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대용량의 동적인 그래프 및 텐서 마이닝 (Mining Large Dynamic Graphs and Tensors)

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NAVER Engineering
NAVER Engineering

발표자: 신기정(CMU 박사 과정) 발표일: 2018.1. 웹, 소셜 미디어, 리뷰 데이터 등 어디서든 쉽게 그래프 데이터를 찾아볼 수 있습니다. 그중 많은 데이터는 테라바이트 혹은 그 이상의 대용량이고, 계속 변화합니다. 또한, 많은 부가 정보를 포함하고 있어 텐서, 즉 다차원 행렬의 형태로 표현됩니다. 본 발표에서는 이러한 대용량의 동적인 그래프 및 텐서 데이터의 구조를 분석하고 이상점을 탐지하기 위한 알고리즘들을 소개합니다. 특히, 그래프 내의 삼각형의 개수를 정확하게 예측하는 기법과 비이상적으로 밀집된 지점을 탐색하는 기법을 중점적으로 소개할 예정입니다. 소개될 알고리즘들은 샘플링과 근사 기법을 활용하며 분산 환경과 스트림 환경에 적합하게 설계되었습니다. 또한, 소셜 미디어의 가짜 영향력자 탐색, 가짜 리뷰 탐색, 네트워크 공격 탐색 등에 적용될 수 있습니다.

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Mining Large Dynamic Graphs and Tensors Kijung Shin (신기정) Ph.D. Student
Graphs are Everywhere! Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 2/87
Graphs are Everywhere! Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 3/87
Properties of Real-world Graphs • Large: many nodes, more edges Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 4/87 2B+ users 0.5B+ products 0.5B+ users 0.3B+ buyers • Dynamic: additions/deletions of nodes and edges
Properties of Real-world Graphs • Rich with Attributes: timestamps, scores, text, etc. Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 5/87 … …
Matrices for Graphs Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 6/87 0 0 1 0 1 1 0 0 Graph Adjacency Matrix
Tensors for Rich Graphs Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 7/87 1 0 0 0 0 • Tensors: multi-dimensional array 3-order tensor (3-dimensional array) + Stars (4-order tensor) + Text (5-order tensor) … 0
Research Goals •Sub-Tasks: ◦ Structure Analysis ◦ Anomaly Detection ◦ Behavior Modeling Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 8/87 To Understand Large Dynamic Graphs and Tensors on User Behavior
Research Goals Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 9/87 Structure Anomaly & Fraud User BehaviorContrast
Overview of My Research Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 10/87 Structure Analysis Anomaly Detection Behavior Modeling Graphs Triangle Count [ICDM17] Bipartite Graph [KDD16][TKDD17] Purchase [IJCAI17]Degeneracy Unipartite Graph [ICDM16][KAIS18] Tensors PARAFAC [ICDM14][TKDE17] Tucker [WSDM17] Static Tensor [PKDD16][TKDD18] [WSDM17][PKDD17] Dynamic Tensor [KDD17] Progression [WWW18] Other work: Similar Node Search [SIGMOD15][TODS16]
Roadmap • Overview • #1: Counting Triangles in Graph Stream << • #2: Detecting Dense Subtensors • Conclusion Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 11/87
WRS: Waiting Room Sampling for Accurate Triangle Counting in Real Graph Streams Kijung Shin
Triangles in a Graph • Graphs are everywhere! ◦ Social Network, Web, Emails, etc. • Triangles are a fundamental primitive ◦ 3 nodes connected to each other • Counting triangles has many applications ◦ Community detection, Spam detection, Query optimization Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 13/87 Introduction Experiments ConclusionAlgorithmPattern
Challenges in Real Graphs • Many algorithms for small and fixed graphs • However, real-world graphs are ◦ Large: may not fit in main memory ◦ Growing: new nodes and edges are added • Need to consider realistic settings Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 14/87 Introduction Experiments ConclusionAlgorithmPattern time 𝑡 time 𝑡+2 time 𝑡+6
Streaming Model • Stream of edges ◦ edges are streamed one by one from sources • Any or adversarial order ◦ edges can be in any order in the stream • Limited memory ◦ not every edge can be stored in memory Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 15/87 Introduction Experiments ConclusionAlgorithmPattern too strong Source Destination
Relaxed Streaming Model • Stream of edges ◦ edges are streamed one by one from sources • Chronological order ◦ natural for dynamic graphs ◦ edges are streamed when they are created • Limited memory ◦ not every edge can be stored in memory Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 16/87 “What temporal patterns do exist?” “How can we exploit them for accurate triangle counting?” Introduction Experiments ConclusionAlgorithmPattern
Roadmap • Overview • #1: Counting Triangles in Graph Stream ◦ Introduction ◦ Temporal Pattern << ◦ Proposed Algorithm ◦ Experiments ◦ Conclusion • #2: Detecting Dense Subtensors • Conclusion Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 17/87
Time Interval of a Triangle • Time interval of a triangle: Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 18/87 Introduction Experiments ConclusionAlgorithmPattern – arrival order of its last edge arrival order of its first edge arrival order 1 2 3 4 5 6 7 8 Time interval
Time Interval Distribution • Temporal Locality: ◦ average time interval is ◦ 2X shorter in the chronological order ◦ than in a random order Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 19/87 random arrival order chronological arrival order random order chronological order
Temporal Locality (cont.) • One interpretation: ◦ edges are more likely to form ◦ triangles with edges close in time ◦ than with edges far in time • Another interpretation: ◦ new edges are more likely to form ◦ triangles with recent edges ◦ than with old edges Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 20/87 Introduction Experiments ConclusionAlgorithmPattern “How can we exploit temporal locality for accurate triangle counting?” chronological order random order
Roadmap • Overview • #1: Counting Triangles in Graph Stream ◦ Introduction ◦ Temporal Pattern ◦ Proposed Algorithm << ◦ Experiments ◦ Conclusion • #2: Detecting Dense Subtensors • Conclusion Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 21/87
Problem Definition • Given: ◦ a graph stream in the chronological order ◦ memory budget: 𝑘 (i.e., up to 𝑘 edges can be stored) • Estimate: counts of global and local triangles • To Minimize: estimation error Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 22/87 Introduction Experiments ConclusionAlgorithmPattern • Global triangles: all triangles in the graph • Local triangles: the triangles incident to each node 3 2 1 2 3 4 1 3 2 1
Algorithm Overview • General approach used in [LK15] [DERU16] ◦ ∆: our estimate of global triangle counts ◦ 𝑝 𝑢𝑣𝑤: probability that triangle (𝑢, 𝑣, 𝑤) is discovered Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 23/87 Introduction Experiments ConclusionAlgorithmPattern
Algorithm Overview • General approach used in [LK15] [DERU16] ◦ ∆: our estimate of global triangle counts ◦ 𝑝 𝑢𝑣𝑤: probability that triangle (𝑢, 𝑣, 𝑤) is discovered Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 24/87 Introduction Experiments ConclusionAlgorithmPattern 𝑢 | 𝑥 𝑢 | 𝑦 𝑣 | 𝑥 𝑣 | 𝑣 𝑢 − 𝑣 memory new edge (1) Edge Arrival
Algorithm Overview • General approach used in [LK15] [DERU16] ◦ ∆: our estimate of global triangle counts ◦ 𝑝 𝑢𝑣𝑤: probability that triangle (𝑢, 𝑣, 𝑤) is discovered Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 25/87 Introduction Experiments ConclusionAlgorithmPattern 𝑢 | 𝑥 𝑢 | 𝑦 𝑣 | 𝑥 𝑣 | 𝑣 𝑢 | 𝑥 𝑢 | 𝑦 𝑣 | 𝑥 𝑣 | 𝑣 𝑢 − 𝑣 𝑢 − 𝑣 𝑥 ∆← ∆ + 1/𝑝 𝑢𝑣𝑥 discover! memory (1) Edge Arrival (2) Counting Step 𝑢 − 𝑣new edge
Algorithm Overview • General approach used in [LK15] [DERU16] ◦ ∆: our estimate of global triangle counts ◦ 𝑝 𝑢𝑣𝑤: probability that triangle (𝑢, 𝑣, 𝑤) is discovered Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 26/87 Introduction Experiments ConclusionAlgorithmPattern 𝑢 | 𝑥 𝑢 | 𝑦 𝑣 | 𝑥 𝑣 | 𝑣 𝑢 − 𝑣 𝑢 − 𝑣 𝑦 ∆← ∆ + 1/𝑝 𝑢𝑣𝑦 discover! (2) Counting Step 𝑢 | 𝑥 𝑢 | 𝑦 𝑣 | 𝑥 𝑣 | 𝑣 memory (1) Edge Arrival 𝑢 − 𝑣new edge
Algorithm Overview • General approach used in [LK15] [DERU16] ◦ ∆: our estimate of global triangle counts ◦ 𝑝 𝑢𝑣𝑤: probability that triangle (𝑢, 𝑣, 𝑤) is discovered Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 27/87 Introduction Experiments ConclusionAlgorithmPattern 𝑢 | 𝑥 𝑢 | 𝑦 𝑣 | 𝑥 𝑣 | 𝑣 𝑢 − 𝑣 𝑢 − 𝑣 𝑦 ∆← ∆ + 1/𝑝 𝑢𝑣𝑦 𝑢 | 𝑥 𝑢 | 𝑣 𝑣 | 𝑥 𝑣 | 𝑣 (2) Counting Step 𝑢 | 𝑥 𝑢 | 𝑦 𝑣 | 𝑥 𝑣 | 𝑣 memory (1) Edge Arrival (3) Sampling Step (to be explained) 𝑢 − 𝑣new edge
Bias and Variance Analyses • Bias and variance analyses results Theorem. Unbiasedness of our estimate: Theorem. Variance of our estimate: • should increase discovering probability 𝑝 𝑢𝑣𝑤 ◦ which depends on sampling algorithms Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 28/87 Bias[∆] = E ∆ −True count = 0 Var ∆ ≈ σ(𝑢,𝑣,𝑤) (1/𝑝 𝑢𝑣𝑤 − 1) Introduction Experiments ConclusionAlgorithmPattern
Increasing Discovering Prob. • Recall Temporal Locality: ◦ new edges are more likely to form ◦ triangles with recent edges ◦ than with old edges • Waiting-Room Sampling (WRS) ◦ exploits temporal locality ◦ by treating recent edges better than old edges Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 29/87 Introduction Experiments ConclusionAlgorithmPattern “How can we increase discovering probabilities of triangles?”
Waiting-Room Sampling (WRS) • Divides memory space into two parts ◦ Waiting Room: stores latest edges ◦ Reservoir: store samples from the remaining edges Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 30/87 𝑒79 𝑒78 𝑒77 𝑒76 Waiting Room (FIFO) Reservoir (Random Replace) 𝛼% of budget 100 − 𝛼 % of budget 𝑒80New edge Introduction Experiments ConclusionAlgorithmPattern 𝑒61 𝑒7 𝑒18 𝑒25 𝑒40 𝑒1 𝑒28
WRS: Sampling Steps (Step 1) Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 31/87 𝑒76Popped edge Introduction Experiments ConclusionAlgorithmPattern 𝑒79 𝑒78 𝑒77 𝑒76 Waiting Room (FIFO) Reservoir (Random Replace) 𝑒80New edge 𝑒61 𝑒7 𝑒18 𝑒25 𝑒40 𝑒1 𝑒28 𝑒80 𝑒79 𝑒78 𝑒77 𝑒61 𝑒7 𝑒18 𝑒25 𝑒40 𝑒1 𝑒28 Waiting Room (FIFO) Reservoir (Random Replace)
WRS: Sampling Steps (Step 2) Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 32/87 Popped edge Introduction Experiments ConclusionAlgorithmPattern 𝑒76 𝑒80 𝑒79 𝑒78 𝑒77 𝑒61 𝑒7 𝑒18 𝑒25 𝑒40 𝑒1 𝑒28 𝑒61 𝑒7 𝑒18 𝑒25 𝑒76 𝑒1 𝑒28 𝑒61 𝑒7 𝑒18 𝑒25 𝑒40 𝑒1 𝑒28 Waiting Room (FIFO) replace! store discard or or Reservoir (Random Replace)
Summary of Algorithm Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 33/87 Introduction Experiments ConclusionAlgorithmPattern 𝑢 | 𝑥 𝑢 | 𝑦 𝑣 | 𝑥 𝑣 | 𝑣 𝑢 − 𝑣 memory new edge (1) Arrival Step 𝑢 | 𝑥 𝑢 | 𝑦 𝑣 | 𝑥 𝑣 | 𝑣 𝑢 − 𝑣 𝑢 − 𝑣 𝑥 ∆← ∆ + 1/𝑝 𝑢𝑣𝑥 discover! (2) Discovery Step 𝑢 | 𝑥 𝑢 | 𝑣 𝑣 | 𝑥 𝑣 | 𝑣 (3) Sampling Step Waiting-Room Sampling!
Roadmap • Overview • #1: Counting Triangles in Graph Stream ◦ Introduction ◦ Temporal Pattern ◦ Proposed Algorithm ◦ Experiments << ◦ Conclusion • #2: Detecting Dense Subtensors • Conclusion Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 34/87
Experimental Settings • Competitors: Mascot [YK15], Triest-IMPR [DERU16] • Datasets: • Memory budget 𝑘: 10% of the edges • Size of waiting-room: 10% of memory budget Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 35/87 Introduction Experiments ConclusionAlgorithmPattern citation email friendship
Distribution of Estimates Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 36/87 • Waiting Room Sampling (WRS) gives ◦ unbiased estimates ◦ with smallest variance Introduction Experiments ConclusionAlgorithmPattern WRS Triest-IMPR MASCOT True Count
Discovering Probability • WRS increases discovering probability 𝑝 𝑢𝑣𝑤 • WRS discovers up to 3 × more triangles Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 37/87 Introduction Experiments ConclusionAlgorithmPattern WRS Triest-IMPR MASCOT
Estimation Errors • WRS is most accurate • WRS reduces estimation error up to 𝟒𝟕% Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 38/87 Introduction Experiments ConclusionAlgorithmPattern
Roadmap • Overview • #1: Counting Triangles in Graph Stream ◦ Introduction ◦ Temporal Pattern ◦ Proposed Algorithm ◦ Experiments ◦ Conclusion << • #2: Detecting Dense Subtensors • Conclusion Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 39/87
Contributions (1st Paper) • Pattern: Temporal Locality ◦ short time interval of triangles in real graph streams • Algorithm: Waiting-Room Sampling (WRS) ◦ exploits temporal locality for accurate triangle counting • Analyses: Bias and Variance Analyses ◦ WRS gives unbiased estimates with small variances Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 40/87 0 . 3 7 0 Introduction Experiments ConclusionAlgorithmPattern
Further Work • Multiple Sources & Workers (Under Review) • Handling Edge Deletions (Under Preparation) Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 41/87 … , , + , , − , , + , , − , …
Roadmap • Overview • #1: Counting Triangles in Graph Stream • #2: Detecting Dense Subtensors << • Conclusion Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 42/87
M-Zoom: Fast Dense-Block Detection in Tensors with Quality Guarantees Kijung Shin Bryan Hooi Christos Faloutsos
Motivation: Review Fraud Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 44/87 Bob’s Carol’s Alice’s Alice Introduction Experiments ConclusionProposed Method
Fraud Forms Dense Blocks Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 45/87 Restaurants AccountsRestaurants Accounts Adjacency Matrix Introduction Experiments ConclusionProposed Method
Problem: Natural Dense Blocks • Question. How can we distinguish them? Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 46/87 Restaurants Accounts Adjacency Matrix suspicious dense block formed by fraudsters natural dense block (core, community, etc.) Introduction Experiments ConclusionProposed Method
Solution: Tensor Modeling • Natural dense blocks are sparse on the time axis (formed gradually) • Suspicious dense blocks are also dense on the time axis (due to synchronous behavior) • Suspicious dense blocks are denser than natural dense blocks in the tensor model Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 47/87 Restaurants Accounts Introduction Experiments ConclusionProposed Method
Solution: Tensor Modeling (Cont.) • Any side information can be used instead of/in addition to time in review datasets • Using multiple side information ⟹ high-order tensors Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 48/87 IP Address keywords Number of stars Introduction Experiments ConclusionProposed Method
Anomaly/Fraud in Tensors • Dense blocks signal anomalies/fraud in many tensor datasets Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 49/87 Src IP DstIP Src User DstUser User Page TCP Dumps Wikipedia Revision History Time-evolving Social Network Introduction Experiments ConclusionProposed Method
Research Question Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 50/87 “Given a large-scale high-order tensor, how can we find dense subtensors in it?” Introduction Experiments ConclusionProposed Method
Road Map Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 51/87 • Overview • #1: Counting Triangles in Graph Stream • #2: Detecting Dense Subtensors ◦ Introduction ◦ Proposed Method: M-Zoom Terminologies and Problem Definition << Algorithm ◦ Experiments ◦ Conclusion • Conclusion
Terminologies • Assume a block (subtensor) 𝑩 in a 3-way tensor 𝑹 ◦ 𝑺𝒊𝒛𝒆(𝐵): 𝐼1 + 𝐼2 + 𝐼3 ◦ 𝑽𝒐𝒍 𝐵 = 𝑉𝑜𝑙𝑢𝑚𝑒(𝐵): 𝐼1 × 𝐼2 × 𝐼3 ◦ 𝑴𝒂𝒔𝒔(𝐵): sum of entries in 𝐵 Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 52/87 Introduction Experiments ConclusionProposed Method 𝑅 𝐼1 𝐼2 𝐼3 𝐵
Density Measures • Density measures: Traditional Density: ρ 𝑇 𝐵 = 𝑀𝑎𝑠𝑠 𝐵 /Vol(B) - Maximized by a single entry with maximum value Arithmetic Avg. Degree: ρ 𝐴 𝐵 = 𝑀𝑎𝑠𝑠 𝐵 /Size(B) Geometric Avg. Degree: ρ 𝐺 𝐵 = 𝑀𝑎𝑠𝑠 𝐵 / 3 Vol B Suspiciousness (Jiang et al. 2015) : ρ 𝑆 𝐵, 𝑅 = 𝑉𝑜𝑙 𝐵 𝐷 𝐾𝐿 𝑀𝑎𝑠𝑠 𝐵 𝑉𝑜𝑙 𝐵 , 𝑀𝑎𝑠𝑠 𝑅 𝑉𝑜𝑙 𝑅 • Note that our method is not limited by specific density measures Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 53/87 Introduction Experiments ConclusionProposed Method
Problem Definition • Given: (1) 𝑹: a tensor, (2) 𝝆: a density measure, (3) 𝒌: the number of blocks we aim to find • Find: 𝒌 distinct dense blocks maximizing 𝝆 Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 54/87 (𝑹 = , 𝝆 = 𝝆 𝑨, 𝒌 = 𝟑) , , }{ Introduction Experiments ConclusionProposed Method
Requirements • Our goal is to design an approximation algorithm • M-Zoom, our proposed method, satisfies all the requirements Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 55/87 Scalable: runs in near-linear time Accurate: provides an accuracy guarantee Flexible: works well with various density metrics Effective: produces meaningful results in practice 0 Introduction Experiments ConclusionProposed Method
Road Map Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 56/87 • Overview • #1: Counting Triangles in Graph Stream • #2: Detecting Dense Subtensors ◦ Introduction ◦ Proposed Method: M-Zoom Terminologies and Problem Definition Algorithm << ◦ Experiments ◦ Conclusion • Conclusion
Single Dense Block Detection • Greedy search method • Starts from the entire tensor Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 57/87 5 3 0 4 6 1 2 0 0 1 0 1 0 0 𝜌 = 2.9 Introduction Experiments ConclusionProposed Method
Single Dense Block Detection (cont.) • Remove a slice to maximize density 𝝆 Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 58/87 5 3 0 4 6 1 2 0 0 𝜌 = 3 Introduction Experiments ConclusionProposed Method
Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 59/87 5 3 4 6 2 0 𝜌 =3.3 • Remove a slice to maximize density 𝝆 Introduction Experiments ConclusionProposed Method Single Dense Block Detection (cont.)
Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 60/87 5 3 4 6 2 0 𝜌 = 3.6 • Remove a slice to maximize density 𝝆 Introduction Experiments ConclusionProposed Method Single Dense Block Detection (cont.)
0 1 2 3 4 0 2 4 6 8 Density Iteration • Until all slices are removed Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 61/87 𝜌 = 0 Introduction Experiments ConclusionProposed Method Single Dense Block Detection (cont.)
• Output: return the densest block so far Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 62/87 5 3 4 6 2 0 𝜌 = 3.6 Introduction Experiments ConclusionProposed Method Single Dense Block Detection (cont.)
Speeding Up Process • Theorem 1 [Remove Minimum Mass First] Among slices in the same mode, removing the slice with minimum mass is always best Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 63/87 12 > 9 > 2 Introduction Experiments ConclusionProposed Method
Accuracy Guarantee • Theorem 2 [Approximation Guarantee] 𝝆 𝑨 𝑩 ≥ 𝟏 𝑵 𝝆 𝑨 𝑩∗ Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 64/87 M-Zoom Result Order Densest Block Scalable: runs in near-linear time Accurate: provides an accuracy guarantee Flexible: works well with various density metrics Effective: produces meaningful results in practice 0 . 3 7 0 Properties of M-Zoom: Introduction Experiments ConclusionProposed Method
(Optional) Post Processing • Local Search: ◦ grow(G) or shrink(S) until we reach a local optimum Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 65/87 Introduction Experiments ConclusionProposed Method
Handling Multiple Blocks • Deflation: Remove found blocks before finding others Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 66/87 Find & Remove Find & Remove Find & Remove Restore Introduction Experiments ConclusionProposed Method
Road Map Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 67/87 • Overview • #1: Counting Triangles in Graph Stream • #2: Detecting Dense Subtensors ◦ Introduction ◦ Proposed Method: M-Zoom ◦ Experiments << ◦ Conclusion • Conclusion
Exp1. Scalability Test • Goal: Measure scalability w.r.t. each input factor • M-Zoom scales almost linearly! Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 68/87 #Non-Zeros Order Dimensionality (#Slices) M-Zoom (with post process) M-Zoom Introduction Experiments ConclusionProposed Method
Exp1. Scalability Test (cont.) Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 69/87 Scalable: runs in near-linear time Accurate: provides an accuracy guarantee Flexible: works well with various density metrics Effective: produces meaningful results in practice 0 . 3 7 0 Properties of M-Zoom: Introduction Experiments ConclusionProposed Method
Exp2. Speed-Accuracy Test Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 70/87 Introduction Experiments ConclusionProposed Method • Goal: compare speed and accuracy of dense-block detection methods with various density measures • Methods Compared: ◦ M-Zoom: Proposed Method ◦ CPD: Tensor Decomposition ◦ CrossSpot (Jiang et al. 2015): Local Search Method
Exp2. Speed-Accuracy Test (cont.) Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 71/87 • Data: Music Review (User X Music X Time X Rate) Arithmetic Average Degree (𝜌 𝐴) Geometric Average Degree (𝜌 𝐺) Suspiciousness (𝜌 𝑆) M-Zoom (with post process) M-Zoom Introduction Experiments ConclusionProposed Method
Exp2. Speed-Accuracy Test (cont.) Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 72/87 Arithmetic Average Degree (𝜌 𝐴) Geometric Average Degree (𝜌 𝐺) Suspiciousness (𝜌 𝑆) M-Zoom (with post process) M-Zoom • Data: Wikipedia Revision History (User X Page X Time) Introduction Experiments ConclusionProposed Method
Exp2. Speed-Accuracy Test (cont.) Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 73/87 • M-Zoom is up to 100× faster than its competitors • M-Zoom shows comparable accuracy with its competitors regardless of density measures Scalable: runs in near-linear time Accurate: provides an accuracy guarantee Flexible: works well with various density metrics Effective: producing meaningful results in practice 0 . 3 7 0 Properties of M-Zoom: Introduction Experiments ConclusionProposed Method
Exp3. Discovery in Practice Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 74/87 First three blocks found by M-Zoom 11 users revised 10 pages 2,305 times within 16 hours → Edit wars Introduction Experiments ConclusionProposed Method • Data: Korean Wikipedia Revision History (User X Page X Time)
Exp3. Discovery in Practice (cont.) Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 75/87 First three blocks found by M-Zoom 8 accounts revised 12 pages 2.5 million times → Bots Introduction Experiments ConclusionProposed Method • Data: English Wikipedia Revision History (User X Page X Time)
Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 76/87 → Network Attacks First three blocks found by M-Zoom Exp3. Discovery in Practice (cont.) Introduction Experiments ConclusionProposed Method • Data: TCP Dump (7-order tensor)
Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 77/87 Exp3. Discovery in Practice (cont.) First three blocks found by M-Zoom Introduction Experiments ConclusionProposed Method 9 accounts gives 1 product 369 reviews to in 22 hours → Spam reviews • Data: App Market Review Data (User X Product X Timestamp X Score)
Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 78/87 Scalable: runs in near-linear time Accurate: provides an accuracy guarantee Flexible: works well with various density metrics Effective: produces meaningful results in practice 0 . 3 7 0 Properties of M-Zoom: Exp3. Discovery in Practice (cont.) Introduction Experiments ConclusionProposed Method
Road Map Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 79/87 • Overview • #1: Counting Triangles in Graph Stream • #2: Detecting Dense Subtensors ◦ Introduction ◦ Proposed Method: M-Zoom ◦ Experiments ◦ Conclusion << • Conclusion
Conclusion (2nd Paper) • M-Zoom (Multi-dimensional Zoom): ◦ Dense-Block Detection in tensors Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 80/87 Introduction Experiments ConclusionProposed Method Scalable #Non-Zeros Accurate [Approximation Guarantee] 𝝆 𝑨 𝑩 ≥ 𝟏 𝑵 𝝆 𝑨 𝑩∗ Flexible Effective
Related Work • Disk-resident or Distributed Tensors [WSDM17] Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 81/87 #Non-Zeros Proposed (Serial) Proposed (Hadoop) M-Zoom
Related Work • Dynamic Tensors [KDD17] Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 82/87 ToePeubot Dvtbot Territory Regionalism Idol Presidential election Religion Disease Politician Dictator Local election TV Show Bot for auto classification Anime Baseball History Edit War Bot Vandalism Bursty edit (informative) Bursty edit (useless)
Roadmap • Overview • #1: Counting Triangles in Graph Stream • #2: Detecting Dense Subtensors • Conclusion << Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 83/87
Conclusion • Goal: to understand Large Dynamic Graphs and Tensors • Sub-Tasks: ◦ Structure Analysis ◦ Anomaly Detection ◦ Behavior Modeling • Approaches: ◦ Streaming Algorithms (Sampling) ◦ Approximation Algorithms ◦ Distributed Algorithms Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 84/87
Thank You • Papers, software, datasets: http://kijungshin.com/ • Email: kijungs@cs.cmu.edu • Thanks to: Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 85/87
References • Structure Analysis (Graphs) ◦ Kijung Shin, Tina Eliassi-Rad, and Christos Faloutsos, “CoreScope: Graph Mining Using k-Core Analysis - Patterns, Anomalies and Algorithms”, ICDM 2016 ◦ Kijung Shin, “Mining Large Dynamic Graphs and Tensors for Accurate Triangle Counting in Real Graph Streams”, ICDM 2017 ◦ Kijung Shin, Tina Eliassi-Rad, and Christos Faloutsos, “Patterns and Anomalies in k-Cores of Real-world Graphs with Applications”, KAIS 2018 • Structure Analysis (Tensors) ◦ Kijung Shin and U Kang, “Distributed Methods for High-dimensional and Large-scale Tensor Factorization”, ICDM 2014 ◦ Kijung Shin, Lee Sael, and U Kang, “Fully Scalable Methods for Distributed Tensor Factorization”, TKDE 2017 ◦ Jinoh Oh, Kijung Shin, Evangelos E. Papalexakis, Christos Faloutsos, and Hwanjo Yu, “S-HOT: Scalable High-Order Tucker Decomposition”, WSDM 2017 • Anomaly Detection Analysis (Graphs) ◦ Bryan Hooi, Hyun Ah Song, Alex Beutel, Neil Shah, Kijung Shin, and Christos Faloutsos, “FRAUDAR: Bounding Graph Fraud in the Face of Camouflage”, KDD 2016 ◦ Bryan Hooi, Kijung Shin, Hyun Ah Song, Alex Beutel, Neil Shah, and Christos Faloutsos, “Graph-Based Fraud Detection in the Face of Camouflage”, TKDD 2017 ◦ Kijung Shin, Tina Eliassi-Rad, and Christos Faloutsos, “CoreScope: Graph Mining Using k-Core Analysis - Patterns, Anomalies and Algorithms”, ICDM 2016 [Duplicated] ◦ Kijung Shin, Tina Eliassi-Rad, and Christos Faloutsos, “Patterns and Anomalies in k-Cores of Real-world Graphs with Applications”, KAIS 2018 [Duplicated] Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 86/87
References • Anomaly Detection Analysis (Tensors) ◦ Kijung Shin, Bryan Hooi, and Christos Faloutsos, “Mining Large Dynamic Graphs and Tensors (by Kijung Shin)”, ECML/PKDD 2016 ◦ Kijung Shin, Bryan Hooi, Jisu Kim, and Christos Faloutsos, “D-Cube: Dense-Block Detection in Terabyte-Scale Tensors”, WSDM 2017 ◦ Kijung Shin, Bryan Hooi, Jisu Kim, and Christos Faloutsos, “DenseAlert: Incremental Dense-Subtensor Detection in Tensor Streams”, KDD 2017 ◦ Hemank Lamba, Bryan Hooi, Kijung Shin, Christos Faloutsos, and J¨urgen Pfeffer, “ZooRank: Ranking Suspicious Entities in Time-Evolving Tensors”, ECML/PKDD 2017 ◦ Kijung Shin, Bryan Hooi, and Christos Faloutsos, “Fast, Accurate and Flexible Algorithms for Dense Subtensor Mining”, TKDD 2018 • Behavior Modeling ◦ Kijung Shin, Mahdi Shafiei, Myunghwan Kim, Aastha Jain, and Hema Raghavan, “Discovering Progression Stages in Trillion-Scale Behavior Logs”, WWW 2018 ◦ Kijung Shin, Euiwoong Lee, Dhivya Eswaran, and Ariel D. Procaccia, “Why You Should Charge Your Friends for Borrowing Your Stuff”, IJCAI 2017 • Similar Node Search ◦ Kijung Shin, Jinhong Jung, Lee Sael, and U Kang, “BEAR: Block Elimination Approach for Random Walk with Restart on Large Graphs”, SIGMOD 2015 ◦ Jinhong Jung, Kijung Shin, Lee Sael, and U Kang, “Random Walk with Restart on Large Graphs Using Block Elimination”, TODS 2016 Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 87/87

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대용량의 동적인 그래프 및 텐서 마이닝 (Mining Large Dynamic Graphs and Tensors)

  • 1. Mining Large Dynamic Graphs and Tensors Kijung Shin (신기정) Ph.D. Student
  • 2. Graphs are Everywhere! Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 2/87
  • 3. Graphs are Everywhere! Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 3/87
  • 4. Properties of Real-world Graphs • Large: many nodes, more edges Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 4/87 2B+ users 0.5B+ products 0.5B+ users 0.3B+ buyers • Dynamic: additions/deletions of nodes and edges
  • 5. Properties of Real-world Graphs • Rich with Attributes: timestamps, scores, text, etc. Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 5/87 … …
  • 6. Matrices for Graphs Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 6/87 0 0 1 0 1 1 0 0 Graph Adjacency Matrix
  • 7. Tensors for Rich Graphs Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 7/87 1 0 0 0 0 • Tensors: multi-dimensional array 3-order tensor (3-dimensional array) + Stars (4-order tensor) + Text (5-order tensor) … 0
  • 8. Research Goals •Sub-Tasks: ◦ Structure Analysis ◦ Anomaly Detection ◦ Behavior Modeling Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 8/87 To Understand Large Dynamic Graphs and Tensors on User Behavior
  • 9. Research Goals Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 9/87 Structure Anomaly & Fraud User BehaviorContrast
  • 10. Overview of My Research Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 10/87 Structure Analysis Anomaly Detection Behavior Modeling Graphs Triangle Count [ICDM17] Bipartite Graph [KDD16][TKDD17] Purchase [IJCAI17]Degeneracy Unipartite Graph [ICDM16][KAIS18] Tensors PARAFAC [ICDM14][TKDE17] Tucker [WSDM17] Static Tensor [PKDD16][TKDD18] [WSDM17][PKDD17] Dynamic Tensor [KDD17] Progression [WWW18] Other work: Similar Node Search [SIGMOD15][TODS16]
  • 11. Roadmap • Overview • #1: Counting Triangles in Graph Stream << • #2: Detecting Dense Subtensors • Conclusion Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 11/87
  • 12. WRS: Waiting Room Sampling for Accurate Triangle Counting in Real Graph Streams Kijung Shin
  • 13. Triangles in a Graph • Graphs are everywhere! ◦ Social Network, Web, Emails, etc. • Triangles are a fundamental primitive ◦ 3 nodes connected to each other • Counting triangles has many applications ◦ Community detection, Spam detection, Query optimization Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 13/87 Introduction Experiments ConclusionAlgorithmPattern
  • 14. Challenges in Real Graphs • Many algorithms for small and fixed graphs • However, real-world graphs are ◦ Large: may not fit in main memory ◦ Growing: new nodes and edges are added • Need to consider realistic settings Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 14/87 Introduction Experiments ConclusionAlgorithmPattern time 𝑡 time 𝑡+2 time 𝑡+6
  • 15. Streaming Model • Stream of edges ◦ edges are streamed one by one from sources • Any or adversarial order ◦ edges can be in any order in the stream • Limited memory ◦ not every edge can be stored in memory Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 15/87 Introduction Experiments ConclusionAlgorithmPattern too strong Source Destination
  • 16. Relaxed Streaming Model • Stream of edges ◦ edges are streamed one by one from sources • Chronological order ◦ natural for dynamic graphs ◦ edges are streamed when they are created • Limited memory ◦ not every edge can be stored in memory Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 16/87 “What temporal patterns do exist?” “How can we exploit them for accurate triangle counting?” Introduction Experiments ConclusionAlgorithmPattern
  • 17. Roadmap • Overview • #1: Counting Triangles in Graph Stream ◦ Introduction ◦ Temporal Pattern << ◦ Proposed Algorithm ◦ Experiments ◦ Conclusion • #2: Detecting Dense Subtensors • Conclusion Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 17/87
  • 18. Time Interval of a Triangle • Time interval of a triangle: Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 18/87 Introduction Experiments ConclusionAlgorithmPattern – arrival order of its last edge arrival order of its first edge arrival order 1 2 3 4 5 6 7 8 Time interval
  • 19. Time Interval Distribution • Temporal Locality: ◦ average time interval is ◦ 2X shorter in the chronological order ◦ than in a random order Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 19/87 random arrival order chronological arrival order random order chronological order
  • 20. Temporal Locality (cont.) • One interpretation: ◦ edges are more likely to form ◦ triangles with edges close in time ◦ than with edges far in time • Another interpretation: ◦ new edges are more likely to form ◦ triangles with recent edges ◦ than with old edges Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 20/87 Introduction Experiments ConclusionAlgorithmPattern “How can we exploit temporal locality for accurate triangle counting?” chronological order random order
  • 21. Roadmap • Overview • #1: Counting Triangles in Graph Stream ◦ Introduction ◦ Temporal Pattern ◦ Proposed Algorithm << ◦ Experiments ◦ Conclusion • #2: Detecting Dense Subtensors • Conclusion Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 21/87
  • 22. Problem Definition • Given: ◦ a graph stream in the chronological order ◦ memory budget: 𝑘 (i.e., up to 𝑘 edges can be stored) • Estimate: counts of global and local triangles • To Minimize: estimation error Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 22/87 Introduction Experiments ConclusionAlgorithmPattern • Global triangles: all triangles in the graph • Local triangles: the triangles incident to each node 3 2 1 2 3 4 1 3 2 1
  • 23. Algorithm Overview • General approach used in [LK15] [DERU16] ◦ ∆: our estimate of global triangle counts ◦ 𝑝 𝑢𝑣𝑤: probability that triangle (𝑢, 𝑣, 𝑤) is discovered Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 23/87 Introduction Experiments ConclusionAlgorithmPattern
  • 24. Algorithm Overview • General approach used in [LK15] [DERU16] ◦ ∆: our estimate of global triangle counts ◦ 𝑝 𝑢𝑣𝑤: probability that triangle (𝑢, 𝑣, 𝑤) is discovered Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 24/87 Introduction Experiments ConclusionAlgorithmPattern 𝑢 | 𝑥 𝑢 | 𝑦 𝑣 | 𝑥 𝑣 | 𝑣 𝑢 − 𝑣 memory new edge (1) Edge Arrival
  • 25. Algorithm Overview • General approach used in [LK15] [DERU16] ◦ ∆: our estimate of global triangle counts ◦ 𝑝 𝑢𝑣𝑤: probability that triangle (𝑢, 𝑣, 𝑤) is discovered Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 25/87 Introduction Experiments ConclusionAlgorithmPattern 𝑢 | 𝑥 𝑢 | 𝑦 𝑣 | 𝑥 𝑣 | 𝑣 𝑢 | 𝑥 𝑢 | 𝑦 𝑣 | 𝑥 𝑣 | 𝑣 𝑢 − 𝑣 𝑢 − 𝑣 𝑥 ∆← ∆ + 1/𝑝 𝑢𝑣𝑥 discover! memory (1) Edge Arrival (2) Counting Step 𝑢 − 𝑣new edge
  • 26. Algorithm Overview • General approach used in [LK15] [DERU16] ◦ ∆: our estimate of global triangle counts ◦ 𝑝 𝑢𝑣𝑤: probability that triangle (𝑢, 𝑣, 𝑤) is discovered Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 26/87 Introduction Experiments ConclusionAlgorithmPattern 𝑢 | 𝑥 𝑢 | 𝑦 𝑣 | 𝑥 𝑣 | 𝑣 𝑢 − 𝑣 𝑢 − 𝑣 𝑦 ∆← ∆ + 1/𝑝 𝑢𝑣𝑦 discover! (2) Counting Step 𝑢 | 𝑥 𝑢 | 𝑦 𝑣 | 𝑥 𝑣 | 𝑣 memory (1) Edge Arrival 𝑢 − 𝑣new edge
  • 27. Algorithm Overview • General approach used in [LK15] [DERU16] ◦ ∆: our estimate of global triangle counts ◦ 𝑝 𝑢𝑣𝑤: probability that triangle (𝑢, 𝑣, 𝑤) is discovered Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 27/87 Introduction Experiments ConclusionAlgorithmPattern 𝑢 | 𝑥 𝑢 | 𝑦 𝑣 | 𝑥 𝑣 | 𝑣 𝑢 − 𝑣 𝑢 − 𝑣 𝑦 ∆← ∆ + 1/𝑝 𝑢𝑣𝑦 𝑢 | 𝑥 𝑢 | 𝑣 𝑣 | 𝑥 𝑣 | 𝑣 (2) Counting Step 𝑢 | 𝑥 𝑢 | 𝑦 𝑣 | 𝑥 𝑣 | 𝑣 memory (1) Edge Arrival (3) Sampling Step (to be explained) 𝑢 − 𝑣new edge
  • 28. Bias and Variance Analyses • Bias and variance analyses results Theorem. Unbiasedness of our estimate: Theorem. Variance of our estimate: • should increase discovering probability 𝑝 𝑢𝑣𝑤 ◦ which depends on sampling algorithms Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 28/87 Bias[∆] = E ∆ −True count = 0 Var ∆ ≈ σ(𝑢,𝑣,𝑤) (1/𝑝 𝑢𝑣𝑤 − 1) Introduction Experiments ConclusionAlgorithmPattern
  • 29. Increasing Discovering Prob. • Recall Temporal Locality: ◦ new edges are more likely to form ◦ triangles with recent edges ◦ than with old edges • Waiting-Room Sampling (WRS) ◦ exploits temporal locality ◦ by treating recent edges better than old edges Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 29/87 Introduction Experiments ConclusionAlgorithmPattern “How can we increase discovering probabilities of triangles?”
  • 30. Waiting-Room Sampling (WRS) • Divides memory space into two parts ◦ Waiting Room: stores latest edges ◦ Reservoir: store samples from the remaining edges Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 30/87 𝑒79 𝑒78 𝑒77 𝑒76 Waiting Room (FIFO) Reservoir (Random Replace) 𝛼% of budget 100 − 𝛼 % of budget 𝑒80New edge Introduction Experiments ConclusionAlgorithmPattern 𝑒61 𝑒7 𝑒18 𝑒25 𝑒40 𝑒1 𝑒28
  • 31. WRS: Sampling Steps (Step 1) Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 31/87 𝑒76Popped edge Introduction Experiments ConclusionAlgorithmPattern 𝑒79 𝑒78 𝑒77 𝑒76 Waiting Room (FIFO) Reservoir (Random Replace) 𝑒80New edge 𝑒61 𝑒7 𝑒18 𝑒25 𝑒40 𝑒1 𝑒28 𝑒80 𝑒79 𝑒78 𝑒77 𝑒61 𝑒7 𝑒18 𝑒25 𝑒40 𝑒1 𝑒28 Waiting Room (FIFO) Reservoir (Random Replace)
  • 32. WRS: Sampling Steps (Step 2) Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 32/87 Popped edge Introduction Experiments ConclusionAlgorithmPattern 𝑒76 𝑒80 𝑒79 𝑒78 𝑒77 𝑒61 𝑒7 𝑒18 𝑒25 𝑒40 𝑒1 𝑒28 𝑒61 𝑒7 𝑒18 𝑒25 𝑒76 𝑒1 𝑒28 𝑒61 𝑒7 𝑒18 𝑒25 𝑒40 𝑒1 𝑒28 Waiting Room (FIFO) replace! store discard or or Reservoir (Random Replace)
  • 33. Summary of Algorithm Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 33/87 Introduction Experiments ConclusionAlgorithmPattern 𝑢 | 𝑥 𝑢 | 𝑦 𝑣 | 𝑥 𝑣 | 𝑣 𝑢 − 𝑣 memory new edge (1) Arrival Step 𝑢 | 𝑥 𝑢 | 𝑦 𝑣 | 𝑥 𝑣 | 𝑣 𝑢 − 𝑣 𝑢 − 𝑣 𝑥 ∆← ∆ + 1/𝑝 𝑢𝑣𝑥 discover! (2) Discovery Step 𝑢 | 𝑥 𝑢 | 𝑣 𝑣 | 𝑥 𝑣 | 𝑣 (3) Sampling Step Waiting-Room Sampling!
  • 34. Roadmap • Overview • #1: Counting Triangles in Graph Stream ◦ Introduction ◦ Temporal Pattern ◦ Proposed Algorithm ◦ Experiments << ◦ Conclusion • #2: Detecting Dense Subtensors • Conclusion Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 34/87
  • 35. Experimental Settings • Competitors: Mascot [YK15], Triest-IMPR [DERU16] • Datasets: • Memory budget 𝑘: 10% of the edges • Size of waiting-room: 10% of memory budget Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 35/87 Introduction Experiments ConclusionAlgorithmPattern citation email friendship
  • 36. Distribution of Estimates Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 36/87 • Waiting Room Sampling (WRS) gives ◦ unbiased estimates ◦ with smallest variance Introduction Experiments ConclusionAlgorithmPattern WRS Triest-IMPR MASCOT True Count
  • 37. Discovering Probability • WRS increases discovering probability 𝑝 𝑢𝑣𝑤 • WRS discovers up to 3 × more triangles Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 37/87 Introduction Experiments ConclusionAlgorithmPattern WRS Triest-IMPR MASCOT
  • 38. Estimation Errors • WRS is most accurate • WRS reduces estimation error up to 𝟒𝟕% Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 38/87 Introduction Experiments ConclusionAlgorithmPattern
  • 39. Roadmap • Overview • #1: Counting Triangles in Graph Stream ◦ Introduction ◦ Temporal Pattern ◦ Proposed Algorithm ◦ Experiments ◦ Conclusion << • #2: Detecting Dense Subtensors • Conclusion Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 39/87
  • 40. Contributions (1st Paper) • Pattern: Temporal Locality ◦ short time interval of triangles in real graph streams • Algorithm: Waiting-Room Sampling (WRS) ◦ exploits temporal locality for accurate triangle counting • Analyses: Bias and Variance Analyses ◦ WRS gives unbiased estimates with small variances Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 40/87 0 . 3 7 0 Introduction Experiments ConclusionAlgorithmPattern
  • 41. Further Work • Multiple Sources & Workers (Under Review) • Handling Edge Deletions (Under Preparation) Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 41/87 … , , + , , − , , + , , − , …
  • 42. Roadmap • Overview • #1: Counting Triangles in Graph Stream • #2: Detecting Dense Subtensors << • Conclusion Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 42/87
  • 43. M-Zoom: Fast Dense-Block Detection in Tensors with Quality Guarantees Kijung Shin Bryan Hooi Christos Faloutsos
  • 44. Motivation: Review Fraud Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 44/87 Bob’s Carol’s Alice’s Alice Introduction Experiments ConclusionProposed Method
  • 45. Fraud Forms Dense Blocks Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 45/87 Restaurants AccountsRestaurants Accounts Adjacency Matrix Introduction Experiments ConclusionProposed Method
  • 46. Problem: Natural Dense Blocks • Question. How can we distinguish them? Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 46/87 Restaurants Accounts Adjacency Matrix suspicious dense block formed by fraudsters natural dense block (core, community, etc.) Introduction Experiments ConclusionProposed Method
  • 47. Solution: Tensor Modeling • Natural dense blocks are sparse on the time axis (formed gradually) • Suspicious dense blocks are also dense on the time axis (due to synchronous behavior) • Suspicious dense blocks are denser than natural dense blocks in the tensor model Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 47/87 Restaurants Accounts Introduction Experiments ConclusionProposed Method
  • 48. Solution: Tensor Modeling (Cont.) • Any side information can be used instead of/in addition to time in review datasets • Using multiple side information ⟹ high-order tensors Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 48/87 IP Address keywords Number of stars Introduction Experiments ConclusionProposed Method
  • 49. Anomaly/Fraud in Tensors • Dense blocks signal anomalies/fraud in many tensor datasets Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 49/87 Src IP DstIP Src User DstUser User Page TCP Dumps Wikipedia Revision History Time-evolving Social Network Introduction Experiments ConclusionProposed Method
  • 50. Research Question Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 50/87 “Given a large-scale high-order tensor, how can we find dense subtensors in it?” Introduction Experiments ConclusionProposed Method
  • 51. Road Map Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 51/87 • Overview • #1: Counting Triangles in Graph Stream • #2: Detecting Dense Subtensors ◦ Introduction ◦ Proposed Method: M-Zoom Terminologies and Problem Definition << Algorithm ◦ Experiments ◦ Conclusion • Conclusion
  • 52. Terminologies • Assume a block (subtensor) 𝑩 in a 3-way tensor 𝑹 ◦ 𝑺𝒊𝒛𝒆(𝐵): 𝐼1 + 𝐼2 + 𝐼3 ◦ 𝑽𝒐𝒍 𝐵 = 𝑉𝑜𝑙𝑢𝑚𝑒(𝐵): 𝐼1 × 𝐼2 × 𝐼3 ◦ 𝑴𝒂𝒔𝒔(𝐵): sum of entries in 𝐵 Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 52/87 Introduction Experiments ConclusionProposed Method 𝑅 𝐼1 𝐼2 𝐼3 𝐵
  • 53. Density Measures • Density measures: Traditional Density: ρ 𝑇 𝐵 = 𝑀𝑎𝑠𝑠 𝐵 /Vol(B) - Maximized by a single entry with maximum value Arithmetic Avg. Degree: ρ 𝐴 𝐵 = 𝑀𝑎𝑠𝑠 𝐵 /Size(B) Geometric Avg. Degree: ρ 𝐺 𝐵 = 𝑀𝑎𝑠𝑠 𝐵 / 3 Vol B Suspiciousness (Jiang et al. 2015) : ρ 𝑆 𝐵, 𝑅 = 𝑉𝑜𝑙 𝐵 𝐷 𝐾𝐿 𝑀𝑎𝑠𝑠 𝐵 𝑉𝑜𝑙 𝐵 , 𝑀𝑎𝑠𝑠 𝑅 𝑉𝑜𝑙 𝑅 • Note that our method is not limited by specific density measures Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 53/87 Introduction Experiments ConclusionProposed Method
  • 54. Problem Definition • Given: (1) 𝑹: a tensor, (2) 𝝆: a density measure, (3) 𝒌: the number of blocks we aim to find • Find: 𝒌 distinct dense blocks maximizing 𝝆 Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 54/87 (𝑹 = , 𝝆 = 𝝆 𝑨, 𝒌 = 𝟑) , , }{ Introduction Experiments ConclusionProposed Method
  • 55. Requirements • Our goal is to design an approximation algorithm • M-Zoom, our proposed method, satisfies all the requirements Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 55/87 Scalable: runs in near-linear time Accurate: provides an accuracy guarantee Flexible: works well with various density metrics Effective: produces meaningful results in practice 0 Introduction Experiments ConclusionProposed Method
  • 56. Road Map Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 56/87 • Overview • #1: Counting Triangles in Graph Stream • #2: Detecting Dense Subtensors ◦ Introduction ◦ Proposed Method: M-Zoom Terminologies and Problem Definition Algorithm << ◦ Experiments ◦ Conclusion • Conclusion
  • 57. Single Dense Block Detection • Greedy search method • Starts from the entire tensor Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 57/87 5 3 0 4 6 1 2 0 0 1 0 1 0 0 𝜌 = 2.9 Introduction Experiments ConclusionProposed Method
  • 58. Single Dense Block Detection (cont.) • Remove a slice to maximize density 𝝆 Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 58/87 5 3 0 4 6 1 2 0 0 𝜌 = 3 Introduction Experiments ConclusionProposed Method
  • 59. Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 59/87 5 3 4 6 2 0 𝜌 =3.3 • Remove a slice to maximize density 𝝆 Introduction Experiments ConclusionProposed Method Single Dense Block Detection (cont.)
  • 60. Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 60/87 5 3 4 6 2 0 𝜌 = 3.6 • Remove a slice to maximize density 𝝆 Introduction Experiments ConclusionProposed Method Single Dense Block Detection (cont.)
  • 61. 0 1 2 3 4 0 2 4 6 8 Density Iteration • Until all slices are removed Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 61/87 𝜌 = 0 Introduction Experiments ConclusionProposed Method Single Dense Block Detection (cont.)
  • 62. • Output: return the densest block so far Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 62/87 5 3 4 6 2 0 𝜌 = 3.6 Introduction Experiments ConclusionProposed Method Single Dense Block Detection (cont.)
  • 63. Speeding Up Process • Theorem 1 [Remove Minimum Mass First] Among slices in the same mode, removing the slice with minimum mass is always best Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 63/87 12 > 9 > 2 Introduction Experiments ConclusionProposed Method
  • 64. Accuracy Guarantee • Theorem 2 [Approximation Guarantee] 𝝆 𝑨 𝑩 ≥ 𝟏 𝑵 𝝆 𝑨 𝑩∗ Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 64/87 M-Zoom Result Order Densest Block Scalable: runs in near-linear time Accurate: provides an accuracy guarantee Flexible: works well with various density metrics Effective: produces meaningful results in practice 0 . 3 7 0 Properties of M-Zoom: Introduction Experiments ConclusionProposed Method
  • 65. (Optional) Post Processing • Local Search: ◦ grow(G) or shrink(S) until we reach a local optimum Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 65/87 Introduction Experiments ConclusionProposed Method
  • 66. Handling Multiple Blocks • Deflation: Remove found blocks before finding others Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 66/87 Find & Remove Find & Remove Find & Remove Restore Introduction Experiments ConclusionProposed Method
  • 67. Road Map Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 67/87 • Overview • #1: Counting Triangles in Graph Stream • #2: Detecting Dense Subtensors ◦ Introduction ◦ Proposed Method: M-Zoom ◦ Experiments << ◦ Conclusion • Conclusion
  • 68. Exp1. Scalability Test • Goal: Measure scalability w.r.t. each input factor • M-Zoom scales almost linearly! Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 68/87 #Non-Zeros Order Dimensionality (#Slices) M-Zoom (with post process) M-Zoom Introduction Experiments ConclusionProposed Method
  • 69. Exp1. Scalability Test (cont.) Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 69/87 Scalable: runs in near-linear time Accurate: provides an accuracy guarantee Flexible: works well with various density metrics Effective: produces meaningful results in practice 0 . 3 7 0 Properties of M-Zoom: Introduction Experiments ConclusionProposed Method
  • 70. Exp2. Speed-Accuracy Test Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 70/87 Introduction Experiments ConclusionProposed Method • Goal: compare speed and accuracy of dense-block detection methods with various density measures • Methods Compared: ◦ M-Zoom: Proposed Method ◦ CPD: Tensor Decomposition ◦ CrossSpot (Jiang et al. 2015): Local Search Method
  • 71. Exp2. Speed-Accuracy Test (cont.) Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 71/87 • Data: Music Review (User X Music X Time X Rate) Arithmetic Average Degree (𝜌 𝐴) Geometric Average Degree (𝜌 𝐺) Suspiciousness (𝜌 𝑆) M-Zoom (with post process) M-Zoom Introduction Experiments ConclusionProposed Method
  • 72. Exp2. Speed-Accuracy Test (cont.) Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 72/87 Arithmetic Average Degree (𝜌 𝐴) Geometric Average Degree (𝜌 𝐺) Suspiciousness (𝜌 𝑆) M-Zoom (with post process) M-Zoom • Data: Wikipedia Revision History (User X Page X Time) Introduction Experiments ConclusionProposed Method
  • 73. Exp2. Speed-Accuracy Test (cont.) Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 73/87 • M-Zoom is up to 100× faster than its competitors • M-Zoom shows comparable accuracy with its competitors regardless of density measures Scalable: runs in near-linear time Accurate: provides an accuracy guarantee Flexible: works well with various density metrics Effective: producing meaningful results in practice 0 . 3 7 0 Properties of M-Zoom: Introduction Experiments ConclusionProposed Method
  • 74. Exp3. Discovery in Practice Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 74/87 First three blocks found by M-Zoom 11 users revised 10 pages 2,305 times within 16 hours → Edit wars Introduction Experiments ConclusionProposed Method • Data: Korean Wikipedia Revision History (User X Page X Time)
  • 75. Exp3. Discovery in Practice (cont.) Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 75/87 First three blocks found by M-Zoom 8 accounts revised 12 pages 2.5 million times → Bots Introduction Experiments ConclusionProposed Method • Data: English Wikipedia Revision History (User X Page X Time)
  • 76. Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 76/87 → Network Attacks First three blocks found by M-Zoom Exp3. Discovery in Practice (cont.) Introduction Experiments ConclusionProposed Method • Data: TCP Dump (7-order tensor)
  • 77. Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 77/87 Exp3. Discovery in Practice (cont.) First three blocks found by M-Zoom Introduction Experiments ConclusionProposed Method 9 accounts gives 1 product 369 reviews to in 22 hours → Spam reviews • Data: App Market Review Data (User X Product X Timestamp X Score)
  • 78. Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 78/87 Scalable: runs in near-linear time Accurate: provides an accuracy guarantee Flexible: works well with various density metrics Effective: produces meaningful results in practice 0 . 3 7 0 Properties of M-Zoom: Exp3. Discovery in Practice (cont.) Introduction Experiments ConclusionProposed Method
  • 79. Road Map Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 79/87 • Overview • #1: Counting Triangles in Graph Stream • #2: Detecting Dense Subtensors ◦ Introduction ◦ Proposed Method: M-Zoom ◦ Experiments ◦ Conclusion << • Conclusion
  • 80. Conclusion (2nd Paper) • M-Zoom (Multi-dimensional Zoom): ◦ Dense-Block Detection in tensors Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 80/87 Introduction Experiments ConclusionProposed Method Scalable #Non-Zeros Accurate [Approximation Guarantee] 𝝆 𝑨 𝑩 ≥ 𝟏 𝑵 𝝆 𝑨 𝑩∗ Flexible Effective
  • 81. Related Work • Disk-resident or Distributed Tensors [WSDM17] Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 81/87 #Non-Zeros Proposed (Serial) Proposed (Hadoop) M-Zoom
  • 82. Related Work • Dynamic Tensors [KDD17] Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 82/87 ToePeubot Dvtbot Territory Regionalism Idol Presidential election Religion Disease Politician Dictator Local election TV Show Bot for auto classification Anime Baseball History Edit War Bot Vandalism Bursty edit (informative) Bursty edit (useless)
  • 83. Roadmap • Overview • #1: Counting Triangles in Graph Stream • #2: Detecting Dense Subtensors • Conclusion << Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 83/87
  • 84. Conclusion • Goal: to understand Large Dynamic Graphs and Tensors • Sub-Tasks: ◦ Structure Analysis ◦ Anomaly Detection ◦ Behavior Modeling • Approaches: ◦ Streaming Algorithms (Sampling) ◦ Approximation Algorithms ◦ Distributed Algorithms Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 84/87
  • 85. Thank You • Papers, software, datasets: http://kijungshin.com/ • Email: kijungs@cs.cmu.edu • Thanks to: Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 85/87
  • 86. References • Structure Analysis (Graphs) ◦ Kijung Shin, Tina Eliassi-Rad, and Christos Faloutsos, “CoreScope: Graph Mining Using k-Core Analysis - Patterns, Anomalies and Algorithms”, ICDM 2016 ◦ Kijung Shin, “Mining Large Dynamic Graphs and Tensors for Accurate Triangle Counting in Real Graph Streams”, ICDM 2017 ◦ Kijung Shin, Tina Eliassi-Rad, and Christos Faloutsos, “Patterns and Anomalies in k-Cores of Real-world Graphs with Applications”, KAIS 2018 • Structure Analysis (Tensors) ◦ Kijung Shin and U Kang, “Distributed Methods for High-dimensional and Large-scale Tensor Factorization”, ICDM 2014 ◦ Kijung Shin, Lee Sael, and U Kang, “Fully Scalable Methods for Distributed Tensor Factorization”, TKDE 2017 ◦ Jinoh Oh, Kijung Shin, Evangelos E. Papalexakis, Christos Faloutsos, and Hwanjo Yu, “S-HOT: Scalable High-Order Tucker Decomposition”, WSDM 2017 • Anomaly Detection Analysis (Graphs) ◦ Bryan Hooi, Hyun Ah Song, Alex Beutel, Neil Shah, Kijung Shin, and Christos Faloutsos, “FRAUDAR: Bounding Graph Fraud in the Face of Camouflage”, KDD 2016 ◦ Bryan Hooi, Kijung Shin, Hyun Ah Song, Alex Beutel, Neil Shah, and Christos Faloutsos, “Graph-Based Fraud Detection in the Face of Camouflage”, TKDD 2017 ◦ Kijung Shin, Tina Eliassi-Rad, and Christos Faloutsos, “CoreScope: Graph Mining Using k-Core Analysis - Patterns, Anomalies and Algorithms”, ICDM 2016 [Duplicated] ◦ Kijung Shin, Tina Eliassi-Rad, and Christos Faloutsos, “Patterns and Anomalies in k-Cores of Real-world Graphs with Applications”, KAIS 2018 [Duplicated] Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 86/87
  • 87. References • Anomaly Detection Analysis (Tensors) ◦ Kijung Shin, Bryan Hooi, and Christos Faloutsos, “Mining Large Dynamic Graphs and Tensors (by Kijung Shin)”, ECML/PKDD 2016 ◦ Kijung Shin, Bryan Hooi, Jisu Kim, and Christos Faloutsos, “D-Cube: Dense-Block Detection in Terabyte-Scale Tensors”, WSDM 2017 ◦ Kijung Shin, Bryan Hooi, Jisu Kim, and Christos Faloutsos, “DenseAlert: Incremental Dense-Subtensor Detection in Tensor Streams”, KDD 2017 ◦ Hemank Lamba, Bryan Hooi, Kijung Shin, Christos Faloutsos, and J¨urgen Pfeffer, “ZooRank: Ranking Suspicious Entities in Time-Evolving Tensors”, ECML/PKDD 2017 ◦ Kijung Shin, Bryan Hooi, and Christos Faloutsos, “Fast, Accurate and Flexible Algorithms for Dense Subtensor Mining”, TKDD 2018 • Behavior Modeling ◦ Kijung Shin, Mahdi Shafiei, Myunghwan Kim, Aastha Jain, and Hema Raghavan, “Discovering Progression Stages in Trillion-Scale Behavior Logs”, WWW 2018 ◦ Kijung Shin, Euiwoong Lee, Dhivya Eswaran, and Ariel D. Procaccia, “Why You Should Charge Your Friends for Borrowing Your Stuff”, IJCAI 2017 • Similar Node Search ◦ Kijung Shin, Jinhong Jung, Lee Sael, and U Kang, “BEAR: Block Elimination Approach for Random Walk with Restart on Large Graphs”, SIGMOD 2015 ◦ Jinhong Jung, Kijung Shin, Lee Sael, and U Kang, “Random Walk with Restart on Large Graphs Using Block Elimination”, TODS 2016 Mining Large Dynamic Graphs and Tensors (by Kijung Shin) 87/87