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CMU SCS 15-826: Multimedia Databases and Data Mining Lecture #26: Graph mining - patterns Christos Faloutsos

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CMU SCS Must-read Material Michalis Faloutsos, Petros Faloutsos and Christos Faloutsos, On Power-Law Relationships of the Internet Topology, SIGCOMM 1999. R. Albert, H. Jeong, and A.-L. Barabasi, Diameter of the World Wide Web Nature, 401, 130-131 (1999). Reka Albert and Albert-Laszlo Barabasi Statistical mechanics of complex networks, Reviews of Modern Physics, 74, 47 (2002). Jure Leskovec, Jon Kleinberg, Christos Faloutsos Graphs over Time: Densification Laws, Shrinking Diameters and Possible Explanations, KDD 2005, Chicago, IL, USA 2 15-826(c) 2012 C. Faloutsos

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CMU SCS Must-read Material (cont’d) D. Chakrabarti and C. Faloutsos, Graph Mining: Laws, Generators and Algorithms, in ACM Computing Surveys, 38(1), 2006 J. Leskovec, D. Chakrabarti, J. Kleinberg, and C. Faloutsos, Realistic, Mathematically Tractable Graph Generation and Evolution, Using Kronecker Multiplication, in PKDD 2005, Porto, Portugal 3 15-826(c) 2012 C. Faloutsos

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CMU SCS (c) 2012 C. Faloutsos 4 Outline Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools Problem#3: Scalability Conclusions 15-826

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CMU SCS (c) 2012 C. Faloutsos 5 Graphs - why should we care? Internet Map [lumeta.com] Food Web [Martinez ’91] Friendship Network [Moody ’01] 15-826

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CMU SCS (c) 2012 C. Faloutsos 6 Graphs - why should we care? IR: bi-partite graphs (doc-terms) web: hyper-text graph... and more: D1D1 DNDN T1T1 TMTM... 15-826

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CMU SCS (c) 2012 C. Faloutsos 7 Graphs - why should we care? ‘viral’ marketing web-log (‘blog’) news propagation computer network security: email/IP traffic and anomaly detection.... 15-826

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CMU SCS (c) 2012 C. Faloutsos 8 Outline Introduction – Motivation Problem#1: Patterns in graphs –Static graphs –Weighted graphs –Time evolving graphs Problem#2: Tools Problem#3: Scalability Conclusions 15-826

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CMU SCS (c) 2012 C. Faloutsos 9 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? 15-826

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CMU SCS (c) 2012 C. Faloutsos 10 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 15-826

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CMU SCS (c) 2012 C. Faloutsos 11 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… 15-826

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CMU SCS 12 Are real graphs random? random (Erdos-Renyi) graph – 100 nodes, avg degree = 2 before layout after layout No obvious patterns (generated with: pajek http://vlado.fmf.uni-lj.si/pub/networks/pajek/ ) 15-826(c) 2012 C. Faloutsos

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CMU SCS (c) 2012 C. Faloutsos 13 Graph mining Are real graphs random? 15-826

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CMU SCS (c) 2012 C. Faloutsos 14 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 15-826

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CMU SCS (c) 2012 C. Faloutsos 15 Solution# S.1 Power law in the degree distribution [SIGCOMM99] log(rank) log(degree) internet domains att.com ibm.com 15-826

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CMU SCS (c) 2012 C. Faloutsos 16 Solution# S.1 Power law in the degree distribution [SIGCOMM99] log(rank) log(degree) -0.82 internet domains att.com ibm.com 15-826

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CMU SCS (c) 2012 C. Faloutsos 17 Solution# S.2: Eigen Exponent E A2: power law in the eigenvalues of the adjacency matrix E = -0.48 Exponent = slope Eigenvalue Rank of decreasing eigenvalue May 2001 15-826

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CMU SCS (c) 2012 C. Faloutsos 18 Solution# S.2: Eigen Exponent E [Mihail, Papadimitriou ’02]: slope is ½ of rank exponent E = -0.48 Exponent = slope Eigenvalue Rank of decreasing eigenvalue May 2001 15-826

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CMU SCS (c) 2012 C. Faloutsos 19 But: How about graphs from other domains? 15-826

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CMU SCS (c) 2012 C. Faloutsos 20 More power laws: web hit counts [w/ A. Montgomery] Web Site Traffic in-degree (log scale) Count (log scale) Zipf users sites ``ebay’’ 15-826

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CMU SCS (c) 2012 C. Faloutsos 21 epinions.com who-trusts-whom [Richardson + Domingos, KDD 2001] (out) degree count trusts-2000-people user 15-826

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CMU SCS And numerous more # of sexual contacts Income [Pareto] –’80-20 distribution’ Duration of downloads [Bestavros+] Duration of UNIX jobs (‘mice and elephants’) Size of files of a user … ‘Black swans’ 15-826(c) 2012 C. Faloutsos 22

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CMU SCS (c) 2012 C. Faloutsos 23 Outline Introduction – Motivation Problem#1: Patterns in graphs –Static graphs degree, diameter, eigen, triangles cliques –Weighted graphs –Time evolving graphs Problem#2: Tools 15-826

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CMU SCS (c) 2012 C. Faloutsos 24 Solution# S.3: Triangle ‘Laws’ Real social networks have a lot of triangles 15-826

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CMU SCS (c) 2012 C. Faloutsos 25 Solution# S.3: Triangle ‘Laws’ Real social networks have a lot of triangles –Friends of friends are friends Any patterns? 15-826

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CMU SCS (c) 2012 C. Faloutsos 26 Triangle Law: #S.3 [Tsourakakis ICDM 2008] ASNHEP-TH Epinions X-axis: # of participating triangles Y: count (~ pdf) 15-826

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CMU SCS (c) 2012 C. Faloutsos 27 Triangle Law: #S.3 [Tsourakakis ICDM 2008] ASNHEP-TH Epinions 15-826 X-axis: # of participating triangles Y: count (~ pdf)

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CMU SCS (c) 2012 C. Faloutsos 28 Triangle Law: #S.4 [Tsourakakis ICDM 2008] SNReuters Epinions X-axis: degree Y-axis: mean # triangles n friends -> ~n 1.6 triangles 15-826

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CMU SCS (c) 2012 C. Faloutsos 29 Triangle Law: Computations [Tsourakakis ICDM 2008] But: triangles are expensive to compute (3-way join; several approx. algos) Q: Can we do that quickly? details 15-826

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CMU SCS (c) 2012 C. Faloutsos 30 Triangle Law: Computations [Tsourakakis ICDM 2008] But: triangles are expensive to compute (3-way join; several approx. algos) Q: Can we do that quickly? A: Yes! #triangles = 1/6 Sum ( i 3 ) (and, because of skewness (S2), we only need the top few eigenvalues! details 15-826

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CMU SCS (c) 2012 C. Faloutsos 31 Triangle Law: Computations [Tsourakakis ICDM 2008] 1000x+ speed-up, >90% accuracy details 15-826

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CMU SCS Triangle counting for large graphs? Anomalous nodes in Twitter(~ 3 billion edges) [U Kang, Brendan Meeder, +, PAKDD’11] 32 15-826 32 (c) 2012 C. Faloutsos

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CMU SCS Triangle counting for large graphs? Anomalous nodes in Twitter(~ 3 billion edges) [U Kang, Brendan Meeder, +, PAKDD’11] 33 15-826 33 (c) 2012 C. Faloutsos

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CMU SCS 34 Any other ‘laws’? Yes! 15-826(c) 2012 C. Faloutsos

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CMU SCS 35 Any other ‘laws’? Yes! Small diameter (~ constant!) – –six degrees of separation / ‘Kevin Bacon’ –small worlds [Watts and Strogatz] 15-826(c) 2012 C. Faloutsos

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CMU SCS 36 Any other ‘laws’? Bow-tie, for the web [Kumar+ ‘99] IN, SCC, OUT, ‘tendrils’ disconnected components 15-826(c) 2012 C. Faloutsos

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CMU SCS 37 Any other ‘laws’? power-laws in communities (bi-partite cores) [Kumar+, ‘99] 2:3 core (m:n core) Log(m) Log(count) n:1 n:2 n:3 15-826(c) 2012 C. Faloutsos

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CMU SCS 38 Any other ‘laws’? “Jellyfish” for Internet [Tauro+ ’01] core: ~clique ~5 concentric layers many 1-degree nodes 15-826(c) 2012 C. Faloutsos

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CMU SCS EigenSpokes B. Aditya Prakash, Mukund Seshadri, Ashwin Sridharan, Sridhar Machiraju and Christos Faloutsos: EigenSpokes: Surprising Patterns and Scalable Community Chipping in Large Graphs, PAKDD 2010, Hyderabad, India, 21-24 June 2010. (c) 2012 C. Faloutsos 39 15-826

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CMU SCS EigenSpokes Eigenvectors of adjacency matrix equivalent to singular vectors (symmetric, undirected graph) 40 (c) 2012 C. Faloutsos15-826

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CMU SCS EigenSpokes Eigenvectors of adjacency matrix equivalent to singular vectors (symmetric, undirected graph) 41 (c) 2012 C. Faloutsos15-826 N N details

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CMU SCS EigenSpokes Eigenvectors of adjacency matrix equivalent to singular vectors (symmetric, undirected graph) 42 (c) 2012 C. Faloutsos15-826 N N details

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CMU SCS EigenSpokes Eigenvectors of adjacency matrix equivalent to singular vectors (symmetric, undirected graph) 43 (c) 2012 C. Faloutsos15-826 N N details

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CMU SCS EigenSpokes Eigenvectors of adjacency matrix equivalent to singular vectors (symmetric, undirected graph) 44 (c) 2012 C. Faloutsos15-826 N N details

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CMU SCS EigenSpokes EE plot: Scatter plot of scores of u1 vs u2 One would expect –Many points @ origin –A few scattered ~randomly (c) 2012 C. Faloutsos 45 u1 u2 15-826 1 st Principal component 2 nd Principal component

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CMU SCS EigenSpokes EE plot: Scatter plot of scores of u1 vs u2 One would expect –Many points @ origin –A few scattered ~randomly (c) 2012 C. Faloutsos 46 u1 u2 90 o 15-826

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CMU SCS EigenSpokes - pervasiveness Present in mobile social graph across time and space Patent citation graph 47 (c) 2012 C. Faloutsos15-826

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CMU SCS EigenSpokes - explanation Near-cliques, or near- bipartite-cores, loosely connected 48 (c) 2012 C. Faloutsos15-826

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CMU SCS EigenSpokes - explanation Near-cliques, or near- bipartite-cores, loosely connected 49 (c) 2012 C. Faloutsos15-826

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CMU SCS EigenSpokes - explanation Near-cliques, or near- bipartite-cores, loosely connected 50 (c) 2012 C. Faloutsos15-826

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CMU SCS EigenSpokes - explanation Near-cliques, or near- bipartite-cores, loosely connected So what? Extract nodes with high scores high connectivity Good “communities” spy plot of top 20 nodes 51 (c) 2012 C. Faloutsos15-826

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CMU SCS Bipartite Communities! magnified bipartite community patents from same inventor(s) `cut-and-paste’ bibliography! 52 (c) 2012 C. Faloutsos15-826

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CMU SCS (c) 2012 C. Faloutsos 53 Outline Introduction – Motivation Problem#1: Patterns in graphs –Static graphs degree, diameter, eigen, triangles cliques –Weighted graphs –Time evolving graphs Problem#2: Tools 15-826

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CMU SCS (c) 2012 C. Faloutsos 54 Observations on weighted graphs? A: yes - even more ‘laws’! M. McGlohon, L. Akoglu, and C. Faloutsos Weighted Graphs and Disconnected Components: Patterns and a Generator. SIG-KDD 2008 15-826

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CMU SCS (c) 2012 C. Faloutsos 55 Observation W.1: Fortification Q: How do the weights of nodes relate to degree? 15-826

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CMU SCS (c) 2012 C. Faloutsos 56 Observation W.1: Fortification More donors, more $ ? $10 $5 15-826 ‘Reagan’ ‘Clinton’ $7

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CMU SCS Edges (# donors) In-weights ($) (c) 2012 C. Faloutsos 57 Observation W.1: fortification: Snapshot Power Law Weight: super-linear on in-degree exponent ‘iw’: 1.01 < iw < 1.26 Orgs-Candidates e.g. John Kerry, $10M received, from 1K donors More donors, even more $ $10 $5 15-826

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CMU SCS (c) 2012 C. Faloutsos 58 Outline Introduction – Motivation Problem#1: Patterns in graphs –Static graphs –Weighted graphs –Time evolving graphs Problem#2: Tools … 15-826

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CMU SCS (c) 2012 C. Faloutsos 59 Problem: Time evolution with Jure Leskovec (CMU -> Stanford) and Jon Kleinberg (Cornell – sabb. @ CMU) 15-826

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CMU SCS (c) 2012 C. Faloutsos 60 T.1 Evolution of the Diameter Prior work on Power Law graphs hints at slowly growing diameter: –diameter ~ O(log N) –diameter ~ O(log log N) What is happening in real data? 15-826

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CMU SCS (c) 2012 C. Faloutsos 61 T.1 Evolution of the Diameter Prior work on Power Law graphs hints at slowly growing diameter: –diameter ~ O(log N) –diameter ~ O(log log N) What is happening in real data? Diameter shrinks over time 15-826

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CMU SCS (c) 2012 C. Faloutsos 62 T.1 Diameter – “Patents” Patent citation network 25 years of data @1999 –2.9 M nodes –16.5 M edges time [years] diameter 15-826

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CMU SCS (c) 2012 C. Faloutsos 63 T.2 Temporal Evolution of the Graphs N(t) … nodes at time t E(t) … edges at time t Suppose that N(t+1) = 2 * N(t) Q: what is your guess for E(t+1) =? 2 * E(t) 15-826

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CMU SCS (c) 2012 C. Faloutsos 64 T.2 Temporal Evolution of the Graphs N(t) … nodes at time t E(t) … edges at time t Suppose that N(t+1) = 2 * N(t) Q: what is your guess for E(t+1) =? 2 * E(t) A: over-doubled! –But obeying the ``Densification Power Law’’ 15-826

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CMU SCS (c) 2012 C. Faloutsos 65 T.2 Densification – Patent Citations Citations among patents granted @1999 –2.9 M nodes –16.5 M edges Each year is a datapoint N(t) E(t) 1.66 15-826

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CMU SCS (c) 2012 C. Faloutsos 66 Outline Introduction – Motivation Problem#1: Patterns in graphs –Static graphs –Weighted graphs –Time evolving graphs Problem#2: Tools … 15-826

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CMU SCS (c) 2012 C. Faloutsos 67 More on Time-evolving graphs M. McGlohon, L. Akoglu, and C. Faloutsos Weighted Graphs and Disconnected Components: Patterns and a Generator. SIG-KDD 2008 15-826

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CMU SCS 68 [ Gelling Point ] Most real graphs display a gelling point After gelling point, they exhibit typical behavior. This is marked by a spike in diameter. Time Diameter IMDB t=1914 15-826(c) 2012 C. Faloutsos

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CMU SCS (c) 2012 C. Faloutsos 69 Observation T.3: NLCC behavior Q: How do NLCC’s emerge and join with the GCC? (``NLCC’’ = non-largest conn. components) –Do they continue to grow in size? – or do they shrink? – or stabilize? 15-826

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CMU SCS (c) 2012 C. Faloutsos 70 Observation T.3: NLCC behavior Q: How do NLCC’s emerge and join with the GCC? (``NLCC’’ = non-largest conn. components) –Do they continue to grow in size? – or do they shrink? – or stabilize? 15-826

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CMU SCS (c) 2012 C. Faloutsos 71 Observation T.3: NLCC behavior Q: How do NLCC’s emerge and join with the GCC? (``NLCC’’ = non-largest conn. components) –Do they continue to grow in size? – or do they shrink? – or stabilize? 15-826 YES

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CMU SCS (c) 2012 C. Faloutsos 72 Observation T.3: NLCC behavior After the gelling point, the GCC takes off, but NLCC’s remain ~constant (actually, oscillate). IMDB CC size Time-stamp 15-826

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CMU SCS (c) 2012 C. Faloutsos 73 Timing for Blogs with Mary McGlohon (CMU->Google) Jure Leskovec (CMU->Stanford) Natalie Glance (now at Google) Mat Hurst (now at MSR) [SDM’07] 15-826

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CMU SCS (c) 2012 C. Faloutsos 74 T.4 : popularity over time Post popularity drops-off – exponentially? lag: days after post # in links 1 2 3 @t @t + lag 15-826

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CMU SCS (c) 2012 C. Faloutsos 75 T.4 : popularity over time Post popularity drops-off – exponentially? POWER LAW! Exponent? # in links (log) days after post (log) 15-826

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CMU SCS (c) 2012 C. Faloutsos 76 T.4 : popularity over time 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 # in links (log) -1.6 days after post (log) 15-826

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CMU SCS 15-826(c) 2012 C. Faloutsos 77 -1.5 slope J. G. Oliveira & A.-L. Barabási Human Dynamics: The Correspondence Patterns of Darwin and Einstein. Nature 437, 1251 (2005). [PDF]PDF

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CMU SCS T.5: duration of phonecalls Surprising Patterns for the Call Duration Distribution of Mobile Phone Users Pedro O. S. Vaz de Melo, Leman Akoglu, Christos Faloutsos, Antonio A. F. Loureiro PKDD 2010 15-826(c) 2012 C. Faloutsos 78

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CMU SCS Probably, power law (?) 15-826(c) 2012 C. Faloutsos 79 ??

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CMU SCS No Power Law! 15-826(c) 2012 C. Faloutsos 80

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CMU SCS ‘TLaC: Lazy Contractor’ The longer a task (phonecall) has taken, The even longer it will take 15-826(c) 2012 C. Faloutsos 81 Odds ratio= Casualties(

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CMU SCS 82 Data Description Data from a private mobile operator of a large city 4 months of data 3.1 million users more than 1 billion phone records Over 96% of ‘talkative’ users obeyed a TLAC distribution (‘talkative’: >30 calls) 15-826(c) 2012 C. Faloutsos

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CMU SCS Outliers: 15-826(c) 2012 C. Faloutsos 83

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CMU SCS (c) 2012 C. Faloutsos 84 Outline Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools –OddBall (anomaly detection) –Belief Propagation –Immunization Problem#3: Scalability Conclusions 15-826

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CMU SCS OddBall: Spotting A n o m a l i e s in Weighted Graphs Leman Akoglu, Mary McGlohon, Christos Faloutsos Carnegie Mellon University School of Computer Science PAKDD 2010, Hyderabad, India

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CMU SCS Main idea For each node, extract ‘ego-net’ (=1-step-away neighbors) Extract features (#edges, total weight, etc etc) Compare with the rest of the population (c) 2012 C. Faloutsos 86 15-826

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CMU SCS What is an egonet? ego 87 egonet (c) 2012 C. Faloutsos15-826

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CMU SCS Selected Features N i : number of neighbors (degree) of ego i E i : number of edges in egonet i W i : total weight of egonet i λ w,i : principal eigenvalue of the weighted adjacency matrix of egonet I 88 (c) 2012 C. Faloutsos15-826

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CMU SCS Near-Clique/Star 89 15-826(c) 2012 C. Faloutsos

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CMU SCS Near-Clique/Star 90 (c) 2012 C. Faloutsos15-826

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CMU SCS Near-Clique/Star 91 (c) 2012 C. Faloutsos15-826

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CMU SCS Andrew Lewis (director) Near-Clique/Star 92 (c) 2012 C. Faloutsos15-826

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CMU SCS (c) 2012 C. Faloutsos 93 Outline Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools –OddBall (anomaly detection) –Belief Propagation –Immunization Problem#3: Scalability Conclusions 15-826

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CMU SCS 15-826(c) 2012 C. Faloutsos 94 E-bay Fraud detection w/ Polo Chau & Shashank Pandit, CMU [www’07]

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CMU SCS 15-826(c) 2012 C. Faloutsos 95 E-bay Fraud detection

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CMU SCS 15-826(c) 2012 C. Faloutsos 96 E-bay Fraud detection

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CMU SCS 15-826(c) 2012 C. Faloutsos 97 E-bay Fraud detection - NetProbe

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CMU SCS Popular press And less desirable attention: E-mail from ‘Belgium police’ (‘copy of your code?’) 15-826(c) 2012 C. Faloutsos 98

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CMU SCS (c) 2012 C. Faloutsos 99 Outline Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools –OddBall (anomaly detection) –Belief Propagation – antivirus app –Immunization Problem#3: Scalability Conclusions 15-826

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CMU SCS Polo Chau Machine Learning Dept Carey Nachenberg Vice President & Fellow Jeffrey Wilhelm Principal Software Engineer Adam Wright Software Engineer Prof. Christos Faloutsos Computer Science Dept Polonium: Tera-Scale Graph Mining and Inference for Malware Detection SDM 2011, Mesa, Arizona

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CMU SCS Polonium: The Data 60+ terabytes of data anonymously contributed by participants of worldwide Norton Community Watch program 50+ million machines 900+ million executable files Constructed a machine-file bipartite graph (0.2 TB+) 1 billion nodes (machines and files) 37 billion edges 15-826 101 (c) 2012 C. Faloutsos

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CMU SCS Polonium: Key Ideas Use Belief Propagation to propagate domain knowledge in machine-file graph to detect malware Use “guilt-by-association” (i.e., homophily) –E.g., files that appear on machines with many bad files are more likely to be bad Scalability: handles 37 billion-edge graph 15-826 102 (c) 2012 C. Faloutsos

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CMU SCS Polonium: One-Interaction Results 84.9% True Positive Rate 1% False Positive Rate True Positive Rate % of malware correctly identified 103 Ideal 15-826(c) 2012 C. Faloutsos False Positive Rate % of non-malware wrongly labeled as malware

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CMU SCS (c) 2012 C. Faloutsos 104 Outline Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools –OddBall (anomaly detection) –Belief propagation –Immunization Problem#3: Scalability -PEGASUS Conclusions 15-826

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CMU SCS Immunization and epidemic thresholds Q1: which nodes to immunize? Q2: will a virus vanish, or will it create an epidemic? 15-826(c) 2012 C. Faloutsos 105

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CMU SCS Q1: Immunization: ? ? Given a network, k vaccines, and the virus details Which nodes to immunize? 15-826 106 (c) 2012 C. Faloutsos

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CMU SCS Q1: Immunization: ? ? Given a network, k vaccines, and the virus details Which nodes to immunize? 15-826 107 (c) 2012 C. Faloutsos

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CMU SCS Q1: Immunization: ? ? Given a network, k vaccines, and the virus details Which nodes to immunize? 15-826 108 (c) 2012 C. Faloutsos

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CMU SCS Q1: Immunization: ? ? Given a network, k vaccines, and the virus details Which nodes to immunize? A: immunize the ones that maximally raise the `epidemic threshold’ [Tong+, ICDM’10] 15-826 109 (c) 2012 C. Faloutsos

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CMU SCS Q2: will a virus take over? Flu-like virus (no immunity, ‘SIS’) Mumps (life-time immunity, ‘SIR’) Pertussis (finite-length immunity, ‘SIRS’) 15-826(c) 2012 C. Faloutsos 110 ? ? : attack prob : heal prob details

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CMU SCS Q2: will a virus take over? Flu-like virus (no immunity, ‘SIS’) Mumps (life-time immunity, ‘SIR’) Pertussis (finite-length immunity, ‘SIRS’) 15-826(c) 2012 C. Faloutsos 111 ? ? : attack prob : heal prob depends on connectivity (avg degree? Max degree? variance? Something else? details

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CMU SCS 15-826(c) 2012 C. Faloutsos 112 Epidemic threshold What should depend on? avg. degree? and/or highest degree? and/or variance of degree? and/or third moment of degree? and/or diameter? details

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CMU SCS 15-826(c) 2012 C. Faloutsos 113 Epidemic threshold [Theorem] We have no epidemic, if β/δ <τ = 1/ λ 1,A details

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CMU SCS 15-826(c) 2012 C. Faloutsos 114 Epidemic threshold [Theorem] We have no epidemic, if β/δ <τ = 1/ λ 1,A largest eigenvalue of adj. matrix A attack prob. recovery prob. epidemic threshold Proof: [Wang+03] (for SIS=flu only) details

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CMU SCS A2: will a virus take over? For all typical virus propagation models (flu, mumps, pertussis, HIV, etc) The only connectivity measure that matters, is 1 the first eigenvalue of the adj. matrix [Prakash+, ‘10, arxiv] 15-826(c) 2012 C. Faloutsos 115 ? ? details

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CMU SCS 116 (c) 2012 C. Faloutsos15-826 Thresholds for some models s = effective strength s < 1 : below threshold Models Effective Strength (s) Threshold (tipping point) SIS, SIR, SIRS, SEIR s = λ. s = 1 SIV, SEIV s = λ. (H.I.V.) s = λ.

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CMU SCS A2: will a virus take over? 15-826(c) 2012 C. Faloutsos 117 Fraction of infected Time ticks Below: exp. extinction Above: take-over Graph: Portland, OR 31M links 1.5M nodes

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CMU SCS Q1: Immunization: ? ? Given a network, k vaccines, and the virus details Which nodes to immunize? A: immunize the ones that maximally raise the `epidemic threshold’ [Tong+, ICDM’10] 15-826 118 (c) 2012 C. Faloutsos

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CMU SCS Q1: Immunization: ? ? Given a network, k vaccines, and the virus details Which nodes to immunize? A: immunize the ones that maximally raise the `epidemic threshold’ [Tong+, ICDM’10] 15-826 119 (c) 2012 C. Faloutsos Max eigen-drop for any virus!

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CMU SCS (c) 2012 C. Faloutsos 120 Outline Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools –OddBall (anomaly detection) –Belief propagation –Immunization Problem#3: Scalability -PEGASUS Conclusions 15-826

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CMU SCS 15-826(c) 2012 C. Faloutsos 121 Scalability Google: > 450,000 processors in clusters of ~2000 processors each [ Barroso, Dean, Hölzle, “Web Search for a Planet: The Google Cluster Architecture” IEEE Micro 2003 ] Yahoo: 5Pb of data [Fayyad, KDD’07] Problem: machine failures, on a daily basis How to parallelize data mining tasks, then? A: map/reduce – hadoop (open-source clone) http://hadoop.apache.org/ http://hadoop.apache.org/

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CMU SCS (c) 2012 C. Faloutsos 122 CentralizedHadoop/PEG ASUS Degree Distr. old Pagerank old Diameter/ANF old HERE Conn. Comp old HERE TrianglesdoneHERE Visualizationstarted Outline – Algorithms & results 15-826

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CMU SCS HADI for diameter estimation Radius Plots for Mining Tera-byte Scale Graphs U Kang, Charalampos Tsourakakis, Ana Paula Appel, Christos Faloutsos, Jure Leskovec, SDM’10 Naively: diameter needs O(N**2) space and up to O(N**3) time – prohibitive (N~1B) Our HADI: linear on E (~10B) –Near-linear scalability wrt # machines –Several optimizations -> 5x faster (c) 2012 C. Faloutsos 123 15-826

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CMU SCS ???? 19+ [Barabasi+] 124 (c) 2012 C. Faloutsos Radius Count 15-826 ~1999, ~1M nodes

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CMU SCS YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) Largest publicly available graph ever studied. ???? 19+ [Barabasi+] 125 (c) 2012 C. Faloutsos Radius Count 15-826 ?? ~1999, ~1M nodes

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CMU SCS YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) Largest publicly available graph ever studied. ???? 19+? [Barabasi+] 126 (c) 2012 C. Faloutsos Radius Count 15-826 14 (dir.) ~7 (undir.)

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CMU SCS YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) 7 degrees of separation (!) Diameter: shrunk ???? 19+? [Barabasi+] 127 (c) 2012 C. Faloutsos Radius Count 15-826 14 (dir.) ~7 (undir.)

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CMU SCS YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) Q: Shape? ???? 128 (c) 2012 C. Faloutsos Radius Count 15-826 ~7 (undir.)

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CMU SCS 129 (c) 2012 C. Faloutsos YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) effective diameter: surprisingly small. Multi-modality (?!) 15-826

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CMU SCS Radius Plot of GCC of YahooWeb. 130 (c) 2012 C. Faloutsos15-826

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CMU SCS 131 (c) 2012 C. Faloutsos YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) effective diameter: surprisingly small. Multi-modality: probably mixture of cores. 15-826

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CMU SCS 132 (c) 2012 C. Faloutsos YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) effective diameter: surprisingly small. Multi-modality: probably mixture of cores. 15-826 EN ~7 Conjecture: DE BR

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CMU SCS 133 (c) 2012 C. Faloutsos YahooWeb graph (120Gb, 1.4B nodes, 6.6 B edges) effective diameter: surprisingly small. Multi-modality: probably mixture of cores. 15-826 ~7 Conjecture:

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CMU SCS Running time - Kronecker and Erdos-Renyi Graphs with billions edges. details

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CMU SCS (c) 2012 C. Faloutsos 135 CentralizedHadoop/PEG ASUS Degree Distr. old Pagerank old Diameter/ANF old HERE Conn. Comp old HERE Triangles HERE Visualizationstarted Outline – Algorithms & results 15-826

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CMU SCS Generalized Iterated Matrix Vector Multiplication (GIMV) (c) 2012 C. Faloutsos 136 PEGASUS: A Peta-Scale Graph Mining System - Implementation and ObservationsSystem - Implementation and Observations. U Kang, Charalampos E. Tsourakakis, and Christos Faloutsos. (ICDM) 2009, Miami, Florida, USA. Best Application Paper (runner-up).ICDM 15-826

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CMU SCS Generalized Iterated Matrix Vector Multiplication (GIMV) (c) 2012 C. Faloutsos 137 PageRank proximity (RWR) Diameter Connected components (eigenvectors, Belief Prop. … ) Matrix – vector Multiplication (iterated) 15-826 details

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CMU SCS 138 Example: GIM-V At Work Connected Components – 4 observations: Size Count (c) 2012 C. Faloutsos 15-826

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CMU SCS 139 Example: GIM-V At Work Connected Components Size Count (c) 2012 C. Faloutsos 15-826 1) 10K x larger than next

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CMU SCS 140 Example: GIM-V At Work Connected Components Size Count (c) 2012 C. Faloutsos 15-826 2) ~0.7B singleton nodes

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CMU SCS 141 Example: GIM-V At Work Connected Components Size Count (c) 2012 C. Faloutsos 15-826 3) SLOPE!

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CMU SCS 142 Example: GIM-V At Work Connected Components Size Count 300-size cmpt X 500. Why? 1100-size cmpt X 65. Why? (c) 2012 C. Faloutsos 15-826 4) Spikes!

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CMU SCS 143 Example: GIM-V At Work Connected Components Size Count suspicious financial-advice sites (not existing now) (c) 2012 C. Faloutsos 15-826

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CMU SCS 144 GIM-V At Work Connected Components over Time LinkedIn: 7.5M nodes and 58M edges Stable tail slope after the gelling point (c) 2012 C. Faloutsos15-826

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CMU SCS (c) 2012 C. Faloutsos 145 Outline Introduction – Motivation Problem#1: Patterns in graphs Problem#2: Tools Problem#3: Scalability Conclusions 15-826

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CMU SCS (c) 2012 C. Faloutsos 146 OVERALL CONCLUSIONS – low level: Several new patterns (fortification, shrinking diameter, triangle-laws, conn. components, etc) New tools: –anomaly detection (OddBall), belief propagation, immunization Scalability: PEGASUS / hadoop 15-826

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CMU SCS (c) 2012 C. Faloutsos 147 OVERALL CONCLUSIONS – high level BIG DATA: Large datasets reveal patterns/outliers that are invisible otherwise 15-826

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CMU SCS (c) 2012 C. Faloutsos 148 References Leman Akoglu, Christos Faloutsos: RTG: A Recursive Realistic Graph Generator Using Random Typing. ECML/PKDD (1) 2009: 13-28 Deepayan Chakrabarti, Christos Faloutsos: Graph mining: Laws, generators, and algorithms. ACM Comput. Surv. 38(1): (2006) 15-826

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CMU SCS (c) 2012 C. Faloutsos 149 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: 1316-1324 15-826

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CMU SCS (c) 2012 C. Faloutsos 150 References Christos Faloutsos, Tamara G. Kolda, Jimeng Sun: Mining large graphs and streams using matrix and tensor tools. Tutorial, SIGMOD Conference 2007: 1174 15-826

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CMU SCS (c) 2012 C. Faloutsos 151 References T. G. Kolda and J. Sun. Scalable Tensor Decompositions for Multi-aspect Data Mining. In: ICDM 2008, pp. 363-372, December 2008. 15-826

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CMU SCS (c) 2012 C. Faloutsos 152 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: 133-145 15-826

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CMU SCS (c) 2012 C. Faloutsos 153 References Jimeng Sun, Yinglian Xie, Hui Zhang, Christos Faloutsos. Less is More: Compact Matrix Decomposition for Large Sparse Graphs, SDM, Minneapolis, Minnesota, Apr 2007. Jimeng Sun, Spiros Papadimitriou, Philip S. Yu, and Christos Faloutsos, GraphScope: Parameter- free Mining of Large Time-evolving Graphs ACM SIGKDD Conference, San Jose, CA, August 2007 15-826

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CMU SCS References Jimeng Sun, Dacheng Tao, Christos Faloutsos: Beyond streams and graphs: dynamic tensor analysis. KDD 2006: 374- 383 15-826(c) 2012 C. Faloutsos 154

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CMU SCS (c) 2012 C. Faloutsos 155 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 15-826

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CMU SCS (c) 2012 C. Faloutsos 156 References Hanghang Tong, Christos Faloutsos, Brian Gallagher, Tina Eliassi-Rad: Fast best-effort pattern matching in large attributed graphs. KDD 2007: 737-746 15-826

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CMU SCS (c) 2012 C. Faloutsos 157 (Project info) Akoglu, Leman Chau, Polo Kang, U McGlohon, Mary Tong, Hanghang Prakash, Aditya 15-826 Thanks to: NSF IIS-0705359, IIS-0534205, CTA-INARC ; Yahoo (M45), LLNL, IBM, SPRINT, Google, INTEL, HP, iLab www.cs.cmu.edu/~pegasus Koutra, Danae

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