CMU SCS Mining Large Graphs and Tensors - Patterns, Tools and Discoveries.
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1 Mining Large Graphs and Tensors - Patterns, Tools and Discoveries. Christos Faloutsos CMU
2 Thank you! Nikos Sidiropoulos Kuo-Chu Chang Zhi (Gerry) Tian C. Faloutsos (CMU) 2
3 Roadmap Introduction Motivation Why big data Why (big) graphs? Problem#1: Patterns in graphs Problem#2: Tools Conclusions C. Faloutsos (CMU) 3
4 Why? Why big data What is the problem definition? C. Faloutsos (CMU) 4
5 Main message: Big data: often > experts Super Crunchers Why Thinking-By-Numbers is the New Way To Be Smart by Ian Ayres, 2008 Google won the machine translation competition mt05eval_official_results_release_ _v3.html C. Faloutsos (CMU) 5
6 Problem definition big picture Tera/Peta-byte data Analytics Insights, outliers C. Faloutsos (CMU) 6
7 Problem definition big picture Tera/Peta-byte data Analytics Insights, outliers Main emphasis in this talk C. Faloutsos (CMU) 7
8 Roadmap Introduction Motivation Why big data Why (big) graphs? Problem#1: Patterns in graphs Problem#2: Tools Problem#3: Scalability Conclusions C. Faloutsos (CMU) 8
9 Graphs - why should we care? >$10B revenue Food Web [Martinez 91] >0.5B users Internet Map [lumeta.com] C. Faloutsos (CMU) 9
10 Graphs - why should we care? IR: bi-partite graphs (doc-terms) D T 1 web: hyper-text graph D N T M... and more: C. Faloutsos (CMU) 10
11 Graphs - why should we care? web-log ( blog ) news propagation computer network security: /ip traffic and anomaly detection viral marketing Supplier-supply business chains (-> instabilities)... Subject-verb-object -> graph Many-to-many db relationship -> graph C. Faloutsos (CMU) 11
12 Outline Introduction Motivation Problem#1: Patterns in graphs Static graphs Time evolving graphs Problem#2: Tools Conclusions C. Faloutsos (CMU) 12
13 Problem #1 - network and graph mining What does the Internet look like? What does FaceBook look like? What is normal / abnormal? which patterns/laws hold? C. Faloutsos (CMU) 13
14 Problem #1 - network and graph mining What does the Internet look like? What does FaceBook look like? What is normal / abnormal? which patterns/laws hold? To spot anomalies (rarities), we have to discover patterns C. Faloutsos (CMU) 14
15 Problem #1 - network and graph mining What does the Internet look like? What does FaceBook look like? What is normal / abnormal? which patterns/laws hold? To spot anomalies (rarities), we have to discover patterns Large datasets reveal patterns/anomalies that may be invisible otherwise C. Faloutsos (CMU) 15
16 Graph mining Are real graphs random? C. Faloutsos (CMU) 16
17 Laws and patterns Are real graphs random? A: NO!! Diameter in- and out- degree distributions other (surprising) patterns So, let s look at the data C. Faloutsos (CMU) 17
18 Solution# S.1 Power law in the degree distribution [SIGCOMM99] log(degree) ibm.com internet domains att.com log(rank) C. Faloutsos (CMU) 18
19 Solution# S.1 Power law in the degree distribution [SIGCOMM99] log(degree) ibm.com internet domains att.com log(rank) C. Faloutsos (CMU) 19
20 Solution# S.1 Q: So what? log(degree) ibm.com internet domains att.com log(rank) C. Faloutsos (CMU) 20
21 Solution# S.1 Q: So what? A1: # of two-step-away pairs: O(d_max ^2) ~ 10M^2 log(degree) ibm.com internet domains att.com ~0.8PB -> a data center(!) log(rank) C. Faloutsos (CMU) 21
22 Solution# S.1 Q: So what? A1: # of two-step-away pairs: O(d_max ^2) ~ 10M^2 log(degree) ibm.com internet domains att.com ~0.8PB -> a data center(!) log(rank) C. Faloutsos (CMU) 22
23 Solution# S.2: Eigen Exponent E Eigenvalue Exponent = slope E = May 2001 Rank of decreasing eigenvalue A2: power law in the eigenvalues of the adjacency matrix C. Faloutsos (CMU) 23
24 Many more power laws # of sexual contacts Income [Pareto] distribution Duration of downloads [Bestavros+] Duration of UNIX jobs ( mice and elephants ) Size of files of a user Black swans C. Faloutsos (CMU) 24
25 Roadmap Introduction Motivation Problem#1: Patterns in graphs Static graphs degree, diameter, eigen, triangles cliques Weighted graphs Time evolving graphs Problem#2: Tools C. Faloutsos (CMU) 25
26 Solution# S.3: Triangle Laws Real social networks have a lot of triangles C. Faloutsos (CMU) 26
27 Solution# S.3: Triangle Laws Real social networks have a lot of triangles Friends of friends are friends Any patterns? C. Faloutsos (CMU) 27
28 Triangle Law: #S.3 [Tsourakakis ICDM 2008] Reuters SN Epinions X-axis: degree Y-axis: mean # triangles n friends -> ~n 1.6 triangles C. Faloutsos (CMU) 28
29 details Triangle Law: Computations [Tsourakakis ICDM 2008] But: triangles are expensive to compute (3-way join; several approx. algos) O(d max2 ) Q: Can we do that quickly? A: C. Faloutsos (CMU) 29
30 details Triangle Law: Computations [Tsourakakis ICDM 2008] But: triangles are expensive to compute (3-way join; several approx. algos) O(d max2 ) Q: Can we do that quickly? A: Yes! #triangles = 1/6 Sum ( λ i 3 ) (and, because of skewness (S2), we only need the top few eigenvalues! - O(E) C. Faloutsos (CMU) 30
31 details Triangle Law: Computations [Tsourakakis ICDM 2008] 1000x+ speed-up, >90% accuracy C. Faloutsos (CMU) 31
32 Triangle counting for large graphs???? Anomalous nodes in Twitter(~ 3 billion edges) [U Kang, Brendan Meeder, +, PAKDD 11] C. Faloutsos (CMU) 32
33 Triangle counting for large graphs? Anomalous nodes in Twitter(~ 3 billion edges) [U Kang, Brendan Meeder, +, PAKDD 11] C. Faloutsos (CMU) 33
34 Triangle counting for large graphs? Anomalous nodes in Twitter(~ 3 billion edges) [U Kang, Brendan Meeder, +, PAKDD 11] C. Faloutsos (CMU) 34
35 Triangle counting for large graphs? Anomalous nodes in Twitter(~ 3 billion edges) [U Kang, Brendan Meeder, +, PAKDD 11] C. Faloutsos (CMU) 35
36 Roadmap Introduction Motivation Problem#1: Patterns in graphs Static graphs Time evolving graphs Problem#2: Tools C. Faloutsos (CMU) 36
37 T.1 : popularity over time # in links lag: days after post Post popularity + lag C. Faloutsos (CMU) 37
38 T.1 : popularity over time # in links (log) Post popularity drops-off exponentially? POWER LAW! Exponent? days after post (log) C. Faloutsos (CMU) 38
39 T.1 : popularity over time # in links (log) -1.6 Post popularity drops-off exponentially? POWER LAW! Exponent? -1.6 close to -1.5: Barabasi s stack model and like the zero-crossings of a random walk days after post (log) C. Faloutsos (CMU) 39
40 -1.5 slope J. G. Oliveira & A.-L. Barabási Human Dynamics: The Correspondence Patterns of Darwin and Einstein. Nature 437, 1251 (2005). [PDF] Prob(RT > x) (log) Response time (log) C. Faloutsos (CMU) 40
41 -1.5 slope J. G. Oliveira & A.-L. Barabási Human Dynamics: The Correspondence Patterns of Darwin and Einstein. Nature 437, 1251 (2005). [PDF] C. Faloutsos (CMU) 41
42 Roadmap Introduction Motivation Problem#1: Patterns in graphs Problem#2: Tools (Belief Propagation) Tensors Spike analysis Conclusions C. Faloutsos (CMU) 42
43 GigaTensor: Scaling Tensor Analysis Up By 100 Times Algorithms and Discoveries U Kang Evangelos Papalexakis Abhay Harpale Christos Faloutsos KDD 12 C. Faloutsos (CMU) 43
44 Background: Tensor Tensors (=multi-dimensional arrays) are everywhere Hyperlinks &anchor text [Kolda+,05] Anchor Text URL 2 URL Java C++ C# C. Faloutsos (CMU) 44
45 Time evolving graphs: Tensors date caller callee C. Faloutsos (CMU) 45
46 Background: Tensor Tensors (=multi-dimensional arrays) are everywhere Sensor stream (time, location, type) Predicates (subject, verb, object) in knowledge base Eric Clapton plays guitar Barrack Obama is the president of U.S. (48M) (26M) (26M) NELL (Never Ending Language Learner) data Nonzeros =144M C. Faloutsos (CMU) 46
47 Background: Tensor Tensors (=multi-dimensional arrays) are everywhere Sensor stream (time, location, type) Predicates (subject, verb, object) in knowledge base Time-stamp IP-source Anomaly Detection in Computer networks C. IP-destination Faloutsos (CMU) 47
48 all I learned on tensors: from Nikos Sidiropoulos UMN Tamara Kolda, Sandia Labs (tensor toolbox) C. Faloutsos (CMU) 48
49 Problem Definition How to decompose a billion-scale tensor? Corresponds to SVD in 2D case C. Faloutsos (CMU) 49
50 Problem Definition How to decompose a billion-scale tensor? Corresponds to SVD in 2D case = soft clustering C. Faloutsos (CMU) 50
51 Problem Definition Q1: Dominant concepts/topics? Q2: Find synonyms to a given noun phrase? (and how to scale up: data > RAM) (48M) (26M) (26M) NELL (Never Ending Language Learner) data Nonzeros =144M C. Faloutsos (CMU) 51
52 Experiments GigaTensor solves 100x larger problem GigaTensor Out of Memory 100x (K) (J) (I) Number of nonzero = I / 50 C. Faloutsos (CMU) 52
53 A1: Concept Discovery Concept Discovery in Knowledge Base C. Faloutsos (CMU) 53
54 A1: Concept Discovery C. Faloutsos (CMU) 54
55 A2: Synonym Discovery C. Faloutsos (CMU) 55
56 Roadmap Introduction Motivation Problem#1: Patterns in graphs Problem#2: Tools Belief propagation Tensors Spike analysis Graph summarization Conclusions C. Faloutsos (CMU) 56
57 Rise and fall patterns in social media Meme (# of mentions in blogs) short phrases Sourced from U.S. politics in 2008 you can put lipstick on a pig yes we can C. Faloutsos (CMU) 57
58 Rise and fall patterns in social media Can we find a unifying model, which includes these patterns? four classes on YouTube [Crane et al. 08] six classes on Meme [Yang et al. 11] C. Faloutsos (CMU) 58
59 Rise and fall patterns in social media Answer: YES! We can represent all patterns by single model C. Faloutsos (CMU) In Matsubara+ SIGKDD
60 Main idea - SpikeM - 1. Un-informed bloggers (uninformed about rumor) - 2. External shock at time nb (e.g, breaking news) - 3. Infection (word-of-mouth) Time n=0 Time n=nb Time n=nb+1 Infectiveness of a blog-post at age n: - Strength of infection (quality of news) - Decay function C. Faloutsos (CMU) 60
61 Main idea - SpikeM - 1. Un-informed bloggers (uninformed about rumor) - 2. External shock at time nb (e.g, breaking news) - 3. Infection (word-of-mouth) Time n=0 Time n=nb Time n=nb+1 Infectiveness of a blog-post at age n: - Strength of infection (quality of news) - Decay function C. Faloutsos (CMU) 61
62 SpikeM - with periodicity Full equation of SpikeM Periodicity Bloggers change their activity over time (e.g., daily, weekly, yearly) activity noon Peak 3am Dip C. Faloutsos (CMU) Time n 62
63 Details Analysis exponential rise and power-raw fall Lin-log Log-log C. Faloutsos (CMU) Rise-part SI -> exponential SpikeM -> exponential 63
64 Details Analysis exponential rise and power-raw fall Fall-part SI -> exponential SpikeM -> power law Lin-log Log-log C. Faloutsos (CMU) 64
65 Tail-part forecasts SpikeM can capture tail part C. Faloutsos (CMU) 65
66 What-if forecasting (1) First spike (2) Release date (3) Two weeks before release?? e.g., given (1) first spike, (2) release date of two sequel movies (3) access volume before the release date C. Faloutsos (CMU) 66
67 What-if forecasting (1) First spike (2) Release date (3) Two weeks before release SpikeM can forecast upcoming spikes C. Faloutsos (CMU) 67
68 Roadmap Introduction Motivation Problem#1: Patterns in graphs Problem#2: Tools Belief Propagation Tensors Spike analysis Graph understanding (through MDL) Conclusions C. Faloutsos (CMU) 68
69 Goal: Summarizing Graphs?? Main Idea: MDL + syllables : star, clique, chain, bi-partite core Koutra, Kang, Vreeken, et al, (subm.)
70 Summarizing Wiki-controversy top-8 stars: admins, bots top-1 and top-2 bipartite cores: edit wars. Left: warring factions ( Kiev vs Kyev ) Right: between vandals
71 Roadmap Introduction Motivation Problem#1: Patterns in graphs Problem#2: Tools Conclusions C. Faloutsos (CMU) 71
72 OVERALL CONCLUSIONS low level: Several new patterns (power laws, trianglelaws, etc) New tools: belief propagation, gigatensor, etc Scalability: PEGASUS / hadoop C. Faloutsos (CMU) 72
73 OVERALL CONCLUSIONS high level BIG DATA: Large datasets reveal patterns/ outliers that are invisible otherwise C. Faloutsos (CMU) 73
74 Theory & Algo. Biology Physics Comp. Systems ML, Stats., DSP (Graph) Analytics Social Science Econ. 74 C. Faloutsos (CMU)
75 Theory & Algo. Biology Physics Comp. Systems ML & Stats. (Graph) Analytics Social Science Econ. 75 C. Faloutsos (CMU)
76 References Leman Akoglu, Christos Faloutsos: RTG: A Recursive Realistic Graph Generator Using Random Typing. ECML/PKDD (1) 2009: Deepayan Chakrabarti, Christos Faloutsos: Graph mining: Laws, generators, and algorithms. ACM Comput. Surv. 38(1): (2006) C. Faloutsos (CMU) 76
77 References D. Chakrabarti, C. Faloutsos: Graph Mining Laws, Tools and Case Studies, Morgan Claypool S00449ED1V01Y201209DMK006 C. Faloutsos (CMU) 77
78 References Deepayan Chakrabarti, Yang Wang, Chenxi Wang, Jure Leskovec, Christos Faloutsos: Epidemic thresholds in real networks. ACM Trans. Inf. Syst. Secur. 10(4): (2008) Deepayan Chakrabarti, Jure Leskovec, Christos Faloutsos, Samuel Madden, Carlos Guestrin, Michalis Faloutsos: Information Survival Threshold in Sensor and P2P Networks. INFOCOM 2007: C. Faloutsos (CMU) 78
79 References Christos Faloutsos, Tamara G. Kolda, Jimeng Sun: Mining large graphs and streams using matrix and tensor tools. Tutorial, SIGMOD Conference 2007: 1174 C. Faloutsos (CMU) 79
80 References T. G. Kolda and J. Sun. Scalable Tensor Decompositions for Multi-aspect Data Mining. In: ICDM 2008, pp , December C. Faloutsos (CMU) 80
81 References Jure Leskovec, Jon Kleinberg and Christos Faloutsos Graphs over Time: Densification Laws, Shrinking Diameters and Possible Explanations, KDD 2005 (Best Research paper award). Jure Leskovec, Deepayan Chakrabarti, Jon M. Kleinberg, Christos Faloutsos: Realistic, Mathematically Tractable Graph Generation and Evolution, Using Kronecker Multiplication. PKDD 2005: C. Faloutsos (CMU) 81
82 References Yasuko Matsubara, Yasushi Sakurai, B. Aditya Prakash, Lei Li, Christos Faloutsos, "Rise and Fall Patterns of Information Diffusion: Model and Implications", KDD 12, pp. 6-14, Beijing, China, August 2012 C. Faloutsos (CMU) 82
83 References Jimeng Sun, Yinglian Xie, Hui Zhang, Christos Faloutsos. Less is More: Compact Matrix Decomposition for Large Sparse Graphs, SDM, Minneapolis, Minnesota, Apr Jimeng Sun, Spiros Papadimitriou, Philip S. Yu, and Christos Faloutsos, GraphScope: Parameterfree Mining of Large Time-evolving Graphs ACM SIGKDD Conference, San Jose, CA, August 2007 C. Faloutsos (CMU) 83
84 References Jimeng Sun, Dacheng Tao, Christos Faloutsos: Beyond streams and graphs: dynamic tensor analysis. KDD 2006: C. Faloutsos (CMU) 84
85 References Hanghang Tong, Christos Faloutsos, and Jia-Yu Pan, Fast Random Walk with Restart and Its Applications, ICDM 2006, Hong Kong. Hanghang Tong, Christos Faloutsos, Center-Piece Subgraphs: Problem Definition and Fast Solutions, KDD 2006, Philadelphia, PA C. Faloutsos (CMU) 85
86 References Hanghang Tong, Christos Faloutsos, Brian Gallagher, Tina Eliassi-Rad: Fast best-effort pattern matching in large attributed graphs. KDD 2007: (Best paper award, CIKM'12) Hanghang Tong, B. Aditya Prakash, Tina Eliassi-Rad, Michalis Faloutsos and Christos Faloutsos Gelling, and Melting, Large Graphs by Edge Manipulation, Maui, Hawaii, USA, NSF, Oct. 3/ C. Faloutsos (CMU) 86
87 References Hanghang Tong, Spiros Papadimitriou, Christos Faloutsos, Philip S. Yu, Tina Eliassi-Rad: Gateway finder in large graphs: problem definitions and fast solutions. Inf. Retr. 15(3-4): (2012) C. Faloutsos (CMU) 87
88 Project info & thanks Thanks to: NSF IIS , IIS , CTA-INARC; Yahoo (M45), LLNL, IBM, SPRINT, C. Faloutsos (CMU) 88 Google, INTEL, HP, ilab
89 Cast Akoglu, Leman Beutel, Alex Chau, Polo Kang, U Koutra, Danai McGlohon, Mary Prakash, Aditya Papalexakis, Vagelis Tong, Hanghang C. Faloutsos (CMU) 89
90 Take-home message Tera/Peta-byte data Analytics Insights, outliers Big data reveal insights that would be invisible otherwise (even to experts) C. Faloutsos (CMU) 90
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