Welcome to DS504/CS586: Big Data Analytics Graph Mining Prof. Yanhua Li Time: 6:00pm –8:50pm R Location: AK 233 Spring 2018
Service Providing Improve urban planning, Ease Traffic Congestion, Save Energy, Reduce Recommender Applications Air Pollution, ... systems Urban Data Analytics Big Graph Big Data Data Mining, Machine Learning, Visualization Data Mining Clustering Management Urban Data Management Spatio-temporal index, streaming, trajectory, and graph data management,... Human Meteorolo Road Air Social Energy Networks POIs Traffic mobility Quality gy Media Acquisition Cleaning Urban Sensing & Data Acquisition Participatory Sensing, Crowd Sensing, Mobile Sensing Urban Computing: concepts, methodologies, and applications . Zheng, Y., et al. ACM transactions on Intelligent Systems and Technology .
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, http:// www.mmds.org
Graph Data: Media Networks Connections between political blogs Polarization of the network [Adamic-Glance, 2005] 4 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
Graph Data: Information Nets Citation networks and Maps of science [Börner et al., 2012] 5 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
Graph Data: Communication Nets domain2 domain1 router domain3 Internet 6 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
Questions? Partial map of the Internet based on the January 15, 2005 data found on opte.org. (from http://atheistuniverse.net/group/internet)
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. 8 J. Leskovec, A. Rajaraman, J. Ullman: Mining of Massive Datasets, http://www.mmds.org
Graph representation of networks Following Friendship One-way road Resistance Co-authorship Wireless channel Undirected links Directed links + Multi-relational links Friend & foe - Hyperlinks + + Trust & distrust … … Repulsion & cohesion Signed links
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
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, http:// www.mmds.org
Sampling graphs R andom sampling c rawling (uniform & independent) } vertex sampling } BFS sampling } random walk sampling } edge sampling 12 12
Random Walks on Graphs Random Walk Random walk sampling Routing Molecule in liquid Influence diffusion
Undirected Graphs Undirected !! 2 3 1 6 4 5
Random Walk v Adjacency matrix 1 2 ! $ ! $ 0 1 1 1 3 0 0 0 # & # & 0 2 0 0 1 0 1 0 Symmetric # & # & A = D = # & # & 1 1 0 1 0 0 3 0 # & # & 0 0 0 2 1 0 1 0 " % " % 4 3 v Transition Probability Matrix Undirected ⎧ 1 if i is not equal to j " % ⎪ 0 1/ 3 1/ 3 1/ 3 P k i $ ' ij = ⎨ 1/ 2 0 1/ 2 0 P = A • D − 1 = $ ' ⎪ $ ' 1/ 3 1/ 3 0 1/ 3 0 if i = j ⎩ $ ' 1/ 2 0 1/ 2 0 # & v |E|: number of links v Stationary Distribution π i = d i 2 E
Metropolis-Hastings Random Walk v Adjacency matrix 1 2 ! $ ! $ 0 1 1 1 3 0 0 0 # & # & 0 2 0 0 1 0 1 0 Symmetric # & # & A = D = # & # & 1 1 0 1 0 0 3 0 # & # & 0 0 0 2 1 0 1 0 " % " % 4 3 v Transition Probability Matrix Undirected ⎧ min(1, k υ 1 " % ) if w neighbor of υ 0 1/ 3 1/ 3 1/ 3 ⎪ $ ' ⎪ k υ k w MH = 1/ 3 1/ 3 1/ 3 0 P = A • D − 1 = $ ' P ⎨ υ , w $ ' MH if w = υ 1/ 3 1/ 3 0 1/ 3 ∑ ⎪ 1 − P $ ' 1/ 3 0 1/ 3 1/ 3 υ , y # & ⎪ ⎩ y ≠ υ 1 2 v |E|: number of links v Stationary Distribution 1 π = υ V 4 3 Undirected
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
Outline v Motivation and Problem Statement v Sampling Methodology v Data Analysis v Conclusion Minas Gjoka, UC Irvine Walking in Facebook 18
Online Social Networks (OSNs) v A network of declared friendships between users v Allows users to maintain relationships F G E H C D B v Many popular OSNs with different focus A § Facebook, LinkedIn, Flickr, … Social Graph Minas Gjoka, UC Irvine Walking in Facebook 19
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
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
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
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
(1) Breadth-First-Search (BFS) v Starting from a seed, explores all neighbor nodes. Process continues F iteratively without replacement. G E v BFS leads to bias towards high H degree nodes C Lee et al, “Statistical properties of Sampled D B Networks”, Phys Review E, 2006 A v Early measurement studies of OSNs use BFS as primary sampling Unexplored technique i.e [Mislove et al], [Ahn et al], [Wilson et al.] Explored Visited 1 ∑ | A | Subset of sampled u A nodes with value i ∈ i p A ( ) i = = i 1 | V | ∑ u V ∈ All sampled nodes Walking in Facebook 24
(2) Random Walk (RW) • Explores graph one node at a time with replacement F 1 G E RW P = , w υ k H C υ Degree of node υ 3 / 1 D B • In the stationary distribution 1/3 1/3 A k υ π = υ 2 E Next candidate ⋅ Current node Number of edges 1 ∑ | A | Subset of sampled u A nodes with value i ∈ i p A ( ) i = = i 1 | V | ∑ u V ∈ All sampled nodes Minas Gjoka, UC Irvine Walking in Facebook 25
(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: 1 ∑ | A | Subset of sampled u A nodes with value i ∈ i p A ( ) i = = i 1 | V | ∑ u V ∈ v Re-Weighted probability distribution : All sampled nodes 1/ k = ∑ Degree of node u u A u ∈ p A ( ) i i 1/ k ∑ u V u ∈ Minas Gjoka, UC Irvine Walking in Facebook 26
(4) Metropolis-Hastings Random Walk (MHRW) F G E v Explore graph one node at a time with replacement H C 1/5 D B 1 min(1, k ⎧ ) if neighbor of w 3 1/3 υ / 1 υ A ⎪ k k ⎪ MH P w = ⎨ υ , w υ 2/15 MH 1 P if = w ⎩ ∑ ⎪ − υ , y υ Next candidate ⎪ y ≠ υ v In the stationary distribution Current node 1 1 1 1 2 π = MH P 1 ( ) = − + + = υ V AA 3 3 5 15 1 1 3 1 ∑ | A | Subset of sampled MH P u A nodes with value i ∈ = ⋅ = p A ( ) i i = = AC 3 5 5 i 1 | V | ∑ All sampled nodes Walking in Facebook 27 u V ∈
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
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 Minas Gjoka, UC Irvine Walking in Facebook 29
Data Collection Basic Node Information v What information do we collect for each sampled node u ? Friend List UserID Name UserID UserID Networks Name Name Networks Privacy Settings Networks Privacy settings Privacy settings u Regional School/Workplace UserID 1 1 1 1 Name Send Message Networks View Friends Privacy settings Profile Photo Add as Friend Minas Gjoka, UC Irvine Walking in Facebook 30
Detecting Convergence • Number of samples (iterations) to loose dependence from starting points? Minas Gjoka, UC Irvine Walking in Facebook 31
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