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: AK 233 Spring 2018

2 Service Providing Improve urban planning, Ease Traffic Congestion, Save Energy, Reduce Air Pollution,... Urban Data Analytics Data Mining, Machine Learning, Visualization Recommender systems Big Graph Data Mining Applications Big Data Clustering Urban Data Management Spatio-temporal index, streaming, trajectory, and graph data management,... Management Human mobility Traffic Air Quality Meteorolo gy Social Media Energy Road Networks POIs Urban Sensing & Data Acquisition Participatory Sensing, Crowd Sensing, Mobile Sensing Acquisition Cleaning Urban Computing: concepts, methodologies, and applications. Zheng, Y., et al. ACM transactions on Intelligent Systems and Technology.

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

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

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

6 Graph Data: Communication Nets domain2 domain1 router domain3 Internet J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive 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 Seven Bridges of Königsberg [Euler, 1735] Return to the starting point by traveling each link of the graph once and only once. J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive 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 v Network Statistic Analysis (this lecture) Network Size Degree distribution. v 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. Leskovec, A. Rajaraman, J. Ullman: 11 Mining of Massive Datasets,

12 Sampling graphs Random sampling (uniform & independent) crawling } vertex 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 v Adjacency matrix! # A = # # # " $ & & & & %! # D = # # # " v Transition Probability Matrix 1 if i is not equal to j P ij = k i 0 if i=j v E : number of links v 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 v Adjacency matrix! # A = # # # " $ & & & & %! # D = # # # " $ & & & & % Symmetric v 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 v E : number of links v Stationary Distribution π υ = 1 V 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/ Undirected 2 % ' ' ' ' &

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

18 Outline v Motivation and Problem Statement v Sampling Methodology v Data Analysis v Conclusion Minas Gjoka, UC Irvine Walking in Facebook 18

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

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) Minas Gjoka, UC Irvine Walking in Facebook 20

21 Problem statement v Obtain a representative sample of users in a given OSN by exploration of the social graph. in this work we sample Facebook (FB) explore graph using various crawling techniques Minas Gjoka, UC Irvine Walking in Facebook 21

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 Minas Gjoka, UC Irvine Walking in Facebook 22

23 Outline v Motivation and Problem Statement v Sampling Methodology crawling methods data collection convergence evaluation method comparisons v Data Analysis v Conclusion Minas Gjoka, UC Irvine Walking in Facebook 23

24 (1) Breadth-First-Search (BFS) v v v pa ( ) Starting from a seed, explores all neighbor nodes. Process continues iteratively without replacement. BFS leads to bias towards high degree nodes Lee et al, Statistical properties of Sampled Networks, Phys Review E, 2006 Early measurement studies of OSNs use BFS as primary sampling technique i.e [Mislove et al], [Ahn et al], [Wilson et al.] i u A1 A i i = = 1 V u V H G D Subset of sampled nodes with value i All sampled nodes F E C B A Unexplored Explored Visited Walking in Facebook 24

25 (2) Random Walk (RW) Explores graph one node at a time with replacement P Degree of node υ RW υ, w = 1 k In the stationary distribution π υ = k 2 υ υ E Number of edges Subset of sampled u A1 A i i nodes with value i pa ( i) = = u V1 V All sampled nodes Minas Gjoka, UC Irvine Walking in Facebook 25 H G D 1/3 C F 1/3 A 1/3 Next candidate Current node E B

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: pa ( ) i u A1 A i i = = 1 V u V v Re-Weighted probability distribution : Subset of sampled nodes with value i All sampled nodes Minas Gjoka, UC Irvine pa ( ) i u A = u V i 1/ 1/ k k u u Degree of node u Walking in Facebook 26

27 (4) Metropolis-Hastings Random Walk (MHRW) v Explore graph one node at a time with replacement H G C F E 1 k min(1, υ ) if w neighbor of υ MH k k P υ w υ, w = MH 1 Pυ, y if w= υ y υ v In the stationary distribution pa ( ) i 1 π υ = V u A1 A i i = = 1 V u V MH P AA Subset of sampled nodes with value i All sampled nodes 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) 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 Minas Gjoka, UC Irvine 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 available at: Minas Gjoka, UC Irvine Walking in Facebook 29

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 u UserID Name Networks Privacy settings 1111 UserID Name Networks Privacy settings Regional School/Workplace Send Message View Friends Profile Photo Add as Friend Minas Gjoka, UC Irvine Walking in Facebook 30

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

32 Online Convergence Diagnostics Geweke v Detects convergence for a single walk. Let X be a sequence of samples for metric of interest. Xa Xb a =10%,b = 50% J. Geweke, Evaluating the accuracy of sampling based approaches to calculate posterior moments in Bayesian Statistics 4, 1992 Minas Gjoka, UC Irvine z = E(X a ) E(X b ) Var(X a ) Var(X b ) [ 1,1] Walking in Facebook 32

33 Online Convergence Diagnostics Gelman-Rubin v Detects convergence for M>1 walks Walk 1 Between walks variance Walk 2 Walk 3 R = N 1 N + M +1 B MN W Within walks variance Convergence is declared when R<1.02 A. Gelman, D. Rubin, Inference from iterative simulation using multiple sequences in Statistical Science Volume 7, 1992 Minas Gjoka, UC Irvine Walking in Facebook 33

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

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

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

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

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 Minas Gjoka, UC Irvine Walking in Facebook 38

39 FB Social Graph Degree Distribution, Power Law Distribution f(k)=b*k -a a 1 =1.32 a 2 =3.38 v Degree distribution not a power law Minas Gjoka, UC Irvine Walking in Facebook 39

40 Outline v Motivation and Problem Statement v Sampling Methodology v Data Analysis v Conclusion Minas Gjoka, UC Irvine Walking in Facebook 40

41 Any Comments & Critiques?

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 Minas Gjoka, UC Irvine Walking in Facebook 42

43 Project 2 starts next week v KDD cup 2018 v Real data, worldwide competition v Will be announced March 1 st, 2018 v Define your own project or KDD cup v v v view/announcing-kdd-cup highway-tollgates-traffic-flowprediction Gjoka, UC Minas Irvine Walking in Facebook 43