random Walk (RWRW)

Hansen-Hurwitz estimator

v Corrects for degree bias at the end of collection v Without re-weighting, the probability distribution for

node property A is:

v Re-Weighted probability distribution :

1/ ( ) 1/

i

u A u i u V u

k p A k

∈ ∈

= ∑

∑

1 | | ( ) 1 | |

i

u A i i u V

A p A V

∈ ∈

= =

∑ ∑

Subset of sampled nodes with value i All sampled nodes Degree of node u

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Walking in Facebook 27

(4) Metropolis-Hastings Random Walk (MHRW)

v Explore graph one node at

a time with replacement

v In the stationary distribution

, ,

1 min(1, ) if neighbor of 1 if =

MH w w MH y y

k w k k P P w

υ υ υ υ υ

υ υ

≠

⎧ ⎪ ⎪ = ⎨ − ⎪ ⎪ ⎩ ∑

C A E G F B D H

1 / 3 1/5 1/3

Next candidate Current node

2/15

1 V

υ

π =

5 1 5 3 3 1 = ⋅ =

MH AC

P 15 2 ) 5 1 3 1 3 1 ( 1 = + + − =

MH AA

P

1 | | ( ) 1 | |

i

u A i i u V

A p A V

∈ ∈

= =

∑ ∑

Subset of sampled nodes with value i All sampled nodes

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Minas Gjoka, UC Irvine Walking in Facebook 28

Uniform userID Sampling (UNI)

v As a basis for comparison, we collect

a uniform sample of Facebook userIDs (UNI)

§ rejection sampling on the 32-bit userID space

v UNI not a general solution for

sampling OSNs

§ userID space must not be sparse § names instead of numbers

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Minas Gjoka, UC Irvine Walking in Facebook 29

Summary of Datasets

Sampling method MHRW RW BFS UNI #Valid Users 28x81K 28x81K 28x81K 984K # Unique Users 957K 2.19M 2.20M 984K

Egonets for a subsample of MHRW
local properties of nodes
Datasets available at:

http://odysseas.calit2.uci.edu/research/osn.html

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Minas Gjoka, UC Irvine Walking in Facebook 30

Data Collection

Basic Node Information

v What information do we collect for each sampled

node u?

UserID Name Networks Privacy settings

Friend List

UserID Name Networks Privacy Settings UserID Name Networks Privacy settings

u

1 1 1 1

Profile Photo Add as Friend Regional School/Workplace

UserID Name Networks Privacy settings

View Friends Send Message

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Minas Gjoka, UC Irvine Walking in Facebook 31

Detecting Convergence

Number of samples (iterations) to loose

dependence from starting points?

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Minas Gjoka, UC Irvine Walking in Facebook 32

Online Convergence Diagnostics

Geweke

v Detects convergence for a single walk. Let X be a

sequence of samples for metric of interest.

J. Geweke, “Evaluating the accuracy of sampling based approaches to calculate

posterior moments“ in Bayesian Statistics 4, 1992

Xa Xb

z = E(X a)− E(X b) Var(X a)−Var(X b) ∈ [−1,1] a =10%,b = 50%

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Minas Gjoka, UC Irvine Walking in Facebook 33

Online Convergence Diagnostics

Gelman-Rubin

v Detects convergence for M>1 walks

Convergence is declared when R<1.02

A. Gelman, D. Rubin, “Inference from iterative simulation using multiple sequences“ in

Statistical Science Volume 7, 1992 Walk 1 Walk 2 Walk 3

R = N −1 N + M +1 MN B W

Between walks variance Within walks variance

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Minas Gjoka, UC Irvine Walking in Facebook 34

When do we reach equilibrium?

Burn-in determined to be 3K

Node Degree

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Minas Gjoka, UC Irvine Walking in Facebook 35

Methods Comparison

Node Degree

v Poor performance

for BFS, RW

v MHRW, RWRW

produce good estimates

§ per chain § overall

28 crawls

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Minas Gjoka, UC Irvine Walking in Facebook 36

Sampling Bias

BFS

v Low degree nodes

under- represented by two orders of magnitude

v BFS is biased

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Minas Gjoka, UC Irvine Walking in Facebook 37

Sampling Bias

MHRW, RW, RWRW

v Degree distribution identical to UNI (MHRW,RWRW) v RW as biased as BFS but with smaller variance in each walk

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Minas Gjoka, UC Irvine Walking in Facebook 38

Practical Recommendations for Sampling Methods

v Use MHRW or RWRW. Do not use BFS, RW. v Use formal convergence diagnostics

§ assess convergence online § use multiple parallel walks

v MHRW vs RWRW

§ RWRW slightly better performance § MHRW provides a “ready-to-use” sample

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Minas Gjoka, UC Irvine Walking in Facebook 39

FB Social Graph

Degree Distribution, Power Law Distribution f(k)=b*k-a

v Degree distribution not a power law

a

2=3.38

a

1=1.32

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Minas Gjoka, UC Irvine Walking in Facebook 40

Outline

v Motivation and Problem Statement v Sampling Methodology v Data Analysis v Conclusion

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Any Comments & Critiques?

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Minas Gjoka, UC Irvine Walking in Facebook 42

Next Week

v 5 team presentations v 30 minutes each team including the

presentation and Q&A

v We will have snacks and soft drinks

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Minas Gjoka, UC Irvine Walking in Facebook 43

Project 2 starts next week

v KDD cup 2018 v Real data, worldwide competition v Will be announced March 1st, 2018 v Define your own project or KDD cup v http://www.kdd.org/News/view/kdd-

cup-2018-call-for-proposals

v http://www.kdd.org/kdd-cup v http://www.kdd.org/kdd2017/News/