DS504/CS586: Big Data Analytics Graph Mining Prof. Yanhua Li

Transcription

1 Welcome to DS504/CS586: Big Data Analytics Graph Mining Prof. Yanhua Li Time: 6:00pm 8:50pm R Location: KH 116 Fall 2017

2 Reiews/Critiques I will choose one reiew to grade this week.

3 Graph Data: Social Networks Facebook social graph 4-degrees of separation [Backstrom-Boldi-Rosa-Ugander-Vigna, 2011] J. Leskoec, A. Rajaraman, J. Ullman: Mining of Massie Datasets, 3

4 Graph Data: Media Networks Connections between political blogs Polarization of the network [Adamic-Glance, 2005] J. Leskoec, A. Rajaraman, J. Ullman: Mining of Massie Datasets, 4

5 Graph Data: Information Nets Citation networks and Maps of science [Börner et al., 2012] J. Leskoec, A. Rajaraman, J. Ullman: Mining of Massie Datasets, 5

6 Graph Data: Communication Nets domain2 domain1 router domain3 Internet J. Leskoec, A. Rajaraman, J. Ullman: Mining of Massie Datasets, 6

7 Questions? Partial map of the Internet based on the January 15, 2005 data found on opte.org. (from

8 Graph Data: Topological Networks Seen Bridges of Königsberg [Euler, 1735] Return to the starting point by traeling each link of the graph once and only once. J. Leskoec, A. Rajaraman, J. Ullman: Mining of Massie Datasets, 8

9 Graph representation of networks Friendship Resistance Co-authorship Following One-way road Wireless channel Undirected links Directed links + Friend & foe Trust & distrust Multi-relational links Hyperlinks Repulsion & cohesion Signed links

10 Mining in Big Graphs Network Statistic Analysis (this lecture) Network Size Degree distribution. Node Ranking (Next lecture) Identifying most influential nodes Viral Marketing, resource allocation

11 Graph Data: Social Networks Facebook social graph 4-degrees of separation [Backstrom-Boldi-Rosa-Ugander-Vigna, 2011] J. Leskoec, A. Rajaraman, J. Ullman: Mining of Massie Datasets, 11

12 Sampling graphs Random sampling (uniform & independent) crawling } ertex sampling } BFS sampling } edge sampling } random walk sampling 12 12

13 Random Walks on Graphs Random walk sampling Random Walk Routing Molecule in liquid Influence diffusion

14 Undirected Graphs Undirected!!

15 Random Walk! # A = # # # " Adjacency matrix $ & & & & %! # D = # # # " Transition Probability Matrix 1 if i is not equal to j P ij = k i 0 if i=j E : number of links Stationary Distribution $ & & & & % Symmetric 1 " $ P = A D 1 = $ $ $ # Undirected 0 1/ 3 1/ 3 1/ 3 1/ 2 0 1/ 2 0 1/ 3 1/ 3 0 1/ 3 1/ 2 0 1/ 2 0 % ' ' ' ' & π i = d i 2 E

16 Metropolis-Hastings Random Walk! # A = # # # " Adjacency matrix $ & & & & %! # D = # # # " Transition Probability Matrix 1 min(1, k υ ) if w neighbor of υ P MH k υ,w = υ k w 1 MH P υ,y if w=υ y υ E : number of links Stationary Distribution p = u 1 V $ & & & & % Symmetric 1 " $ P = A D 1 = $ $ $ # Undirected 0 1/ 3 1/ 3 1/ 3 1/ 3 1/ 3 1/ 3 0 1/ 3 1/ 3 0 1/ 3 1/ 3 0 1/ 3 1/ 3 % ' ' ' ' &

17 Walking in Facebook: A Case Study of Unbiased Sampling of OSNs Minas Gjoka, Maciej Kurant, Carter Butts, Athina Markopoulou UC Irine, EPFL Minas Gjoka, UC Irine Walking in Facebook 17

18 Outline Motiation and Problem Statement Sampling Methodology Data Analysis Conclusion Minas Gjoka, UC Irine Walking in Facebook 18

19 Online Social Networks (OSNs) A network of declared friendships between users Allows users to maintain relationships G F E H C Many popular OSNs with different focus Facebook, LinkedIn, Flickr, D A Social Graph B Minas Gjoka, UC Irine Walking in Facebook 19

20 Why Sample OSNs? Representatie samples desirable study properties test algorithms Obtaining complete dataset difficult companies usually unwilling to share data tremendous oerhead to measure all (~100TB for Facebook) Minas Gjoka, UC Irine Walking in Facebook 20

21 Problem statement Obtain a representatie sample of users in a gien OSN by exploration of the social graph. in this work we sample Facebook (FB) explore graph using arious crawling techniques Minas Gjoka, UC Irine Walking in Facebook 21

22 Related Work Graph traersal (BFS) A. Misloe et al, IMC 2007 Y. Ahn et al, WWW 2007 C. Wilson, Eurosys 2009 Random walks (MHRW, RDS) M. Henzinger et al, WWW 2000 D. Stutbach et al, IMC 2006 A. Rasti et al, Mini Infocom 2009 Minas Gjoka, UC Irine Walking in Facebook 22

23 Outline Motiation and Problem Statement Sampling Methodology crawling methods data collection conergence ealuation method comparisons Data Analysis Conclusion Minas Gjoka, UC Irine Walking in Facebook 23

24 (1) Breadth-First-Search (BFS) Starting from a seed, explores all neighbor nodes. Process continues iteratiely without replacement. G F E H C BFS leads to bias towards high degree nodes Lee et al, Statistical properties of Sampled Networks, Phys Reiew E, 2006 D A B Unexplored Early measurement studies of OSNs use BFS as primary sampling technique i.e [Misloe et al], [Ahn et al], [Wilson et al.] Explored Visited Minas Gjoka, UC Irine Walking in Facebook 24

25 (2) Random Walk (RW) Explores graph one node at a time with replacement RW u, w P Degree of node υ In the stationary distribution Number of edges = 1 k u k p = u u 2 E H G D 1/3 F E C 1/3 A 1/3 Next candidate Current node B Minas Gjoka, UC Irine Walking in Facebook 25

26 (3) Re-Weighted Random Walk (RWRW) Hansen-Hurwitz estimator Corrects for degree bias at the end of collection Without re-weighting, the probability distribution for node property A is: pa ( ) i å å 1 A uîa = i = i 1 V uîv Re-Weighted probability distribution : Subset of sampled nodes with alue i All sampled nodes Minas Gjoka, UC Irine pa ( ) i uîa = å å Î u V i 1/ 1/ k k u u Degree of node u Walking in Facebook 26

27 (4) Metropolis-Hastings Random Walk (MHRW) Explore graph one node at a time with replacement H G C F E ì 1 k min(1, u ) if w neighbor of u MH ïk k P u w u, w = í ï MH 1 - å Pu, y if w= u ïî y¹ u In the stationary distribution Minas Gjoka, UC Irine p = u 1 V MH P AA D 1/3 1/5 A 2/15 1/3 Walking in Facebook 27 B Next candidate Current node = 1 - ( + + ) = MH 1 3 P AC = =

28 Uniform userid Sampling (UNI) As a basis for comparison, we collect a uniform sample of Facebook userids (UNI) rejection sampling on the 32-bit userid space UNI not a general solution for sampling OSNs userid space must not be sparse names instead of numbers Minas Gjoka, UC Irine Walking in Facebook 28

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 aailable at: Minas Gjoka, UC Irine Walking in Facebook 29

30 Data Collection Basic Node Information What information do we collect for each sampled node u? UserID Name Networks Priacy settings Friend List UserID Name Networks Priacy Settings u UserID Name Networks Priacy settings 1111 UserID Name Networks Priacy settings Regional School/Workplace Send Message View Friends Profile Photo Add as Friend Minas Gjoka, UC Irine Walking in Facebook 30

31 Detecting Conergence Number of samples (iterations) to loose dependence from starting points? Minas Gjoka, UC Irine Walking in Facebook 31

32 Online Conergence Diagnostics Geweke Detects conergence for a single walk. Let X be a sequence of samples for metric of interest. Xa Xb EX ( a) - EX ( b) z = Var( X )-Var( X ) a b J. Geweke, Ealuating the accuracy of sampling based approaches to calculate posterior moments in Bayesian Statistics 4, 1992 Minas Gjoka, UC Irine Walking in Facebook 32

33 Online Conergence Diagnostics Gelman-Rubin Detects conergence for m>1 walks Walk 1 Between walks ariance Walk 2 Walk 3 R æn- 1 m+ 1 B ö = ç + è n mn W ø Within walks ariance A. Gelman, D. Rubin, Inference from iteratie simulation using multiple sequences in Statistical Science Volume 7, 1992 Minas Gjoka, UC Irine Walking in Facebook 33

34 When do we reach equilibrium? Node Degree Burn-in determined to be 3K Minas Gjoka, UC Irine Walking in Facebook 34

35 Methods Comparison Node Degree Poor performance for BFS, RW 28 crawls MHRW, RWRW produce good estimates per chain oerall Minas Gjoka, UC Irine Walking in Facebook 35

36 Sampling Bias BFS Low degree nodes underrepresented by two orders of magnitude BFS is biased Minas Gjoka, UC Irine Walking in Facebook 36

37 Sampling Bias MHRW, RW, RWRW Degree distribution identical to UNI (MHRW,RWRW) RW as biased as BFS but with smaller ariance in each walk Minas Gjoka, UC Irine Walking in Facebook 37

38 Practical Recommendations for Sampling Methods Use MHRW or RWRW. Do not use BFS, RW. Use formal conergence diagnostics assess conergence online use multiple parallel walks MHRW s RWRW RWRW slightly better performance MHRW proides a ready-to-use sample Minas Gjoka, UC Irine Walking in Facebook 38

39 Outline Motiation and Problem Statement Sampling Methodology Data Analysis Conclusion Minas Gjoka, UC Irine Walking in Facebook 39

40 FB Social Graph Degree Distribution, Power Law Distribution f(k)=b*k -a a 1 =1.32 a 2 =3.38 Degree distribution not a power law Minas Gjoka, UC Irine Walking in Facebook 40

41 Any Comments & Critiques?

42 Next Class: Graph Mining (II) Do assigned readings before class Submit reiews/critiques Attend in-class discussions 42